Finite Group Databasebeta

← #29586318

Subgroups — #29586318

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519705C211yesmaximal, characteristic
2C22×#14519703C211yesmaximal, characteristic
3C2×M16×Dic61C211yesmaximal, characteristic, complemented
4#21448307C211yesmaximal, characteristic
5C2×#16839521C211yesmaximal, characteristic, complemented
6C4×#14519684C211yesmaximal, characteristic, complemented
7C2×C61×#339C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C22×C4×Dic61C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519705—22no—
13C2×#14519705—22no—
14C2×#14519705C414yescomplemented
15C2×#14519703—22no—
16C2×#14519703—22no—
17C2×#14519703C414yescomplemented
18M16×Dic61—22no—
19M16×Dic61—22no—
20C2×#14519687—21no—
21C2×C8×Dic61—21no—
22#16839667C412yescomplemented
23#16839663C412yescomplemented
24#16839639—22no—
25C4×#11758198—21no—
26C4×#11758198—21no—
27C2×#14519684—22no—
28#16839521—22no—
29C22×M16×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C22×C42×C61C2211yescharacteristic
32C61×#339—22no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222C2×C412yes—
44C2×#11758222C2×C412yes—
45C22×Dic122C2×C411yescharacteristic
46C22×D244C2×C411yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61—21no—
49C23×Dic61D412yes—
50C22×C4×D61—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194C2×C412yes—
55C2×#11758194D412yes—
56C2×#11758194D412yes—
57C22×#92418—21no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61C2×C412yes—
70C2×C4×Dic61C2×C412yes—
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73#14519705—28no—
74#14519703—28no—
75#14519687—24no—
76C8×Dic61—24no—
77#14519684—28no—
78#14519684—22no—
79#14519684—22no—
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61C2×C412yes—
86C2×M16×C61D412yes—
87C2×M16×C61D412yes—
88C22×C8×C61—21no—
89#117×C61—21no—
90#117×C61—21no—
91#117×C61—21no—
92#117×C61—21no—
93C2×C4⋊C4×C61C2×C412yes—
94C2×C22⋊C4×C61—21no—
95C2×C42×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61—21no—
101C2×C42×C61C2×C412yes—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C22×Dic61—24no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—24no—
136C22×Dic61—42no—
137C22×Dic61—42no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140C22×Dic61C22⋊C412yes—
141C22×Dic61C22⋊C412yes—
142#11758222—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149#11758222—22no—
150#11758222—41no—
151#11758222—41no—
152#11758222—41no—
153#11758222—41no—
154#11758222—41no—
155#11758222—41no—
156#11758222—41no—
157#11758222—41no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C4212yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—41no—
167C2×C4×D61—41no—
168C2×C4×D61—41no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C4212yes—
176M16×C61—22no—
177M16×C61—22no—
178M16×C61—22no—
179M16×C61C22⋊C414yescomplemented
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—41no—
190C42×C61—41no—
191C42×C61—41no—
192C42×C61—41no—
193#11758198—44no—
194#11758198—44no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—41no—
204C4×Dic61—41no—
205#11758194—22no—
206#11758194—22no—
207#11758194—22no—
208#11758194C22⋊C414yescomplemented
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—22no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61C2×C22⋊C411yescharacteristic
215C23×C61C4×D411yescharacteristic
216C23×C61C4×D411yescharacteristic
217C23×C61#12111yescharacteristic
218C23×C61#12111yescharacteristic
219C23×C61#12411yescharacteristic
220C23×C61#12711yescharacteristic
221C22×D61—22no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—42no—
225C22×D61—42no—
226C2×C4×C61—24no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61C2×C4211yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61#11711yescharacteristic
238C2×C4×C61#11711yescharacteristic
239C2×C4×C61#11711yescharacteristic
240C2×C4×C61C4×D411yescharacteristic
241C2×C4×C61C4×D411yescharacteristic
242C2×C4×C61C4×D411yescharacteristic
243C2×C4×C61C4×D411yescharacteristic
244C2×C4×C61C4×D411yescharacteristic
245C2×C4×C61C4×D411yescharacteristic
246C2×C4×C61C4×D411yescharacteristic
247C2×C4×C61C4×D411yescharacteristic
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C4×D411yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12111yescharacteristic
252C2×C4×C61#12111yescharacteristic
253C2×C4×C61#12111yescharacteristic
254C2×C4×C61#12111yescharacteristic
255C2×C4×C61#12311yescharacteristic
256C2×C4×C61#12311yescharacteristic
257C2×C4×C61#12311yescharacteristic
258C2×C4×C61#12411yescharacteristic
259C2×C4×C61#12411yescharacteristic
260C2×C4×C61#12611yescharacteristic
261C2×C4×C61#12611yescharacteristic
262C22⋊D61—42no—
263C22⋊D61—42no—
264C22⋊D61—42no—
265C22⋊D61—42no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—24no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—44no—
274C2×Dic61—44no—
275C2×Dic61—44no—
276C2×Dic61—44no—
277C2×Dic61—42no—
278C2×Dic61—41no—
279C2×Dic61—42no—
280C2×Dic61—41no—
281C2×Dic61#10414yescomplemented
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286C4×D61—42no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—42no—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—24no—
295#92418—24no—
296#10933—611nomaximal, Sylow
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—21no—
305C22×C61C4×C22⋊C411yescharacteristic
306C22×C61#38611yescharacteristic
307C22×C61#38811yescharacteristic
308C22×C61#38811yescharacteristic
309C22×C61#39011yescharacteristic
310C22×C61#39011yescharacteristic
311C22×C61#39011yescharacteristic
312C22×C61C2×#10412yes—
313C22×C61#42112yes—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—82no—
318C2×D61—82no—
319C4×C61—44no—
320C4×C61—82no—
321C4×C61C4×C22⋊C411yescharacteristic
322C4×C61C4×C22⋊C411yescharacteristic
323C4×C61#38611yescharacteristic
324C4×C61#38611yescharacteristic
325C4×C61#38811yescharacteristic
326C4×C61#38811yescharacteristic
327C4×C61#38811yescharacteristic
328C4×C61#38811yescharacteristic
329Dic61—24no—
330Dic61—44no—
331Dic61—42no—
332Dic61—42no—
333C22×#104—611no—
334C22×#104—611no—
335C2×C4×M16—611no—
336#2297—611no—
337C2×#339—611no—
338C2×#339D6111yescharacteristic, complemented
339C4×#95—611no—
340C2×C61—21no—
341C2×C61—21no—
342C2×C61—21no—
343C2×C61—21no—
344C2×C61#143711yescharacteristic
345C2×C61C4×#10411yescharacteristic
346C2×C61#175911yescharacteristic
347C2×C61#187112yes—
348C2×C61#198012yes—
349D61—82no—
350C22×#41—611no—
351C22×M16—611no—
352C22×M16C2×D6111yescharacteristic
353C2×#117C2×D6111yescharacteristic
354C22×C42—611no—
355C22×C42—611no—
356C22×C42C2×D6111yescharacteristic
357C2×#104—614no—
358C2×#104—614no—
359C2×#104—1222no—
360C2×#104—1222no—
361C2×#104—1222no—
362C2×#104—1222no—
363C4×M16—1222no—
364C4×M16—1222no—
365C2×#97—1221no—
366C2×C4×C8—1221no—
367#391—612no—
368#390—612no—
369#386—1222no—
370C4×C22⋊C4—1221no—
371C4×C22⋊C4—1221no—
372C2×#95—1222no—
373#339—22no—
374#339—1222no—
375C61#1093311yescharacteristic, complemented, Sylow
376C2×#41—612no—
377C2×#41—612no—
378C2×#41—1221no—
379C2×#41—1221no—
380C2×#41—1221no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41—1221no—
384C2×#41—1221no—
385C2×#41—1221no—
386C2×#41—1221no—
387C2×#41—1221no—
388C22×Q8—611no—
389C22×D4—611no—
390C22×D4—1221no—
391C23×C4—21no—
392C23×C4—612no—
393C23×C4—1221no—
394C23×C4—1221no—
395C23×C4C22×D6111yescharacteristic
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—21no—
399C2×M16—612no—
400C2×M16—612no—
401C2×M16—612no—
402C2×M16—1221no—
403C2×M16—1221no—
404C2×M16—1221no—
405C2×M16C4×D6112yes—
406C2×M16D24412yes—
407C2×M16C22⋊D6112yes—
408C22×C8—21no—
409C22×C8—1221no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117—21no—
414C2×C4⋊C4C4×D6112yes—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1222no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—612no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—612no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42C4×D6112yes—
438#104—1228no—
439#104—1228no—
440#97—1224no—
441C4×C8—1224no—
442#95—1222no—
443#95—1222no—
444#95—1228no—
445C24—1221no—
446C24D4×D6111yescharacteristic
447#41—1222no—
448#41—1222no—
449#41—1222no—
450#41—1222no—
451#41—1222no—
452#41—1222no—
453#41—1222no—
454#41—1222no—
455#41—2441no—
456#41—2441no—
457#41—2441no—
458#41—2441no—
459#41—2441no—
460#41—2441no—
461#41—2441no—
462#41—2441no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×D4—612no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—2441no—
478C2×D4—2441no—
479C2×D4—2441no—
480C2×D4—2441no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—1224no—
490C22×C4—1224no—
491C22×C4—1221no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4—2441no—
500C22×C4—2441no—
501C22×C4—2441no—
502C22×C4—2441no—
503C22×C4—2441no—
504C22×C4—2441no—
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4C2×C4×D6111yescharacteristic
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4#1175822211yescharacteristic
510C22×C4#1175822211yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4D4×D6111yescharacteristic
513C22×C4D4×D6111yescharacteristic
514C22×C4#1175822411yescharacteristic
515C22×C4#1175822411yescharacteristic
516C22×C4#1175822611yescharacteristic
517C22×C4#1175822611yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519M16—22no—
520M16—22no—
521M16—22no—
522M16—614no—
523M16—1222no—
524M16—1222no—
525M16—1222no—
526M16#1175819814yescomplemented
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—2444no—
538C22⋊C4—2444no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—1222no—
546C42—1222no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1222no—
552C42—1222no—
553C42—2441no—
554C42—2441no—
555C23—22no—
556C23—22no—
557C23—1222no—
558C23—2442no—
559C23—2442no—
560C23—2441no—
561C23—2441no—
562C23C22⋊C4×D6111yescharacteristic
563C23#1451973711yescharacteristic
564C23#1451973911yescharacteristic
565C23#1451975011yescharacteristic
566C23#1451975211yescharacteristic
567C23#1451976311yescharacteristic
568C23#1451977311yescharacteristic
569Q8—1222no—
570Q8—1222no—
571Q8—2441no—
572Q8—2441no—
573D4—1222no—
574D4—1222no—
575D4—2442no—
576D4—2442no—
577D4—2442no—
578D4—2442no—
579D4—2441no—
580D4—2441no—
581C2×C4—24no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—614no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4—2442no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4C42×D6111yescharacteristic
610C2×C4C22⋊C4×D6111yescharacteristic
611C2×C4C2×#1175819811yescharacteristic
612C2×C4#1451973411yescharacteristic
613C2×C4#1451973511yescharacteristic
614C2×C4#1451973511yescharacteristic
615C2×C4#1451973711yescharacteristic
616C2×C4#1451973711yescharacteristic
617C2×C4#1451973711yescharacteristic
618C2×C4C4×C22⋊D6111yescharacteristic
619C2×C4C4×C22⋊D6111yescharacteristic
620C2×C4#1451973911yescharacteristic
621C2×C4#1451973911yescharacteristic
622C2×C4#1451973911yescharacteristic
623C2×C4C4×D24411yescharacteristic
624C2×C4C4×D24411yescharacteristic
625C2×C4#1451974711yescharacteristic
626C2×C4#1451974911yescharacteristic
627C2×C4#1451974911yescharacteristic
628C2×C4#1451975011yescharacteristic
629C2×C4#1451975211yescharacteristic
630C2×C4#1451976111yescharacteristic
631C2×C4#1451976211yescharacteristic
632C2×C4#1451976211yescharacteristic
633C2×C4#1451976311yescharacteristic
634C2×C4#1451976611yescharacteristic
635C2×C4#1451977211yescharacteristic
636C2×C4#1451977211yescharacteristic
637C8—24no—
638C8—1224no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—4882no—
648C22—4882no—
649C22—2442no—
650C22—2442no—
651C22—2442no—
652C22#1683960911yescharacteristic
653C22#1683964011yescharacteristic
654C22#1683965111yescharacteristic
655C22#1683965411yescharacteristic
656C22#1683966211yescharacteristic
657C22#1683966211yescharacteristic
658C22#1683966311yescharacteristic
659C22#104×D6112yes—
660C22#1683977712yes—
661C4—44no—
662C4—82no—
663C4—1224no—
664C4—2444no—
665C4—2442no—
666C4—2442no—
667C4#1683960911yescharacteristic
668C4C4×#1175819811yescharacteristic
669C4#1683963911yescharacteristic
670C4#1683964011yescharacteristic
671C4#1683965111yescharacteristic
672C4#1683965211yescharacteristic
673C4#1683965211yescharacteristic
674C4#1683965411yescharacteristic
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—4882no—
680C2#2144405511yescharacteristic
681C2#2144562311yescharacteristic
682C2#2144562811yescharacteristic
683C2#2144612012yes—
684C2#2144673612yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×#339C2
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#339D61
hypocenterC61#10933
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×M16×Dic61C2semidirect
5. C2×#16839521C2semidirect
6. C4×#14519684C2semidirect
7. C2×C61×#339C2semidirect
14. C2×#14519705C4semidirect
17. C2×#14519703C4semidirect
22. #16839667C4semidirect
23. #16839663C4semidirect
179. M16×C61C22⋊C4semidirect
208. #11758194C22⋊C4semidirect
281. C2×Dic61#104semidirect
338. C2×#339D61semidirect
375. C61#10933semidirect
526. M16#11758198semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145197052
C22×#145197032
C2×M16×Dic612
#214483072
C2×#168395212
C4×#145196842
C2×C61×#3392
#1093361