Finite Group Databasebeta

← #29585480

Subgroups — #29585480

Order 15616.

1298 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2C2×#16839778C211yesmaximal, characteristic
3#21454339C211yesmaximal, characteristic
4#21445551C211yesmaximal, characteristic, complemented
5C2×#16839523C211yesmaximal, characteristic, complemented
6C2×#16839523C211yesmaximal, characteristic, complemented
7C2×C61×#344C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9#16840230—22no—
10#16840230—22no—
11#16840230—22no—
12#16840230C414yescomplemented
13C2×#14519742—21no—
14C2×#14519739—21no—
15C2×#14519737—22no—
16C2×#14519736C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C2×#14519734—21no—
19C22×C4×Dic61C2211yescharacteristic
20C2×C42×D61—21no—
21#16839778—22no—
22#16839778C414yescomplemented
23C2×#14519703—22no—
24#16839772—21no—
25#16839768—21no—
26#16839764C414yescomplemented
27#16839759—22no—
28#16839528—22no—
29#16839523—21no—
30#16839523—21no—
31#16839523—22no—
32C22×M16×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C61×#344—22no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222C2×C412yes—
44C2×#11758222C2×C412yes—
45C2×#11758222D412yes—
46C2×#11758222D412yes—
47C2×#11758222Q812yes—
48C22×Dic122D411yescharacteristic
49C22×D244D411yescharacteristic
50C22×C22⋊D61—21no—
51C23×Dic61—21no—
52C22×C4×D61—21no—
53C22×C4×D61—22no—
54#14519742—24no—
55#14519742—24no—
56#14519739—24no—
57#14519739—24no—
58#14519737—28no—
59#14519737—28no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736C2×C412yescomplemented
68#14519734—24no—
69#14519734—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2×C412yescomplemented
72C2×#11758196—22no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2×C412yes—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D412yes—
86C42×D61—24no—
87C42×D61—24no—
88#14519703—28no—
89#14519702—24no—
90#14519699—24no—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—21no—
93C2×M16×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61—21no—
96C2×M16×C61—21no—
97C2×M16×C61—21no—
98C2×M16×C61—21no—
99C2×M16×C61—21no—
100C2×M16×C61—21no—
101C2×M16×C61—21no—
102C2×M16×C61C2×C412yescomplemented
103C2×M16×C61C2×C412yescomplemented
104C22×C8×C61—21no—
105C22×C8×C61—21no—
106#117×C61—21no—
107#117×C61—21no—
108#117×C61C2×C412yes—
109#117×C61D412yes—
110C2×C4⋊C4×C61—21no—
111C2×C22⋊C4×C61D412yes—
112C2×C42×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×Q811yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—44no—
143C22×Dic61—41no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149#11758222—22no—
150#11758222—22no—
151#11758222—22no—
152#11758222—22no—
153#11758222—22no—
154#11758222—22no—
155#11758222—22no—
156#11758222C4214yes—
157#11758222C22⋊C414yes—
158#11758222C4⋊C414yes—
159#11758222C4⋊C414yes—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22⋊C412yes—
164C2×D244C22⋊C412yes—
165C2×D244C4⋊C412yes—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—28no—
173C2×C4×D61—48no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122C22⋊C412yes—
178C2×Dic122C22⋊C412yes—
179C2×Dic122C4⋊C412yes—
180M16×C61—22no—
181M16×C61—22no—
182M16×C61—22no—
183M16×C61—22no—
184M16×C61—41no—
185M16×C61—41no—
186M16×C61—41no—
187M16×C61—41no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C2×C8×C61—41no—
191C2×C8×C61—41no—
192C2×C8×C61—41no—
193C2×C8×C61—41no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61C22⋊C414yes—
198C42×C61—22no—
199C42×C61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—41no—
202C22⋊Dic61—41no—
203C22⋊Dic61—41no—
204#11758198—28no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61—41no—
209#11758196—28no—
210C4×Dic61—28no—
211C4×Dic61—22no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—41no—
217C4×Dic61—41no—
218C4×Dic61—41no—
219C4×Dic61C22⋊C412yes—
220C4×Dic61C22⋊C412yes—
221C23×C61—42no—
222C23×C61C2×C22⋊C411yescharacteristic
223C23×C61C4×D411yescharacteristic
224C23×C61C4×D411yescharacteristic
225C23×C61#12011yescharacteristic
226C23×C61#12011yescharacteristic
227C23×C61#12111yescharacteristic
228C23×C61C22⋊Q811yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—44no—
233C2×C4×C61—24no—
234C2×C4×C61—44no—
235C2×C4×C61C2×C4211yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C4⋊C411yescharacteristic
241C2×C4×C61C2×C4⋊C411yescharacteristic
242C2×C4×C61C2×C4⋊C411yescharacteristic
243C2×C4×C61#11711yescharacteristic
244C2×C4×C61#11711yescharacteristic
245C2×C4×C61C4×D411yescharacteristic
246C2×C4×C61C4×D411yescharacteristic
247C2×C4×C61C4×D411yescharacteristic
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C4×D411yescharacteristic
250C2×C4×C61C4×D411yescharacteristic
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12111yescharacteristic
254C2×C4×C61#12111yescharacteristic
255C2×C4×C61#12111yescharacteristic
256C2×C4×C61C22⋊Q811yescharacteristic
257C2×C4×C61C22⋊Q811yescharacteristic
258C2×C4×C61C22⋊Q811yescharacteristic
259C2×C4×C61#12311yescharacteristic
260C2×C4×C61#12311yescharacteristic
261C2×C4×C61#12311yescharacteristic
262C2×C4×C61#12311yescharacteristic
263C22⋊D61—24no—
264C22⋊D61—24no—
265C22⋊D61—24no—
266C22⋊D61—24no—
267C2×Dic61—24no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—44no—
274C2×Dic61—44no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—82no—
279C2×Dic61—82no—
280D244—22no—
281D244—22no—
282D244—22no—
283D244#9514yes—
284C4×D61—24no—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—24no—
288C4×D61—416no—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122#9514yes—
293C8×C61—42no—
294C8×C61—42no—
295#10772—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61C4×C22⋊C411yescharacteristic
304C22×C61#37911yescharacteristic
305C22×C61#38011yescharacteristic
306C22×C61#38511yescharacteristic
307C22×C61#38511yescharacteristic
308C22×C61#38611yescharacteristic
309C22×C61#38611yescharacteristic
310C2×D61—24no—
311C2×D61—44no—
312C2×D61—44no—
313C2×D61—44no—
314C4×C61—44no—
315C4×C61C2×#9511yescharacteristic
316C4×C61C4×C22⋊C411yescharacteristic
317C4×C61#38011yescharacteristic
318C4×C61#38111yescharacteristic
319C4×C61#38511yescharacteristic
320C4×C61#38511yescharacteristic
321C4×C61#38611yescharacteristic
322C4×C61#38611yescharacteristic
323Dic61—24no—
324Dic61—24no—
325Dic61—42no—
326Dic61—82no—
327Dic61—82no—
328C2×C4×#41—611no—
329C2×#421—611no—
330#2896—611no—
331C2×#344D6111yescharacteristic, complemented
332#1740—611no—
333C2×#339—611no—
334C2×#339—611no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61—21no—
341C2×C61C22⋊#9511yescharacteristic
342C2×C61#190612yes—
343D61—44no—
344C22×#41—611no—
345C22×M16C2×D6111yescharacteristic
346C22×M16C2×D6111yescharacteristic
347C4×#41—614no—
348C4×#41—1222no—
349C4×#41—1222no—
350C4×#41—1222no—
351C2×C4×Q8—1221no—
352C2×C4×D4—1221no—
353C2×C4×D4—1222no—
354C2×#117—611no—
355C2×#117—611no—
356C2×#117—1221no—
357C2×#117C2×D6111yescharacteristic
358C22×C42—611no—
359C22×C42—1221no—
360#421—614no—
361#421—1222no—
362C2×#104—1222no—
363#419—1221no—
364#418—1221no—
365#417—614no—
366#416—1222no—
367#344—22no—
368#342—1222no—
369#339—1221no—
370#339—1222no—
371#339—1221no—
372C61#1077211yescharacteristic, complemented, Sylow
373C2×#41—612no—
374C2×#41—612no—
375C2×#41—612no—
376C2×#41—612no—
377C2×#41—612no—
378C2×#41—612no—
379C2×#41—1221no—
380C2×#41—1221no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41—1221no—
384C2×#41—1221no—
385C22×Q8—611no—
386C22×D4—611no—
387C22×D4—1221no—
388C23×C4—1222no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C2×M16—21no—
393C2×M16—21no—
394C2×M16—21no—
395C2×M16—21no—
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—21no—
399C2×M16—21no—
400C2×M16—21no—
401C2×M16—21no—
402C2×M16C4×D6112yescomplemented
403C2×M16C4×D6112yescomplemented
404C22×C8—21no—
405C22×C8—21no—
406C4×Q8—1224no—
407C4×Q8—1224no—
408C4×D4—1228no—
409C4×D4—1224no—
410C4×D4—1224no—
411C4×D4—1228no—
412#117—21no—
413#117—21no—
414#117—612no—
415#117—1224no—
416#117—1224no—
417#117—1221no—
418#117—1221no—
419#117—1221no—
420#117—1221no—
421#117—1221no—
422#117—1221no—
423#117—1221no—
424#117C4×D6112yes—
425#117C22⋊D6112yes—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1221no—
430C2×C22⋊C4—1222no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4D24412yes—
434C2×C42—21no—
435C2×C42—612no—
436C2×C42—611no—
437C2×C42—612no—
438C2×C42—611no—
439C2×C42—1224no—
440C2×C42—1224no—
441C2×C42—1222no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447#104—1228no—
448#103—1224no—
449#102—1224no—
450C24—1221no—
451C24C2×D24411yescharacteristic
452#41—614no—
453#41—614no—
454#41—614no—
455#41—614no—
456#41—1222no—
457#41—1222no—
458#41—1222no—
459#41—1222no—
460#41—1222no—
461#41—1222no—
462#41—1222no—
463#41—1222no—
464#41—1222no—
465#41—1222no—
466#41—1222no—
467#41—1222no—
468C2×Q8—612no—
469C2×Q8—612no—
470C2×Q8—612no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×D4—612no—
475C2×D4—612no—
476C2×D4—612no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—1222no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C22×C4—22no—
487C22×C4—1228no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—2448no—
501C22×C4—2444no—
502C22×C4—2441no—
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4#1175822211yescharacteristic
507C22×C4#1175822211yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4D4×D6111yescharacteristic
513C22×C4#1175822411yescharacteristic
514C22×C4Q8×D6111yescharacteristic
515C22×C4#1175822611yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517M16—22no—
518M16—22no—
519M16—22no—
520M16—22no—
521M16—41no—
522M16—41no—
523M16—41no—
524M16—41no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—1228no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—2441no—
538C22⋊C4—22no—
539C22⋊C4—1228no—
540C22⋊C4—1222no—
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C22⋊C4—2441no—
544C22⋊C4#1175819814yes—
545C42—22no—
546C42—22no—
547C42—612no—
548C42—612no—
549C42—1228no—
550C42—1222no—
551C42—1221no—
552C42—1221no—
553C42—1222no—
554C42—1222no—
555C42—2441no—
556C42—2441no—
557C42—2441no—
558C23—42no—
559C23—1222no—
560C23—1222no—
561C23—1222no—
562C23—2444no—
563C23C2×#1175819811yescharacteristic
564C23C4×D24411yescharacteristic
565C23C4×D24411yescharacteristic
566C23#1451974711yescharacteristic
567C23#1451974711yescharacteristic
568C23#1451975211yescharacteristic
569C23#1451975711yescharacteristic
570Q8—614no—
571Q8—1222no—
572Q8—1222no—
573Q8—1222no—
574D4—614no—
575D4—1224no—
576D4—1224no—
577D4—1224no—
578D4—1224no—
579D4—1222no—
580D4—1222no—
581D4—1222no—
582C2×C4—24no—
583C2×C4—44no—
584C2×C4—1224no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—4882no—
595C2×C4—4882no—
596C2×C4—24416no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4C42×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C4⋊C4×D6111yescharacteristic
608C2×C4C4⋊C4×D6111yescharacteristic
609C2×C4C4⋊C4×D6111yescharacteristic
610C2×C4#1451973411yescharacteristic
611C2×C4#1451973511yescharacteristic
612C2×C4#1451973711yescharacteristic
613C2×C4#1451973711yescharacteristic
614C2×C4C4×C22⋊D6111yescharacteristic
615C2×C4C4×C22⋊D6111yescharacteristic
616C2×C4#1451973911yescharacteristic
617C2×C4#1451973911yescharacteristic
618C2×C4#1451974511yescharacteristic
619C2×C4#1451974511yescharacteristic
620C2×C4#1451974811yescharacteristic
621C2×C4#1451975011yescharacteristic
622C2×C4#1451975011yescharacteristic
623C2×C4#1451975311yescharacteristic
624C2×C4#1451975511yescharacteristic
625C2×C4#1451975511yescharacteristic
626C2×C4#1451975911yescharacteristic
627C2×C4#1451975911yescharacteristic
628C2×C4#1451976211yescharacteristic
629C2×C4#1451976211yescharacteristic
630C8—42no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—1224no—
640C22—2444no—
641C22—2444no—
642C22—2444no—
643C22C4×#1175819811yescharacteristic
644C22#1683961711yescharacteristic
645C22#1683961911yescharacteristic
646C22#1683963611yescharacteristic
647C22#1683963611yescharacteristic
648C22#1683964011yescharacteristic
649C22#1683964011yescharacteristic
650C4—44no—
651C4—1224no—
652C4—1224no—
653C4—4882no—
654C4—4882no—
655C4—2442no—
656C4#95×D6111yescharacteristic
657C4#1683960911yescharacteristic
658C4#1683962111yescharacteristic
659C4#1683962411yescharacteristic
660C4#1683963311yescharacteristic
661C4#1683963311yescharacteristic
662C4#1683963811yescharacteristic
663C4#1683963811yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—2444no—
671C2#2144404011yescharacteristic
672C2#2144632112yes—
subgroupG/H
centerC2×C4#14519750
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×#344C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#344D61
hypocenterC61#10772
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. #21445551C2semidirect
5. C2×#16839523C2semidirect
6. C2×#16839523C2semidirect
7. C2×C61×#344C2semidirect
12. #16840230C4semidirect
22. #16839778C4semidirect
26. #16839764C4semidirect
67. #14519736C2×C4semidirect
71. C2×C4⋊Dic61C2×C4semidirect
102. C2×M16×C61C2×C4semidirect
103. C2×M16×C61C2×C4semidirect
331. C2×#344D61semidirect
372. C61#10772semidirect
402. C2×M16C4×D61semidirect
403. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
C2×#168397782
#214543392
#214455512
C2×#168395232
C2×C61×#3442
#1077261