Finite Group Databasebeta

← #11675915

Subgroups — #11675915

Order 768.

1742 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86280C211yesmaximal, characteristic
2#86258C211yesmaximal, characteristic, complemented
3C3×#2961C211yesmaximal, characteristic, complemented
4#79978C214yesmaximal, complemented
5#79978C214yesmaximal, complemented
6#79977C214yesmaximal, complemented
7#79976C214yesmaximal, complemented
8#79101C212yesmaximal, complemented
9#79090C212yesmaximal
10#79068C211yesmaximal, characteristic, complemented
11#79059C211yesmaximal, characteristic
12#78572C212yesmaximal
13#78546C212yesmaximal, complemented
14C2×#4710C211yesmaximal, characteristic
15#78479C211yesmaximal, characteristic, complemented
16#61377—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C22×#204C2211yescharacteristic
19#5730—21no—
20#5727—21no—
21#5722—21no—
22#5721—21no—
23#5690—22no—
24#5685—22no—
25#5634—22no—
26D3×C22⋊Q8—22no—
27#5623—22no—
28#5622—22no—
29#5579C2214yes—
30#5579C2212yes—
31Q8×D12C2212yes—
32#5567C2214yescomplemented
33#5567C2212yescomplemented
34Q8×Dic6C2212yescomplemented
35#5521C2212yes—
36#5519C2212yes—
37#5519C2212yes—
38#5516C2212yes—
39C2×C4×D12C2211yescharacteristic
40#5470C2211yescharacteristic
41#5470C2211yescharacteristic
42C2×C4×Dic6C2211yescharacteristic
43C3×#446C2214yescomplemented
44C3×#446C2214yescomplemented
45C3×#424C2212yes—
46C3×#423C2211yescharacteristic
47C3×#405C2212yes—
48C2×C3×#97C2211yescharacteristic
49#5131C2212yes—
50#5121C2212yes—
51#5121C2212yes—
52#5120C2212yes—
53C2×#821C2211yescharacteristic
54#5106C2212yes—
55#5106C2212yes—
56C2×#818C2211yescharacteristic
57C2×#816C2211yescharacteristic
58#4851C2214yescomplemented
59#4851C2212yescomplemented
60#4850C2214yescomplemented
61#4850C2212yescomplemented
62#4847C2214yes—
63#4847C2214yes—
64#4847C2214yes—
65#4846C2212yes—
66#4846C2214yes—
67#4842C2214yes—
68#4842C2212yes—
69#4829C2214yescomplemented
70#4829C2214yescomplemented
71#4829C2214yescomplemented
72#4710C2214yes—
73#4710C2214yes—
74#4710C2214yes—
75#4710C2214yes—
76#4702C2212yescomplemented
77#4702C2214yescomplemented
78#4702C2212yescomplemented
79#4702C2214yescomplemented
80#4702C2212yescomplemented
81#4702C2212yescomplemented
82C4×#204C2214yes—
83C4×#204C2214yes—
84#4683C2214yescomplemented
85#4682C2214yescomplemented
86#3502—31no—
87#3501—31no—
88#3388—34no—
89#3388—34no—
90#3386—34no—
91#3385—34no—
92#3164—32no—
93#3162—31no—
94#3082—32no—
95#3081—32no—
96#3078—31no—
97#3077—31no—
98#2961D311yescharacteristic, complemented
99#2949—32no—
100C2×#440—31no—
101C2×D3×Q8—22no—
102C22×D12C2311yescharacteristic
103C22×C4×D3—21no—
104C22×Dic6C2311yescharacteristic
105C2×C3×M16C2312yes—
106C22×C3×C8C2311yescharacteristic
107C22×C3×C8C2311yescharacteristic
108C3×C4⋊Q8D411yescharacteristic
109C3×C4⋊Q8D411yescharacteristic
110C3×#125D412yes—
111C3×C22⋊Q8D412yes—
112C3×C22⋊Q8D412yes—
113C3×C4×Q8C2312yes—
114C3×C4×Q8C2312yes—
115C3×#117C2312yes—
116C2×C3×C42C2311yescharacteristic
117#946—24no—
118#946—22no—
119Q8×Dic3—22no—
120#944—22no—
121#944—24no—
122C2×(C2×D3)⋊C4—21no—
123#926C2312yes—
124#926C2312yes—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2311yescharacteristic
127#924—21no—
128#924—21no—
129#924—21no—
130#924—21no—
131C2×Dic3⋊C4—21no—
132C2×C4×Dic3—21no—
133C2×#204Q818yescomplemented
134C2×#204C2314yes—
135#897—24no—
136#897—22no—
137#896—24no—
138#896—24no—
139#896—24no—
140#893—22no—
141#893—22no—
142D3×C4⋊C4—24no—
143D3×C4⋊C4—22no—
144#890—22no—
145#890—22no—
146#889—24no—
147#889—24no—
148#889—24no—
149#888—22no—
150#888—24no—
151#888—22no—
152#888—22no—
153#887—22no—
154#887—22no—
155#887—24no—
156#887—22no—
157D3×C22⋊C4—24no—
158#878—24no—
159#878—24no—
160#877—24no—
161C4×D12—21no—
162C4×D12—21no—
163C4×D12C2314yes—
164C4×D12C2314yes—
165C4×D12C2312yes—
166C4×D12C2312yes—
167#870C2314yes—
168#870C2314yes—
169#870C2312yes—
170#870C2312yes—
171#869C2314yes—
172#869C2314yes—
173#869C2311yescharacteristic
174#869C2311yescharacteristic
175#869C2311yescharacteristic
176#869C2311yescharacteristic
177C4×Dic6—21no—
178C4×Dic6—21no—
179C4×Dic6—21no—
180C4×Dic6—21no—
181C4×Dic6—21no—
182C4×Dic6—21no—
183C4×Dic6C2314yes—
184C4×Dic6C2314yes—
185C4×Dic6C2312yes—
186C4×Dic6C2312yes—
187C3×C4⋊C8C2312yes—
188C3×C4⋊C8C2312yes—
189C3×C4⋊C8C2314yes—
190C3×C4⋊C8C2314yes—
191C3×#97C2312yes—
192C3×#97C2312yes—
193C3×#97C2314yes—
194C3×C4×C8C2314yes—
195#821C2314yes—
196#821C2314yes—
197#821C2314yes—
198#818C2314yes—
199#818C2314yes—
200#818C2312yes—
201#818C2314yes—
202#818C2312yes—
203#818C2312yes—
204#818C2312yes—
205#817C2314yes—
206#817C2314yes—
207#817C2312yes—
208#817C2312yes—
209#817C2312yes—
210#817C2312yes—
211#816C2314yes—
212#816C2314yes—
213#816C2314yes—
214C22×SD16—31no—
215Q82—32no—
216Q82—61no—
217#557—34no—
218#557—32no—
219#557—61no—
220#557—62no—
221#554—34no—
222#554—32no—
223#554—61no—
224#554—62no—
225D4×Q8—32no—
226D4×Q8—61no—
227#544—32no—
228#544—32no—
229#544—62no—
230#543—62no—
231#533—31no—
232#533—31no—
233#533—62no—
234#533C2×D311yescharacteristic
235C2×C22⋊Q8—62no—
236#519—32no—
237#518—32no—
238C2×C4×Q8—31no—
239C2×C4×D4—31no—
240#501—32no—
241#501—34no—
242#501—32no—
243#501—34no—
244#501—32no—
245#501—32no—
246#500—34no—
247#499—34no—
248#479—34no—
249#479—34no—
250#479—34no—
251#478—34no—
252#478—34no—
253#478—34no—
254#477—32no—
255#477—34no—
256#476—32no—
257#476—34no—
258#475—32no—
259#475—34no—
260#474—32no—
261#474—34no—
262#446C2×D314yescomplemented
263#446C2×D314yescomplemented
264#440—34no—
265#440—34no—
266#440—34no—
267#440—34no—
268C4×SD16—34no—
269C4×SD16—34no—
270#428—32no—
271#428—32no—
272#427—32no—
273#427—32no—
274C2×#107—31no—
275#424C2×D312yes—
276#423C2×D311yescharacteristic
277#419—32no—
278#418—32no—
279C2×#103—31no—
280C2×#102—31no—
281#405C2×D312yes—
282C2×#97C2×D311yescharacteristic
283C23×D3—21no—
284C2×C3×Q8C2×D412yes—
285C2×C3×Q8C2×D412yes—
286C22×C3×C4C2×D411yescharacteristic
287C22×C3×C4C2×D411yescharacteristic
288C22×C3×C4C2411yescharacteristic
289C22×Dic3—21no—
290C22×Dic3—21no—
291C22×Dic3—21no—
292D3×Q8—84no—
293C2×D12—21no—
294C2×D12—22no—
295C2×D12C2×Q814yes—
296C2×D12C2412yes—
297C2×C4×D3—22no—
298C2×C4×D3—22no—
299C2×C4×D3—24no—
300C2×C4×D3—24no—
301C2×C4×D3—82no—
302C2×Dic6—21no—
303C2×Dic6—22no—
304C2×Dic6—21no—
305C2×Dic6—21no—
306C2×Dic6—21no—
307C2×Dic6—21no—
308C2×Dic6—24no—
309C2×Dic6—24no—
310C2×Dic6C2×Q814yescomplemented
311C2×Dic6C2412yes—
312C3×M16—24no—
313C2×C3×C8—22no—
314C2×C3×C8C2×Q814yescomplemented
315C2×C3×C8C2414yes—
316C2×C3×C8C2412yes—
317C2×C3×C8C2412yes—
318C3×C4⋊C4C2×D412yes—
319C3×C4⋊C4C2×D412yes—
320C3×C4⋊C4C2×D414yes—
321C3×C4⋊C4C2×D412yes—
322C3×C4⋊C4C2×D414yes—
323C3×C4⋊C4C2×D412yes—
324C3×C22⋊C4C2×D414yes—
325C3×C42C2×D411yescharacteristic
326C3×C42C2×D411yescharacteristic
327C3×C42C2414yes—
328C3×C42C2411yescharacteristic
329C3×C42C2411yescharacteristic
330#217—22no—
331#217—22no—
332(C2×D3)⋊C4—22no—
333(C2×D3)⋊C4—22no—
334(C2×D3)⋊C4—24no—
335(C2×D3)⋊C4—24no—
336C4⋊Dic3—21no—
337C4⋊Dic3—21no—
338C4⋊Dic3—21no—
339C4⋊Dic3—21no—
340C4⋊Dic3—21no—
341C4⋊Dic3—24no—
342C4⋊Dic3—24no—
343C4⋊Dic3—21no—
344C4⋊Dic3C2412yes—
345C4⋊Dic3C2412yes—
346C4⋊Dic3C2412yes—
347C4⋊Dic3C2412yes—
348C4⋊Dic3C2412yes—
349C4⋊Dic3C2412yes—
350Dic3⋊C4—22no—
351Dic3⋊C4—22no—
352Dic3⋊C4—22no—
353Dic3⋊C4—22no—
354Dic3⋊C4—22no—
355Dic3⋊C4—22no—
356Dic3⋊C4—24no—
357Dic3⋊C4—24no—
358Dic3⋊C4—24no—
359Dic3⋊C4—24no—
360Dic3⋊C4—24no—
361Dic3⋊C4—24no—
362C4×Dic3—22no—
363C4×Dic3—22no—
364C4×Dic3—22no—
365C4×Dic3—22no—
366C4×Dic3—24no—
367C4×Dic3—24no—
368#204C2×Q8116yescomplemented
369C22×Q8—31no—
370C22×Q8—62no—
371C22×D4—31no—
372C23×C4—61no—
373C2×SD16—38no—
374C2×SD16—34no—
375C2×M16C22×D312yes—
376C22×C8C22×D311yescharacteristic
377C22×C8C22×D311yescharacteristic
378C4⋊Q8—31no—
379C4⋊Q8—31no—
380C4⋊Q8—31no—
381C4⋊Q8—31no—
382C4⋊Q8—34no—
383C4⋊Q8—34no—
384C4⋊Q8—62no—
385C4⋊Q8—64no—
386C4⋊Q8—62no—
387C4⋊Q8—62no—
388C4⋊Q8—62no—
389C4⋊Q8—64no—
390C4⋊Q8D1211yescharacteristic
391C4⋊Q8D1211yescharacteristic
392#125—32no—
393#125—32no—
394#125—34no—
395#125—34no—
396#125—64no—
397#125—62no—
398#125—62no—
399#125—64no—
400#125—64no—
401#125D1212yes—
402C22⋊Q8—62no—
403C22⋊Q8—64no—
404C22⋊Q8—61no—
405C22⋊Q8—61no—
406C22⋊Q8—61no—
407C22⋊Q8—61no—
408C22⋊Q8—62no—
409C22⋊Q8—64no—
410C22⋊Q8—64no—
411C22⋊Q8—64no—
412C22⋊Q8—64no—
413C22⋊Q8—64no—
414C22⋊Q8—64no—
415C22⋊Q8D1212yes—
416C22⋊Q8D1212yes—
417C4×Q8—32no—
418C4×Q8—32no—
419C4×Q8—34no—
420C4×Q8—34no—
421C4×Q8—62no—
422C4×Q8—62no—
423C4×Q8—62no—
424C4×Q8—64no—
425C4×Q8—62no—
426C4×Q8—61no—
427C4×Q8—61no—
428C4×Q8—61no—
429C4×Q8—61no—
430C4×Q8—61no—
431C4×Q8—61no—
432C4×Q8C22×D312yes—
433C4×Q8C22×D312yes—
434C4×D4—32no—
435C4×D4—32no—
436C4×D4—34no—
437C4×D4—34no—
438C4×D4—62no—
439C4×D4—62no—
440C4×D4—61no—
441C4×D4—61no—
442#117—32no—
443#117—32no—
444#117—64no—
445#117C22×D312yes—
446C2×C4⋊C4—31no—
447C2×C4⋊C4—31no—
448C2×C4⋊C4—61no—
449C2×C4⋊C4—64no—
450C2×C4⋊C4—62no—
451C2×C22⋊C4—61no—
452C2×C22⋊C4—64no—
453C2×C42—61no—
454C2×C42C22×D311yescharacteristic
455#107—34no—
456#107—32no—
457#107—32no—
458#107—32no—
459#107—32no—
460#107—34no—
461#107—34no—
462#106—32no—
463#106—32no—
464#106—32no—
465#106—32no—
466#106—34no—
467#106—34no—
468C4⋊C8C22×D312yes—
469C4⋊C8C22×D312yes—
470C4⋊C8C22×D314yes—
471C4⋊C8C22×D314yes—
472#103—34no—
473#103—34no—
474#103—34no—
475#102—34no—
476#102—34no—
477#102—34no—
478#97C22×D312yes—
479#97C22×D312yes—
480#97C22×D314yes—
481C4×C8C22×D314yes—
482C23×C3C22×D411yescharacteristic
483C22×D3—22no—
484C22×D3—24no—
485C22×D3—81no—
486C22×D3—81no—
487C3×Q8—42no—
488C3×Q8—42no—
489C2×C3×C4—41no—
490C2×C3×C4—41no—
491C2×C3×C4C22×D411yescharacteristic
492C2×C3×C4C22×D411yescharacteristic
493C2×C3×C4C22×D414yes—
494C2×C3×C4C22×D414yes—
495C2×C3×C4C22×D411yescharacteristic
496C2×C3×C4C22×D411yescharacteristic
497C2×C3×C4C22×D411yescharacteristic
498C2×C3×C4C22×Q812yes—
499C2×C3×C42−1+412yes—
500C2×C3×C4C2511yescharacteristic
501C2×Dic3—24no—
502C2×Dic3—22no—
503C2×Dic3—22no—
504C2×Dic3—22no—
505C2×Dic3—22no—
506C2×Dic3—22no—
507C2×Dic3—22no—
508C2×Dic3—22no—
509C2×Dic3—81no—
510C2×Dic3—81no—
511D12—24no—
512D12—22no—
513D12C22×Q814yes—
514C4×D3—44no—
515C4×D3—84no—
516C4×D3—82no—
517C4×D3—84no—
518Dic6—24no—
519Dic6—22no—
520Dic6—42no—
521Dic6—42no—
522Dic6—81no—
523Dic6—84no—
524Dic6—81no—
525Dic6—84no—
526Dic6C22×Q814yes—
527C3×C8—22no—
528C3×C8—24no—
529C3×C8C22×Q814yes—
530C24—61no—
531C2×Q8—34no—
532C2×Q8—32no—
533C2×Q8—61no—
534C2×Q8—61no—
535C2×Q8—61no—
536C2×Q8—61no—
537C2×Q8—61no—
538C2×Q8—64no—
539C2×Q8—64no—
540C2×Q8—62no—
541C2×Q8—244no—
542C2×Q8C2×D1212yes—
543C2×Q8C2×D1212yes—
544C2×D4—34no—
545C2×D4—32no—
546C2×D4—61no—
547C2×D4—62no—
548C22×C4—61no—
549C22×C4—61no—
550C22×C4—61no—
551C22×C4—62no—
552C22×C4—64no—
553C22×C4—64no—
554C22×C4—62no—
555C22×C4—242no—
556C22×C4C2×D1211yescharacteristic
557C22×C4C2×D1211yescharacteristic
558C22×C4C23×D311yescharacteristic
559SD16—316no—
560M16—24no—
561C2×C8—22no—
562C2×C8D3×Q814yescomplemented
563C2×C8C23×D314yes—
564C2×C8C23×D312yes—
565C2×C8C23×D312yes—
566C4⋊C4—32no—
567C4⋊C4—32no—
568C4⋊C4—32no—
569C4⋊C4—32no—
570C4⋊C4—32no—
571C4⋊C4—32no—
572C4⋊C4—61no—
573C4⋊C4—61no—
574C4⋊C4—62no—
575C4⋊C4—61no—
576C4⋊C4—62no—
577C4⋊C4—62no—
578C4⋊C4—64no—
579C4⋊C4—64no—
580C4⋊C4—64no—
581C4⋊C4—64no—
582C4⋊C4—64no—
583C4⋊C4—62no—
584C4⋊C4—61no—
585C4⋊C4—62no—
586C4⋊C4—61no—
587C4⋊C4—64no—
588C4⋊C4—62no—
589C4⋊C4—64no—
590C4⋊C4—61no—
591C4⋊C4—64no—
592C4⋊C4C2×D1212yes—
593C4⋊C4C2×D1214yes—
594C4⋊C4C2×D1212yes—
595C4⋊C4C2×D1212yes—
596C4⋊C4C2×D1214yes—
597C4⋊C4C2×D1212yes—
598C22⋊C4—62no—
599C22⋊C4—62no—
600C22⋊C4—64no—
601C22⋊C4—64no—
602C22⋊C4—62no—
603C22⋊C4—62no—
604C22⋊C4C2×D1214yes—
605C42—62no—
606C42—62no—
607C42—62no—
608C42—62no—
609C42—64no—
610C42—64no—
611C42C2×D1211yescharacteristic
612C42C2×D1211yescharacteristic
613C42C23×D314yes—
614C42C23×D311yescharacteristic
615C42C23×D311yescharacteristic
616C22×C3—21no—
617C22×C3—21no—
618C22×C3D4×Q812yes—
619C22×C3C23×D411yescharacteristic
620C2×D3—24no—
621C2×D3—42no—
622C2×D3—42no—
623C2×D3—84no—
624C3×C4—21no—
625C3×C4—24no—
626C3×C4—21no—
627C3×C4—21no—
628C3×C4—21no—
629C3×C4—24no—
630C3×C4D4×Q812yes—
631C3×C4C23×Q811yescharacteristic
632C3×C4C2×2−1+411yescharacteristic
633Dic3—24no—
634Dic3—42no—
635Dic3—42no—
636Dic3—42no—
637Dic3—42no—
638Dic3—42no—
639Dic3—42no—
640C23—62no—
641C23—64no—
642C23—241no—
643C23—241no—
644C23C22×D1211yescharacteristic
645Q8—34no—
646Q8—42no—
647Q8—42no—
648Q8—62no—
649Q8—64no—
650Q8—122no—
651Q8—122no—
652Q8—244no—
653Q8—241no—
654Q8—244no—
655Q8—241no—
656D4—34no—
657D4—62no—
658D4—64no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—62no—
662C2×C4—64no—
663C2×C4—62no—
664C2×C4—62no—
665C2×C4—62no—
666C2×C4—62no—
667C2×C4—62no—
668C2×C4—62no—
669C2×C4—124no—
670C2×C4—242no—
671C2×C4—244no—
672C2×C4—244no—
673C2×C4—241no—
674C2×C4—241no—
675C2×C4C22×D1211yescharacteristic
676C2×C4C22×D1211yescharacteristic
677C2×C4C22×D1214yes—
678C2×C4C22×D1214yes—
679C2×C4C22×D1211yescharacteristic
680C2×C4C22×D1211yescharacteristic
681C2×C4C22×D1211yescharacteristic
682C2×C4C2×D3×Q812yes—
683C2×C4#101012yes—
684C2×C4C24×D311yescharacteristic
685C8—22no—
686C8—24no—
687C8C2×D3×Q814yes—
688C2×C3—22no—
689C2×C3C2×D4×Q811yescharacteristic
690C2×C3#358511yescharacteristic
691C2×C3#358611yescharacteristic
692D3—44no—
693C22—21no—
694C22—21no—
695C22—64no—
696C22—122no—
697C22—122no—
698C22—244no—
699C22Q8×D1212yes—
700C22C23×D1211yescharacteristic
701C4—21no—
702C4—21no—
703C4—24no—
704C4—21no—
705C4—21no—
706C4—24no—
707C4—64no—
708C4—122no—
709C4—122no—
710C4—122no—
711C4—122no—
712C4—122no—
713C4—122no—
714C4Q8×D1212yes—
715C4C22×D3×Q811yescharacteristic
716C4C2×#101011yescharacteristic
717C3#6137711yescharacteristic, complemented, Sylow
718C2—22no—
719C2—124no—
720C2C2×Q8×D1211yescharacteristic
721C2#8716711yescharacteristic
722C2#8716811yescharacteristic
subgroupG/H
centerC22C23×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2961C2
socleC22×C3C23×D4
prime radicalC22×D12C23
hypercenter#2961D3
hypocenterC3#61377
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #86258C2semidirect
3. C3×#2961C2semidirect
4. #79978C2semidirect
5. #79978C2semidirect
6. #79977C2semidirect
7. #79976C2semidirect
8. #79101C2semidirect
10. #79068C2semidirect
13. #78546C2semidirect
15. #78479C2semidirect
32. #5567C22semidirect
33. #5567C22semidirect
34. Q8×Dic6C22semidirect
43. C3×#446C22semidirect
44. C3×#446C22semidirect
58. #4851C22semidirect
59. #4851C22semidirect
60. #4850C22semidirect
61. #4850C22semidirect
69. #4829C22semidirect
70. #4829C22semidirect
71. #4829C22semidirect
76. #4702C22semidirect
77. #4702C22semidirect
78. #4702C22semidirect
79. #4702C22semidirect
80. #4702C22semidirect
81. #4702C22semidirect
84. #4683C22semidirect
85. #4682C22semidirect
98. #2961D3semidirect
133. C2×#204Q8semidirect
262. #446C2×D3semidirect
263. #446C2×D3semidirect
310. C2×Dic6C2×Q8semidirect
314. C2×C3×C8C2×Q8semidirect
368. #204C2×Q8semidirect
562. C2×C8D3×Q8semidirect
717. C3#61377semidirect

Maximal subgroups

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

subgroupindex
#862802
#862582
C3×#29612
#799782
#799772
#799762
#791012
#790902
#790682
#790592
#785722
#785462
C2×#47102
#784792
#613773