Finite Group Databasebeta

← #79776

Subgroups — #79776

Order 384.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#915C211yesmaximal, characteristic
2#5572C211yesmaximal, characteristic, complemented
3#5546C211yesmaximal, characteristic
4C3×#440C211yesmaximal, characteristic, complemented
5Dic3×SD16C211yesmaximal, characteristic, complemented
6C4×#216C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4881C211yesmaximal, characteristic, complemented
9#4873C211yesmaximal, characteristic, complemented
10D3×#103C211yesmaximal, characteristic, complemented
11#4789C211yesmaximal, characteristic, complemented
12#4772C211yesmaximal, characteristic
13#4757C211yesmaximal, characteristic, complemented
14#4716C211yesmaximal, characteristic, complemented
15#4708C211yesmaximal, characteristic, complemented
16#2947—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23C2×#216C2211yescharacteristic
24D4×Dic3C2211yescharacteristic
25C2×#214C2211yescharacteristic
26C4×#67—21no—
27C2×Dic3⋊C4—21no—
28#915C418yescomplemented
29C2×Dic12C2211yescharacteristic
30C2×ΓL(1,𝔽25)C2211yescharacteristic
31C2×D3×C8C2211yescharacteristic
32#892—21no—
33#892C2211yescharacteristic
34D3×C4⋊C4C2211yescharacteristic
35#887—21no—
36#887C2211yescharacteristic
37#887C2211yescharacteristic
38#67⋊C4—21no—
39#877—21no—
40#872—21no—
41#872C2211yescharacteristic
42C4×Dic6—21no—
43C4×Dic6C2211yescharacteristic
44C4×Dic6C2211yescharacteristic
45C3×#107C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×#97C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60C4×C3⋊C8C2211yescharacteristic
61C2×#137—31no—
62#520—31no—
63#519—31no—
64#441—31no—
65#441—31no—
66#440—31no—
67#440D311yescharacteristic, complemented
68C4×Q16—31no—
69C4×Q16—31no—
70C4×SD16—31no—
71C4×SD16—31no—
72#428—31no—
73#416—31no—
74C2×#103—31no—
75#405—31no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81C22×Dic3—21no—
82D3×Q8—22no—
83D3×Q8C2×C414yescomplemented
84#237—22no—
85#237C2×C414yes—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C3×SD16C2×C414yescomplemented
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99C3×C42—21no—
100C3×C42C2311yescharacteristic
101#217—21no—
102#216C2×C414yescomplemented
103#214C2×C414yescomplemented
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4D411yescharacteristic
106(C2×D3)⋊C4D411yescharacteristic
107C4⋊Dic3—21no—
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116Dic3⋊C4D411yescharacteristic
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126Dic12C2×C414yescomplemented
127ΓL(1,𝔽25)C2×C414yescomplemented
128D3×C8—22no—
129C2×#41—31no—
130C22×Q8—31no—
131#137—38no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16C2×D311yescharacteristic
136C2×M16—31no—
137C22×C8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×Q8—31no—
142C4×Q8—31no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×Q8C2×D311yescharacteristic
146C4×D4—31no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#117—31no—
152#117—61no—
153#117—61no—
154#117—61no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157#107—31no—
158#107C2×D311yescharacteristic
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103C2×D311yescharacteristic
167#102—31no—
168#102C2×D311yescharacteristic
169#97—31no—
170#97C2×D311yescharacteristic
171C4×C8—31no—
172C4×C8—31no—
173C23×C3—21no—
174C22×D3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8C22×C412yes—
177C3×D4—21no—
178C3×D4C22×C412yes—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—81no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—24no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—24no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—81no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C4×D3—24no—
197C4×D3—81no—
198C4×D3C22×C412yes—
199C4×D3#4112yes—
200Dic6—21no—
201Dic6—21no—
202Dic6—24no—
203Dic6C22×C412yes—
204Dic6C22×C412yes—
205C3×C8—21no—
206C3×C8C22×C412yes—
207C3⋊C8—21no—
208C3⋊C8C22×C412yes—
209#41—34no—
210#41—62no—
211C2×Q8—34no—
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—61no—
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225Q16—34no—
226Q16—34no—
227SD16—34no—
228SD16C4×D314yescomplemented
229M16—34no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—62no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—31no—
250C22⋊C4—31no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—21no—
254C42—31no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42—61no—
260C42—61no—
261C42C22×D311yescharacteristic
262C22×C3—22no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C22×D411yescharacteristic
266C2×D3—21no—
267C2×D3—21no—
268C2×D3C4×D412yes—
269C3×C4—22no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4C23×C411yescharacteristic
275C3×C4C2×#4111yescharacteristic
276Dic3—22no—
277Dic3—22no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3C4×D412yes—
283C23—21no—
284C23—31no—
285Q8—21no—
286Q8—32no—
287Q8—32no—
288Q8—64no—
289Q8—61no—
290Q8—61no—
291Q8C2×C4×D312yes—
292D4—21no—
293D4—64no—
294D4C2×C4×D312yes—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—64no—
304C2×C4—64no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—81no—
310C2×C4—241no—
311C2×C4—241no—
312C2×C4D3×D411yescharacteristic
313C2×C4D3×D411yescharacteristic
314C2×C4C23×D311yescharacteristic
315C8—21no—
316C8—32no—
317C8—61no—
318C8C2×C4×D312yes—
319C2×C3—42no—
320C2×C3C2×C4×D411yescharacteristic
321C2×C3#57711yescharacteristic
322C2×C3#57811yescharacteristic
323D3—22no—
324C22—22no—
325C22—32no—
326C22—41no—
327C22—41no—
328C22—61no—
329C22—61no—
330C22C2×D3×D411yescharacteristic
331C4—21no—
332C4—21no—
333C4—22no—
334C4—32no—
335C4—41no—
336C4—41no—
337C4—61no—
338C4—61no—
339C4—62no—
340C4—62no—
341C4—121no—
342C4—121no—
343C4C22×C4×D311yescharacteristic
344C4D3×#4111yescharacteristic
345C3#294711yescharacteristic, complemented, Sylow
346C2—42no—
347C2—62no—
348C2C4×D3×D411yescharacteristic
349C2#577511yescharacteristic
350C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radicalC2×#237C22
cosocleC2×C3×C4C24
hypercenter#440D3
hypocenterC3#2947
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #5572C2semidirect
4. C3×#440C2semidirect
5. Dic3×SD16C2semidirect
6. C4×#216C2semidirect
7. C4×#214C2semidirect
8. #4881C2semidirect
9. #4873C2semidirect
10. D3×#103C2semidirect
11. #4789C2semidirect
13. #4757C2semidirect
14. #4716C2semidirect
15. #4708C2semidirect
28. #915C4semidirect
67. #440D3semidirect
83. D3×Q8C2×C4semidirect
92. C3×SD16C2×C4semidirect
102. #216C2×C4semidirect
103. #214C2×C4semidirect
126. Dic12C2×C4semidirect
127. ΓL(1,𝔽25)C2×C4semidirect
228. SD16C4×D3semidirect
345. C3#2947semidirect

Maximal subgroups

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

subgroupindex
C2×#9152
#55722
#55462
C3×#4402
Dic3×SD162
C4×#2162
C4×#2142
#48812
#48732
D3×#1032
#47892
#47722
#47572
#47162
#47082
#29473