Finite Group Databasebeta

← #10762163

Subgroups — #10762163

Order 768.

1096 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86239C211yesmaximal, characteristic
2#79051C211yesmaximal, characteristic, complemented
3C4×#901C211yesmaximal, characteristic, complemented
4C3×#1492C211yesmaximal, characteristic, complemented
5#71801C211yesmaximal, characteristic, complemented
6#70497C211yesmaximal, characteristic, complemented
7#69734C211yesmaximal, characteristic, complemented
8#68177C212yesmaximal, complemented
9#68175C212yesmaximal, complemented
10#68170C212yesmaximal, complemented
11D3×D4⋊C8C212yesmaximal, complemented
12#17517—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14#5828—22no—
15C2×#901—21no—
16#5730C411yescharacteristic
17#5730C411yescharacteristic
18#5717C411yescharacteristic
19#5713C411yescharacteristic
20#5679—22no—
21#5614—21no—
22#5614C412yes—
23#5612—21no—
24#5612C412yes—
25#5557—21no—
26#5557C2212yes—
27C4×D3×D4—21no—
28C4×D3×D4C2212yes—
29C4×#235C2211yescharacteristic
30C4×#235C2211yescharacteristic
31C3×#423C2211yescharacteristic, complemented
32C2×C3×C4×C8C2211yescharacteristic, complemented
33#5127—22no—
34#5101—21no—
35#4925C2211yescharacteristic, complemented
36C4×#208C2211yescharacteristic, complemented
37#4835C2212yescomplemented
38#4834—21no—
39D3×C4⋊C8C2212yescomplemented
40#4831—21no—
41#4689—22no—
42C4×ΓL(1,𝔽25)C2212yescomplemented
43C4×D3×C8C2212yescomplemented
44C3×D4⋊C8C2212yescomplemented
45C3×D4⋊C8C2212yescomplemented
46D4⋊(C3⋊C8)C2212yescomplemented
47D4⋊(C3⋊C8)C2212yescomplemented
48#4484C2212yescomplemented
49#4483C2212yes—
50#4459C2211yescharacteristic
51#4459C2211yescharacteristic
52#4456C2211yescharacteristic, complemented
53#4456C2211yescharacteristic, complemented
54#3481—31no—
55#2990—31no—
56C4×#131—31no—
57#1494—32no—
58#1492D311yescharacteristic, complemented
59#1490—32no—
60#1489—31no—
61#1484—31no—
62#1483—32no—
63#1476—31no—
64C2×D4⋊C8—32no—
65C2×C3×#41—21no—
66C2×D3×D4—21no—
67C2×D3×D4C2×C412yes—
68C2×#235C2×C411yescharacteristic
69C2×#235D411yescharacteristic
70C22×C4×D3—22no—
71C2×C3×M16—21no—
72C22×C3×C8—21no—
73C3×C4⋊Q8C2×C411yescharacteristic
74C3×#127C2×C411yescharacteristic
75C3×#124—21no—
76C3×#121—21no—
77C3×#121C2×C412yes—
78C3×C4×D4—21no—
79C3×C4×D4C2312yes—
80C2×C3×C42C2311yescharacteristic
81#946—22no—
82#946—22no—
83#938—22no—
84#938—22no—
85#938—21no—
86#938C2×C412yes—
87#937—22no—
88#937—22no—
89#936—21no—
90#936C2×C412yes—
91#935—22no—
92#935—22no—
93D4×Dic3—21no—
94D4×Dic3C2×C412yes—
95C4×#67—21no—
96C4×#67—21no—
97C4×#67—22no—
98C4×#67—22no—
99C4×#67C2×C412yes—
100C4×#67D412yes—
101#926—22no—
102#926C2×C411yescharacteristic
103#926D411yescharacteristic
104C2×#208—21no—
105#901—28no—
106C2×ΓL(1,𝔽25)—22no—
107C2×D3×C8—22no—
108C2×D3×C8—22no—
109#897—21no—
110#897C2×C412yes—
111#893—21no—
112#893C2×C412yes—
113D3×C4⋊C4—21no—
114D3×C4⋊C4C2×C412yes—
115#890—21no—
116#890C2×C412yes—
117#883—22no—
118#883—22no—
119#67⋊C4—22no—
120#67⋊C4—22no—
121D3×C22⋊C4—22no—
122D3×C22⋊C4—22no—
123#879—22no—
124#879—22no—
125C4×D12C2×C411yescharacteristic
126C4×D12C2×C411yescharacteristic
127C4×D12D411yescharacteristic
128C4×D12C2311yescharacteristic
129#872—21no—
130#872D412yes—
131C42×D3—21no—
132C42×D3C2312yes—
133C4×Dic6C2×C411yescharacteristic
134C4×Dic6C2×C411yescharacteristic
135C4×Dic6D411yescharacteristic
136C4×Dic6C2311yescharacteristic
137C3×C4⋊C8—21no—
138C3×C4⋊C8C2312yes—
139C3×C4×C8—21no—
140C3×C4×C8C2311yescharacteristic
141C3×C4×C8C2311yescharacteristic
142#830—21no—
143(C2×D3)⋊C8—22no—
144#815—22no—
145#815—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150#803—21no—
151C4×C3⋊C8—21no—
152C4×C3⋊C8C2312yes—
153C2×#131—61no—
154#559—31no—
155#554—31no—
156#554—31no—
157#553—32no—
158#553—61no—
159#551—62no—
160#546—32no—
161#546—61no—
162D42—31no—
163#532C2×D311yescharacteristic
164#529—32no—
165#529—61no—
166#525—62no—
167C4×#41—31no—
168C4×#41—31no—
169C2×C4×D4—32no—
170C2×C4×D4—61no—
171#446—61no—
172#435—62no—
173C8×D4—61no—
174#432—32no—
175#431—31no—
176#423C2×D311yescharacteristic, complemented
177C2×C4⋊C8—32no—
178#405—62no—
179#405—61no—
180C4×M16—31no—
181C4×M16—32no—
182C2×C4×C8—32no—
183C2×C4×C8C2×D311yescharacteristic, complemented
184Q8⋊C8—32no—
185Q8⋊C8—31no—
186Q8⋊C8—31no—
187D4⋊C8—32no—
188D4⋊C8—32no—
189D4⋊C8—32no—
190D4⋊C8—31no—
191D4⋊C8—31no—
192D4⋊C8C2×D312yescomplemented
193D4⋊C8C2×D312yescomplemented
194C23×D3—22no—
195C3×#41—81no—
196C3×#41—81no—
197C2×C3×Q8—21no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C22×C412yes—
202C22×C3×C4—22no—
203C22×C3×C4C22×C411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C2×#67—21no—
207C2×#67—22no—
208C2×#67—22no—
209C2×#67C22⋊C412yes—
210C22×Dic3—22no—
211D3×D4—24no—
212D3×D4—42no—
213D3×D4—44no—
214#235—24no—
215#235—22no—
216#235—41no—
217C2×D12C22⋊C411yescharacteristic
218C2×D12C22×C411yescharacteristic
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—22no—
223C2×C4×D3—22no—
224C2×C4×D3—82no—
225C2×C4×D3—82no—
226C2×C4×D3C22×C412yes—
227C2×C4×D3C2×D412yes—
228C2×C4×D3C2×D412yes—
229C2×Dic6C22⋊C411yescharacteristic
230C2×Dic6C22×C411yescharacteristic
231C3×M16—81no—
232C2×C3×C8—24no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C22×C412yes—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C42C22×C411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2411yescharacteristic
244#217—21no—
245#217—22no—
246#217—22no—
247#217—22no—
248#217—22no—
249#217C22⋊C412yes—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4C22⋊C412yes—
255(C2×D3)⋊C4C22⋊C412yes—
256C4⋊Dic3—22no—
257C4⋊Dic3—22no—
258C4⋊Dic3C22⋊C411yescharacteristic
259C4⋊Dic3C22⋊C411yescharacteristic
260C4⋊Dic3C22×C411yescharacteristic
261C4⋊Dic3C22×C411yescharacteristic
262Dic3⋊C4—21no—
263Dic3⋊C4—21no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4C22⋊C412yes—
267Dic3⋊C4C22⋊C412yes—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3C22×C412yes—
272C4×Dic3C2×D412yes—
273C4×Dic3C2×D412yes—
274#208—24no—
275C2×C3⋊C8—22no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278ΓL(1,𝔽25)—28no—
279D3×C8—28no—
280D3×C8—82no—
281C2×#41—21no—
282C2×#41—31no—
283C2×#41—31no—
284C22×D4—32no—
285C22×D4—61no—
286C23×C4—62no—
287#131—68no—
288C2×M16—21no—
289C2×M16—62no—
290C2×M16—61no—
291C22×C8—21no—
292C22×C8—62no—
293C22×C8—62no—
294C4⋊Q8C4×D311yescharacteristic
295#127C4×D311yescharacteristic
296#126—62no—
297#126—62no—
298#125—32no—
299#125—61no—
300#124—21no—
301#124—32no—
302#124—61no—
303#123—62no—
304#123—62no—
305#123—62no—
306#123—62no—
307C22⋊Q8—32no—
308C22⋊Q8—62no—
309C22⋊Q8—61no—
310C22⋊Q8—62no—
311#121—21no—
312#121—32no—
313#121—62no—
314#121—61no—
315#121—62no—
316#121C4×D312yes—
317#120—62no—
318#120—62no—
319C4×Q8—31no—
320C4×Q8—31no—
321C4×Q8—31no—
322C4×Q8—31no—
323C4×D4—21no—
324C4×D4—32no—
325C4×D4—32no—
326C4×D4—32no—
327C4×D4—32no—
328C4×D4—31no—
329C4×D4—31no—
330C4×D4—31no—
331C4×D4—31no—
332C4×D4—62no—
333C4×D4—62no—
334C4×D4—62no—
335C4×D4—61no—
336C4×D4—61no—
337C4×D4—62no—
338C4×D4—61no—
339C4×D4—61no—
340C4×D4C22×D312yes—
341#117—31no—
342#117—32no—
343#117—31no—
344#117—62no—
345#117—61no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—61no—
348C2×C22⋊C4—62no—
349C2×C22⋊C4—62no—
350C2×C42—32no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353C4⋊C8—21no—
354C4⋊C8—31no—
355C4⋊C8—31no—
356C4⋊C8—62no—
357C4⋊C8C22×D312yes—
358C22⋊C8—62no—
359C22⋊C8—61no—
360#97—62no—
361#97—62no—
362#97—61no—
363C4×C8—21no—
364C4×C8—32no—
365C4×C8—62no—
366C4×C8—61no—
367C4×C8C22×D311yescharacteristic
368C4×C8C22×D311yescharacteristic
369C23×C3—22no—
370C23×C3C2×C22⋊C411yescharacteristic
371C22×D3—21no—
372C22×D3—44no—
373C22×D3—44no—
374C22×D3—82no—
375C22×D3C2×C22⋊C412yes—
376C3×Q8—81no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—42no—
380C3×D4—81no—
381C3×D4—82no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4C2×C22⋊C411yescharacteristic
391C2×C3×C4C2×C22⋊C411yescharacteristic
392C2×C3×C4C2×C22⋊C411yescharacteristic
393C2×C3×C4C23×C411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396#67—24no—
397#67—44no—
398#67—42no—
399#67—44no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—82no—
404C2×Dic3C2×C22⋊C412yes—
405C2×Dic3C2×C22⋊C412yes—
406C2×Dic3C2×C22⋊C412yes—
407D12—21no—
408D12#10414yes—
409D12#13112yescomplemented
410C4×D3—24no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—42no—
416C4×D3—41no—
417C4×D3—41no—
418C4×D3—42no—
419C4×D3—82no—
420C4×D3—82no—
421Dic6—22no—
422Dic6—21no—
423Dic6—21no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—42no—
427C3⋊C8—24no—
428C3⋊C8—41no—
429C3⋊C8—41no—
430C24—62no—
431#41—64no—
432#41—62no—
433#41—81no—
434#41—81no—
435#41—121no—
436C2×Q8—21no—
437C2×Q8—31no—
438C2×Q8—31no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—32no—
443C2×D4—31no—
444C2×D4—31no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4—62no—
448C2×D4—61no—
449C2×D4—122no—
450C2×D4—124no—
451C2×D4C2×C4×D312yes—
452C22×C4—22no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—32no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—242no—
463C22×C4—242no—
464C22×C4C2×C4×D311yescharacteristic
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467M16—68no—
468M16—64no—
469M16—81no—
470C2×C8—24no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—68no—
475C2×C8—62no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—242no—
479C4⋊C4—21no—
480C4⋊C4—31no—
481C4⋊C4—31no—
482C4⋊C4—32no—
483C4⋊C4—32no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4C2×C4×D312yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—32no—
496C22⋊C4—32no—
497C22⋊C4—32no—
498C22⋊C4—62no—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C22⋊C4—62no—
504C22⋊C4—61no—
505C22⋊C4—61no—
506C22⋊C4—61no—
507C42—32no—
508C42—32no—
509C42—32no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42C2×C4×D311yescharacteristic
514C42D3×D411yescharacteristic
515C42D3×D411yescharacteristic
516C42C23×D311yescharacteristic
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3C22×C22⋊C411yescharacteristic
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—88no—
530C2×D3C2×#10414yes—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—42no—
535C3×C4C2×#10412yes—
536C3×C4C2×#13111yescharacteristic
537C3×C4#56811yescharacteristic
538C3×C4#57611yescharacteristic
539C3×C4#57711yescharacteristic
540Dic3—22no—
541Dic3—22no—
542Dic3—22no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546C23—22no—
547C23—32no—
548C23—61no—
549C23—124no—
550C23—124no—
551C23—242no—
552C23D3×C22⋊C411yescharacteristic
553Q8—62no—
554Q8—61no—
555Q8—61no—
556Q8—81no—
557D4—22no—
558D4—22no—
559D4—32no—
560D4—34no—
561D4—42no—
562D4—64no—
563D4—61no—
564D4—81no—
565D4—82no—
566D4—124no—
567D4—122no—
568D4—124no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—64no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—122no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—122no—
592C2×C4—242no—
593C2×C4—242no—
594C2×C4—242no—
595C2×C4D3×C22⋊C411yescharacteristic
596C2×C4D3×C22⋊C411yescharacteristic
597C2×C4D3×C22⋊C411yescharacteristic
598C2×C4C22×C4×D311yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C8—22no—
602C8—22no—
603C8—42no—
604C8—64no—
605C8—121no—
606C8—121no—
607C2×C3—41no—
608C2×C3—82no—
609C2×C3#288011yescharacteristic
610C2×C3#289711yescharacteristic
611C2×C3C22×#10411yescharacteristic
612D3—24no—
613D3—42no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—34no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—62no—
622C22—62no—
623C22—62no—
624C22—121no—
625C22—121no—
626C22—248no—
627C22C2×D3×C22⋊C411yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—42no—
632C4—62no—
633C4—62no—
634C4—62no—
635C4—121no—
636C4—121no—
637C4—121no—
638C4D3×#10412yes—
639C4D3×#13111yescharacteristic
640C4#575011yescharacteristic
641C4#575711yescharacteristic
642C4#576211yescharacteristic
643C3#1751711yescharacteristic, complemented, Sylow
644C2—41no—
645C2—64no—
646C2—82no—
647C2—122no—
648C2#7866311yescharacteristic
649C2#7874211yescharacteristic
650C2C2×D3×#10411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1492C2
socleC22×C3C22×C22⋊C4
prime radical#86239C2
cosocleC3×C42C24
hypercenter#1492D3
hypocenterC3#17517
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #79051C2semidirect
3. C4×#901C2semidirect
4. C3×#1492C2semidirect
5. #71801C2semidirect
6. #70497C2semidirect
7. #69734C2semidirect
8. #68177C2semidirect
9. #68175C2semidirect
10. #68170C2semidirect
11. D3×D4⋊C8C2semidirect
31. C3×#423C22semidirect
32. C2×C3×C4×C8C22semidirect
35. #4925C22semidirect
36. C4×#208C22semidirect
37. #4835C22semidirect
39. D3×C4⋊C8C22semidirect
42. C4×ΓL(1,𝔽25)C22semidirect
43. C4×D3×C8C22semidirect
44. C3×D4⋊C8C22semidirect
45. C3×D4⋊C8C22semidirect
46. D4⋊(C3⋊C8)C22semidirect
47. D4⋊(C3⋊C8)C22semidirect
48. #4484C22semidirect
52. #4456C22semidirect
53. #4456C22semidirect
58. #1492D3semidirect
176. #423C2×D3semidirect
183. C2×C4×C8C2×D3semidirect
192. D4⋊C8C2×D3semidirect
193. D4⋊C8C2×D3semidirect
409. D12#131semidirect
643. C3#17517semidirect

Maximal subgroups

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

subgroupindex
#862392
#790512
C4×#9012
C3×#14922
#718012
#704972
#697342
#681772
#681752
#681702
D3×D4⋊C82
#175173