Finite Group Databasebeta

← #10762164

Subgroups — #10762164

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85826C211yesmaximal, characteristic
2#79294C211yesmaximal, characteristic, complemented
3C4×#901C211yesmaximal, characteristic, complemented
4C3×#1476C211yesmaximal, characteristic, complemented
5#71641C211yesmaximal, characteristic, complemented
6#70769C211yesmaximal, characteristic, complemented
7#69738C211yesmaximal, characteristic, complemented
8#68240C212yesmaximal, complemented
9#68235C212yesmaximal, complemented
10#68174C212yesmaximal, complemented
11#68165C212yesmaximal, complemented
12#17518—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5824C412yes—
15C2×#901—21no—
16#5584C411yescharacteristic
17#5578C2212yes—
18#5567C411yescharacteristic
19#5555C411yescharacteristic
20#5552—21no—
21#5541—21no—
22#5538—22no—
23D4×Dic6C411yescharacteristic
24C4×#237C2212yes—
25#5533C412yes—
26#5531—22no—
27#5479—21no—
28C4×#235C2211yescharacteristic
29#5470C2211yescharacteristic
30C2×C4×Dic6—21no—
31C3×#431C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33#5101—21no—
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4837C2212yes—
37#4833C2212yes—
38#4729—22no—
39#4689—22no—
40#4688—21no—
41C4×ΓL(1,𝔽25)C2212yes—
42C4×D3×C8C2212yes—
43C8×Dic6—21no—
44C3×Q8⋊C8C2211yescharacteristic
45C3×Q8⋊C8C2211yescharacteristic
46C3×D4⋊C8C2211yescharacteristic
47C3×D4⋊C8C2211yescharacteristic
48#4535C2212yes—
49D4⋊(C3⋊C8)C2212yes—
50#4484C2212yes—
51#4484C2212yes—
52#4456C2212yes—
53#4456C2212yes—
54#3482—31no—
55#2991—31no—
56C4×#131—31no—
57#1493—31no—
58#1493—32no—
59#1489—32no—
60#1489—31no—
61#1485—31no—
62#1485—32no—
63#1476—32no—
64#1476D311yescharacteristic, complemented
65C2×C3×#41C2×C411yescharacteristic
66C2×#237C2×C412yes—
67C2×#235D411yescharacteristic
68C22×Dic6—21no—
69C22×C3×C8—21no—
70C3×C4×Q8C2×C411yescharacteristic
71C3×C4×Q8C2311yescharacteristic
72C3×C4×D4—21no—
73C3×C4×D4—21no—
74C3×C4×D4C2×C411yescharacteristic
75C3×C4×D4C2311yescharacteristic
76C3×#117—21no—
77C3×#117C2×C411yescharacteristic
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80#946C2×C412yes—
81#944C2×C412yes—
82#939—22no—
83#935—22no—
84D4×Dic3C2×C412yes—
85#930—21no—
86#929—22no—
87C4×#67—22no—
88C4×#67C2×C412yes—
89C4×#67D412yes—
90#926—21no—
91#926D411yescharacteristic
92C2×C4⋊Dic3—21no—
93#924—21no—
94#924—21no—
95#924C2×C412yes—
96C2×Dic3⋊C4—22no—
97C2×C4×Dic3—22no—
98C2×#208—21no—
99C2×#208—21no—
100#901—28no—
101C2×ΓL(1,𝔽25)—22no—
102C2×ΓL(1,𝔽25)—22no—
103C2×D3×C8—22no—
104#898—22no—
105#897C2×C412yes—
106#896—22no—
107#892C2×C412yes—
108#889—22no—
109#889—22no—
110#888C2×C412yes—
111#887C2×C412yes—
112#886—22no—
113#885—22no—
114#67⋊C4—22no—
115#879—22no—
116#878—22no—
117#877—22no—
118#876—22no—
119#875—21no—
120C4×D12C2×C411yescharacteristic
121C4×D12D411yescharacteristic
122#872—22no—
123#872D412yes—
124C42×D3C2312yes—
125#870—21no—
126#870—21no—
127#869C2×C411yescharacteristic
128#869C2×C411yescharacteristic
129C4×Dic6—21no—
130C4×Dic6—21no—
131C4×Dic6—21no—
132C4×Dic6—22no—
133C4×Dic6—22no—
134C4×Dic6C2×C411yescharacteristic
135C4×Dic6D411yescharacteristic
136C4×Dic6C2312yes—
137C3×C4⋊C8C2311yescharacteristic
138C3×C4⋊C8C2311yescharacteristic
139C3×C22⋊C8—21no—
140C3×C4×C8—21no—
141C3×C4×C8—21no—
142C3×C4×C8C2311yescharacteristic
143C3×C4×C8C2311yescharacteristic
144(C2×D3)⋊C8—22no—
145#815—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148C8×Dic3—22no—
149C4⋊(C3⋊C8)—21no—
150C4⋊(C3⋊C8)C2312yes—
151#803—21no—
152C4×C3⋊C8C2312yes—
153C2×#131—61no—
154#557—31no—
155#556—31no—
156#556—32no—
157#552—62no—
158#552—61no—
159D4×Q8—31no—
160#548—61no—
161#547—32no—
162#547—31no—
163#533—32no—
164#533—31no—
165#529—61no—
166#529—62no—
167C4×#41—31no—
168C4×#41—32no—
169C4×#41C2×D311yescharacteristic
170C2×C4×Q8—61no—
171C8×Q8—61no—
172#436—61no—
173#436—62no—
174#431—32no—
175#431C2×D311yescharacteristic
176#423—31no—
177#423—32no—
178#405—62no—
179#405—61no—
180C4×M16—31no—
181C4×M16—32no—
182C2×C4×C8—32no—
183C2×C4×C8C2×D311yescharacteristic
184Q8⋊C8—32no—
185Q8⋊C8—32no—
186Q8⋊C8—32no—
187Q8⋊C8—32no—
188Q8⋊C8—32no—
189Q8⋊C8C2×D311yescharacteristic
190Q8⋊C8C2×D311yescharacteristic
191D4⋊C8—32no—
192D4⋊C8C2×D311yescharacteristic
193D4⋊C8C2×D311yescharacteristic
194C3×#41—22no—
195C3×#41—41no—
196C2×C3×Q8C22×C411yescharacteristic
197C2×C3×D4—21no—
198C2×C3×D4C22×C411yescharacteristic
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C22×C411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C2×#67—22no—
206C2×#67C22⋊C412yes—
207C22×Dic3—22no—
208C22×Dic3—22no—
209#237—24no—
210#237—42no—
211#235—24no—
212C2×D12C22⋊C411yescharacteristic
213C2×C4×D3C22×C412yes—
214C2×C4×D3C2×D412yes—
215C2×C4×D3C2×D412yes—
216C2×Dic6—21no—
217C2×Dic6—22no—
218C2×Dic6—44no—
219C2×Dic6C22⋊C411yescharacteristic
220C2×Dic6C22×C412yes—
221C2×C3×C8—24no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C22×C411yescharacteristic
229C3×C4⋊C4C22×C411yescharacteristic
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42C22×C411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2411yescharacteristic
240#217—22no—
241#217—22no—
242#217C22⋊C412yes—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4C22⋊C412yes—
247(C2×D3)⋊C4C22⋊C412yes—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—22no—
252C4⋊Dic3C22⋊C411yescharacteristic
253C4⋊Dic3C22⋊C411yescharacteristic
254C4⋊Dic3C22×C412yes—
255Dic3⋊C4—22no—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262Dic3⋊C4C22⋊C412yes—
263Dic3⋊C4C22⋊C412yes—
264C4×Dic3—22no—
265C4×Dic3—22no—
266C4×Dic3—22no—
267C4×Dic3C22×C412yes—
268C4×Dic3C2×D412yes—
269C4×Dic3C2×D412yes—
270#208—24no—
271#208—81no—
272C2×C3⋊C8—22no—
273C2×C3⋊C8—22no—
274ΓL(1,𝔽25)—28no—
275ΓL(1,𝔽25)—82no—
276D3×C8—28no—
277C2×#41—31no—
278C2×#41—32no—
279C2×#41C4×D311yescharacteristic
280C22×Q8—61no—
281#131—68no—
282C2×M16—62no—
283C2×M16—62no—
284C2×M16—61no—
285C2×M16—61no—
286C22×C8—21no—
287C22×C8—62no—
288C4⋊Q8—32no—
289C4⋊Q8—32no—
290C4⋊Q8—31no—
291C4⋊Q8—31no—
292#126—62no—
293#126—62no—
294#126—62no—
295#125—61no—
296#125—62no—
297#125—62no—
298#125—61no—
299#124—61no—
300#124—62no—
301#123—62no—
302#123—62no—
303#123—62no—
304C22⋊Q8—32no—
305C22⋊Q8—32no—
306C22⋊Q8—32no—
307C22⋊Q8—62no—
308C22⋊Q8—62no—
309C22⋊Q8—61no—
310C22⋊Q8—61no—
311#121—61no—
312#121—62no—
313C4×Q8—31no—
314C4×Q8—31no—
315C4×Q8—32no—
316C4×Q8—32no—
317C4×Q8—62no—
318C4×Q8—62no—
319C4×Q8—61no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×Q8C4×D311yescharacteristic
323C4×Q8C22×D311yescharacteristic
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—32no—
327C4×D4—32no—
328C4×D4—32no—
329C4×D4—31no—
330C4×D4—31no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4C4×D311yescharacteristic
334C4×D4C22×D311yescharacteristic
335#117—21no—
336#117—31no—
337#117—32no—
338#117—32no—
339#117—62no—
340#117—62no—
341#117—61no—
342#117C4×D311yescharacteristic
343C2×C4⋊C4—62no—
344C2×C4⋊C4—61no—
345C2×C42—21no—
346C2×C42—32no—
347C2×C42—62no—
348C2×C42C22×D311yescharacteristic
349C4⋊C8—32no—
350C4⋊C8—62no—
351C4⋊C8—61no—
352C4⋊C8C22×D311yescharacteristic
353C4⋊C8C22×D311yescharacteristic
354C22⋊C8—21no—
355C22⋊C8—62no—
356#97—62no—
357#97—61no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—32no—
361C4×C8—62no—
362C4×C8—62no—
363C4×C8C22×D311yescharacteristic
364C4×C8C22×D311yescharacteristic
365C23×C3—21no—
366C23×C3C2×C22⋊C411yescharacteristic
367C22×D3C2×C22⋊C412yes—
368C3×Q8—21no—
369C3×Q8—21no—
370C3×D4—21no—
371C3×D4—42no—
372C3×D4#13112yescomplemented
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4C2×C22⋊C411yescharacteristic
386C2×C3×C4C2×C22⋊C411yescharacteristic
387C2×C3×C4C2×C22⋊C411yescharacteristic
388C2×C3×C4C23×C411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391#67—24no—
392#67—44no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—44no—
396C2×Dic3—44no—
397C2×Dic3C2×C22⋊C412yes—
398C2×Dic3C2×C22⋊C412yes—
399C2×Dic3C2×C22⋊C412yes—
400D12—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—24no—
406Dic6—22no—
407Dic6—22no—
408Dic6—42no—
409Dic6—44no—
410Dic6#10414yes—
411C3×C8—22no—
412C3×C8—22no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—24no—
416C3⋊C8—42no—
417#41—22no—
418#41—41no—
419#41—64no—
420#41—64no—
421#41—122no—
422C2×Q8—31no—
423C2×Q8—32no—
424C2×Q8—62no—
425C2×Q8—61no—
426C2×Q8—124no—
427C2×Q8C2×C4×D311yescharacteristic
428C2×D4—21no—
429C2×D4—32no—
430C2×D4—31no—
431C2×D4—62no—
432C2×D4C2×C4×D311yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4C2×C4×D311yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4D3×D411yescharacteristic
444M16—68no—
445M16—64no—
446M16—242no—
447M16—241no—
448C2×C8—24no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—68no—
454C2×C8—62no—
455C2×C8—62no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—31no—
460C4⋊C4—32no—
461C4⋊C4—31no—
462C4⋊C4—32no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—62no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—62no—
474C4⋊C4C2×C4×D311yescharacteristic
475C4⋊C4C2×C4×D311yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—32no—
480C22⋊C4—32no—
481C22⋊C4—32no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—32no—
491C42—32no—
492C42—32no—
493C42—62no—
494C42—62no—
495C42—62no—
496C42C2×C4×D311yescharacteristic
497C42D3×D411yescharacteristic
498C42D3×D411yescharacteristic
499C42C23×D311yescharacteristic
500C22×C3—22no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3C22×C22⋊C411yescharacteristic
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—22no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4C2×#10412yes—
517C3×C4C2×#13111yescharacteristic
518C3×C4#56811yescharacteristic
519C3×C4#57711yescharacteristic
520C3×C4#57811yescharacteristic
521Dic3—22no—
522Dic3—22no—
523Dic3—42no—
524Dic3—42no—
525Dic3C2×#10414yes—
526C23—21no—
527C23—32no—
528C23D3×C22⋊C411yescharacteristic
529Q8—21no—
530Q8—21no—
531Q8—34no—
532Q8—62no—
533Q8—62no—
534Q8—124no—
535Q8—122no—
536D4—21no—
537D4—42no—
538D4—64no—
539D4—62no—
540D4—124no—
541D4#90112yescomplemented
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—124no—
565C2×C4—124no—
566C2×C4D3×C22⋊C411yescharacteristic
567C2×C4D3×C22⋊C411yescharacteristic
568C2×C4D3×C22⋊C411yescharacteristic
569C2×C4C22×C4×D311yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C8—64no—
577C8—122no—
578C2×C3—41no—
579C2×C3—42no—
580C2×C3#288011yescharacteristic
581C2×C3#289811yescharacteristic
582C2×C3C22×#10411yescharacteristic
583D3—42no—
584C22—21no—
585C22—22no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22C2×D3×C22⋊C411yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—34no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—62no—
602C4—62no—
603C4—122no—
604C4—122no—
605C4D3×#10412yes—
606C4C2×#90111yescharacteristic
607C4#574511yescharacteristic
608C4#577011yescharacteristic
609C4#577111yescharacteristic
610C3#1751811yescharacteristic, complemented, Sylow
611C2—41no—
612C2—42no—
613C2—122no—
614C2#7860011yescharacteristic
615C2#7883211yescharacteristic
616C2C2×D3×#10411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1476C2
socleC22×C3C22×C22⋊C4
prime radical#85826C2
cosocleC3×C42C24
hypercenter#1476D3
hypocenterC3#17518
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #79294C2semidirect
3. C4×#901C2semidirect
4. C3×#1476C2semidirect
5. #71641C2semidirect
6. #70769C2semidirect
7. #69738C2semidirect
8. #68240C2semidirect
9. #68235C2semidirect
10. #68174C2semidirect
11. #68165C2semidirect
64. #1476D3semidirect
372. C3×D4#131semidirect
541. D4#901semidirect
610. C3#17518semidirect

Maximal subgroups

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

subgroupindex
#858262
#792942
C4×#9012
C3×#14762
#716412
#707692
#697382
#682402
#682352
#681742
#681652
#175183