Finite Group Databasebeta

← #11675912

Subgroups — #11675912

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#85803C211yesmaximal, characteristic, complemented
3C3×#3078C211yesmaximal, characteristic, complemented
4#83340C211yesmaximal, characteristic, complemented
5C2×#5026C211yesmaximal, characteristic
6#81225C214yesmaximal, complemented
7#81223C214yesmaximal, complemented
8#81103C214yesmaximal, complemented
9#81008C214yesmaximal, complemented
10#79242C212yesmaximal
11#79206C212yesmaximal, complemented
12#79059C211yesmaximal, characteristic
13#79045C211yesmaximal, characteristic, complemented
14#78950C212yesmaximal, complemented
15#78947C212yesmaximal
16#61377—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19#5812C2212yes—
20C22×#215C2211yescharacteristic
21#5730C2212yes—
22#5727C2212yes—
23#5690C2214yes—
24#5634—22no—
25D3×C22⋊Q8—22no—
26#5579—22no—
27#5579—21no—
28Q8×D12—21no—
29#5570—21no—
30#5570C2212yescomplemented
31#5569—22no—
32#5569C2214yescomplemented
33Q8×Dic6—21no—
34Q8×Dic6C2212yescomplemented
35#5521C2212yes—
36#5519C2212yes—
37#5500—22no—
38#5500C2212yes—
39C2×C4×D12C2211yescharacteristic
40#5470—22no—
41#5470C2211yescharacteristic
42#5470C2211yescharacteristic
43C3×#479C2214yescomplemented
44C3×#477C2212yescomplemented
45C3×#475C2212yescomplemented
46C3×#428C2212yes—
47C3×#423C2211yescharacteristic
48C3×#419C2212yes—
49C2×#835C2211yescharacteristic
50#5131C2212yes—
51#5122C2212yes—
52#5121C2212yes—
53#5119C2212yes—
54#5081C2212yescomplemented
55#5081C2212yescomplemented
56#5064C2212yescomplemented
57#5064C2212yescomplemented
58#5026C2214yes—
59#5026C2214yes—
60#4998C2212yes—
61#4998C2212yes—
62C2×#809C2211yescharacteristic
63C2×#807C2211yescharacteristic
64C2×#803C2211yescharacteristic
65#4891C2214yes—
66#4890C2214yes—
67#4876C2214yescomplemented
68#4874C2214yescomplemented
69#4871C2214yes—
70#4870C2214yes—
71#4858C2214yescomplemented
72#4856C2214yescomplemented
73#4847C2214yes—
74#4846C2212yes—
75#4842C2212yes—
76#4828C2214yescomplemented
77#4828C2214yescomplemented
78#4817C2214yes—
79#4817C2214yes—
80#4799C2214yescomplemented
81#4796C2214yescomplemented
82#4793C2214yescomplemented
83#4791C2214yescomplemented
84#4788C2214yes—
85#4788C2214yes—
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#3078D311yescharacteristic, complemented
97#3077—31no—
98#2961—31no—
99#2949—32no—
100C2×#440—31no—
101C22×C3×Q8C2311yescharacteristic
102C2×D3×Q8—22no—
103C22×D12C2311yescharacteristic
104C22×C4×D3—21no—
105C2×C3×M16C2312yes—
106C3×C4⋊Q8C2311yescharacteristic
107C3×C4⋊Q8C2311yescharacteristic
108C3×#125C2312yes—
109C3×C22⋊Q8—21no—
110C3×C22⋊Q8—21no—
111C3×C4×Q8—21no—
112C3×C4×Q8—21no—
113C3×C4×Q8—21no—
114C3×C4×Q8—21no—
115C3×C4×Q8C2312yes—
116C3×C4×Q8C2312yes—
117C3×#117C2312yes—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C42—21no—
121C2×C3×C42C2311yescharacteristic
122#946—22no—
123#946—24no—
124Q8×Dic3—22no—
125Q8×Dic3—22no—
126Q8×Dic3C2314yes—
127Q8×Dic3C2314yes—
128C2×#215Q818yescomplemented
129C2×#215C2314yes—
130C2×(C2×D3)⋊C4—21no—
131#926—24no—
132#926C2312yes—
133#926C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135#924—21no—
136#924—21no—
137#924—24no—
138#924D412yes—
139#924D412yes—
140C22×C3⋊C8C2311yescharacteristic
141C22×C3⋊C8C2311yescharacteristic
142#897—22no—
143#897—24no—
144#896—24no—
145#896—24no—
146#896—24no—
147#893C2314yes—
148#893C2314yes—
149D3×C4⋊C4—22no—
150D3×C4⋊C4—24no—
151#890—24no—
152#890—22no—
153#890—24no—
154#890C2314yes—
155#890C2314yes—
156#888—24no—
157#888—22no—
158#888—24no—
159#888C2314yes—
160#888C2314yes—
161D3×C22⋊C4—24no—
162#878—24no—
163C4×D12—21no—
164C4×D12—21no—
165C4×D12—22no—
166C4×D12—22no—
167C4×D12C2312yes—
168C4×D12C2312yes—
169#870—22no—
170#870—24no—
171#870D412yes—
172#870C2312yes—
173#869—22no—
174#869—22no—
175#869—22no—
176#869D411yescharacteristic
177#869D411yescharacteristic
178#869C2311yescharacteristic
179#869C2311yescharacteristic
180C4×Dic6—22no—
181C4×Dic6—21no—
182C4×Dic6—21no—
183C4×Dic6—22no—
184C4×Dic6—24no—
185C4×Dic6C2312yes—
186C4×Dic6C2312yes—
187C3×#107C2314yes—
188C3×#106C2314yes—
189C3×C4⋊C8C2312yes—
190C3×C4⋊C8C2312yes—
191C3×#103C2314yes—
192C3×#103C2314yes—
193#835C2314yes—
194#821C2314yes—
195#821C2314yes—
196#818C2314yes—
197#817C2314yes—
198#815C2314yes—
199Dic3⋊C8C2314yes—
200Dic3⋊C8C2314yes—
201C8×Dic3C2314yes—
202#809C2314yes—
203#808C2312yes—
204#808C2312yes—
205#808C2312yes—
206#808C2312yes—
207#807C2312yes—
208#807C2312yes—
209#807C2314yes—
210#807C2312yes—
211#807C2312yes—
212#803C2312yes—
213#803C2312yes—
214C22×SD16—31no—
215Q82—32no—
216Q82—61no—
217#557—34no—
218#557—32no—
219#557—62no—
220#557—61no—
221#554—32no—
222#554—34no—
223#554—62no—
224#554—61no—
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×Q8C2×D311yescharacteristic
239C2×C4×D4—31no—
240#501—32no—
241#501—32no—
242#501—32no—
243#501—32no—
244#501—34no—
245#501—34no—
246#500—34no—
247#499—34no—
248#479—34no—
249#479—34no—
250#479C2×D314yescomplemented
251#478—34no—
252#478—34no—
253#478—34no—
254#477—34no—
255#477C2×D312yescomplemented
256#476—34no—
257#476—32no—
258#475—34no—
259#475C2×D312yescomplemented
260#474—34no—
261#474—32no—
262#446—34no—
263#446—34no—
264#440—34no—
265#440—34no—
266#440—34no—
267#440—34no—
268C4×SD16—34no—
269C4×SD16—34no—
270#428—32no—
271#428C2×D312yes—
272#427—32no—
273#427—32no—
274C2×#107—31no—
275#424—32no—
276#423C2×D311yescharacteristic
277#419C2×D312yes—
278#418—32no—
279C2×#103—31no—
280C2×#102—31no—
281#405—32no—
282C2×#97—31no—
283C23×D3—21no—
284C2×C3×Q8—21no—
285C2×C3×Q8—21no—
286C2×C3×Q8—21no—
287C2×C3×Q8—22no—
288C2×C3×Q8C2×Q814yescomplemented
289C2×C3×Q8C2412yes—
290C22×C3×C4—21no—
291C22×C3×C4—21no—
292C22×C3×C4C2×D411yescharacteristic
293C22×C3×C4C2×D411yescharacteristic
294C22×C3×C4C2411yescharacteristic
295C22×Dic3—21no—
296D3×Q8—84no—
297C2×D12—21no—
298C2×D12—22no—
299C2×D12C2×Q814yes—
300C2×D12C2412yes—
301C2×C4×D3—22no—
302C2×C4×D3—22no—
303C2×C4×D3—24no—
304C2×C4×D3—24no—
305C2×C4×D3—82no—
306C2×Dic6—21no—
307C2×Dic6—21no—
308C2×Dic6—24no—
309C2×Dic6—24no—
310C2×Dic6C2×D412yes—
311C2×Dic6C2×D412yes—
312C3×M16—24no—
313C2×C3×C8C2414yes—
314C3×C4⋊C4—22no—
315C3×C4⋊C4—22no—
316C3×C4⋊C4—21no—
317C3×C4⋊C4—22no—
318C3×C4⋊C4—21no—
319C3×C4⋊C4—21no—
320C3×C4⋊C4—22no—
321C3×C4⋊C4C2412yes—
322C3×C4⋊C4C2412yes—
323C3×C4⋊C4C2412yes—
324C3×C22⋊C4—22no—
325C3×C42—22no—
326C3×C42—22no—
327C3×C42—22no—
328C3×C42C2×D411yescharacteristic
329C3×C42C2×D411yescharacteristic
330C3×C42C2411yescharacteristic
331C3×C42C2411yescharacteristic
332#217—22no—
333#217C2×D414yes—
334#215C2×Q8116yescomplemented
335(C2×D3)⋊C4—22no—
336(C2×D3)⋊C4—22no—
337(C2×D3)⋊C4—24no—
338(C2×D3)⋊C4—24no—
339C4⋊Dic3—21no—
340C4⋊Dic3—21no—
341C4⋊Dic3—21no—
342C4⋊Dic3—24no—
343C4⋊Dic3—24no—
344C4⋊Dic3—24no—
345C4⋊Dic3—24no—
346C4⋊Dic3C2×D412yes—
347C4⋊Dic3C2×D412yes—
348C4⋊Dic3C2×D412yes—
349C4⋊Dic3C2×D412yes—
350C4⋊Dic3C2412yes—
351C4⋊Dic3C2412yes—
352C4⋊Dic3C2412yes—
353Dic3⋊C4—22no—
354Dic3⋊C4—22no—
355Dic3⋊C4—24no—
356Dic3⋊C4—24no—
357Dic3⋊C4—24no—
358Dic3⋊C4—24no—
359Dic3⋊C4C2×D414yes—
360Dic3⋊C4C2×D414yes—
361C4×Dic3—22no—
362C4×Dic3—24no—
363C4×Dic3—24no—
364C4×Dic3C2414yes—
365C2×C3⋊C8—22no—
366C2×C3⋊C8C2×Q814yescomplemented
367C2×C3⋊C8C2412yes—
368C2×C3⋊C8C2412yes—
369C22×Q8—62no—
370C22×Q8C22×D311yescharacteristic
371C22×D4—31no—
372C23×C4—61no—
373C2×SD16—38no—
374C2×SD16—34no—
375C2×M16C22×D312yes—
376C22×C8—31no—
377C22×C8—31no—
378C4⋊Q8—34no—
379C4⋊Q8—34no—
380C4⋊Q8—31no—
381C4⋊Q8—31no—
382C4⋊Q8—31no—
383C4⋊Q8—31no—
384C4⋊Q8—62no—
385C4⋊Q8—62no—
386C4⋊Q8—62no—
387C4⋊Q8—62no—
388C4⋊Q8—64no—
389C4⋊Q8—64no—
390C4⋊Q8C22×D311yescharacteristic
391C4⋊Q8C22×D311yescharacteristic
392#125—34no—
393#125—34no—
394#125—32no—
395#125—32no—
396#125—64no—
397#125—62no—
398#125—62no—
399#125—64no—
400#125—64no—
401#125C22×D312yes—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—32no—
405C22⋊Q8—32no—
406C22⋊Q8—64no—
407C22⋊Q8—64no—
408C22⋊Q8—64no—
409C22⋊Q8—64no—
410C22⋊Q8—64no—
411C22⋊Q8—62no—
412C22⋊Q8—64no—
413C22⋊Q8—62no—
414C22⋊Q8—64no—
415C22⋊Q8—61no—
416C22⋊Q8—61no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—32no—
422C4×Q8—32no—
423C4×Q8—34no—
424C4×Q8—34no—
425C4×Q8—64no—
426C4×Q8—62no—
427C4×Q8—62no—
428C4×Q8—62no—
429C4×Q8—62no—
430C4×Q8—61no—
431C4×Q8—61no—
432C4×Q8C22×D312yes—
433C4×Q8C22×D312yes—
434C4×D4—34no—
435C4×D4—34no—
436C4×D4—32no—
437C4×D4—32no—
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—21no—
447C2×C4⋊C4—31no—
448C2×C4⋊C4—64no—
449C2×C4⋊C4—62no—
450C2×C4⋊C4C22×D311yescharacteristic
451C2×C22⋊C4—64no—
452C2×C22⋊C4—61no—
453C2×C42—21no—
454C2×C42C22×D311yescharacteristic
455#107—34no—
456#107—34no—
457#107—32no—
458#107—32no—
459#107—32no—
460#107—32no—
461#107C22×D314yes—
462#106—34no—
463#106—32no—
464#106—32no—
465#106—32no—
466#106—32no—
467#106C22×D314yes—
468C4⋊C8—34no—
469C4⋊C8—34no—
470C4⋊C8C22×D312yes—
471C4⋊C8C22×D312yes—
472#103—34no—
473#103C22×D314yes—
474#103C22×D314yes—
475#102—34no—
476#102—34no—
477#102—34no—
478#97—34no—
479#97—32no—
480#97—32no—
481C4×C8—34no—
482C23×C3C22×D411yescharacteristic
483C22×D3—22no—
484C22×D3—24no—
485C22×D3—81no—
486C22×D3—81no—
487C3×Q8—24no—
488C3×Q8—22no—
489C3×Q8—42no—
490C3×Q8—81no—
491C3×Q8C22×Q814yes—
492C2×C3×C4—24no—
493C2×C3×C4—22no—
494C2×C3×C4—22no—
495C2×C3×C4—22no—
496C2×C3×C4—22no—
497C2×C3×C4—41no—
498C2×C3×C4—41no—
499C2×C3×C4—81no—
500C2×C3×C4C22×D411yescharacteristic
501C2×C3×C4C22×D411yescharacteristic
502C2×C3×C4C22×D411yescharacteristic
503C2×C3×C4C22×D411yescharacteristic
504C2×C3×C4C22×D411yescharacteristic
505C2×C3×C4C22×Q812yes—
506C2×C3×C42−1+412yes—
507C2×C3×C4C2511yescharacteristic
508C2×Dic3—22no—
509C2×Dic3—22no—
510C2×Dic3—22no—
511C2×Dic3—81no—
512C2×Dic3C22×D414yes—
513C2×Dic3C22×D414yes—
514D12—24no—
515D12—22no—
516D12C22×Q814yes—
517C4×D3—44no—
518C4×D3—84no—
519C4×D3—84no—
520C4×D3—82no—
521Dic6—42no—
522Dic6—42no—
523Dic6—42no—
524Dic6—84no—
525Dic6—84no—
526Dic6—81no—
527C3×C8—24no—
528C3⋊C8—22no—
529C3⋊C8C22×Q814yes—
530C24—61no—
531C2×Q8—21no—
532C2×Q8—21no—
533C2×Q8—21no—
534C2×Q8—22no—
535C2×Q8—32no—
536C2×Q8—32no—
537C2×Q8—64no—
538C2×Q8—64no—
539C2×Q8—61no—
540C2×Q8—61no—
541C2×Q8—244no—
542C2×Q8D3×Q814yescomplemented
543C2×Q8C23×D312yes—
544C2×D4—34no—
545C2×D4—32no—
546C2×D4—61no—
547C2×D4—62no—
548C22×C4—21no—
549C22×C4—21no—
550C22×C4—64no—
551C22×C4—64no—
552C22×C4—61no—
553C22×C4—62no—
554C22×C4—62no—
555C22×C4—242no—
556C22×C4D3×D411yescharacteristic
557C22×C4D3×D411yescharacteristic
558C22×C4C23×D311yescharacteristic
559SD16—316no—
560M16—24no—
561C2×C8—34no—
562C2×C8—32no—
563C2×C8—32no—
564C2×C8—62no—
565C2×C8C23×D314yes—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—34no—
574C4⋊C4—32no—
575C4⋊C4—32no—
576C4⋊C4—32no—
577C4⋊C4—34no—
578C4⋊C4—32no—
579C4⋊C4—32no—
580C4⋊C4—32no—
581C4⋊C4—32no—
582C4⋊C4—64no—
583C4⋊C4—64no—
584C4⋊C4—64no—
585C4⋊C4—64no—
586C4⋊C4—64no—
587C4⋊C4—64no—
588C4⋊C4—64no—
589C4⋊C4—64no—
590C4⋊C4—61no—
591C4⋊C4—62no—
592C4⋊C4—61no—
593C4⋊C4—62no—
594C4⋊C4—61no—
595C4⋊C4C23×D312yes—
596C4⋊C4C23×D312yes—
597C4⋊C4C23×D312yes—
598C22⋊C4—22no—
599C22⋊C4—34no—
600C22⋊C4—64no—
601C22⋊C4—64no—
602C22⋊C4—62no—
603C22⋊C4—62no—
604C22⋊C4—62no—
605C42—22no—
606C42—22no—
607C42—22no—
608C42—34no—
609C42—64no—
610C42—64no—
611C42—62no—
612C42D3×D411yescharacteristic
613C42D3×D411yescharacteristic
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—21no—
626C3×C4—21no—
627C3×C4—21no—
628C3×C4—24no—
629C3×C4—42no—
630C3×C4—42no—
631C3×C4—42no—
632C3×C4D4×Q812yes—
633C3×C4C23×Q811yescharacteristic
634C3×C4C2×2−1+411yescharacteristic
635Dic3—24no—
636Dic3—24no—
637Dic3—42no—
638Dic3—42no—
639Dic3—42no—
640C23—62no—
641C23—64no—
642C23—241no—
643C23—241no—
644C23C2×D3×D411yescharacteristic
645Q8—24no—
646Q8—22no—
647Q8—42no—
648Q8—81no—
649Q8—122no—
650Q8—122no—
651Q8—122no—
652Q8—241no—
653Q8—244no—
654Q8—244no—
655Q8C2×D3×Q814yes—
656D4—34no—
657D4—64no—
658D4—62no—
659C2×C4—24no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—34no—
665C2×C4—34no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—62no—
669C2×C4—62no—
670C2×C4—62no—
671C2×C4—81no—
672C2×C4—124no—
673C2×C4—242no—
674C2×C4—244no—
675C2×C4—244no—
676C2×C4—241no—
677C2×C4C2×D3×D411yescharacteristic
678C2×C4C2×D3×D411yescharacteristic
679C2×C4C2×D3×D411yescharacteristic
680C2×C4C2×D3×D411yescharacteristic
681C2×C4C2×D3×D411yescharacteristic
682C2×C4C2×D3×Q812yes—
683C2×C4#101012yes—
684C2×C4C24×D311yescharacteristic
685C8—24no—
686C8—34no—
687C8—62no—
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—
699C22#563212yes—
700C22C22×D3×D411yescharacteristic
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—24no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—64no—
710C4—64no—
711C4—122no—
712C4—122no—
713C4—122no—
714C4#563212yes—
715C4C22×D3×Q811yescharacteristic
716C4C2×#101011yescharacteristic
717C3#6137711yescharacteristic, complemented, Sylow
718C2—22no—
719C2—124no—
720C2C2×#563211yescharacteristic
721C2#8719711yescharacteristic
722C2#8720011yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3078C2
socleC22×C3C23×D4
prime radicalC22×D12C23
hypercenter#3078D3
hypocenterC3#61377
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
2. #85803C2semidirect
3. C3×#3078C2semidirect
4. #83340C2semidirect
6. #81225C2semidirect
7. #81223C2semidirect
8. #81103C2semidirect
9. #81008C2semidirect
11. #79206C2semidirect
13. #79045C2semidirect
14. #78950C2semidirect
30. #5570C22semidirect
32. #5569C22semidirect
34. Q8×Dic6C22semidirect
43. C3×#479C22semidirect
44. C3×#477C22semidirect
45. C3×#475C22semidirect
54. #5081C22semidirect
55. #5081C22semidirect
56. #5064C22semidirect
57. #5064C22semidirect
67. #4876C22semidirect
68. #4874C22semidirect
71. #4858C22semidirect
72. #4856C22semidirect
76. #4828C22semidirect
77. #4828C22semidirect
80. #4799C22semidirect
81. #4796C22semidirect
82. #4793C22semidirect
83. #4791C22semidirect
96. #3078D3semidirect
128. C2×#215Q8semidirect
250. #479C2×D3semidirect
255. #477C2×D3semidirect
259. #475C2×D3semidirect
288. C2×C3×Q8C2×Q8semidirect
334. #215C2×Q8semidirect
366. C2×C3⋊C8C2×Q8semidirect
542. C2×Q8D3×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
#858032
C3×#30782
#833402
C2×#50262
#812252
#812232
#811032
#810082
#792422
#792062
#790592
#790452
#789502
#789472
#613773