Finite Group Databasebeta

← #11664880

Subgroups — #11664880

Order 768.

1834 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86281C211yesmaximal, characteristic
2#86215C211yesmaximal, characteristic
3C3×#3165C211yesmaximal, characteristic, complemented
4#83339C211yesmaximal, characteristic, complemented
5C2×#5078C211yesmaximal, characteristic, complemented
6#81975C218yesmaximal, complemented
7#81940C212yesmaximal, complemented
8D3×#500C212yesmaximal, complemented
9#81893C212yesmaximal, complemented
10#81891C212yesmaximal, complemented
11#79210C214yesmaximal
12#79206C214yesmaximal, complemented
13C4×#907C211yesmaximal, characteristic
14#78506C211yesmaximal, characteristic, complemented
15#60667—31nomaximal, Sylow
16C3×#533C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18Q8×#67—22no—
19C2×#907C2212yes—
20#5730—21no—
21#5729C2214yes—
22#5728C2212yes—
23#5728C2212yes—
24#5727—21no—
25D3×C4⋊Q8C2212yes—
26D3×C4⋊Q8C2212yes—
27#5722—21no—
28#5721—21no—
29#5689C2214yes—
30#5635—22no—
31#5632—22no—
32#5579—21no—
33Q8×D12—21no—
34#5567—21no—
35Q8×Dic6—21no—
36#5523C2212yes—
37#5519C2212yes—
38#5511—22no—
39#5504C2212yes—
40#5500C2212yes—
41C4×#235C2211yescharacteristic
42#5470C2211yescharacteristic
43C2×#869C2211yescharacteristic
44C3×#501C2214yes—
45C3×#500C2212yes—
46C3×#498C2212yes—
47C3×#428C2214yes—
48C3×C4×M16C2211yescharacteristic
49#5122C2212yes—
50#5121C2214yes—
51#5080C2211yescharacteristic
52#5080C2211yescharacteristic
53#5078C2212yes—
54#5078C2212yes—
55#5078C2214yes—
56#5078C2211yescharacteristic
57#5078C2211yescharacteristic
58#5065C2214yes—
59#4998C2214yes—
60C2×#807C2214yes—
61C2×C4×C3⋊C8C2211yescharacteristic
62#4881C2218yes—
63D3×#107C2218yes—
64#4876C2218yes—
65#4874C2218yes—
66#4861C2218yes—
67#4860C2218yes—
68#4858C2218yes—
69#4856C2218yes—
70#4708C2214yes—
71#4702C2214yes—
72C4×ΓL(1,𝔽25)C2212yes—
73C4×D3×C8C2212yes—
74#4682C2212yes—
75#4680C2212yes—
76#3503—31no—
77#3503—31no—
78#3165—32no—
79#3165—32no—
80#3165—31no—
81#3165D311yescharacteristic, complemented
82#3164—38no—
83#3164—34no—
84#3160—32no—
85#3160—31no—
86C2×#500—32no—
87C2×#500—31no—
88#2913—34no—
89C4×#131—31no—
90C22×C3×Q8—21no—
91C2×D3×Q8—22no—
92C2×D3×Q8—22no—
93C2×#235C2311yescharacteristic
94C22×Dic6—21no—
95C2×C3×M16C2312yes—
96C3×C4⋊Q8—21no—
97C3×C4⋊Q8—21no—
98C3×C4⋊Q8C2311yescharacteristic
99C3×C4⋊Q8C2312yes—
100C3×C4⋊Q8C2311yescharacteristic
101C3×C4⋊Q8C2312yes—
102C3×#125C2312yes—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×#117C2312yes—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C42C2311yescharacteristic
111#946—22no—
112#946—22no—
113#946—24no—
114Q8×Dic3—22no—
115Q8×Dic3—22no—
116#944—24no—
117#944—22no—
118#944—22no—
119C4×#67—24no—
120C4×#67D412yes—
121C4×#67D412yes—
122#926C2312yes—
123#926C2311yescharacteristic
124C2×C4⋊Dic3—21no—
125C2×C4⋊Dic3C2312yes—
126#924—21no—
127#924—21no—
128#924—24no—
129C22×C3⋊C8C2312yes—
130#907Q8116yescomplemented
131C2×ΓL(1,𝔽25)C2314yes—
132C2×D3×C8C2314yes—
133#897—22no—
134#897—22no—
135#897—24no—
136#896—24no—
137#896—24no—
138#893—22no—
139#892C2314yes—
140#892C2314yes—
141D3×C4⋊C4—22no—
142D3×C4⋊C4—22no—
143D3×C4⋊C4C2314yes—
144D3×C4⋊C4C2314yes—
145#890—22no—
146#890C2314yes—
147#890C2314yes—
148#889—24no—
149#889—24no—
150#888—22no—
151#888—24no—
152#888—22no—
153#888—22no—
154#888C2314yes—
155#888C2314yes—
156#887—22no—
157#887—22no—
158#887—22no—
159#67⋊C4—24no—
160#878—24no—
161C4×D12—22no—
162C4×D12D411yescharacteristic
163C4×D12D411yescharacteristic
164#872C2312yes—
165C42×D3C2312yes—
166#870—22no—
167#870C2312yes—
168#869—21no—
169#869—21no—
170#869—22no—
171#869C2312yes—
172#869C2311yescharacteristic
173#869C2312yes—
174#869C2311yescharacteristic
175C4×Dic6—22no—
176C4×Dic6—21no—
177C4×Dic6—21no—
178C4×Dic6D411yescharacteristic
179C4×Dic6D411yescharacteristic
180C3×#107C2318yes—
181C3×#106C2318yes—
182C3×#97C2312yes—
183C3×C4×C8C2312yes—
184#818C2318yes—
185#817C2318yes—
186#815C2314yes—
187C8×Dic3C2314yes—
188#808C2314yes—
189#808C2314yes—
190#807C2314yes—
191#807C2318yes—
192#807C2318yes—
193#807C2314yes—
194C4×C3⋊C8C2311yescharacteristic
195C4×C3⋊C8C2311yescharacteristic
196C4×C3⋊C8C2312yes—
197C2×#131—32no—
198Q82—61no—
199Q82—61no—
200#557—61no—
201#557—61no—
202#554—62no—
203#554—61no—
204#554—62no—
205#554—61no—
206D4×Q8—61no—
207D4×Q8—62no—
208D4×Q8—61no—
209D4×Q8—62no—
210#544—34no—
211#544—34no—
212#544—32no—
213#544—32no—
214#533—32no—
215#533—32no—
216#533—31no—
217#533C2×D311yescharacteristic
218C2×C4⋊Q8—32no—
219C2×C4⋊Q8—32no—
220C2×C4⋊Q8—31no—
221C2×C4⋊Q8C2×D311yescharacteristic
222#520—32no—
223#520—32no—
224C4×#41—31no—
225#501—38no—
226#501—38no—
227#501—34no—
228#501C2×D314yes—
229#500—31no—
230#500—32no—
231#500—34no—
232#500—32no—
233#500—31no—
234#500—38no—
235#500—32no—
236#500C2×D312yes—
237#499—34no—
238#499—38no—
239#498—31no—
240#498—31no—
241#498—32no—
242#498C2×D312yes—
243#428—34no—
244#428—38no—
245#428—38no—
246#428C2×D314yes—
247#427—34no—
248#427—38no—
249C2×#107—34no—
250C2×#107—38no—
251#405—32no—
252#405—34no—
253C4×M16—32no—
254C4×M16C2×D311yescharacteristic
255C2×C4×C8—31no—
256C2×C4×C8—32no—
257C2×C3×Q8—21no—
258C2×C3×Q8—22no—
259C2×C3×Q8—21no—
260C2×C3×Q8—22no—
261C2×C3×Q8—82no—
262C22×C3×C4—22no—
263C22×C3×C4C2×D411yescharacteristic
264C22×C3×C4C2×D411yescharacteristic
265C22×C3×C4C2411yescharacteristic
266C2×#67C2×D412yes—
267C22×Dic3—22no—
268D3×Q8—84no—
269D3×Q8—84no—
270#235C2×Q818yes—
271C2×D12C2×D411yescharacteristic
272C2×C4×D3—24no—
273C2×C4×D3—24no—
274C2×C4×D3C2×D412yes—
275C2×C4×D3C2×D412yes—
276C2×C4×D3C2412yes—
277C2×Dic6—21no—
278C2×Dic6—21no—
279C2×Dic6—22no—
280C2×Dic6—22no—
281C2×Dic6—24no—
282C2×Dic6—24no—
283C2×Dic6—82no—
284C2×Dic6C2×D411yescharacteristic
285C3×M16C2×Q818yescomplemented
286C2×C3×C8C2414yes—
287C3×C4⋊C4—21no—
288C3×C4⋊C4—22no—
289C3×C4⋊C4—22no—
290C3×C4⋊C4—21no—
291C3×C4⋊C4—22no—
292C3×C4⋊C4—22no—
293C3×C4⋊C4—22no—
294C3×C4⋊C4C2414yes—
295C3×C4⋊C4C2412yes—
296C3×C4⋊C4C2412yes—
297C3×C22⋊C4—22no—
298C3×C42—22no—
299C3×C42C2×D411yescharacteristic
300C3×C42C2×D411yescharacteristic
301C3×C42C2411yescharacteristic
302C3×C42C2411yescharacteristic
303#217—22no—
304#217C2×D412yes—
305(C2×D3)⋊C4—24no—
306(C2×D3)⋊C4—24no—
307(C2×D3)⋊C4C2×D412yes—
308(C2×D3)⋊C4C2×D412yes—
309C4⋊Dic3—22no—
310C4⋊Dic3—21no—
311C4⋊Dic3—21no—
312C4⋊Dic3—22no—
313C4⋊Dic3—22no—
314C4⋊Dic3—24no—
315C4⋊Dic3—24no—
316C4⋊Dic3C2×D411yescharacteristic
317C4⋊Dic3C2×D411yescharacteristic
318C4⋊Dic3C2414yes—
319C4⋊Dic3C2412yes—
320C4⋊Dic3C2412yes—
321Dic3⋊C4—22no—
322Dic3⋊C4—22no—
323Dic3⋊C4—24no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4C2×D412yes—
330Dic3⋊C4C2×D412yes—
331C4×Dic3—22no—
332C4×Dic3—24no—
333C4×Dic3—24no—
334C4×Dic3C2×D412yes—
335C4×Dic3C2×D412yes—
336C4×Dic3C2412yes—
337C2×C3⋊C8C2×Q818yescomplemented
338C2×C3⋊C8C2412yes—
339C2×C3⋊C8C2412yes—
340ΓL(1,𝔽25)C2×Q8116yescomplemented
341D3×C8C2×Q8116yescomplemented
342C2×#41—31no—
343C22×Q8—21no—
344C22×Q8—62no—
345C22×Q8—62no—
346C22×Q8—61no—
347#131—316no—
348C2×M16—34no—
349C2×M16C22×D312yes—
350C22×C8—32no—
351C22×C8—34no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—34no—
355C4⋊Q8—34no—
356C4⋊Q8—31no—
357C4⋊Q8—32no—
358C4⋊Q8—32no—
359C4⋊Q8—31no—
360C4⋊Q8—62no—
361C4⋊Q8—62no—
362C4⋊Q8—62no—
363C4⋊Q8—62no—
364C4⋊Q8—62no—
365C4⋊Q8—64no—
366C4⋊Q8—62no—
367C4⋊Q8—64no—
368C4⋊Q8—61no—
369C4⋊Q8—61no—
370C4⋊Q8C22×D311yescharacteristic
371C4⋊Q8C22×D312yes—
372C4⋊Q8C22×D312yes—
373C4⋊Q8C22×D311yescharacteristic
374#125—34no—
375#125—34no—
376#125—32no—
377#125—62no—
378#125—64no—
379#125—64no—
380#125—62no—
381#125C22×D312yes—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—64no—
385C22⋊Q8—64no—
386C22⋊Q8—64no—
387C22⋊Q8—64no—
388C22⋊Q8—64no—
389C22⋊Q8—64no—
390C22⋊Q8—62no—
391C22⋊Q8—62no—
392C22⋊Q8—62no—
393C22⋊Q8—62no—
394C22⋊Q8—61no—
395C22⋊Q8—61no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—31no—
399C4×Q8—31no—
400C4×Q8—62no—
401C4×Q8—62no—
402C4×Q8—62no—
403C4×Q8—62no—
404C4×Q8—62no—
405C4×Q8—62no—
406C4×Q8—61no—
407C4×Q8—61no—
408C4×D4—32no—
409C4×D4—32no—
410C4×D4—31no—
411C4×D4—31no—
412C4×D4—62no—
413C4×D4—64no—
414C4×D4—62no—
415C4×D4—64no—
416#117—31no—
417#117—32no—
418#117—32no—
419#117—34no—
420#117—34no—
421#117C22×D312yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—32no—
424C2×C4⋊C4—34no—
425C2×C4⋊C4—34no—
426C2×C4⋊C4—62no—
427C2×C4⋊C4—62no—
428C2×C4⋊C4—61no—
429C2×C4⋊C4C22×D312yes—
430C2×C42—32no—
431C2×C42C22×D311yescharacteristic
432#107—38no—
433#107—34no—
434#107—38no—
435#107—38no—
436#107—34no—
437#107C22×D318yes—
438#106—38no—
439#106—34no—
440#106—34no—
441#106C22×D318yes—
442#97—34no—
443#97C22×D312yes—
444C4×C8—34no—
445C4×C8—31no—
446C4×C8—31no—
447C4×C8—32no—
448C4×C8C22×D312yes—
449C23×C3C22×D411yescharacteristic
450C22×D3C22×D412yes—
451C3×Q8—42no—
452C3×Q8—44no—
453C3×Q8—42no—
454C3×Q8—82no—
455C2×C3×C4—22no—
456C2×C3×C4—22no—
457C2×C3×C4—22no—
458C2×C3×C4—24no—
459C2×C3×C4—22no—
460C2×C3×C4—82no—
461C2×C3×C4C22×D411yescharacteristic
462C2×C3×C4C22×D411yescharacteristic
463C2×C3×C4C22×D411yescharacteristic
464C2×C3×C4C22×D411yescharacteristic
465C2×C3×C4C22×D411yescharacteristic
466C2×C3×C4C22×Q814yes—
467C2×C3×C4C2511yescharacteristic
468#67C4⋊Q818yes—
469C2×Dic3—22no—
470C2×Dic3—22no—
471C2×Dic3—24no—
472C2×Dic3—82no—
473C2×Dic3C22×D412yes—
474C2×Dic3C22×D412yes—
475C2×Dic3C22×D412yes—
476D12C4⋊Q814yes—
477C4×D3—24no—
478C4×D3—24no—
479C4×D3—84no—
480C4×D3—84no—
481C4×D3C22×Q818yes—
482Dic6—44no—
483Dic6—42no—
484Dic6—42no—
485Dic6—84no—
486Dic6—84no—
487Dic6—82no—
488Dic6C4⋊Q814yes—
489C3×C8C22×Q818yes—
490C3⋊C8C22×Q814yes—
491C3⋊C8C22×Q814yes—
492#41—38no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—31no—
498C2×Q8—64no—
499C2×Q8—64no—
500C2×Q8—61no—
501C2×Q8—62no—
502C2×Q8—61no—
503C2×Q8—62no—
504C2×Q8—82no—
505C2×Q8—242no—
506C2×Q8—244no—
507C2×Q8—244no—
508C2×D4—32no—
509C2×D4—31no—
510C22×C4—22no—
511C22×C4—32no—
512C22×C4—32no—
513C22×C4—32no—
514C22×C4—64no—
515C22×C4—64no—
516C22×C4—62no—
517C22×C4D3×D411yescharacteristic
518C22×C4D3×D411yescharacteristic
519C22×C4C23×D311yescharacteristic
520M16—316no—
521M16D3×Q818yescomplemented
522C2×C8—38no—
523C2×C8—32no—
524C2×C8—32no—
525C2×C8—316no—
526C2×C8C23×D314yes—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—32no—
535C4⋊C4—31no—
536C4⋊C4—32no—
537C4⋊C4—31no—
538C4⋊C4—32no—
539C4⋊C4—34no—
540C4⋊C4—32no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4—64no—
544C4⋊C4—64no—
545C4⋊C4—64no—
546C4⋊C4—64no—
547C4⋊C4—64no—
548C4⋊C4—64no—
549C4⋊C4—62no—
550C4⋊C4—61no—
551C4⋊C4—62no—
552C4⋊C4—62no—
553C4⋊C4—61no—
554C4⋊C4—62no—
555C4⋊C4—62no—
556C4⋊C4C23×D312yes—
557C4⋊C4C23×D312yes—
558C4⋊C4C23×D314yes—
559C22⋊C4—22no—
560C22⋊C4—32no—
561C22⋊C4—32no—
562C22⋊C4—32no—
563C22⋊C4—64no—
564C22⋊C4—64no—
565C22⋊C4—62no—
566C42—22no—
567C42—32no—
568C42—32no—
569C42—32no—
570C42—64no—
571C42—64no—
572C42—62no—
573C42D3×D411yescharacteristic
574C42D3×D411yescharacteristic
575C42C23×D311yescharacteristic
576C42C23×D311yescharacteristic
577C22×C3—21no—
578C22×C3—21no—
579C22×C3C2×C4⋊Q812yes—
580C22×C3C23×D411yescharacteristic
581C2×D3—22no—
582C2×D3—22no—
583C2×D3C2×C4⋊Q814yes—
584C3×C4—21no—
585C3×C4—21no—
586C3×C4—42no—
587C3×C4—44no—
588C3×C4—42no—
589C3×C4C2×C4⋊Q812yes—
590C3×C4#57612yes—
591C3×C4#57812yes—
592C3×C4C23×Q812yes—
593Dic3—22no—
594Dic3—22no—
595Dic3—42no—
596Dic3—44no—
597Dic3—42no—
598Dic3C2×C4⋊Q814yes—
599C23—32no—
600C23C2×D3×D411yescharacteristic
601Q8—34no—
602Q8—44no—
603Q8—42no—
604Q8—42no—
605Q8—82no—
606Q8—124no—
607Q8—122no—
608Q8—122no—
609Q8—242no—
610Q8—244no—
611Q8—244no—
612D4—38no—
613D4—34no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—32no—
620C2×C4—38no—
621C2×C4—32no—
622C2×C4—32no—
623C2×C4—64no—
624C2×C4—64no—
625C2×C4—62no—
626C2×C4—64no—
627C2×C4—62no—
628C2×C4—82no—
629C2×C4—244no—
630C2×C4—244no—
631C2×C4—242no—
632C2×C4C2×D3×D411yescharacteristic
633C2×C4C2×D3×D411yescharacteristic
634C2×C4C2×D3×D411yescharacteristic
635C2×C4C2×D3×D411yescharacteristic
636C2×C4C2×D3×D411yescharacteristic
637C2×C4C2×D3×Q814yes—
638C2×C4C24×D311yescharacteristic
639C8—34no—
640C8—34no—
641C8C2×D3×Q818yes—
642C2×C3—22no—
643C2×C3C22×C4⋊Q811yescharacteristic
644C2×C3C2×#57611yescharacteristic
645C2×C3C2×#57811yescharacteristic
646D3—24no—
647C22—21no—
648C22—21no—
649C22—34no—
650C22—62no—
651C22—62no—
652C22D3×C4⋊Q812yes—
653C22C22×D3×D411yescharacteristic
654C4—21no—
655C4—21no—
656C4—34no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—62no—
661C4—62no—
662C4—122no—
663C4—124no—
664C4—122no—
665C4D3×C4⋊Q812yes—
666C4#577412yes—
667C4#577912yes—
668C4C22×D3×Q812yes—
669C3#6066711yescharacteristic, complemented, Sylow
670C2—22no—
671C2—64no—
672C2C2×D3×C4⋊Q811yescharacteristic
673C2C2×#577411yescharacteristic
674C2C2×#577911yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3165C2
socleC22×C3C23×D4
prime radicalC2×#235C23
hypercenter#3165D3
hypocenterC3#60667
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. C3×#3165C2semidirect
4. #83339C2semidirect
5. C2×#5078C2semidirect
6. #81975C2semidirect
7. #81940C2semidirect
8. D3×#500C2semidirect
9. #81893C2semidirect
10. #81891C2semidirect
12. #79206C2semidirect
14. #78506C2semidirect
81. #3165D3semidirect
130. #907Q8semidirect
285. C3×M16C2×Q8semidirect
337. C2×C3⋊C8C2×Q8semidirect
340. ΓL(1,𝔽25)C2×Q8semidirect
341. D3×C8C2×Q8semidirect
521. M16D3×Q8semidirect
669. C3#60667semidirect

Maximal subgroups

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

subgroupindex
#862812
#862152
C3×#31652
#833392
C2×#50782
#819752
#819402
D3×#5002
#818932
#818912
#792102
#792062
C4×#9072
#785062
#606673