Finite Group Databasebeta

← #11675901

Subgroups — #11675901

Order 768.

1706 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85890C212yesmaximal, complemented
2C3×#2951C211yesmaximal, characteristic
3#83898C211yesmaximal, characteristic
4Dic3×#135C211yesmaximal, characteristic
5#82560C212yesmaximal, complemented
6#79762C214yesmaximal, complemented
7#79761C214yesmaximal, complemented
8#79707C212yesmaximal, complemented
9#79706C212yesmaximal, complemented
10#79575C212yesmaximal, complemented
11#79574C212yesmaximal, complemented
12C2×#4878C211yesmaximal, characteristic, complemented
13#79139C211yesmaximal, characteristic, complemented
14#78873C212yesmaximal, complemented
15#78871C212yesmaximal, complemented
16#78546C211yesmaximal, characteristic, complemented
17#78542C211yesmaximal, characteristic, complemented
18#61376—31nomaximal, Sylow
19C2×C3×#135C2211yescharacteristic
20C3×#520C2212yes—
21#5827C2212yes—
22Dic3×#41C2212yes—
23#5570C2214yes—
24#5569C2214yes—
25#5543—24no—
26#5543C2214yes—
27#5539—24no—
28#5539C2214yes—
29#5519—22no—
30#5519C2212yes—
31#5517—22no—
32#5517C2211yescharacteristic
33#5500—22no—
34#5500C2212yes—
35C2×#890—22no—
36C2×#890C2212yes—
37C3×#440C2214yes—
38C3×C4×Q16C2212yes—
39C3×C4×D8C2212yes—
40C2×C3×#107C2211yescharacteristic
41C3×#417C2212yes—
42C3×#405C2211yescharacteristic
43#5233C2212yes—
44#5232C2212yes—
45Dic3×Q16C2211yescharacteristic
46Dic3×Q16C2212yes—
47Dic3×Q16C2211yescharacteristic
48#5164C2214yes—
49Dic3×SD16C2212yes—
50Dic3×SD16C2212yes—
51Dic3×D8C2211yescharacteristic
52Dic3×D8C2211yescharacteristic
53Dic3×D8C2212yes—
54#5121C2212yes—
55#5119C2211yescharacteristic
56#5118C2211yescharacteristic
57#5106C2211yescharacteristic
58C2×#818C2211yescharacteristic
59#5101C2211yescharacteristic
60C2×C8×Dic3C2211yescharacteristic
61#5022C2214yes—
62#5020C2214yes—
63#5009C2214yes—
64#5008C2214yes—
65#4999C2211yescharacteristic
66#4964C2212yes—
67C2×#808C2212yes—
68#4878C2212yes—
69#4878C2212yes—
70#4878C2212yes—
71#4878C2212yes—
72#4878C2212yes—
73#4876C2214yes—
74#4874C2212yes—
75#4799C2214yes—
76#4796C2214yes—
77#4796C2214yes—
78#4791C2214yes—
79#4766C2214yes—
80#4766C2214yes—
81#4763C2214yes—
82#4761C2214yes—
83#4702C2214yes—
84#4701C2212yes—
85#4701C2212yes—
86#4683C2212yes—
87#4682C2212yes—
88C8×Dic6C2212yes—
89C8×Dic6C2212yes—
90#3504—32no—
91#3388—34no—
92#3388—34no—
93#3383—32no—
94#3383—32no—
95#3380—32no—
96#3380—32no—
97#3164—31no—
98#3164—31no—
99C2×#499—31no—
100#3084—32no—
101#3084—32no—
102#3080—32no—
103#2960—31no—
104#2951—31no—
105#2951D311yescharacteristic
106C4×#135—31no—
107C2×C3×#41C2312yes—
108C3×#135Q818yescomplemented
109C2×C3×Q16C2311yescharacteristic
110C2×C3×SD16C2312yes—
111C2×C3×D8C2311yescharacteristic
112C2×C3×M16C2311yescharacteristic
113C22×C3×C8C2311yescharacteristic
114C3×C4×Q8C2314yes—
115C3×C4×D4—24no—
116C3×C4×D4C2314yes—
117C3×#117—22no—
118C3×#117C2311yescharacteristic
119C2×C3×C4⋊C4—22no—
120C2×C3×C4⋊C4C2312yes—
121Q8×Dic3C2312yes—
122Q8×Dic3C2312yes—
123Q8×Dic3C2314yes—
124D4×Dic3—22no—
125D4×Dic3—22no—
126D4×Dic3—24no—
127D4×Dic3C2312yes—
128D4×Dic3C2312yes—
129D4×Dic3C2314yes—
130#926—22no—
131#926—22no—
132#926C2312yes—
133#926C2311yescharacteristic
134C2×C4⋊Dic3—22no—
135C2×C4⋊Dic3—22no—
136C2×C4⋊Dic3—24no—
137C2×C4⋊Dic3C2312yes—
138#924—24no—
139#924—24no—
140#924—24no—
141C2×Dic3⋊C4—22no—
142C2×Dic3⋊C4—24no—
143C2×C4×Dic3—22no—
144C2×C4×Dic3C2311yescharacteristic
145C2×#208C2311yescharacteristic
146C22×C3⋊C8C2311yescharacteristic
147#890—22no—
148#890—22no—
149#890—24no—
150#890—24no—
151#890—24no—
152#890—22no—
153#890—22no—
154#890D411yescharacteristic
155#890D411yescharacteristic
156#890C2314yes—
157#890C2314yes—
158#890C2314yes—
159#888—24no—
160#888—22no—
161#888—22no—
162#888D411yescharacteristic
163#888D411yescharacteristic
164#888C2314yes—
165#878—24no—
166#878—24no—
167#878—24no—
168#878—24no—
169#878—24no—
170#878D412yes—
171#878D412yes—
172#870—24no—
173#870C2314yes—
174#869—24no—
175#869C2314yes—
176C4×Dic6—24no—
177C4×Dic6—24no—
178C4×Dic6C2312yes—
179C4×Dic6C2312yes—
180C3×#107C2312yes—
181C3×#107C2312yes—
182C3×#103C2314yes—
183C3×#102C2314yes—
184C3×#97C2312yes—
185C3×C4×C8C2312yes—
186#835C2312yes—
187#835C2312yes—
188#835C2314yes—
189#832C2312yes—
190#832C2312yes—
191#832C2314yes—
192#818C2314yes—
193#818C2311yescharacteristic
194#818C2311yescharacteristic
195#818C2312yes—
196#818C2312yes—
197#817C2314yes—
198#817C2312yes—
199#815C2312yes—
200Dic3⋊C8C2312yes—
201Dic3⋊C8C2312yes—
202Dic3⋊C8C2312yes—
203Dic3⋊C8C2312yes—
204C8×Dic3C2312yes—
205C8×Dic3C2311yescharacteristic
206C8×Dic3C2311yescharacteristic
207C8×Dic3C2311yescharacteristic
208C8×Dic3C2311yescharacteristic
209#808C2314yes—
210#808C2314yes—
211#808C2314yes—
212#807C2314yes—
213C4⋊(C3⋊C8)C2312yes—
214C4⋊(C3⋊C8)C2312yes—
215C2×#135C2×D311yescharacteristic
216#557—34no—
217#557—34no—
218#554—34no—
219#554—34no—
220#554—64no—
221#554—64no—
222#544—32no—
223#544—32no—
224#544—62no—
225#544—62no—
226#533—31no—
227#533—62no—
228C2×#125—32no—
229C2×#125—62no—
230#520—32no—
231#520C2×D312yes—
232C4×#41—32no—
233#501—34no—
234#501—34no—
235#500—32no—
236#500—32no—
237#499—32no—
238#499—32no—
239#499—32no—
240#499—32no—
241#499—32no—
242#499—32no—
243#479—34no—
244#479—34no—
245#479—34no—
246#479—34no—
247#478—34no—
248#478—34no—
249#478—34no—
250#478—34no—
251#476—34no—
252#476—34no—
253#475—34no—
254#475—34no—
255#446—32no—
256#446—32no—
257C8×Q8—32no—
258C8×Q8—32no—
259#440—34no—
260#440C2×D314yes—
261C4×Q16—31no—
262C4×Q16—31no—
263C4×Q16—32no—
264C4×Q16C2×D312yes—
265C4×SD16—32no—
266C4×SD16—32no—
267C4×D8—31no—
268C4×D8—31no—
269C4×D8—32no—
270C4×D8C2×D312yes—
271#428—32no—
272#428—32no—
273#427—31no—
274C2×#107—31no—
275C2×#107C2×D311yescharacteristic
276C2×#106—32no—
277#424—31no—
278#424—31no—
279#423—31no—
280#417—32no—
281#417C2×D312yes—
282#416—32no—
283#405—31no—
284#405C2×D311yescharacteristic
285C2×C4×C8—31no—
286C3×#41—24no—
287C3×#41C2×Q818yes—
288C2×C3×Q8C2412yes—
289C2×C3×D4—22no—
290C2×C3×D4C2412yes—
291C22×C3×C4—22no—
292C22×C3×C4—22no—
293C22×C3×C4—24no—
294C22×C3×C4C2411yescharacteristic
295C22×Dic3—22no—
296C22×Dic3—22no—
297C22×Dic3—22no—
298C22×Dic3—24no—
299C22×Dic3C2×D411yescharacteristic
300C22×Dic3C2×D411yescharacteristic
301C2×Dic6—24no—
302C2×Dic6—24no—
303C2×Dic6C2×D412yes—
304C2×Dic6C2×D412yes—
305C3×Q16C2×Q814yes—
306C3×SD16C2×Q818yes—
307C3×D8C2×Q814yes—
308C3×M16—22no—
309C2×C3×C8C2×Q814yes—
310C2×C3×C8C2412yes—
311C2×C3×C8C2411yescharacteristic
312C2×C3×C8C2411yescharacteristic
313C3×C4⋊C4—22no—
314C3×C4⋊C4—24no—
315C3×C4⋊C4C2×D411yescharacteristic
316C3×C4⋊C4C2×D411yescharacteristic
317C3×C4⋊C4C2414yes—
318C3×C22⋊C4—24no—
319C3×C22⋊C4C2×D412yes—
320C3×C42—24no—
321C3×C42C2412yes—
322#217—22no—
323#217—22no—
324#217—22no—
325#217—24no—
326#217C2×D412yes—
327C4⋊Dic3—24no—
328C4⋊Dic3—22no—
329C4⋊Dic3—24no—
330C4⋊Dic3—22no—
331C4⋊Dic3—24no—
332C4⋊Dic3—24no—
333C4⋊Dic3—24no—
334C4⋊Dic3—22no—
335C4⋊Dic3C2×D412yes—
336C4⋊Dic3C2×D412yes—
337C4⋊Dic3C2×D411yescharacteristic
338C4⋊Dic3C2×D411yescharacteristic
339C4⋊Dic3C2412yes—
340C4⋊Dic3C2412yes—
341C4⋊Dic3C2414yes—
342Dic3⋊C4—24no—
343Dic3⋊C4—24no—
344Dic3⋊C4—24no—
345Dic3⋊C4—24no—
346Dic3⋊C4—24no—
347Dic3⋊C4—24no—
348Dic3⋊C4C2×D412yes—
349Dic3⋊C4C2×D412yes—
350Dic3⋊C4C2×D412yes—
351Dic3⋊C4C2×D412yes—
352C4×Dic3—22no—
353C4×Dic3—24no—
354C4×Dic3—22no—
355C4×Dic3—22no—
356C4×Dic3C2×D411yescharacteristic
357C4×Dic3C2×D411yescharacteristic
358C4×Dic3C2411yescharacteristic
359C4×Dic3C2411yescharacteristic
360C4×Dic3C2412yes—
361#208—22no—
362C2×C3⋊C8—22no—
363C2×C3⋊C8C2412yes—
364C2×C3⋊C8C2411yescharacteristic
365C2×C3⋊C8C2411yescharacteristic
366C2×#41C22×D312yes—
367#135Dic618yescomplemented
368C2×Q16C22×D311yescharacteristic
369C2×SD16C22×D312yes—
370C2×D8C22×D311yescharacteristic
371C2×M16—31no—
372C2×M16C22×D311yescharacteristic
373C22×C8—31no—
374C22×C8C22×D311yescharacteristic
375C4⋊Q8—31no—
376C4⋊Q8—31no—
377C4⋊Q8—34no—
378C4⋊Q8—34no—
379C4⋊Q8—64no—
380C4⋊Q8—62no—
381C4⋊Q8—64no—
382C4⋊Q8—62no—
383#125—31no—
384#125—31no—
385#125—34no—
386#125—34no—
387#125—34no—
388#125—34no—
389#125—64no—
390#125—62no—
391#125—62no—
392#125—62no—
393#125—64no—
394#125—64no—
395#125—64no—
396#125—62no—
397C22⋊Q8—32no—
398C22⋊Q8—32no—
399C22⋊Q8—64no—
400C22⋊Q8—64no—
401C22⋊Q8—64no—
402C22⋊Q8—64no—
403C22⋊Q8—64no—
404C22⋊Q8—64no—
405C22⋊Q8—64no—
406C22⋊Q8—64no—
407C4×Q8—32no—
408C4×Q8—32no—
409C4×Q8—32no—
410C4×Q8—32no—
411C4×Q8—34no—
412C4×Q8—64no—
413C4×Q8—64no—
414C4×Q8C22×D314yes—
415C4×D4—24no—
416C4×D4—32no—
417C4×D4—32no—
418C4×D4—34no—
419C4×D4—64no—
420C4×D4—62no—
421C4×D4—62no—
422C4×D4C22×D314yes—
423#117—22no—
424#117—31no—
425#117—32no—
426#117—62no—
427#117—62no—
428#117C22×D311yescharacteristic
429C2×C4⋊C4—22no—
430C2×C4⋊C4—32no—
431C2×C4⋊C4—62no—
432C2×C4⋊C4—64no—
433C2×C4⋊C4—64no—
434C2×C4⋊C4—62no—
435C2×C4⋊C4—62no—
436C2×C4⋊C4C22×D312yes—
437C2×C42—31no—
438C2×C42—62no—
439#107—32no—
440#107—32no—
441#107—31no—
442#107—31no—
443#107—34no—
444#107—34no—
445#107C22×D312yes—
446#107C22×D312yes—
447#106—32no—
448#106—34no—
449#106—34no—
450#106—34no—
451#106—34no—
452C4⋊C8—32no—
453C4⋊C8—32no—
454C4⋊C8—32no—
455C4⋊C8—32no—
456C4⋊C8—32no—
457C4⋊C8—32no—
458#103—32no—
459#103—32no—
460#103—34no—
461#103C22×D314yes—
462#102—32no—
463#102—32no—
464#102—34no—
465#102C22×D314yes—
466#97—32no—
467#97C22×D312yes—
468C4×C8—31no—
469C4×C8—31no—
470C4×C8—31no—
471C4×C8—31no—
472C4×C8—32no—
473C4×C8C22×D312yes—
474C23×C3—22no—
475C23×C3C22×D411yescharacteristic
476C3×Q8—22no—
477C3×Q8C22×Q814yes—
478C3×D4—28no—
479C3×D4—22no—
480C3×D4C22×Q814yes—
481C2×C3×C4—22no—
482C2×C3×C4—28no—
483C2×C3×C4—24no—
484C2×C3×C4—84no—
485C2×C3×C4—82no—
486C2×C3×C4C22×D412yes—
487C2×C3×C4C22×D412yes—
488C2×C3×C4C22×D411yescharacteristic
489C2×C3×C4C22×Q812yes—
490C2×C3×C42−1+412yes—
491C2×C3×C4C2511yescharacteristic
492C2×Dic3—22no—
493C2×Dic3—22no—
494C2×Dic3—24no—
495C2×Dic3—41no—
496C2×Dic3—41no—
497C2×Dic3—84no—
498C2×Dic3—82no—
499C2×Dic3—82no—
500C2×Dic3—82no—
501C2×Dic3C22×D411yescharacteristic
502C2×Dic3C22×D411yescharacteristic
503C2×Dic3C22×D412yes—
504C2×Dic3C22×D411yescharacteristic
505C2×Dic3C22×D411yescharacteristic
506C2×Dic3C22×D412yes—
507Dic6—42no—
508Dic6—42no—
509Dic6—84no—
510Dic6—84no—
511C3×C8—22no—
512C3×C8C22×Q812yes—
513C3×C8C22×Q812yes—
514C3⋊C8—21no—
515C3⋊C8—21no—
516C3⋊C8—22no—
517#41—24no—
518#41C2×Dic618yes—
519C2×Q8—32no—
520C2×Q8—32no—
521C2×Q8—64no—
522C2×Q8—64no—
523C2×Q8C23×D312yes—
524C2×D4—22no—
525C2×D4C23×D312yes—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—24no—
529C22×C4—31no—
530C22×C4—31no—
531C22×C4—62no—
532C22×C4—62no—
533C22×C4—64no—
534C22×C4—62no—
535C22×C4C23×D311yescharacteristic
536Q16C2×Dic614yes—
537SD16C2×Dic618yes—
538D8C2×Dic614yes—
539M16—22no—
540M16—62no—
541C2×C8—31no—
542C2×C8—31no—
543C2×C8—32no—
544C2×C8—62no—
545C2×C8C2×Dic614yes—
546C2×C8C23×D312yes—
547C2×C8C23×D311yescharacteristic
548C2×C8C23×D311yescharacteristic
549C4⋊C4—22no—
550C4⋊C4—24no—
551C4⋊C4—31no—
552C4⋊C4—32no—
553C4⋊C4—32no—
554C4⋊C4—31no—
555C4⋊C4—32no—
556C4⋊C4—32no—
557C4⋊C4—32no—
558C4⋊C4—32no—
559C4⋊C4—34no—
560C4⋊C4—32no—
561C4⋊C4—32no—
562C4⋊C4—64no—
563C4⋊C4—64no—
564C4⋊C4—64no—
565C4⋊C4—64no—
566C4⋊C4—64no—
567C4⋊C4—64no—
568C4⋊C4—64no—
569C4⋊C4—62no—
570C4⋊C4—62no—
571C4⋊C4—64no—
572C4⋊C4—62no—
573C4⋊C4—64no—
574C4⋊C4—64no—
575C4⋊C4—64no—
576C4⋊C4D3×D411yescharacteristic
577C4⋊C4D3×D411yescharacteristic
578C4⋊C4C23×D314yes—
579C22⋊C4—24no—
580C22⋊C4—32no—
581C22⋊C4—62no—
582C22⋊C4—62no—
583C22⋊C4—64no—
584C22⋊C4—62no—
585C22⋊C4D3×D412yes—
586C42—24no—
587C42—31no—
588C42—31no—
589C42—31no—
590C42—31no—
591C42—32no—
592C42—62no—
593C42—62no—
594C42—64no—
595C42—62no—
596C42C23×D312yes—
597C22×C3—21no—
598C22×C3—21no—
599C22×C3—24no—
600C22×C3—42no—
601C22×C3—42no—
602C22×C3D4×Q812yes—
603C22×C3C23×D411yescharacteristic
604C3×C4—22no—
605C3×C4—22no—
606C3×C4—24no—
607C3×C4—44no—
608C3×C4D4×Q812yes—
609C3×C4C23×Q811yescharacteristic
610C3×C4C2×2−1+411yescharacteristic
611Dic3—21no—
612Dic3—21no—
613Dic3—22no—
614Dic3—22no—
615Dic3—21no—
616Dic3—21no—
617Dic3—42no—
618Dic3—42no—
619Dic3—44no—
620C23—22no—
621C23C2×D3×D411yescharacteristic
622Q8—22no—
623Q8—122no—
624Q8—122no—
625Q8—244no—
626Q8—244no—
627Q8C22×Dic614yes—
628D4—28no—
629D4—22no—
630D4C22×Dic614yes—
631C2×C4—22no—
632C2×C4—28no—
633C2×C4—24no—
634C2×C4—31no—
635C2×C4—31no—
636C2×C4—31no—
637C2×C4—31no—
638C2×C4—32no—
639C2×C4—32no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—64no—
643C2×C4—84no—
644C2×C4—82no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—242no—
648C2×C4—242no—
649C2×C4—244no—
650C2×C4—242no—
651C2×C4C22×Dic612yes—
652C2×C4C2×D3×D412yes—
653C2×C4C2×D3×D412yes—
654C2×C4C2×D3×D411yescharacteristic
655C2×C4#101012yes—
656C2×C4C24×D311yescharacteristic
657C8—22no—
658C8—61no—
659C8—61no—
660C8—62no—
661C8C22×Dic612yes—
662C8C22×Dic612yes—
663C2×C3—22no—
664C2×C3—44no—
665C2×C3C2×D4×Q811yescharacteristic
666C2×C3#358411yescharacteristic
667C2×C3#358411yescharacteristic
668C22—21no—
669C22—21no—
670C22—24no—
671C22—42no—
672C22—42no—
673C22D4×Dic612yes—
674C22C22×D3×D411yescharacteristic
675C4—22no—
676C4—22no—
677C4—24no—
678C4—44no—
679C4—61no—
680C4—61no—
681C4—62no—
682C4—61no—
683C4—61no—
684C4—62no—
685C4—122no—
686C4—124no—
687C4—122no—
688C4D4×Dic612yes—
689C4C23×Dic611yescharacteristic
690C4C2×#101011yescharacteristic
691C3#6137611yescharacteristic, complemented, Sylow
692C2—22no—
693C2—44no—
694C2C2×D4×Dic611yescharacteristic
695C2#8720311yescharacteristic
696C2#8721011yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2951C2
socleC22×C3C23×D4
prime radicalC2×C3×#135C22
hypercenter#2951D3
hypocenterC3#61376
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #85890C2semidirect
5. #82560C2semidirect
6. #79762C2semidirect
7. #79761C2semidirect
8. #79707C2semidirect
9. #79706C2semidirect
10. #79575C2semidirect
11. #79574C2semidirect
12. C2×#4878C2semidirect
13. #79139C2semidirect
14. #78873C2semidirect
15. #78871C2semidirect
16. #78546C2semidirect
17. #78542C2semidirect
108. C3×#135Q8semidirect
367. #135Dic6semidirect
691. C3#61376semidirect

Maximal subgroups

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

subgroupindex
#858902
C3×#29512
#838982
Dic3×#1352
#825602
#797622
#797612
#797072
#797062
#795752
#795742
C2×#48782
#791392
#788732
#788712
#785462
#785422
#613763