Finite Group Databasebeta

← #10764133

Subgroups — #10764133

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86283C211yesmaximal, characteristic
2#79340C211yesmaximal, characteristic
3#79051C211yesmaximal, characteristic
4C3×#1559C211yesmaximal, characteristic, complemented
5#71828C211yesmaximal, characteristic, complemented
6#70821C211yesmaximal, characteristic, complemented
7#70509C211yesmaximal, characteristic, complemented
8#68488C211yesmaximal, characteristic, complemented
9#68487C211yesmaximal, characteristic, complemented
10#68481C211yesmaximal, characteristic, complemented
11D3×#333C211yesmaximal, characteristic, complemented
12#68427C211yesmaximal, characteristic, complemented
13#68425C211yesmaximal, characteristic, complemented
14#68424C211yesmaximal, characteristic, complemented
15#68423C211yesmaximal, characteristic, complemented
16#17780—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18#5728C2211yescharacteristic
19#5728C2211yescharacteristic
20#5728C2211yescharacteristic
21D3×C4⋊Q8C2211yescharacteristic
22#5694—21no—
23#5685—21no—
24#5641—21no—
25#5639—21no—
26#5628C412yes—
27#5625C412yes—
28#5583C412yes—
29#5581—21no—
30#5572C412yes—
31#5569—21no—
32#5533—21no—
33#5529—21no—
34C4×#235C2211yescharacteristic
35#5470C2211yescharacteristic
36C3×#432C2211yescharacteristic
37C3×#423C2211yescharacteristic
38#5127—21no—
39#5127—21no—
40#4925C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42#4837C2211yescharacteristic
43#4835C2211yescharacteristic
44#4835C2211yescharacteristic
45#4835C2211yescharacteristic
46#4834—21no—
47#4833C2211yescharacteristic
48D3×C4⋊C8C2211yescharacteristic
49D3×C4⋊C8C2211yescharacteristic
50D3×C4⋊C8C2211yescharacteristic
51#4831—21no—
52#4730—21no—
53#67⋊C8—21no—
54#4706—21no—
55#4701—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59C3×#332C2211yescharacteristic
60#4547C2211yescharacteristic
61#4547C2211yescharacteristic
62#4546C2211yescharacteristic
63#4546C2211yescharacteristic
64#4486C2211yescharacteristic
65#4486C2211yescharacteristic
66#4482C2211yescharacteristic
67#4482C2211yescharacteristic
68#4481C2211yescharacteristic
69#4481C2211yescharacteristic
70#4481C2211yescharacteristic
71#4481C2211yescharacteristic
72#3509—31no—
73#2990—31no—
74#2990—31no—
75#1559—31no—
76#1559D311yescharacteristic, complemented
77#1558—31no—
78#1554—31no—
79#1554—31no—
80#1552—31no—
81#1552—31no—
82#1550—31no—
83#1550—31no—
84#1542—31no—
85#1542—31no—
86C2×#333—31no—
87C2×D3×Q8C2×C412yes—
88C2×#235C2×C411yescharacteristic
89C2×C3×M16—21no—
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×#125—21no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8C2×C412yes—
95C3×C4×Q8—21no—
96C3×C4×Q8C2×C412yes—
97C3×#117—21no—
98C2×C3×C42C2311yescharacteristic
99#946—21no—
100#946—21no—
101#946C2×C412yes—
102Q8×Dic3—21no—
103Q8×Dic3—21no—
104Q8×Dic3C2×C412yes—
105#944C2×C412yes—
106C4×#67D411yescharacteristic
107C4×#67D411yescharacteristic
108C4×#67D411yescharacteristic
109C4×#67D411yescharacteristic
110#926—21no—
111#926C2×C411yescharacteristic
112#924—21no—
113#924C2×C412yes—
114C22×C3⋊C8—21no—
115C2×ΓL(1,𝔽25)—21no—
116C2×D3×C8—21no—
117C2×D3×C8—21no—
118C2×D3×C8—21no—
119#898—22no—
120#897—21no—
121#897—21no—
122#897C2×C412yes—
123#894—22no—
124#892—21no—
125#892—21no—
126#892—22no—
127#892C2×C412yes—
128D3×C4⋊C4—21no—
129D3×C4⋊C4—21no—
130#890—22no—
131#890—21no—
132#890—21no—
133#890C2×C412yes—
134#889—22no—
135#889—22no—
136#888—22no—
137#888—21no—
138#888—21no—
139#887—22no—
140#887—21no—
141#887—21no—
142#887C2×C412yes—
143#885—22no—
144#883—22no—
145#879—22no—
146#879—22no—
147#876—22no—
148C4×D12D411yescharacteristic
149C4×D12D411yescharacteristic
150#872—22no—
151#872C2×C411yescharacteristic
152#872C2×C411yescharacteristic
153C42×D3C2311yescharacteristic
154C42×D3C2311yescharacteristic
155#870—21no—
156#870—22no—
157#869C2311yescharacteristic
158#869C2311yescharacteristic
159C4×Dic6—21no—
160C4×Dic6—22no—
161C4×Dic6C2×C412yes—
162C4×Dic6D411yescharacteristic
163C4×Dic6D411yescharacteristic
164C3×C4⋊C8—21no—
165C3×C4⋊C8C2311yescharacteristic
166C3×C4⋊C8C2311yescharacteristic
167C3×C4⋊C8C2311yescharacteristic
168C3×C4⋊C8C2311yescharacteristic
169C3×C22⋊C8—21no—
170C3×#97—21no—
171#830—21no—
172(C2×D3)⋊C8—21no—
173(C2×D3)⋊C8—21no—
174(C2×D3)⋊C8—21no—
175(C2×D3)⋊C8—21no—
176#815—21no—
177#815—21no—
178#815—21no—
179Dic3⋊C8—21no—
180Dic3⋊C8—21no—
181Dic3⋊C8—21no—
182Dic3⋊C8—21no—
183C8×Dic3—21no—
184C4⋊(C3⋊C8)—21no—
185C4⋊(C3⋊C8)C2311yescharacteristic
186C4⋊(C3⋊C8)C2311yescharacteristic
187C4⋊(C3⋊C8)C2311yescharacteristic
188C4⋊(C3⋊C8)C2311yescharacteristic
189C4×C3⋊C8—21no—
190#557—61no—
191#557—61no—
192#556—61no—
193#556—61no—
194#552—61no—
195#552—61no—
196#552—61no—
197#552—61no—
198#541—32no—
199#541—32no—
200#536—32no—
201#533—31no—
202#533—31no—
203#533—31no—
204#533—31no—
205#533C2×D311yescharacteristic
206C2×C4⋊Q8—31no—
207#519—32no—
208C4×#41—31no—
209#446—61no—
210#446—61no—
211#435—61no—
212#435—61no—
213#435—61no—
214#435—61no—
215C8×D4—61no—
216C8×D4—61no—
217#432—31no—
218#432—31no—
219#432—31no—
220#432C2×D311yescharacteristic
221#431—31no—
222#431—31no—
223#423—31no—
224#423C2×D311yescharacteristic
225C2×C4⋊C8—31no—
226C2×C4⋊C8—31no—
227C2×C4⋊C8—31no—
228C2×C4⋊C8—31no—
229#333—31no—
230#333—31no—
231#333—31no—
232#333—31no—
233#333—31no—
234#333—31no—
235#333C2×D311yescharacteristic
236#333C2×D311yescharacteristic
237#332—31no—
238#332—31no—
239#332—31no—
240#332—31no—
241#332—31no—
242#332—31no—
243#332C2×D311yescharacteristic
244#332C2×D311yescharacteristic
245C2×C3×Q8—21no—
246C2×C3×Q8C22×C412yes—
247C22×C3×C4C22×C411yescharacteristic
248C22×C3×C4C2×D411yescharacteristic
249C22×C3×C4C2×D411yescharacteristic
250C2×#67C22⋊C411yescharacteristic
251C2×#67C22⋊C411yescharacteristic
252D3×Q8—44no—
253#235—42no—
254C2×D12C22⋊C411yescharacteristic
255C2×C4×D3—22no—
256C2×C4×D3C22×C411yescharacteristic
257C2×C4×D3C22×C411yescharacteristic
258C2×C4×D3C2×D411yescharacteristic
259C2×C4×D3C2×D411yescharacteristic
260C2×C4×D3C2×D411yescharacteristic
261C2×C4×D3C2×D411yescharacteristic
262C2×Dic6—21no—
263C2×Dic6—22no—
264C2×Dic6C22⋊C411yescharacteristic
265C2×Dic6C22×C412yes—
266C3×M16—81no—
267C2×C3×C8—21no—
268C2×C3×C8—21no—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—22no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—21no—
275C3×C4⋊C4—21no—
276C3×C4⋊C4C22×C412yes—
277C3×C22⋊C4—22no—
278C3×C42—22no—
279C3×C42C22×C411yescharacteristic
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282C3×C42C2411yescharacteristic
283#217—22no—
284#217C22⋊C411yescharacteristic
285#217C22⋊C411yescharacteristic
286(C2×D3)⋊C4—22no—
287(C2×D3)⋊C4—22no—
288(C2×D3)⋊C4—22no—
289(C2×D3)⋊C4C22⋊C411yescharacteristic
290(C2×D3)⋊C4C22⋊C411yescharacteristic
291(C2×D3)⋊C4C22⋊C411yescharacteristic
292(C2×D3)⋊C4C22⋊C411yescharacteristic
293C4⋊Dic3—21no—
294C4⋊Dic3—21no—
295C4⋊Dic3—21no—
296C4⋊Dic3—22no—
297C4⋊Dic3—22no—
298C4⋊Dic3—22no—
299C4⋊Dic3C22⋊C411yescharacteristic
300C4⋊Dic3C22⋊C411yescharacteristic
301C4⋊Dic3C22×C412yes—
302Dic3⋊C4—22no—
303Dic3⋊C4—22no—
304Dic3⋊C4—22no—
305Dic3⋊C4—22no—
306Dic3⋊C4—22no—
307Dic3⋊C4—22no—
308Dic3⋊C4—22no—
309Dic3⋊C4C22⋊C411yescharacteristic
310Dic3⋊C4C22⋊C411yescharacteristic
311Dic3⋊C4C22⋊C411yescharacteristic
312Dic3⋊C4C22⋊C411yescharacteristic
313C4×Dic3—22no—
314C4×Dic3—22no—
315C4×Dic3—22no—
316C4×Dic3—22no—
317C4×Dic3C22×C411yescharacteristic
318C4×Dic3C22×C411yescharacteristic
319C4×Dic3C2×D411yescharacteristic
320C4×Dic3C2×D411yescharacteristic
321C4×Dic3C2×D411yescharacteristic
322C4×Dic3C2×D411yescharacteristic
323C2×C3⋊C8—21no—
324C2×C3⋊C8—21no—
325C2×C3⋊C8—21no—
326C2×C3⋊C8—21no—
327C2×C3⋊C8—81no—
328ΓL(1,𝔽25)—81no—
329D3×C8—81no—
330D3×C8—81no—
331D3×C8—81no—
332C2×#41—31no—
333C22×Q8—32no—
334C2×M16—21no—
335C2×M16—61no—
336C22×C8—61no—
337C22×C8—61no—
338C22×C8—61no—
339C22×C8—61no—
340C4⋊Q8—32no—
341C4⋊Q8—31no—
342C4⋊Q8—31no—
343C4⋊Q8—61no—
344C4⋊Q8—61no—
345C4⋊Q8—62no—
346C4⋊Q8C22×D311yescharacteristic
347C4⋊Q8C22×D311yescharacteristic
348#126—62no—
349#126—62no—
350#126—62no—
351#126—62no—
352#125—21no—
353#125—32no—
354#125—62no—
355#125—61no—
356#125—61no—
357#125—62no—
358#125—62no—
359#125—62no—
360#125—61no—
361#124—62no—
362#123—62no—
363#123—62no—
364C22⋊Q8—21no—
365C22⋊Q8—32no—
366C22⋊Q8—32no—
367C22⋊Q8—32no—
368C22⋊Q8—61no—
369C22⋊Q8—61no—
370C22⋊Q8—61no—
371C22⋊Q8—61no—
372C22⋊Q8—61no—
373C22⋊Q8C4×D312yes—
374C4×Q8—21no—
375C4×Q8—31no—
376C4×Q8—31no—
377C4×Q8—32no—
378C4×Q8—32no—
379C4×Q8—32no—
380C4×Q8—62no—
381C4×Q8—61no—
382C4×Q8—61no—
383C4×Q8—61no—
384C4×Q8—61no—
385C4×Q8—62no—
386C4×Q8—61no—
387C4×Q8C4×D312yes—
388C4×D4—31no—
389C4×D4—31no—
390C4×D4—31no—
391C4×D4—31no—
392C4×D4—31no—
393C4×D4—31no—
394#117—21no—
395#117—31no—
396#117—31no—
397#117—31no—
398#117—32no—
399#117—62no—
400#117—62no—
401#117—61no—
402#117—61no—
403#117—61no—
404C2×C4⋊C4—61no—
405C2×C4⋊C4—61no—
406C2×C42—31no—
407C2×C42—31no—
408C2×C42C22×D311yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—31no—
411C4⋊C8—31no—
412C4⋊C8—31no—
413C4⋊C8—31no—
414C4⋊C8—61no—
415C4⋊C8—61no—
416C4⋊C8—61no—
417C4⋊C8—61no—
418C4⋊C8—61no—
419C4⋊C8C22×D311yescharacteristic
420C4⋊C8C22×D311yescharacteristic
421C4⋊C8C22×D311yescharacteristic
422C4⋊C8C22×D311yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—61no—
425C22⋊C8—61no—
426C22⋊C8—61no—
427C22⋊C8—61no—
428C22⋊C8—61no—
429#97—21no—
430#97—61no—
431#97—61no—
432#97—61no—
433C4×C8—61no—
434C4×C8—61no—
435C23×C3C2×C22⋊C411yescharacteristic
436C22×D3C2×C22⋊C411yescharacteristic
437C22×D3C2×C22⋊C411yescharacteristic
438C3×Q8—42no—
439C3×Q8—41no—
440C3×Q8—41no—
441C2×C3×C4—21no—
442C2×C3×C4—21no—
443C2×C3×C4—21no—
444C2×C3×C4—21no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—41no—
448C2×C3×C4C2×C22⋊C411yescharacteristic
449C2×C3×C4C2×C22⋊C411yescharacteristic
450C2×C3×C4C2×C22⋊C411yescharacteristic
451C2×C3×C4C23×C411yescharacteristic
452C2×C3×C4C22×D411yescharacteristic
453C2×C3×C4C22×D411yescharacteristic
454#67—22no—
455#67—41no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3C2×C22⋊C411yescharacteristic
459C2×Dic3C2×C22⋊C411yescharacteristic
460C2×Dic3C2×C22⋊C411yescharacteristic
461C2×Dic3C2×C22⋊C411yescharacteristic
462C2×Dic3C2×C22⋊C411yescharacteristic
463C2×Dic3C2×C22⋊C411yescharacteristic
464D12—22no—
465C4×D3—22no—
466C4×D3—21no—
467C4×D3—21no—
468C4×D3—21no—
469C4×D3—21no—
470C4×D3—21no—
471C4×D3—21no—
472C4×D3—21no—
473C4×D3—21no—
474C4×D3—41no—
475C4×D3—44no—
476Dic6—41no—
477Dic6—44no—
478Dic6—41no—
479Dic6—41no—
480Dic6—42no—
481C3×C8—41no—
482C3×C8—41no—
483C3×C8—41no—
484C3×C8—41no—
485C3⋊C8—41no—
486C3⋊C8—41no—
487C3⋊C8—41no—
488C3⋊C8—41no—
489#41—122no—
490C2×Q8—21no—
491C2×Q8—31no—
492C2×Q8—32no—
493C2×Q8—62no—
494C2×Q8—61no—
495C2×Q8—124no—
496C2×Q8C2×C4×D312yes—
497C2×D4—31no—
498C2×D4—31no—
499C2×D4—31no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—31no—
504C22×C4—31no—
505C22×C4—31no—
506C22×C4—62no—
507C22×C4C2×C4×D311yescharacteristic
508C22×C4D3×D411yescharacteristic
509C22×C4D3×D411yescharacteristic
510M16—81no—
511M16—241no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—241no—
521C2×C8—241no—
522C2×C8—241no—
523C2×C8—241no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—31no—
530C4⋊C4—31no—
531C4⋊C4—31no—
532C4⋊C4—31no—
533C4⋊C4—31no—
534C4⋊C4—31no—
535C4⋊C4—32no—
536C4⋊C4—62no—
537C4⋊C4—62no—
538C4⋊C4—62no—
539C4⋊C4—62no—
540C4⋊C4—62no—
541C4⋊C4—62no—
542C4⋊C4—62no—
543C4⋊C4—62no—
544C4⋊C4—62no—
545C4⋊C4—62no—
546C4⋊C4—61no—
547C4⋊C4—61no—
548C4⋊C4—61no—
549C4⋊C4C2×C4×D312yes—
550C22⋊C4—22no—
551C22⋊C4—31no—
552C22⋊C4—31no—
553C22⋊C4—31no—
554C22⋊C4—31no—
555C22⋊C4—31no—
556C22⋊C4—31no—
557C22⋊C4—62no—
558C22⋊C4—62no—
559C22⋊C4—62no—
560C22⋊C4—62no—
561C42—22no—
562C42—31no—
563C42—31no—
564C42—31no—
565C42—31no—
566C42—31no—
567C42—31no—
568C42—62no—
569C42—62no—
570C42—62no—
571C42—62no—
572C42C2×C4×D311yescharacteristic
573C42D3×D411yescharacteristic
574C42D3×D411yescharacteristic
575C42C23×D311yescharacteristic
576C22×C3—21no—
577C22×C3—21no—
578C22×C3—21no—
579C22×C3C22×C22⋊C411yescharacteristic
580C2×D3—21no—
581C2×D3—21no—
582C2×D3—21no—
583C2×D3—21no—
584C2×D3—21no—
585C2×D3#41312yes—
586C3×C4—21no—
587C3×C4—21no—
588C3×C4—21no—
589C3×C4—21no—
590C3×C4—42no—
591C3×C4—42no—
592C3×C4#57711yescharacteristic
593C3×C4#57711yescharacteristic
594C3×C4#57811yescharacteristic
595C3×C4#57811yescharacteristic
596Dic3—21no—
597Dic3—21no—
598Dic3—21no—
599Dic3—21no—
600Dic3—21no—
601Dic3—21no—
602Dic3—42no—
603Dic3—42no—
604C23—31no—
605C23—31no—
606C23D3×C22⋊C411yescharacteristic
607Q8—42no—
608Q8—41no—
609Q8—41no—
610Q8—121no—
611Q8—122no—
612Q8—124no—
613Q8—121no—
614Q8—121no—
615D4—62no—
616D4—62no—
617D4—121no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—31no—
625C2×C4—31no—
626C2×C4—31no—
627C2×C4—31no—
628C2×C4—31no—
629C2×C4—31no—
630C2×C4—41no—
631C2×C4—62no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—121no—
643C2×C4—124no—
644C2×C4D3×C22⋊C411yescharacteristic
645C2×C4D3×C22⋊C411yescharacteristic
646C2×C4D3×C22⋊C411yescharacteristic
647C2×C4C22×C4×D311yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×D3×D411yescharacteristic
650C8—41no—
651C8—41no—
652C8—41no—
653C8—41no—
654C8—121no—
655C8—121no—
656C8—121no—
657C8—121no—
658C2×C3—41no—
659C2×C3C2×#41311yescharacteristic
660C2×C3#289811yescharacteristic
661C2×C3#289811yescharacteristic
662D3—22no—
663D3—41no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—32no—
668C22—61no—
669C22—61no—
670C22—61no—
671C22—61no—
672C22—61no—
673C22C2×D3×C22⋊C411yescharacteristic
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—42no—
679C4—42no—
680C4—61no—
681C4—61no—
682C4—61no—
683C4—61no—
684C4—61no—
685C4—61no—
686C4—122no—
687C4—122no—
688C4#577511yescharacteristic
689C4#577811yescharacteristic
690C4#577911yescharacteristic
691C4#577911yescharacteristic
692C3#1778011yescharacteristic, complemented, Sylow
693C2—41no—
694C2—62no—
695C2—121no—
696C2D3×#41311yescharacteristic
697C2#7888311yescharacteristic
698C2#7895711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1559C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1559D3
hypocenterC3#17780
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1559

Complements

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

NH ≅ G/Ntype
4. C3×#1559C2semidirect
5. #71828C2semidirect
6. #70821C2semidirect
7. #70509C2semidirect
8. #68488C2semidirect
9. #68487C2semidirect
10. #68481C2semidirect
11. D3×#333C2semidirect
12. #68427C2semidirect
13. #68425C2semidirect
14. #68424C2semidirect
15. #68423C2semidirect
76. #1559D3semidirect
692. C3#17780semidirect

Maximal subgroups

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

subgroupindex
#862832
#793402
#790512
C3×#15592
#718282
#708212
#705092
#684882
#684872
#684812
D3×#3332
#684272
#684252
#684242
#684232
#177803