Finite Group Databasebeta

← #79237

Subgroups — #79237

Order 384.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5533C211yesmaximal, characteristic
2#5511C211yesmaximal, characteristic
3C3×#428C211yesmaximal, characteristic, complemented
4#5137C211yesmaximal, characteristic, complemented
5#5128C211yesmaximal, characteristic
6#4996C211yesmaximal, characteristic, complemented
7#4971C211yesmaximal, characteristic, complemented
8#4888C211yesmaximal, characteristic, complemented
9#4881C211yesmaximal, characteristic, complemented
10#4875C211yesmaximal, characteristic, complemented
11#4873C211yesmaximal, characteristic, complemented
12#4867C211yesmaximal, characteristic, complemented
13#4862C211yesmaximal, characteristic, complemented
14#4855C211yesmaximal, characteristic, complemented
15#4854C211yesmaximal, characteristic, complemented
16#3356—31nomaximal, Sylow
17C2×C3×M16C2211yescharacteristic
18C3×#117C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C4×#67—21no—
21C4×#67C2211yescharacteristic
22#924—21no—
23#924C2211yescharacteristic
24#924C2211yescharacteristic
25C2×Dic12C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27#897C2211yescharacteristic
28#897C2211yescharacteristic
29#896—21no—
30#892C2211yescharacteristic
31D3×C4⋊C4C2211yescharacteristic
32#889—21no—
33#888C2211yescharacteristic
34#888C2211yescharacteristic
35#887C2211yescharacteristic
36#887C2211yescharacteristic
37#885—21no—
38#879—21no—
39#876—21no—
40C4×Dic6—21no—
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#106C2211yescharacteristic
45#830C2211yescharacteristic
46(C2×D3)⋊C8C2211yescharacteristic
47#817C2211yescharacteristic
48#816C2211yescharacteristic
49#816C2211yescharacteristic
50Dic3⋊C8C2211yescharacteristic
51C8×Dic3C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#556—31no—
61#554—31no—
62#498—31no—
63#485—31no—
64#485—31no—
65#483—31no—
66#483—31no—
67#477—31no—
68#475—31no—
69#470—31no—
70#470—31no—
71C4×Q16—31no—
72#436—31no—
73#428—31no—
74#428D311yescharacteristic, complemented
75C22×C3×C4—21no—
76C22×C3×C4C2311yescharacteristic
77C2×#67D411yescharacteristic
78C2×C4×D3—21no—
79C2×C4×D3C2311yescharacteristic
80C2×Dic6—21no—
81C2×Dic6—21no—
82C2×Dic6C2311yescharacteristic
83C2×Dic6C2311yescharacteristic
84C3×M16—22no—
85C2×C3×C8C2311yescharacteristic
86C2×C3×C8C2311yescharacteristic
87C3×C4⋊C4—21no—
88C3×C4⋊C4C2311yescharacteristic
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93C3×C42—21no—
94#217—21no—
95#217—21no—
96#217D411yescharacteristic
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4D411yescharacteristic
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104C4⋊Dic3C2311yescharacteristic
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4D411yescharacteristic
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119Dic12—22no—
120ΓL(1,𝔽25)—22no—
121C2×Q16—31no—
122C2×M16—31no—
123C2×M16C2×D311yescharacteristic
124C4⋊Q8—31no—
125C4⋊Q8—31no—
126#126—61no—
127#126—61no—
128#125—61no—
129#124—61no—
130C22⋊Q8—31no—
131C22⋊Q8—31no—
132C22⋊Q8—31no—
133C22⋊Q8—31no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×Q8—61no—
139C4×D4—31no—
140C4×D4—61no—
141#117—31no—
142#117C2×D311yescharacteristic
143C2×C4⋊C4—31no—
144C2×C4⋊C4C2×D311yescharacteristic
145#107—31no—
146#107C2×D311yescharacteristic
147#107C2×D311yescharacteristic
148#106—31no—
149#106—31no—
150#106—31no—
151#106—31no—
152#106C2×D311yescharacteristic
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161C22⋊C8—31no—
162C22⋊C8—31no—
163C4×C8—31no—
164C23×C3C2×D411yescharacteristic
165C22×D3C2×D411yescharacteristic
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—81no—
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2411yescharacteristic
175#67—22no—
176C2×Dic3—21no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3C2×D411yescharacteristic
181C2×Dic3C2×D411yescharacteristic
182C2×Dic3C2×D411yescharacteristic
183C4×D3—21no—
184C4×D3—21no—
185C4×D3—81no—
186Dic6—21no—
187Dic6—21no—
188Dic6—21no—
189Dic6—21no—
190Dic6—42no—
191Dic6—81no—
192C3×C8—21no—
193C3×C8#4112yes—
194C3⋊C8—21no—
195C3⋊C8—21no—
196C2×Q8—31no—
197C2×Q8—31no—
198C2×Q8—61no—
199C2×Q8—61no—
200C2×D4—31no—
201C22×C4—21no—
202C22×C4—31no—
203C22×C4—61no—
204C22×C4C22×D311yescharacteristic
205Q16—62no—
206M16—22no—
207M16—62no—
208C2×C8—31no—
209C2×C8—31no—
210C2×C8C22×D311yescharacteristic
211C2×C8C22×D311yescharacteristic
212C4⋊C4—21no—
213C4⋊C4—31no—
214C4⋊C4—31no—
215C4⋊C4—31no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4C22×D311yescharacteristic
226C4⋊C4C22×D311yescharacteristic
227C4⋊C4C22×D311yescharacteristic
228C4⋊C4C22×D311yescharacteristic
229C22⋊C4—21no—
230C22⋊C4—31no—
231C22⋊C4—31no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C42—21no—
238C42—31no—
239C42—61no—
240C42—61no—
241C42—61no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3C22×D411yescharacteristic
246C2×D3—21no—
247C2×D3—21no—
248C2×D3—21no—
249C3×C4—21no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4C2×#4111yescharacteristic
255C3×C42−1+411yescharacteristic
256Dic3—21no—
257Dic3—21no—
258Dic3—41no—
259Dic3—41no—
260Dic3—41no—
261Dic3—41no—
262Dic3#13712yes—
263C23—31no—
264C23D3×D411yescharacteristic
265Q8—61no—
266Q8—61no—
267Q8—61no—
268Q8—61no—
269Q8—122no—
270Q8—241no—
271D4—62no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—81no—
288C2×C4—241no—
289C2×C4D3×D411yescharacteristic
290C2×C4C23×D311yescharacteristic
291C8—21no—
292C8—61no—
293C8—61no—
294C8#23512yes—
295C2×C3—41no—
296C2×C3#54711yescharacteristic
297C2×C3C2×#13711yescharacteristic
298C2×C3#57711yescharacteristic
299D3—41no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—61no—
304C22—61no—
305C22—61no—
306C22C2×D3×D411yescharacteristic
307C4—21no—
308C4—32no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—61no—
314C4—61no—
315C4—121no—
316C4—121no—
317C4—121no—
318C4—121no—
319C4C2×#23511yescharacteristic
320C4#100711yescharacteristic
321C3#335611yescharacteristic, complemented, Sylow
322C2—41no—
323C2—121no—
324C2#550711yescharacteristic
325C2#577511yescharacteristic
326C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC2×#67D4
hypercenter#428D3
hypocenterC3#3356
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. C3×#428C2semidirect
4. #5137C2semidirect
6. #4996C2semidirect
7. #4971C2semidirect
8. #4888C2semidirect
9. #4881C2semidirect
10. #4875C2semidirect
11. #4873C2semidirect
12. #4867C2semidirect
13. #4862C2semidirect
14. #4855C2semidirect
15. #4854C2semidirect
74. #428D3semidirect
321. C3#3356semidirect

Maximal subgroups

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

subgroupindex
#55332
#55112
C3×#4282
#51372
#51282
#49962
#49712
#48882
#48812
#48752
#48732
#48672
#48622
#48552
#48542
#33563