Finite Group Databasebeta

← #10782821

Subgroups — #10782821

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81571C211yesmaximal, characteristic, complemented
2#81115C211yesmaximal, characteristic, complemented
3#80401C211yesmaximal, characteristic
4#79909C211yesmaximal, characteristic, complemented
5C3×#1678C211yesmaximal, characteristic, complemented
6#71202C211yesmaximal, characteristic, complemented
7#71153C211yesmaximal, characteristic, complemented
8#71026C211yesmaximal, characteristic, complemented
9#70945C211yesmaximal, characteristic, complemented
10#68543C211yesmaximal, characteristic, complemented
11#68534C211yesmaximal, characteristic, complemented
12#68497C211yesmaximal, characteristic, complemented
13D3×#333C211yesmaximal, characteristic, complemented
14#68396C211yesmaximal, characteristic, complemented
15#68385C211yesmaximal, characteristic, complemented
16#19591—31nomaximal, Sylow
17D3×C4⋊Q8C2211yescharacteristic
18#5716C2211yescharacteristic
19#5580C2211yescharacteristic
20C3×#486C2211yescharacteristic
21C3×#477C2211yescharacteristic
22C3×#460C2211yescharacteristic
23C3×C8×Q8C2211yescharacteristic
24#5171—21no—
25#5165—21no—
26#5084C2211yescharacteristic
27#5078C2211yescharacteristic
28#5072C2211yescharacteristic
29#5070C2211yescharacteristic
30#5024C2211yescharacteristic
31#5021C2211yescharacteristic
32#5021C2211yescharacteristic
33#4882—21no—
34#4876—21no—
35#4842C2211yescharacteristic
36#4842C2211yescharacteristic
37#4839—21no—
38#4836—21no—
39#4835C2211yescharacteristic
40#4834C2211yescharacteristic
41D3×C4⋊C8C2211yescharacteristic
42D3×C4⋊C8C2211yescharacteristic
43#4805—21no—
44#4794—21no—
45#4775—21no—
46#4774—21no—
47#4774—21no—
48#4763—21no—
49#4759—21no—
50#4759—21no—
51#4690—21no—
52#4688C2211yescharacteristic
53C8×D12—21no—
54#4685C2211yescharacteristic
55#4682C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4546C2211yescharacteristic
60#4544C2211yescharacteristic
61#4481C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3247—31no—
69#3238—31no—
70#3226—31no—
71#2994—31no—
72#1698—31no—
73#1697—31no—
74#1696—31no—
75#1695—31no—
76#1678D311yescharacteristic, complemented
77#1677—31no—
78#1676—31no—
79#1675—31no—
80#1573—31no—
81#1551—31no—
82C2×#333—31no—
83C2×D3×Q8—21no—
84C2×#237—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×#127C2311yescharacteristic
88C3×C4×Q8C2311yescharacteristic
89#947D411yescharacteristic
90#946D411yescharacteristic
91#944—21no—
92#938—21no—
93#938—21no—
94#936—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97C2×#213—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101C2×D3×C8—21no—
102#898—21no—
103#895D411yescharacteristic
104#894—21no—
105D3×C4⋊C4—21no—
106D3×C4⋊C4—21no—
107#890D411yescharacteristic
108#888—21no—
109#888—21no—
110C4×D12—21no—
111C4×D12C2311yescharacteristic
112#872—21no—
113C42×D3C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C3×#107—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×#103—21no—
122C3×#102—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×C4×C8—21no—
126C3×C4×C8C2311yescharacteristic
127#835—21no—
128#835—21no—
129#832—21no—
130#832—21no—
131#832—21no—
132#821—21no—
133#821—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8D411yescharacteristic
138(C2×D3)⋊C8D411yescharacteristic
139#818—21no—
140#818—21no—
141#817—21no—
142#817—21no—
143#815—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8D411yescharacteristic
147Dic3⋊C8D411yescharacteristic
148#809—21no—
149#808—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#500—31no—
165#500—31no—
166#489—61no—
167#486—31no—
168#486C2×D311yescharacteristic
169#483—61no—
170#480—61no—
171#480—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#477C2×D311yescharacteristic
177#476—31no—
178#476—31no—
179#476—31no—
180#474—61no—
181#474—61no—
182#466—61no—
183#465—61no—
184#463—61no—
185#460C2×D311yescharacteristic
186#459—31no—
187#446—31no—
188C8×Q8C2×D311yescharacteristic
189#436—31no—
190#435—61no—
191C8×D4—31no—
192C8×D4—61no—
193#433—61no—
194#432—31no—
195#431—31no—
196#424—61no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#333—31no—
208#333C2×D311yescharacteristic
209#331—31no—
210#331C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×#67—41no—
217C22×Dic3—41no—
218D3×Q8—82no—
219#237—82no—
220C2×D12—21no—
221C2×D12C2×D411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—41no—
224C2×C4×D3C2×D411yescharacteristic
225C2×C4×D3C2×D411yescharacteristic
226C2×C4×D3#4111yescharacteristic
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C3×SD16—81no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8C2×D411yescharacteristic
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C42—21no—
241C3×C42C2411yescharacteristic
242#217—41no—
243#217—41no—
244#213—81no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260C4×Dic3C2×D411yescharacteristic
261C4×Dic3C2×D411yescharacteristic
262C4×Dic3#4111yescharacteristic
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268ΓL(1,𝔽25)—81no—
269D3×C8—81no—
270D3×C8—81no—
271D3×C8—81no—
272C2×#41—61no—
273C22×Q8—61no—
274C2×SD16—21no—
275C2×D8—61no—
276C2×M16—61no—
277C22×C8—61no—
278C22×C8—61no—
279C22×C8—61no—
280C4⋊Q8—31no—
281C4⋊Q8—31no—
282C4⋊Q8—61no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8C22×D311yescharacteristic
286#127C22×D311yescharacteristic
287#126—61no—
288#125—31no—
289#124—31no—
290#124—61no—
291#123—61no—
292C22⋊Q8—31no—
293#121—31no—
294#121—61no—
295#121—61no—
296C4×Q8C22×D311yescharacteristic
297C4×D4—31no—
298C4×D4—61no—
299C4×D4—61no—
300C4×D4—61no—
301#117—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C42—31no—
305#107—21no—
306#107—61no—
307#107—61no—
308#107—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#106—61no—
313#106—61no—
314#106—61no—
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—21no—
328#103—61no—
329#103—61no—
330#102—21no—
331#102—21no—
332#102—21no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—31no—
340C22⋊C8—31no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—61no—
345#97—61no—
346C4×C8—21no—
347C4×C8—31no—
348C4×C8C22×D311yescharacteristic
349C23×C3—41no—
350C22×D3—21no—
351C22×D3#12111yescharacteristic
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—81no—
355C3×D4—41no—
356C3×D4—41no—
357C3×D4—41no—
358C3×D4—41no—
359C3×D4—81no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366#67—82no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—82no—
371C2×Dic3#12111yescharacteristic
372C2×Dic3#12111yescharacteristic
373C2×Dic3#12111yescharacteristic
374D12—21no—
375D12—21no—
376D12—42no—
377C4×D3—22no—
378C4×D3—21no—
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—82no—
385Dic6—82no—
386Dic6—81no—
387Dic6—81no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—242no—
399C2×Q8—21no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—121no—
403C2×Q8—242no—
404C2×Q8D3×D411yescharacteristic
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—31no—
409C2×D4—61no—
410C2×D4—121no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—61no—
415C22×C4—121no—
416C22×C4—121no—
417SD16—81no—
418D8—241no—
419M16—241no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—241no—
429C2×C8—241no—
430C2×C8—241no—
431C2×C8D3×D411yescharacteristic
432C2×C8D3×D411yescharacteristic
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—31no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4D3×D411yescharacteristic
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—21no—
455C42—31no—
456C42—31no—
457C42—31no—
458C42C23×D311yescharacteristic
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3C2×#12111yescharacteristic
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—41no—
468C2×D3#47212yes—
469C3×C4—21no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4#57711yescharacteristic
474C3×C4#57711yescharacteristic
475C3×C4#57811yescharacteristic
476C3×C4#57811yescharacteristic
477Dic3—21no—
478Dic3—21no—
479Dic3—21no—
480Dic3—41no—
481Dic3—81no—
482Dic3—81no—
483C23—31no—
484C23—41no—
485C23—61no—
486Q8—21no—
487Q8—21no—
488Q8—81no—
489Q8—241no—
490Q8—241no—
491Q8—242no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—61no—
497D4—61no—
498D4—81no—
499D4—122no—
500D4—242no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—41no—
507C2×C4—62no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—242no—
518C2×C4—242no—
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C2×C4D3×#4111yescharacteristic
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—61no—
528C8—61no—
529C8—121no—
530C8—121no—
531C8—121no—
532C2×C3—161no—
533C2×C3#304611yescharacteristic
534C2×C3#306311yescharacteristic
535C2×C3C2×#47211yescharacteristic
536D3—22no—
537D3—81no—
538C22—32no—
539C22—61no—
540C22—61no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22D3×#12111yescharacteristic
548C4—21no—
549C4—41no—
550C4—41no—
551C4—61no—
552C4—61no—
553C4—61no—
554C4—81no—
555C4—121no—
556C4—241no—
557C4—241no—
558C4#576211yescharacteristic
559C4#577011yescharacteristic
560C4#577111yescharacteristic
561C4#577911yescharacteristic
562C3#1959111yescharacteristic, complemented, Sylow
563C2—62no—
564C2—161no—
565C2—241no—
566C2#8060311yescharacteristic
567C2#8066211yescharacteristic
568C2D3×#47211yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1678C2
socleC22×C3C2×#121
prime radical#80401C2
hypercenter#1678D3
hypocenterC3#19591
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
1. #81571C2semidirect
2. #81115C2semidirect
4. #79909C2semidirect
5. C3×#1678C2semidirect
6. #71202C2semidirect
7. #71153C2semidirect
8. #71026C2semidirect
9. #70945C2semidirect
10. #68543C2semidirect
11. #68534C2semidirect
12. #68497C2semidirect
13. D3×#333C2semidirect
14. #68396C2semidirect
15. #68385C2semidirect
76. #1678D3semidirect
562. C3#19591semidirect

Maximal subgroups

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

subgroupindex
#815712
#811152
#804012
#799092
C3×#16782
#712022
#711532
#710262
#709452
#685432
#685342
#684972
D3×#3332
#683962
#683852
#195913