Finite Group Databasebeta

← #29707447

Subgroups — #29707447

Order 15616.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic, complemented
2C2×#16839779C211yesmaximal, characteristic
3C2×#16839779C211yesmaximal, characteristic
4#21446993C218yesmaximal, complemented
5C2×#16839532C211yesmaximal, characteristic, complemented
6#21445528C211yesmaximal, characteristic, complemented
7C2×#16839518C211yesmaximal, characteristic, complemented
8C2×C61×#410C211yesmaximal, characteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840393—22no—
12#16840390—22no—
13#16840390—22no—
14#16840388—22no—
15#16840319—22no—
16#16840315—22no—
17#16840315C2214yescomplemented
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839779—22no—
21#16839779—22no—
22#16839779—22no—
23#16839779C2214yes—
24C2×#14519704—22no—
25C2×#14519704C2214yes—
26#16839532C2214yescomplemented
27#16839521C2214yescomplemented
28#16839518C2214yescomplemented
29C22×#41×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C2×#117×C61C2211yescharacteristic
32C61×#410—22no—
33C61×#410C2214yes—
34C2×#99×C61—22no—
35C2×#11758194—21no—
36C2×#11758194—21no—
37C2×#11758194—21no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40C2×#11758194—21no—
41C2×#11758194—21no—
42C2×#11758194—21no—
43C2×#11758194—21no—
44C2×#11758194—21no—
45C2×#11758194C2312yes—
46C2×#11758194C2312yes—
47C22×#92418—21no—
48C22×#92418—21no—
49#14519777—24no—
50#14519776—24no—
51#14519776—24no—
52#14519769—24no—
53#14519768—24no—
54#14519768—24no—
55#14519756—28no—
56#14519754—28no—
57#14519754—28no—
58C4×Dic122—28no—
59#14519736—21no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736D412yes—
64#14519736D412yes—
65#14519736D412yes—
66#14519736C2312yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×C4×Dic61D412yes—
72C2×C4×Dic61C2312yes—
73#14519704—28no—
74#14519704—28no—
75C2×#41×C61—21no—
76C2×#41×C61—21no—
77C2×#41×C61—21no—
78C2×#41×C61—21no—
79C2×#41×C61—21no—
80C2×#41×C61—21no—
81C2×#41×C61—21no—
82C2×#41×C61—21no—
83C2×#41×C61—21no—
84C2×#41×C61—21no—
85C2×#41×C61Q812yescomplemented
86C2×#41×C61C2312yes—
87C22×Q8×C61Q811yescharacteristic, complemented
88C22×D4×C61—21no—
89C22×D4×C61Q811yescharacteristic, complemented
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61D412yes—
98#117×C61C2312yes—
99#117×C61C2312yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61—21no—
106#99×C61—28no—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109#41×C61—22no—
110#41×C61—22no—
111#41×C61—22no—
112#41×C61—22no—
113#41×C61—22no—
114#41×C61—22no—
115#41×C61—22no—
116#41×C61—22no—
117#41×C61—41no—
118#41×C61—41no—
119#41×C61—41no—
120#41×C61—41no—
121#41×C61—41no—
122#41×C61—41no—
123#41×C61—41no—
124#41×C61—41no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C2×Q812yes—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—41no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61—41no—
142C2×D4×C61C2×Q812yes—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—41no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61—41no—
151C22×C4×C61—41no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×Q811yescharacteristic
158C22×C4×C61C2×Q811yescharacteristic
159C22×C4×C61#4111yescharacteristic
160C22×C4×C61#4111yescharacteristic
161C22×C4×C61#4111yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C22×C4×C61#4111yescharacteristic
164C22×C4×C61#4111yescharacteristic
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—22no—
167C22×Dic61—22no—
168C2×Dic122—28no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—41no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61C2×D414yes—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61—41no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C4⋊Dic61—28no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190#11758196—28no—
191#11758196—28no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61C2×D414yes—
195C4×Dic61C2×D414yes—
196#11758194—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194—22no—
200#11758194—41no—
201#11758194—41no—
202#11758194—41no—
203#11758194—41no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—41no—
207C2×#92418—41no—
208C2×#92418—41no—
209C2×#92418—41no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—81no—
214C23×C61—82no—
215C23×C61#12011yescharacteristic
216C23×C61#12011yescharacteristic
217C23×C61C22⋊Q811yescharacteristic
218C23×C61C22⋊Q811yescharacteristic
219C23×C61#12411yescharacteristic
220C23×C61#12411yescharacteristic
221C23×C61C22×D411yescharacteristic
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—82no—
244C2×C4×C61—82no—
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61C22⋊Q811yescharacteristic
248C2×C4×C61C22⋊Q811yescharacteristic
249C2×C4×C61C22⋊Q811yescharacteristic
250C2×C4×C61C22⋊Q811yescharacteristic
251C2×C4×C61C22⋊Q811yescharacteristic
252C2×C4×C61C22⋊Q811yescharacteristic
253C2×C4×C61C22⋊Q811yescharacteristic
254C2×C4×C61C22⋊Q811yescharacteristic
255C2×C4×C61C22⋊Q811yescharacteristic
256C2×C4×C61C22⋊Q811yescharacteristic
257C2×C4×C61#12411yescharacteristic
258C2×C4×C61#12411yescharacteristic
259C2×C4×C61#12411yescharacteristic
260C2×C4×C61#12411yescharacteristic
261C2×C4×C61#12411yescharacteristic
262C2×C4×C61#12411yescharacteristic
263C2×C4×C61#12411yescharacteristic
264C2×C4×C61#12411yescharacteristic
265C2×C4×C61#12411yescharacteristic
266C2×C4×C61#12411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×Q811yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—82no—
276C2×Dic61—82no—
277Dic122—88no—
278#92418—42no—
279#92418—42no—
280#23472—611nomaximal, Sylow
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—42no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61—82no—
292C22×C61#39311yescharacteristic
293C22×C61#39311yescharacteristic
294C22×C61#39311yescharacteristic
295C22×C61#39311yescharacteristic
296C22×C61C2×#12011yescharacteristic
297C22×C61C2×C22⋊Q811yescharacteristic
298C22×C61C2×#12411yescharacteristic
299C4×C61—44no—
300C4×C61—42no—
301C4×C61—82no—
302C4×C61#39311yescharacteristic
303C4×C61#39311yescharacteristic
304C4×C61#39311yescharacteristic
305C4×C61#39311yescharacteristic
306C4×C61C2×C22⋊Q811yescharacteristic
307C4×C61C2×C22⋊Q811yescharacteristic
308C4×C61C2×#12411yescharacteristic
309C4×C61C2×#12411yescharacteristic
310Dic61—44no—
311Dic61—44no—
312#3458—611no—
313C2×#421—611no—
314C2×#421—611no—
315C2×#410D6111yescharacteristic
316#2031—618no—
317C2×#344—611no—
318#1733—611no—
319C2×H(3,ℤ/4)—611no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—82no—
328C2×C61C2×#39311yescharacteristic
329C22×#41C2×D6111yescharacteristic
330C22×M16—611no—
331C22×M16—611no—
332#544—1222no—
333#544—1222no—
334#544—1222no—
335#543—1222no—
336#533—614no—
337#533—1222no—
338#533—1222no—
339C2×#117—611no—
340C2×#117—611no—
341C2×#117C2×D6111yescharacteristic
342C2×#117C2×D6111yescharacteristic
343#421—614no—
344#421—1222no—
345#421—1222no—
346#421—1222no—
347C2×#104—614no—
348C2×#104—1222no—
349#410—22no—
350#410C2×D6114yes—
351C2×#99—22no—
352#344—614no—
353#339—614no—
354H(3,ℤ/4)—614no—
355C61#2347211yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41Dic12212yescomplemented
367C2×#41C22×D6112yes—
368C22×Q8Dic12211yescharacteristic, complemented
369C22×D4—21no—
370C22×D4Dic12211yescharacteristic, complemented
371C23×C4—21no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—612no—
374C2×M16—612no—
375C2×M16—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—1221no—
380C2×M16—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C22×C8—1221no—
386C22×C8—1221no—
387C4⋊Q8—1224no—
388C4⋊Q8—1224no—
389C4⋊Q8—1224no—
390#125—1224no—
391#125—1224no—
392#125—1224no—
393C22⋊Q8—1228no—
394C22⋊Q8—1228no—
395C22⋊Q8—1228no—
396C4×Q8—1228no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—612no—
403#117—612no—
404#117—612no—
405#117—612no—
406#117—1221no—
407#117—1221no—
408#117—1221no—
409#117—1221no—
410#117C22⋊D6112yes—
411#117C22×D6112yes—
412#117C22×D6112yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4C22⋊D6112yes—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—612no—
424C2×C42—612no—
425#104—1228no—
426#104—1228no—
427#99—28no—
428C24—21no—
429C24C2×C22⋊D6111yescharacteristic
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—1228no—
452C2×Q8C2×Dic12212yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×Dic12212yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4C2×Dic12211yescharacteristic
477C22×C4C2×Dic12211yescharacteristic
478C22×C4#1175822211yescharacteristic
479C22×C4#1175822211yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4#1175822411yescharacteristic
485C22×C4#1175822411yescharacteristic
486C22×C4#1175822411yescharacteristic
487C22×C4#1175822411yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490M16—1222no—
491M16—1222no—
492M16—1222no—
493M16—1222no—
494M16—2441no—
495M16—2441no—
496M16—2441no—
497M16—2441no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—2441no—
501C2×C8—2441no—
502C2×C8—2441no—
503C2×C8—2441no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—1228no—
509C4⋊C4—1228no—
510C4⋊C4—1228no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—41no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4C2×C22⋊D6114yes—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—41no—
527C42—614no—
528C42—614no—
529C42—1222no—
530C42—1222no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—81no—
535C23—82no—
536C23#1451974511yescharacteristic
537C23#1451974511yescharacteristic
538C23#1451975611yescharacteristic
539C23#1451975611yescharacteristic
540C23#1451976411yescharacteristic
541C23#1451976411yescharacteristic
542C23C22×C22⋊D6111yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—4888no—
548D4—22no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—41no—
555D4—41no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—4882no—
570C2×C4—4882no—
571C2×C4#1451974511yescharacteristic
572C2×C4#1451974511yescharacteristic
573C2×C4#1451975411yescharacteristic
574C2×C4#1451975411yescharacteristic
575C2×C4#1451975411yescharacteristic
576C2×C4#1451975411yescharacteristic
577C2×C4#1451975411yescharacteristic
578C2×C4#1451975411yescharacteristic
579C2×C4#1451975411yescharacteristic
580C2×C4#1451975411yescharacteristic
581C2×C4#1451975611yescharacteristic
582C2×C4#1451975611yescharacteristic
583C2×C4#1451976311yescharacteristic
584C2×C4#1451976311yescharacteristic
585C2×C4#1451976311yescharacteristic
586C2×C4#1451976311yescharacteristic
587C2×C4#1451976311yescharacteristic
588C2×C4#1451976311yescharacteristic
589C2×C4#1451976311yescharacteristic
590C2×C4#1451976311yescharacteristic
591C2×C4#1451976411yescharacteristic
592C2×C4#1451976411yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C22×Dic12211yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822411yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C8—2442no—
600C8—2442no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22#1683967411yescharacteristic
613C22#1683967411yescharacteristic
614C22#1683967411yescharacteristic
615C22#1683967411yescharacteristic
616C22C2×#1451974511yescharacteristic
617C22C2×#1451975611yescharacteristic
618C22C2×#1451976411yescharacteristic
619C4—44no—
620C4—42no—
621C4—82no—
622C4—2444no—
623C4—2444no—
624C4#1683967411yescharacteristic
625C4#1683967411yescharacteristic
626C4#1683967411yescharacteristic
627C4#1683967411yescharacteristic
628C4C2×#1451975411yescharacteristic
629C4C2×#1451975411yescharacteristic
630C4C2×#1451976311yescharacteristic
631C4C2×#1451976311yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2C2×#1683967411yescharacteristic
subgroupG/H
centerC4C2×#14519763
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#410C2
socleC2×C61C2×#393
prime radicalC22×#41×C61C22
hypercenterC2×#410D61
hypocenterC61#23472
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #21459224C2semidirect
4. #21446993C2semidirect
5. C2×#16839532C2semidirect
6. #21445528C2semidirect
7. C2×#16839518C2semidirect
17. #16840315C22semidirect
26. #16839532C22semidirect
27. #16839521C22semidirect
28. #16839518C22semidirect
85. C2×#41×C61Q8semidirect
87. C22×Q8×C61Q8semidirect
89. C22×D4×C61Q8semidirect
355. C61#23472semidirect
366. C2×#41Dic122semidirect
368. C22×Q8Dic122semidirect
370. C22×D4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214592242
C2×#168397792
#214469932
C2×#168395322
#214455282
C2×#168395182
C2×C61×#4102
#2347261