Finite Group Databasebeta

← #29586321

Subgroups — #29586321

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×#14519704C211yesmaximal, characteristic
2C2×M16×Dic61C211yesmaximal, characteristic, complemented
3#21448310C211yesmaximal, characteristic
4C2×#16839523C211yesmaximal, characteristic, complemented
5C2×#16839522C211yesmaximal, characteristic, complemented
6C4×#14519684C211yesmaximal, characteristic, complemented
7C22×#104×C61C211yesmaximal, characteristic
8C22×#11758194C2211yescharacteristic
9C2×#14519736C2211yescharacteristic
10C22×C4×Dic61C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519704—22no—
13C2×#14519704—22no—
14C2×#14519704C414yescomplemented
15M16×Dic61—22no—
16M16×Dic61—22no—
17C2×#14519687—21no—
18C2×C8×Dic61—21no—
19#16839669C412yescomplemented
20#16839665C412yescomplemented
21#16839642—22no—
22C4×C22⋊Dic61—21no—
23C4×C22⋊Dic61—21no—
24C2×#14519684—22no—
25#16839523—22no—
26#16839522—22no—
27C22×#41×C61C2211yescharacteristic
28C22×M16×C61C2211yescharacteristic
29C22×C42×C61C2211yescharacteristic
30C2×#104×C61—22no—
31C2×#104×C61—22no—
32C2×#104×C61C414yescomplemented
33C23×Dic61—21no—
34C23×Dic61D412yes—
35C2×#11758194—21no—
36C2×#11758194—21no—
37C2×#11758194—21no—
38C2×#11758194C2×C412yes—
39C2×#11758194D412yes—
40C2×#11758194D412yes—
41C22×#92418—21no—
42#14519736—21no—
43#14519736—21no—
44#14519736—21no—
45#14519736—21no—
46C2×C4⋊Dic61C2×C412yes—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—22no—
49C2×C22⋊Dic61—22no—
50C2×C4×Dic61—21no—
51C2×C4×Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—21no—
54C2×C4×Dic61—21no—
55C2×C4×Dic61—21no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61—21no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61C2×C412yes—
61C2×C4×Dic61C2×C412yes—
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64#14519704—28no—
65#14519687—24no—
66C8×Dic61—24no—
67#14519684—28no—
68#14519684—22no—
69#14519684—22no—
70C2×#41×C61—21no—
71C2×#41×C61—21no—
72C2×#41×C61—21no—
73C2×#41×C61—21no—
74C2×#41×C61—21no—
75C2×#41×C61—21no—
76C2×#41×C61—21no—
77C2×#41×C61—21no—
78C2×#41×C61—21no—
79C2×#41×C61—21no—
80C2×#41×C61C2×C412yes—
81C2×#41×C61C2×C412yes—
82C22×Q8×C61C2×C411yescharacteristic
83C22×D4×C61—21no—
84C22×D4×C61C2×C411yescharacteristic
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61C2×C412yes—
92C2×M16×C61D412yes—
93C2×M16×C61D412yes—
94C22×C8×C61—21no—
95C2×C42×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61C2×C412yes—
101#104×C61—28no—
102C24×C61—21no—
103C24×C61C2×D411yescharacteristic
104#41×C61—22no—
105#41×C61—22no—
106#41×C61—22no—
107#41×C61—22no—
108#41×C61—22no—
109#41×C61—22no—
110#41×C61—22no—
111#41×C61—22no—
112#41×C61—41no—
113#41×C61—41no—
114#41×C61—41no—
115#41×C61—41no—
116#41×C61—41no—
117#41×C61—41no—
118#41×C61—41no—
119#41×C61—41no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61C4212yes—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—41no—
134C2×D4×C61—41no—
135C2×D4×C61—41no—
136C2×D4×C61—41no—
137C2×D4×C61C4212yes—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—42no—
144C22×C4×C61—41no—
145C22×C4×C61—41no—
146C22×C4×C61—41no—
147C22×C4×C61—41no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4111yescharacteristic
159C22×C4×C61#4111yescharacteristic
160C22×C4×C61#4111yescharacteristic
161C22×C4×C61#4111yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C22×C4×C61#4111yescharacteristic
164C22×Dic61—24no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—24no—
169C22×Dic61—22no—
170C22×Dic61—42no—
171C22×Dic61—42no—
172C22×Dic61—41no—
173C22×Dic61—41no—
174C22×Dic61C22⋊C412yes—
175C22×Dic61C22⋊C412yes—
176M16×C61—22no—
177M16×C61—22no—
178M16×C61—22no—
179M16×C61C22⋊C414yescomplemented
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61—41no—
186C42×C61—41no—
187C22⋊Dic61—44no—
188C22⋊Dic61—41no—
189C22⋊Dic61—41no—
190C22⋊Dic61—44no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—41no—
202C4×Dic61—41no—
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×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61C2×C22⋊C411yescharacteristic
220C23×C61C4×D411yescharacteristic
221C23×C61C4×D411yescharacteristic
222C23×C61#12111yescharacteristic
223C23×C61#12111yescharacteristic
224C23×C61#12411yescharacteristic
225C23×C61#12711yescharacteristic
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—41no—
237D4×C61—41no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—44no—
241C2×C4×C61—44no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C2×C4211yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61#11711yescharacteristic
254C2×C4×C61#11711yescharacteristic
255C2×C4×C61#11711yescharacteristic
256C2×C4×C61C4×D411yescharacteristic
257C2×C4×C61C4×D411yescharacteristic
258C2×C4×C61C4×D411yescharacteristic
259C2×C4×C61C4×D411yescharacteristic
260C2×C4×C61C4×D411yescharacteristic
261C2×C4×C61C4×D411yescharacteristic
262C2×C4×C61C4×D411yescharacteristic
263C2×C4×C61C4×D411yescharacteristic
264C2×C4×C61C4×D411yescharacteristic
265C2×C4×C61C4×D411yescharacteristic
266C2×C4×C61#12011yescharacteristic
267C2×C4×C61#12111yescharacteristic
268C2×C4×C61#12111yescharacteristic
269C2×C4×C61#12111yescharacteristic
270C2×C4×C61#12111yescharacteristic
271C2×C4×C61#12311yescharacteristic
272C2×C4×C61#12311yescharacteristic
273C2×C4×C61#12311yescharacteristic
274C2×C4×C61#12411yescharacteristic
275C2×C4×C61#12411yescharacteristic
276C2×C4×C61#12611yescharacteristic
277C2×C4×C61#12611yescharacteristic
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—24no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—44no—
285C2×Dic61—44no—
286C2×Dic61—44no—
287C2×Dic61—44no—
288C2×Dic61—42no—
289C2×Dic61—42no—
290C2×Dic61—42no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61#10414yescomplemented
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×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—82no—
309C22×C61—82no—
310C22×C61C4×C22⋊C411yescharacteristic
311C22×C61#38611yescharacteristic
312C22×C61#38811yescharacteristic
313C22×C61#38811yescharacteristic
314C22×C61#39011yescharacteristic
315C22×C61#39011yescharacteristic
316C22×C61#39011yescharacteristic
317C22×C61C2×#10412yes—
318C22×C61#42112yes—
319C4×C61—44no—
320C4×C61—42no—
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—82no—
333C22×#104—611no—
334C22×#104D6111yescharacteristic
335C2×C4×M16—611no—
336#2297—611no—
337C2×#339—611no—
338C2×#339—611no—
339C4×#95—611no—
340C2×C61—21no—
341C2×C61—21no—
342C2×C61—21no—
343C2×C61—21no—
344C2×C61—82no—
345C2×C61#143711yescharacteristic
346C2×C61C4×#10411yescharacteristic
347C2×C61#175911yescharacteristic
348C2×C61#187112yes—
349C2×C61#198012yes—
350C22×#41C2×D6111yescharacteristic
351C22×M16—611no—
352C22×M16C2×D6111yescharacteristic
353C2×#117—611no—
354C22×C42—611no—
355C22×C42—611no—
356C22×C42C2×D6111yescharacteristic
357C2×#104—22no—
358C2×#104—22no—
359C2×#104—614no—
360C2×#104—1222no—
361C2×#104—1222no—
362C2×#104Dic6114yescomplemented
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—1222no—
374#339—1222no—
375C61#1093311yescharacteristic, complemented, Sylow
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41C4×D6112yes—
387C2×#41C2×Dic6112yes—
388C22×Q8C4×D6111yescharacteristic
389C22×D4—21no—
390C22×D4C4×D6111yescharacteristic
391C23×C4—21no—
392C23×C4—21no—
393C23×C4—612no—
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×M16C2×Dic6112yes—
406C2×M16C22⋊D6112yes—
407C2×M16C22⋊D6112yes—
408C22×C8—21no—
409C22×C8—1221no—
410#117—1221no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414C2×C4⋊C4—612no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1221no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—612no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—612no—
427C2×C42—1221no—
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×C42C2×Dic6112yes—
438#104—28no—
439#104—1228no—
440#97—1224no—
441C4×C8—1224no—
442#95—1222no—
443#95—1222no—
444#95—1228no—
445C24—21no—
446C24D4×D6111yescharacteristic
447#41—22no—
448#41—22no—
449#41—22no—
450#41—22no—
451#41—22no—
452#41—22no—
453#41—22no—
454#41—22no—
455#41—41no—
456#41—41no—
457#41—41no—
458#41—41no—
459#41—41no—
460#41—41no—
461#41—41no—
462#41—41no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C4×Dic6112yes—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4C4×Dic6112yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2441no—
504C22×C4—2441no—
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4C2×C4×D6111yescharacteristic
507C22×C4#1175822211yescharacteristic
508C22×C4#1175822211yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4#1175822411yescharacteristic
513C22×C4#1175822411yescharacteristic
514C22×C4#1175822411yescharacteristic
515C22×C4#1175822411yescharacteristic
516C22×C4C22×Dic6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519M16—22no—
520M16—22no—
521M16—22no—
522M16—614no—
523M16—1222no—
524M16—1222no—
525M16—1222no—
526M16C22⋊Dic6114yescomplemented
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C22⋊C4—2444no—
536C22⋊C4—2444no—
537C22⋊C4—2441no—
538C22⋊C4—2441no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—1222no—
544C42—1222no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1222no—
550C42—1222no—
551C42—2441no—
552C42—2441no—
553C42—2441no—
554C42—2441no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23C22⋊C4×D6111yescharacteristic
563C23#1451973711yescharacteristic
564C23D4×Dic6111yescharacteristic
565C23#1451974811yescharacteristic
566C23#1451975011yescharacteristic
567C23#1451976311yescharacteristic
568C23#1451977311yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—22no—
574D4—22no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—41no—
580D4—41no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—614no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1224no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—2444no—
601C2×C4—2444no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2444no—
606C2×C4—2444no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4C2×C4×Dic6111yescharacteristic
610C2×C4C22⋊C4×D6111yescharacteristic
611C2×C4C2×C22⋊Dic6111yescharacteristic
612C2×C4#1451973311yescharacteristic
613C2×C4#1451973311yescharacteristic
614C2×C4#1451973611yescharacteristic
615C2×C4#1451973711yescharacteristic
616C2×C4#1451973711yescharacteristic
617C2×C4#1451973711yescharacteristic
618C2×C4C4×C22⋊D6111yescharacteristic
619C2×C4C4×C22⋊D6111yescharacteristic
620C2×C4C4×C22⋊D6111yescharacteristic
621C2×C4C4×C22⋊D6111yescharacteristic
622C2×C4D4×Dic6111yescharacteristic
623C2×C4D4×Dic6111yescharacteristic
624C2×C4D4×Dic6111yescharacteristic
625C2×C4#1451974511yescharacteristic
626C2×C4#1451974811yescharacteristic
627C2×C4#1451974911yescharacteristic
628C2×C4#1451974911yescharacteristic
629C2×C4#1451975011yescharacteristic
630C2×C4#1451975811yescharacteristic
631C2×C4#1451975911yescharacteristic
632C2×C4#1451975911yescharacteristic
633C2×C4#1451976311yescharacteristic
634C2×C4#1451976411yescharacteristic
635C2×C4#1451977011yescharacteristic
636C2×C4#1451977011yescharacteristic
637C8—24no—
638C8—1224no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—82no—
651C22—82no—
652C22C22⋊C4×Dic6111yescharacteristic
653C22#1683963811yescharacteristic
654C22#1683965011yescharacteristic
655C22#1683965511yescharacteristic
656C22#1683966211yescharacteristic
657C22#1683966211yescharacteristic
658C22#1683966511yescharacteristic
659C22#104×D6112yes—
660C22#1683977712yes—
661C4—44no—
662C4—42no—
663C4—1224no—
664C4—4882no—
665C4—2444no—
666C4—2442no—
667C4C4×C22⋊Dic6111yescharacteristic
668C4C22⋊C4×Dic6111yescharacteristic
669C4#1683963811yescharacteristic
670C4#1683964211yescharacteristic
671C4#1683965011yescharacteristic
672C4#1683965511yescharacteristic
673C4#1683965611yescharacteristic
674C4#1683965611yescharacteristic
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—82no—
680C2#2144405611yescharacteristic
681C2#104×Dic6111yescharacteristic
682C2#2144563111yescharacteristic
683C2#2144611812yes—
684C2#2144673612yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC22×#104×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC22×#41×C61C22
cosocleC23×C4×C61C23
hypercenterC22×#104D61
hypocenterC61#10933
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic61C2semidirect
4. C2×#16839523C2semidirect
5. C2×#16839522C2semidirect
6. C4×#14519684C2semidirect
14. C2×#14519704C4semidirect
19. #16839669C4semidirect
20. #16839665C4semidirect
32. C2×#104×C61C4semidirect
179. M16×C61C22⋊C4semidirect
208. #11758194C22⋊C4semidirect
293. C2×Dic61#104semidirect
362. C2×#104Dic61semidirect
375. C61#10933semidirect
526. M16C22⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197042
C2×M16×Dic612
#214483102
C2×#168395232
C2×#168395222
C4×#145196842
C22×#104×C612
#1093361