Finite Group Databasebeta

← #80177

Subgroups — #80177

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5812C211yesmaximal, characteristic
2#5625C211yesmaximal, characteristic
3C3×#451C211yesmaximal, characteristic
4#5228C211yesmaximal, characteristic
5#5184C211yesmaximal, characteristic, complemented
6#5182C211yesmaximal, characteristic, complemented
7#5182C211yesmaximal, characteristic, complemented
8Dic3×Q16C211yesmaximal, characteristic, complemented
9#5045C211yesmaximal, characteristic
10#4799C211yesmaximal, characteristic, complemented
11#4796C211yesmaximal, characteristic, complemented
12#4790C211yesmaximal, characteristic, complemented
13#4789C211yesmaximal, characteristic, complemented
14#4723C211yesmaximal, characteristic
15#4720C211yesmaximal, characteristic
16#3127—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×C3×Q16C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21Q8×Dic3—21no—
22Q8×Dic3—21no—
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic
27Q8×Dic3C2211yescharacteristic
28#944C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#216C2211yescharacteristic
31#926—21no—
32#926—21no—
33#926C2211yescharacteristic
34#926C2211yescharacteristic
35#924C2211yescharacteristic
36#890—21no—
37#890C2211yescharacteristic
38#889—21no—
39#887C2211yescharacteristic
40#879—21no—
41#879—21no—
42C3×#103C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#830C2211yescharacteristic
50#818C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#541—31no—
61#519—31no—
62#485—31no—
63#485—31no—
64#479—31no—
65#479—31no—
66#462—31no—
67#462—31no—
68#451—31no—
69#451D311yescharacteristic
70#441—31no—
71#441—31no—
72C4×Q16—31no—
73C4×Q16—31no—
74#433—31no—
75C2×C3×Q8—22no—
76C2×C3×Q8—22no—
77C2×C3×Q8—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×Dic6C2311yescharacteristic
84C3×Q16—22no—
85C3×Q16—22no—
86C2×C3×C8C2311yescharacteristic
87C2×C3×C8C2311yescharacteristic
88C3×C4⋊C4—21no—
89C3×C4⋊C4C2311yescharacteristic
90C3×C22⋊C4—21no—
91#217—21no—
92#217—21no—
93#217D411yescharacteristic
94#217D411yescharacteristic
95#216—22no—
96#216—22no—
97C4⋊Dic3—21no—
98C4⋊Dic3—21no—
99C4⋊Dic3—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3D411yescharacteristic
104C4⋊Dic3D411yescharacteristic
105C4⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C22×Q8C2×D311yescharacteristic
125C2×Q16—31no—
126C2×Q16—31no—
127C2×Q16C2×D311yescharacteristic
128C2×Q16C2×D311yescharacteristic
129C4⋊Q8—31no—
130#126—61no—
131#126—61no—
132#125—31no—
133#125—61no—
134#125—61no—
135C22⋊Q8—31no—
136C22⋊Q8C2×D311yescharacteristic
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×Q8—31no—
142C4×Q8—31no—
143C4×Q8—61no—
144C4×Q8—61no—
145#117—31no—
146#117—31no—
147#117—61no—
148#117—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103C2×D311yescharacteristic
162#103C2×D311yescharacteristic
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3C2×D411yescharacteristic
168C3×Q8—24no—
169C3×Q8—21no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8—21no—
173C3×Q8#4112yes—
174C3×Q8#4112yes—
175C2×C3×C4—24no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192Dic6—21no—
193Dic6—21no—
194C3×C8—21no—
195C3×C8—21no—
196C3⋊C8—21no—
197C3⋊C8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—21no—
201C2×Q8—31no—
202C2×Q8C22×D311yescharacteristic
203C2×Q8C22×D311yescharacteristic
204C2×Q8C22×D311yescharacteristic
205C22×C4—21no—
206C22×C4C22×D311yescharacteristic
207Q16—22no—
208Q16—22no—
209Q16—62no—
210Q16—62no—
211C2×C8—31no—
212C2×C8—31no—
213C2×C8C22×D311yescharacteristic
214C2×C8C22×D311yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4C22×D311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—31no—
235C22⋊C4—31no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C42—31no—
239C42—31no—
240C42—61no—
241C42—61no—
242C42—61no—
243C42—61no—
244C42—61no—
245C42—61no—
246C42—61no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3C22×D411yescharacteristic
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4C2×#4111yescharacteristic
257C3×C4C2×#4111yescharacteristic
258Dic3—21no—
259Dic3—21no—
260Dic3—21no—
261Dic3—21no—
262Dic3—41no—
263Dic3—41no—
264Dic3—41no—
265Dic3—41no—
266C23D3×D411yescharacteristic
267Q8—24no—
268Q8—21no—
269Q8—21no—
270Q8—21no—
271Q8—21no—
272Q8—61no—
273Q8—61no—
274Q8#23712yes—
275Q8#23712yes—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4D3×D411yescharacteristic
292C2×C4C23×D311yescharacteristic
293C8—21no—
294C8—21no—
295C8—61no—
296C8—61no—
297C2×C3—41no—
298C2×C3#52511yescharacteristic
299C2×C3#57811yescharacteristic
300C2×C3#57811yescharacteristic
301C22—21no—
302C22—21no—
303C22—21no—
304C22C2×D3×D411yescharacteristic
305C4—22no—
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—61no—
311C4—61no—
312C4—61no—
313C4—61no—
314C4—121no—
315C4—121no—
316C4—121no—
317C4—121no—
318C4C2×#23711yescharacteristic
319C4C2×#23711yescharacteristic
320C3#312711yescharacteristic, complemented, Sylow
321C2—41no—
322C2#558611yescharacteristic
323C2#576611yescharacteristic
324C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC23×C3C2×D4
hypercenter#451D3
hypocenterC3#3127
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #5184C2semidirect
6. #5182C2semidirect
7. #5182C2semidirect
8. Dic3×Q16C2semidirect
10. #4799C2semidirect
11. #4796C2semidirect
12. #4790C2semidirect
13. #4789C2semidirect
320. C3#3127semidirect

Maximal subgroups

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

subgroupindex
#58122
#56252
C3×#4512
#52282
#51842
#51822
Dic3×Q162
#50452
#47992
#47962
#47902
#47892
#47232
#47202
#31273