Finite Group Databasebeta

← #10764142

Subgroups — #10764142

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86225C211yesmaximal, characteristic
2#79052C211yesmaximal, characteristic, complemented
3#79051C211yesmaximal, characteristic, complemented
4C3×#1560C211yesmaximal, characteristic, complemented
5#71805C211yesmaximal, characteristic, complemented
6#70508C211yesmaximal, characteristic, complemented
7#70506C211yesmaximal, characteristic, complemented
8#68488C212yesmaximal, complemented
9#68481C212yesmaximal, complemented
10#68386C212yesmaximal, complemented
11#68384C212yesmaximal, complemented
12#17781—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14#5833C412yes—
15#5827C412yes—
16#5728C2212yes—
17#5716C2212yes—
18#5682C414yes—
19#5618—22no—
20#5598—22no—
21#5563—21no—
22#5555—21no—
23#5543—21no—
24#5542—21no—
25C4×#235C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C3×#423C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5128—22no—
30#5127—22no—
31#4925C2211yescharacteristic
32#4925C2211yescharacteristic
33#4838—21no—
34#4837C2212yes—
35#4835C2212yes—
36#4834—21no—
37#4833C2212yes—
38D3×C4⋊C8C2212yes—
39#4831—21no—
40#4830—21no—
41C3×#333C2212yes—
42C3×#331C2212yes—
43#4546C2211yescharacteristic
44#4546C2211yescharacteristic
45#4544C2211yescharacteristic
46#4544C2211yescharacteristic
47#4486C2212yes—
48#4486C2212yes—
49#4482C2212yes—
50#4482C2212yes—
51#3511—31no—
52#2991—31no—
53#2990—31no—
54#1560D311yescharacteristic, complemented
55#1559—32no—
56#1553—32no—
57#1549—31no—
58#1547—32no—
59#1546—31no—
60#1543—31no—
61#1542—32no—
62C2×C3×#41C2×C412yes—
63C2×#237—22no—
64C2×#235C2×C411yescharacteristic
65C22×Dic6—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C3×C4⋊Q8C2311yescharacteristic
69C3×#127C2311yescharacteristic
70C3×#124C2×C412yes—
71C3×#121—21no—
72C3×#121—21no—
73C3×C4×D4—21no—
74C3×C4×D4—21no—
75C2×C3×C42C2311yescharacteristic
76#946C2×C414yes—
77Q8×Dic3C2×C414yes—
78#939—24no—
79#938—22no—
80#938C2×C414yes—
81#936—22no—
82D4×Dic3—24no—
83D4×Dic3—22no—
84D4×Dic3C2×C414yes—
85#930—22no—
86C4×#67D412yes—
87C4×#67D412yes—
88#926—22no—
89#926C2×C411yescharacteristic
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3C2×C412yes—
93#924—22no—
94C2×ΓL(1,𝔽25)—22no—
95C2×D3×C8—22no—
96#897—22no—
97#892—22no—
98#890—22no—
99#887—22no—
100#886—24no—
101#885—24no—
102#879—24no—
103#878—24no—
104C4×D12D411yescharacteristic
105C4×D12D411yescharacteristic
106#872C2×C412yes—
107C42×D3C2312yes—
108#869—21no—
109#869—21no—
110#869C2×C412yes—
111#869C2312yes—
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#556—61no—
133#555—61no—
134#554—61no—
135#553—62no—
136#548—62no—
137#547—61no—
138#542—34no—
139#537—32no—
140#533—32no—
141#532—32no—
142#532C2×D311yescharacteristic
143C2×C4⋊Q8—31no—
144#520—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#333—31no—
161#333—31no—
162#333C2×D312yes—
163#332—32no—
164#332—32no—
165#332—32no—
166#332—32no—
167#331—31no—
168#331—31no—
169#331C2×D312yes—
170C3×#41—44no—
171C2×C3×Q8C22×C412yes—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4C22×C412yes—
176C22×C3×C4—22no—
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C2×#67—24no—
181C2×#67C22⋊C412yes—
182C22×Dic3—22no—
183C22×Dic3—24no—
184#237—82no—
185#237—82no—
186#235—42no—
187C2×D12C22⋊C411yescharacteristic
188C2×C4×D3C22×C412yes—
189C2×C4×D3C2×D412yes—
190C2×C4×D3C2×D412yes—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—81no—
194C2×Dic6—81no—
195C2×Dic6C22⋊C411yescharacteristic
196C3×M16—81no—
197C3×M16—81no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C42C22×C411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207C3×C42C2411yescharacteristic
208#217—24no—
209#217—24no—
210#217C22⋊C412yes—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4C22⋊C412yes—
213(C2×D3)⋊C4C22⋊C412yes—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—24no—
218C4⋊Dic3C22⋊C411yescharacteristic
219C4⋊Dic3C22⋊C411yescharacteristic
220C4⋊Dic3C22×C414yes—
221Dic3⋊C4—24no—
222Dic3⋊C4C22⋊C412yes—
223Dic3⋊C4C22⋊C412yes—
224C4×Dic3—24no—
225C4×Dic3C22×C412yes—
226C4×Dic3C2×D412yes—
227C4×Dic3C2×D412yes—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232ΓL(1,𝔽25)—82no—
233D3×C8—82no—
234C2×#41—31no—
235C2×#41—62no—
236C2×#41C4×D312yes—
237C22×Q8—61no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—62no—
241C22×C8—62no—
242C4⋊Q8—32no—
243C4⋊Q8—32no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8C22×D311yescharacteristic
247#127C22×D311yescharacteristic
248#126—64no—
249#125—62no—
250#124—62no—
251#124—64no—
252#124C4×D312yes—
253#123—64no—
254C22⋊Q8—34no—
255C22⋊Q8—64no—
256C22⋊Q8—62no—
257C22⋊Q8—62no—
258#121—21no—
259#121—21no—
260#121—34no—
261#121—62no—
262#121—62no—
263#121—64no—
264C4×Q8—31no—
265C4×Q8—31no—
266C4×Q8—34no—
267C4×Q8—62no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—32no—
271C4×D4—32no—
272C4×D4—34no—
273C4×D4—31no—
274C4×D4—31no—
275C4×D4—62no—
276C4×D4—64no—
277#117—31no—
278#117—32no—
279#117—62no—
280#117—62no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C42—32no—
285C2×C42C22×D311yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—31no—
289C4⋊C8—31no—
290C4⋊C8—31no—
291C4⋊C8—31no—
292C4⋊C8—62no—
293C4⋊C8—62no—
294C4⋊C8C22×D312yes—
295C4⋊C8C22×D312yes—
296C22⋊C8—62no—
297C22⋊C8—62no—
298C22⋊C8—61no—
299C22⋊C8—61no—
300#97—62no—
301C4×C8—62no—
302C4×C8—61no—
303C4×C8—61no—
304C23×C3—22no—
305C23×C3C2×C22⋊C411yescharacteristic
306C22×D3C2×C22⋊C412yes—
307C3×Q8—42no—
308C3×D4—42no—
309C3×D4—44no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—44no—
321C2×C3×C4C2×C22⋊C411yescharacteristic
322C2×C3×C4C2×C22⋊C411yescharacteristic
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C23×C411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327#67—42no—
328#67—84no—
329C2×Dic3—24no—
330C2×Dic3—84no—
331C2×Dic3—82no—
332C2×Dic3C2×C22⋊C412yes—
333C2×Dic3C2×C22⋊C412yes—
334C2×Dic3C2×C22⋊C412yes—
335D12—22no—
336C4×D3—22no—
337C4×D3—22no—
338C4×D3—22no—
339C4×D3—22no—
340C4×D3—42no—
341Dic6—22no—
342Dic6—42no—
343Dic6—42no—
344Dic6—82no—
345C3×C8—42no—
346C3×C8—42no—
347C3⋊C8—41no—
348C3⋊C8—41no—
349C3⋊C8—41no—
350C3⋊C8—41no—
351#41—44no—
352#41—122no—
353#41—242no—
354#41—242no—
355C2×Q8—31no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—241no—
359C2×Q8—241no—
360C2×Q8C2×C4×D312yes—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—32no—
365C2×D4—31no—
366C2×D4—64no—
367C2×D4C2×C4×D312yes—
368C22×C4—22no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—64no—
373C22×C4—62no—
374C22×C4C2×C4×D311yescharacteristic
375C22×C4D3×D411yescharacteristic
376C22×C4D3×D411yescharacteristic
377M16—81no—
378M16—81no—
379M16—242no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—242no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—31no—
390C4⋊C4—31no—
391C4⋊C4—32no—
392C4⋊C4—32no—
393C4⋊C4—34no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—32no—
402C22⋊C4—32no—
403C22⋊C4—32no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C42—32no—
408C42—32no—
409C42—32no—
410C42—64no—
411C42C2×C4×D311yescharacteristic
412C42D3×D411yescharacteristic
413C42D3×D411yescharacteristic
414C42C23×D311yescharacteristic
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3C22×C22⋊C411yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—42no—
429C3×C4#41312yes—
430C3×C4#57611yescharacteristic
431C3×C4#57711yescharacteristic
432C3×C4#57711yescharacteristic
433C3×C4#57811yescharacteristic
434Dic3—22no—
435Dic3—22no—
436Dic3—22no—
437Dic3—44no—
438C23—22no—
439C23—32no—
440C23D3×C22⋊C411yescharacteristic
441Q8—42no—
442Q8—62no—
443Q8—122no—
444Q8—122no—
445Q8—242no—
446D4—42no—
447D4—44no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—62no—
453D4—122no—
454D4—244no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—44no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—64no—
470C2×C4—122no—
471C2×C4—244no—
472C2×C4—242no—
473C2×C4D3×C22⋊C411yescharacteristic
474C2×C4D3×C22⋊C411yescharacteristic
475C2×C4D3×C22⋊C411yescharacteristic
476C2×C4C22×C4×D311yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C8—42no—
480C8—42no—
481C8—121no—
482C8—121no—
483C8—121no—
484C8—121no—
485C2×C3—41no—
486C2×C3—82no—
487C2×C3C2×#41311yescharacteristic
488C2×C3#289711yescharacteristic
489C2×C3#289811yescharacteristic
490D3—42no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—62no—
498C22—62no—
499C22—62no—
500C22C2×D3×C22⋊C411yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—62no—
506C4—62no—
507C4—62no—
508C4—124no—
509C4#474512yes—
510C4#577411yescharacteristic
511C4#577511yescharacteristic
512C4#577811yescharacteristic
513C4#577911yescharacteristic
514C3#1778111yescharacteristic, complemented, Sylow
515C2—41no—
516C2—82no—
517C2—122no—
518C2C2×#474511yescharacteristic
519C2#7888211yescharacteristic
520C2#7888311yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1560C2
socleC22×C3C22×C22⋊C4
prime radical#86225C2
cosocleC3×C42C24
hypercenter#1560D3
hypocenterC3#17781
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
2. #79052C2semidirect
3. #79051C2semidirect
4. C3×#1560C2semidirect
5. #71805C2semidirect
6. #70508C2semidirect
7. #70506C2semidirect
8. #68488C2semidirect
9. #68481C2semidirect
10. #68386C2semidirect
11. #68384C2semidirect
54. #1560D3semidirect
514. C3#17781semidirect

Maximal subgroups

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

subgroupindex
#862252
#790522
#790512
C3×#15602
#718052
#705082
#705062
#684882
#684812
#683862
#683842
#177813