Finite Group Databasebeta

← #10764096

Subgroups — #10764096

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86261C211yesmaximal, characteristic
2#79052C211yesmaximal, characteristic, complemented
3#79051C211yesmaximal, characteristic, complemented
4C3×#1561C211yesmaximal, characteristic, complemented
5#71829C211yesmaximal, characteristic, complemented
6#70502C211yesmaximal, characteristic, complemented
7#70502C211yesmaximal, characteristic, complemented
8#68429C212yesmaximal, complemented
9#68426C212yesmaximal, complemented
10#68421C212yesmaximal, complemented
11#68417C212yesmaximal, complemented
12#17776—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14#5725C2212yes—
15#5725C2212yes—
16#5697C414yes—
17#5630—22no—
18#5630—22no—
19#5585—21no—
20#5585—21no—
21#5577—21no—
22#5577—21no—
23#5525C412yes—
24#5521C412yes—
25C2×#874C2211yescharacteristic
26C4×#235C2211yescharacteristic
27C3×#423C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5128—22no—
30#5127—22no—
31#4925C2211yescharacteristic
32#4925C2211yescharacteristic
33#4838—21no—
34#4837C2212yescomplemented
35#4835C2212yescomplemented
36#4834—21no—
37#4833C2212yescomplemented
38D3×C4⋊C8C2212yescomplemented
39#4831—21no—
40#4830—21no—
41C3×#332C2212yescomplemented
42C3×#332C2212yescomplemented
43#4547C2211yescharacteristic, complemented
44#4547C2211yescharacteristic, complemented
45#4547C2211yescharacteristic, complemented
46#4547C2211yescharacteristic, complemented
47#4485C2212yes—
48#4485C2212yes—
49#4485C2212yes—
50#4485C2212yes—
51#3508—31no—
52#2991—31no—
53#2990—31no—
54#1561D311yescharacteristic, complemented
55#1557—32no—
56#1553—32no—
57#1547—32no—
58#1545—31no—
59#1545—31no—
60#1544—31no—
61#1540—32no—
62C2×#239—22no—
63C2×#239—22no—
64C2×#235C2×C411yescharacteristic
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2×C412yes—
68C2×C3×M16—21no—
69C2×C3×M16—21no—
70C3×C4⋊Q8C2311yescharacteristic
71C3×C4⋊Q8C2311yescharacteristic
72C3×#125C2×C412yes—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—21no—
75C3×C4×Q8—21no—
76C3×C4×Q8—21no—
77C3×#117C2×C412yes—
78C2×C3×C42C2311yescharacteristic
79#947—22no—
80#947—22no—
81C4×#67D412yes—
82C4×#67D412yes—
83#926C2×C411yescharacteristic
84C2×ΓL(1,𝔽25)—22no—
85C2×D3×C8—22no—
86#895—22no—
87#895—22no—
88#895—24no—
89#895—24no—
90#895C2×C414yes—
91#895C2×C414yes—
92#894—24no—
93#894—24no—
94#893—22no—
95#893—22no—
96#893C2×C414yes—
97#893C2×C414yes—
98#882—24no—
99D3×C22⋊C4—24no—
100#875—22no—
101#874—22no—
102#874—21no—
103#874—21no—
104#874C2×C412yes—
105#874C2312yes—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12D411yescharacteristic
109C4×D12D411yescharacteristic
110#872C2×C412yes—
111C42×D3C2312yes—
112C4×Dic6D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2312yes—
117C3×C4⋊C8C2312yes—
118#830—21no—
119#830—21no—
120(C2×D3)⋊C8—22no—
121(C2×D3)⋊C8—22no—
122#815—22no—
123Dic3⋊C8—22no—
124Dic3⋊C8—22no—
125C8×Dic3—22no—
126C4⋊(C3⋊C8)C2311yescharacteristic
127C4⋊(C3⋊C8)C2311yescharacteristic
128C4⋊(C3⋊C8)C2311yescharacteristic
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4×C3⋊C8—21no—
131C4×C3⋊C8—21no—
132#559—61no—
133#559—61no—
134#550—61no—
135#550—61no—
136#546—62no—
137#546—62no—
138#540—34no—
139#535—32no—
140#533C2×D311yescharacteristic
141#532—32no—
142#532—32no—
143C2×#127—31no—
144#518—32no—
145C4×#41—31no—
146#446—61no—
147C8×Q8—61no—
148#436—61no—
149#436—62no—
150#435—62no—
151C8×D4—61no—
152#432—32no—
153#431—31no—
154#431—31no—
155#431—32no—
156#423—32no—
157#423C2×D311yescharacteristic
158#423C2×D311yescharacteristic
159C2×C4⋊C8—32no—
160#332—31no—
161#332—31no—
162#332—31no—
163#332—31no—
164#332C2×D312yescomplemented
165#332C2×D312yescomplemented
166#331—32no—
167#331—32no—
168#331—32no—
169#331—32no—
170C23×D3—22no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C22×C3×C4C22×C411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C2×#67C22⋊C412yes—
177#239—82no—
178#239—82no—
179#239—82no—
180#239—82no—
181#235—42no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—24no—
186C2×D12—24no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—44no—
190C2×D12—81no—
191C2×D12—81no—
192C2×D12—81no—
193C2×D12—81no—
194C2×D12C22⋊C411yescharacteristic
195C2×D12C22×C414yes—
196C2×C4×D3—24no—
197C2×C4×D3—24no—
198C2×C4×D3C22×C412yes—
199C2×C4×D3C2×D412yes—
200C2×C4×D3C2×D412yes—
201C2×Dic6C22⋊C411yescharacteristic
202C3×M16—81no—
203C3×M16—81no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4C22×C412yes—
211C3×C4⋊C4C22×C412yes—
212C3×C22⋊C4—22no—
213C3×C42—22no—
214C3×C42C22×C411yescharacteristic
215C3×C42C2×D411yescharacteristic
216C3×C42C2×D411yescharacteristic
217C3×C42C2411yescharacteristic
218#217C22⋊C412yes—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4C22⋊C412yes—
224(C2×D3)⋊C4C22⋊C412yes—
225C4⋊Dic3C22⋊C411yescharacteristic
226C4⋊Dic3C22⋊C411yescharacteristic
227Dic3⋊C4C22⋊C412yes—
228Dic3⋊C4C22⋊C412yes—
229C4×Dic3C22×C412yes—
230C4×Dic3C2×D412yes—
231C4×Dic3C2×D412yes—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236ΓL(1,𝔽25)—82no—
237D3×C8—82no—
238C2×#41—31no—
239C2×#41—62no—
240C2×#41—62no—
241C22×D4—32no—
242C22×D4—61no—
243C22×D4—61no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—62no—
247C22×C8—62no—
248C4⋊Q8C22×D311yescharacteristic
249C4⋊Q8C22×D311yescharacteristic
250#127—32no—
251#127—32no—
252#127—62no—
253#127—61no—
254#127—61no—
255#125C4×D312yes—
256#124—62no—
257#124—62no—
258#124—62no—
259#123—64no—
260#123—64no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263#121—34no—
264#121—34no—
265#121—64no—
266#121—64no—
267#121—62no—
268#121—62no—
269#120—64no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—31no—
273C4×Q8—31no—
274C4×D4—32no—
275C4×D4—32no—
276C4×D4—34no—
277C4×D4—34no—
278C4×D4—31no—
279C4×D4—31no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—62no—
284#117—31no—
285#117—32no—
286#117C4×D312yes—
287C2×C22⋊C4—64no—
288C2×C42—32no—
289C2×C42C22×D311yescharacteristic
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—31no—
293C4⋊C8—31no—
294C4⋊C8—31no—
295C4⋊C8—31no—
296C4⋊C8—62no—
297C4⋊C8—62no—
298C4⋊C8C22×D312yes—
299C4⋊C8C22×D312yes—
300C22⋊C8—62no—
301C22⋊C8—62no—
302C22⋊C8—61no—
303C22⋊C8—61no—
304#97—62no—
305C4×C8—62no—
306C4×C8—61no—
307C4×C8—61no—
308C23×C3C2×C22⋊C411yescharacteristic
309C22×D3—24no—
310C22×D3—44no—
311C22×D3—82no—
312C22×D3—82no—
313C22×D3C2×C22⋊C412yes—
314C3×Q8—41no—
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—41no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4C2×C22⋊C411yescharacteristic
327C2×C3×C4C2×C22⋊C411yescharacteristic
328C2×C3×C4C23×C411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331#67—42no—
332C2×Dic3C2×C22⋊C412yes—
333C2×Dic3C2×C22⋊C412yes—
334C2×Dic3C2×C22⋊C412yes—
335D12—22no—
336D12—44no—
337D12—42no—
338D12—42no—
339D12—42no—
340D12—42no—
341D12—44no—
342D12—82no—
343D12—82no—
344D12—84no—
345D12—84no—
346C4×D3—22no—
347C4×D3—22no—
348C4×D3—22no—
349C4×D3—22no—
350C4×D3—42no—
351C4×D3—84no—
352C4×D3—84no—
353Dic6—22no—
354C3×C8—42no—
355C3×C8—42no—
356C3⋊C8—41no—
357C3⋊C8—41no—
358C3⋊C8—41no—
359C3⋊C8—41no—
360C24—62no—
361#41—122no—
362#41—242no—
363#41—242no—
364#41—242no—
365#41—242no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—31no—
369C2×D4—32no—
370C2×D4—31no—
371C2×D4—34no—
372C2×D4—64no—
373C2×D4—64no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—124no—
380C2×D4—241no—
381C2×D4—241no—
382C2×D4—241no—
383C2×D4—241no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—64no—
388C22×C4—64no—
389C22×C4C2×C4×D311yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4D3×D411yescharacteristic
392M16—81no—
393M16—81no—
394M16—242no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—242no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—31no—
407C4⋊C4—31no—
408C4⋊C4—32no—
409C4⋊C4—32no—
410C4⋊C4C2×C4×D312yes—
411C4⋊C4C2×C4×D312yes—
412C22⋊C4—22no—
413C22⋊C4—32no—
414C22⋊C4—32no—
415C22⋊C4—32no—
416C22⋊C4—64no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C42—22no—
421C42—32no—
422C42—32no—
423C42—32no—
424C42C2×C4×D311yescharacteristic
425C42D3×D411yescharacteristic
426C42D3×D411yescharacteristic
427C42C23×D311yescharacteristic
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3C22×C22⋊C411yescharacteristic
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—44no—
436C2×D3—44no—
437C2×D3—44no—
438C2×D3—164no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4#41312yes—
445C3×C4#57611yescharacteristic
446C3×C4#57611yescharacteristic
447C3×C4#57711yescharacteristic
448C3×C4#57711yescharacteristic
449Dic3—22no—
450Dic3—22no—
451Dic3—22no—
452C23—32no—
453C23—64no—
454C23—124no—
455C23—242no—
456C23—242no—
457C23D3×C22⋊C411yescharacteristic
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—62no—
463D4—62no—
464D4—124no—
465D4—122no—
466D4—122no—
467D4—122no—
468D4—124no—
469D4—122no—
470D4—122no—
471D4—242no—
472D4—242no—
473D4—244no—
474D4—244no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—244no—
491C2×C4—244no—
492C2×C4D3×C22⋊C411yescharacteristic
493C2×C4D3×C22⋊C411yescharacteristic
494C2×C4D3×C22⋊C411yescharacteristic
495C2×C4C22×C4×D311yescharacteristic
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C8—42no—
499C8—42no—
500C8—121no—
501C8—121no—
502C8—121no—
503C8—121no—
504C2×C3—41no—
505C2×C3C2×#41311yescharacteristic
506C2×C3#289711yescharacteristic
507C2×C3#289711yescharacteristic
508D3—42no—
509D3—84no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—62no—
514C22—62no—
515C22—62no—
516C22—124no—
517C22—124no—
518C22—124no—
519C22—484no—
520C22C2×D3×C22⋊C411yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—62no—
527C4—62no—
528C4—62no—
529C4#475112yes—
530C4#577411yescharacteristic
531C4#577411yescharacteristic
532C4#577811yescharacteristic
533C4#577811yescharacteristic
534C3#1777611yescharacteristic, complemented, Sylow
535C2—41no—
536C2—122no—
537C2—244no—
538C2C2×#475111yescharacteristic
539C2#7888211yescharacteristic
540C2#7888211yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1561C2
socleC22×C3C22×C22⋊C4
prime radical#86261C2
cosocleC3×C42C24
hypercenter#1561D3
hypocenterC3#17776
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #79052C2semidirect
3. #79051C2semidirect
4. C3×#1561C2semidirect
5. #71829C2semidirect
6. #70502C2semidirect
7. #70502C2semidirect
8. #68429C2semidirect
9. #68426C2semidirect
10. #68421C2semidirect
11. #68417C2semidirect
34. #4837C22semidirect
35. #4835C22semidirect
37. #4833C22semidirect
38. D3×C4⋊C8C22semidirect
41. C3×#332C22semidirect
42. C3×#332C22semidirect
43. #4547C22semidirect
44. #4547C22semidirect
45. #4547C22semidirect
46. #4547C22semidirect
54. #1561D3semidirect
164. #332C2×D3semidirect
165. #332C2×D3semidirect
534. C3#17776semidirect

Maximal subgroups

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

subgroupindex
#862612
#790522
#790512
C3×#15612
#718292
#705022
#684292
#684262
#684212
#684172
#177763