Finite Group Databasebeta

← #10764117

Subgroups — #10764117

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86204C211yesmaximal, characteristic
2#79294C211yesmaximal, characteristic
3#79051C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70511C211yesmaximal, characteristic, complemented
8#68488C212yesmaximal, complemented
9#68486C212yesmaximal, complemented
10#68484C212yesmaximal, complemented
11#68481C212yesmaximal, complemented
12#17778—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14#5816C412yes—
15#5812C412yes—
16#5732—21no—
17#5732—21no—
18#5731C414yes—
19#5728C2212yes—
20#5728C2212yes—
21#5721—21no—
22#5721—21no—
23#5513—22no—
24#5513—22no—
25C4×#235C2211yescharacteristic
26#5470C2211yescharacteristic
27C3×#431C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5127—22no—
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4837C2212yes—
33#4835C2212yes—
34#4834—21no—
35#4833C2212yes—
36D3×C4⋊C8C2212yes—
37#4831—21no—
38#4729—22no—
39#4688—21no—
40C8×Dic6—21no—
41C3×#333C2212yes—
42C3×#333C2212yes—
43#4546C2212yes—
44#4546C2212yes—
45#4486C2212yes—
46#4486C2211yescharacteristic
47#4486C2211yescharacteristic
48#4482C2212yes—
49#4482C2211yescharacteristic
50#4482C2211yescharacteristic
51#3509—31no—
52#2991—31no—
53#2990—31no—
54#1561—31no—
55#1559—32no—
56#1554—32no—
57#1550—32no—
58#1548—31no—
59#1548D311yescharacteristic, complemented
60#1544—31no—
61#1542—32no—
62C22×C3×Q8C2×C412yes—
63C2×#235C2×C411yescharacteristic
64C2×C3×M16—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8C2×C412yes—
68C3×C4⋊Q8C2312yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72#947—22no—
73#946C2×C414yes—
74#946C2×C414yes—
75Q8×Dic3—22no—
76Q8×Dic3—22no—
77Q8×Dic3C2×C414yes—
78Q8×Dic3C2×C414yes—
79#944—22no—
80#929—24no—
81C4×#67D412yes—
82C4×#67D412yes—
83#926—24no—
84#926C2×C412yes—
85#926C2×C411yescharacteristic
86#924—21no—
87#924—21no—
88C2×#208—21no—
89C2×ΓL(1,𝔽25)—22no—
90C2×D3×C8—22no—
91#898—24no—
92#898—24no—
93#897—22no—
94#897—22no—
95#892—22no—
96#892—22no—
97#890—22no—
98#890—22no—
99#889—24no—
100#889—24no—
101#887—22no—
102#887—22no—
103C4×D12D411yescharacteristic
104C4×D12D411yescharacteristic
105#872C2×C412yes—
106C42×D3C2312yes—
107#870C2×C412yes—
108#869C2311yescharacteristic
109#869C2311yescharacteristic
110C4×Dic6—21no—
111C4×Dic6—21no—
112C4×Dic6D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8C2312yes—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C22⋊C8—21no—
119C3×C4×C8—21no—
120#830—21no—
121(C2×D3)⋊C8—22no—
122(C2×D3)⋊C8—22no—
123#815—22no—
124Dic3⋊C8—22no—
125Dic3⋊C8—22no—
126C8×Dic3—22no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)C2312yes—
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4⋊(C3⋊C8)C2311yescharacteristic
131C4×C3⋊C8—21no—
132#557—61no—
133#557—61no—
134#556—61no—
135#556—61no—
136#552—62no—
137#552—62no—
138#541—34no—
139#536—32no—
140#533—32no—
141#533—32no—
142#533—31no—
143C2×C4⋊Q8C2×D311yescharacteristic
144#519—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—32no—
155#431C2×D311yescharacteristic
156#423—31no—
157#423—32no—
158#423C2×D311yescharacteristic
159C2×C4⋊C8—32no—
160#333—32no—
161#333—32no—
162#333C2×D312yes—
163#333C2×D312yes—
164#332—31no—
165#332—32no—
166#332—31no—
167#332—31no—
168#332—31no—
169#332—32no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8C22×C414yes—
173C22×C3×C4—22no—
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C2×#67C22⋊C412yes—
178#235—42no—
179C2×D12C22⋊C411yescharacteristic
180C2×C4×D3C22×C412yes—
181C2×C4×D3C2×D412yes—
182C2×C4×D3C2×D412yes—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C22⋊C411yescharacteristic
186C3×M16—81no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C42C22×C411yescharacteristic
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D411yescharacteristic
197C3×C42C2411yescharacteristic
198#217—22no—
199#217C22⋊C412yes—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4C22⋊C412yes—
203(C2×D3)⋊C4C22⋊C412yes—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—24no—
207C4⋊Dic3—24no—
208C4⋊Dic3C22⋊C411yescharacteristic
209C4⋊Dic3C22⋊C411yescharacteristic
210C4⋊Dic3C22×C412yes—
211C4⋊Dic3C22×C412yes—
212Dic3⋊C4—22no—
213Dic3⋊C4—22no—
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216Dic3⋊C4C22⋊C412yes—
217Dic3⋊C4C22⋊C412yes—
218C4×Dic3—22no—
219C4×Dic3—24no—
220C4×Dic3—24no—
221C4×Dic3C22×C412yes—
222C4×Dic3C2×D412yes—
223C4×Dic3C2×D412yes—
224#208—81no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228ΓL(1,𝔽25)—82no—
229D3×C8—82no—
230C2×#41—31no—
231C22×Q8C4×D312yes—
232C2×M16—21no—
233C2×M16—62no—
234C2×M16—61no—
235C22×C8—62no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—31no—
239C4⋊Q8—31no—
240C4⋊Q8—62no—
241C4⋊Q8C4×D312yes—
242C4⋊Q8C22×D312yes—
243#126—64no—
244#126—64no—
245#125—32no—
246#125—62no—
247#125—62no—
248#125—64no—
249#125—64no—
250#124—62no—
251#123—64no—
252C22⋊Q8—34no—
253C22⋊Q8—34no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256C22⋊Q8—61no—
257C22⋊Q8—61no—
258C4×Q8—31no—
259C4×Q8—31no—
260C4×Q8—34no—
261C4×Q8—34no—
262C4×Q8—62no—
263C4×Q8—62no—
264C4×Q8—62no—
265C4×Q8—62no—
266C4×Q8—61no—
267C4×Q8—61no—
268C4×D4—32no—
269C4×D4—32no—
270C4×D4—31no—
271C4×D4—31no—
272#117—31no—
273#117—32no—
274#117—32no—
275#117—64no—
276#117—62no—
277#117—62no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42—32no—
281C2×C42C22×D311yescharacteristic
282C4⋊C8—21no—
283C4⋊C8—31no—
284C4⋊C8—32no—
285C4⋊C8—31no—
286C4⋊C8—62no—
287C4⋊C8—62no—
288C4⋊C8—61no—
289C4⋊C8C22×D312yes—
290C4⋊C8C22×D311yescharacteristic
291C4⋊C8C22×D311yescharacteristic
292C22⋊C8—21no—
293C22⋊C8—62no—
294C22⋊C8—62no—
295C22⋊C8—61no—
296#97—62no—
297C4×C8—21no—
298C4×C8—62no—
299C4×C8—61no—
300C23×C3C2×C22⋊C411yescharacteristic
301C22×D3C2×C22⋊C412yes—
302C3×Q8—44no—
303C3×Q8—44no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—44no—
311C2×C3×C4C2×C22⋊C411yescharacteristic
312C2×C3×C4C2×C22⋊C411yescharacteristic
313C2×C3×C4C2×C22⋊C411yescharacteristic
314C2×C3×C4C23×C411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317#67—42no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3C2×C22⋊C412yes—
321C2×Dic3C2×C22⋊C412yes—
322C2×Dic3C2×C22⋊C412yes—
323D12—22no—
324C4×D3—22no—
325C4×D3—22no—
326C4×D3—22no—
327C4×D3—22no—
328C4×D3—42no—
329Dic6—22no—
330Dic6—41no—
331Dic6—41no—
332Dic6—41no—
333Dic6—41no—
334C3×C8—42no—
335C3×C8—41no—
336C3×C8—41no—
337C3⋊C8—41no—
338C3⋊C8—42no—
339C3⋊C8—41no—
340#41—122no—
341C2×Q8—22no—
342C2×Q8—31no—
343C2×Q8—44no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8C2×C4×D314yes—
347C2×D4—32no—
348C2×D4—31no—
349C22×C4—22no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4C2×C4×D311yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4D3×D411yescharacteristic
356M16—81no—
357M16—242no—
358M16—241no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—62no—
365C2×C8—242no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—31no—
371C4⋊C4—31no—
372C4⋊C4—32no—
373C4⋊C4—32no—
374C4⋊C4—32no—
375C4⋊C4—32no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—64no—
379C4⋊C4—64no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C22⋊C4—32no—
385C22⋊C4—32no—
386C22⋊C4—32no—
387C22⋊C4—64no—
388C22⋊C4—64no—
389C22⋊C4—62no—
390C42—32no—
391C42—32no—
392C42—32no—
393C42—64no—
394C42—64no—
395C42—62no—
396C42C2×C4×D311yescharacteristic
397C42D3×D411yescharacteristic
398C42D3×D411yescharacteristic
399C42C23×D311yescharacteristic
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3C22×C22⋊C411yescharacteristic
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—22no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—44no—
411C3×C4#41312yes—
412C3×C4#57711yescharacteristic
413C3×C4#57711yescharacteristic
414C3×C4#57811yescharacteristic
415C3×C4#57811yescharacteristic
416Dic3—22no—
417Dic3—22no—
418Dic3—22no—
419Dic3—42no—
420Dic3—42no—
421C23—32no—
422C23D3×C22⋊C411yescharacteristic
423Q8—44no—
424Q8—44no—
425Q8—62no—
426Q8—121no—
427Q8—121no—
428Q8—121no—
429Q8—121no—
430D4—62no—
431D4—122no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—32no—
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—44no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—122no—
449C2×C4D3×C22⋊C411yescharacteristic
450C2×C4D3×C22⋊C411yescharacteristic
451C2×C4D3×C22⋊C411yescharacteristic
452C2×C4C22×C4×D311yescharacteristic
453C2×C4C2×D3×D411yescharacteristic
454C2×C4C2×D3×D411yescharacteristic
455C8—42no—
456C8—41no—
457C8—41no—
458C8—121no—
459C8—121no—
460C8—122no—
461C2×C3—41no—
462C2×C3C2×#41311yescharacteristic
463C2×C3#289811yescharacteristic
464C2×C3#289811yescharacteristic
465D3—42no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—62no—
470C22—62no—
471C22—62no—
472C22C2×D3×C22⋊C411yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—44no—
477C4—62no—
478C4—62no—
479C4—62no—
480C4—122no—
481C4—122no—
482C4#474512yes—
483C4#576211yescharacteristic
484C4#576611yescharacteristic
485C4#577811yescharacteristic
486C4#577911yescharacteristic
487C3#1777811yescharacteristic, complemented, Sylow
488C2—41no—
489C2—122no—
490C2C2×#474511yescharacteristic
491C2#7879011yescharacteristic
492C2#7895711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1548C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1548D3
hypocenterC3#17778
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
4. C3×#1548C2semidirect
5. #71774C2semidirect
6. #70777C2semidirect
7. #70511C2semidirect
8. #68488C2semidirect
9. #68486C2semidirect
10. #68484C2semidirect
11. #68481C2semidirect
59. #1548D3semidirect
487. C3#17778semidirect

Maximal subgroups

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

subgroupindex
#862042
#792942
#790512
C3×#15482
#717742
#707772
#705112
#684882
#684862
#684842
#684812
#177783