Finite Group Databasebeta

← #79124

Subgroups — #79124

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5511C211yesmaximal, characteristic, complemented
2#5507C211yesmaximal, characteristic
3C2×C3×#106C211yesmaximal, characteristic, complemented
4#5114C211yesmaximal, characteristic, complemented
5C8×#67C211yesmaximal, characteristic, complemented
6#4971C211yesmaximal, characteristic, complemented
7#4968C211yesmaximal, characteristic, complemented
8#4870C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4863C211yesmaximal, characteristic, complemented
11#4862C211yesmaximal, characteristic, complemented
12D3×#106C211yesmaximal, characteristic, complemented
13#4855C211yesmaximal, characteristic, complemented
14#4854C211yesmaximal, characteristic, complemented
15#4852C211yesmaximal, characteristic, complemented
16#3335—31nomaximal, Sylow
17C2×#235—21no—
18C22×C3×C8C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#930C2211yescharacteristic
22#929—21no—
23C4×#67—21no—
24C4×#67C2211yescharacteristic
25#924—21no—
26#924C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29#897C2211yescharacteristic
30#896—21no—
31#896—21no—
32#895C2211yescharacteristic
33#894—21no—
34#893C2211yescharacteristic
35D3×C4⋊C4C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic
37#889—21no—
38#888C2211yescharacteristic
39#888C2211yescharacteristic
40#887C2211yescharacteristic
41C3×#106C2211yescharacteristic
42C3×#106C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#106C2211yescharacteristic
45#830C2211yescharacteristic
46#821C2211yescharacteristic
47(C2×D3)⋊C8C2211yescharacteristic
48#817C2211yescharacteristic
49#816C2211yescharacteristic
50Dic3⋊C8C2211yescharacteristic
51C8×Dic3C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#554—31no—
61#547—31no—
62#498—31no—
63#483—31no—
64#483—31no—
65#481—31no—
66#481—31no—
67#476—31no—
68#475—31no—
69#465—31no—
70#465—31no—
71C4×SD16—31no—
72C8×D4—31no—
73C2×#106—31no—
74C2×#106D311yescharacteristic, complemented
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67D411yescharacteristic
80#235—44no—
81C2×D12—21no—
82C2×D12C2311yescharacteristic
83C2×C4×D3—21no—
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—21no—
88C2×Dic6—21no—
89C2×Dic6C2311yescharacteristic
90C2×C3×C8—22no—
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99#217—21no—
100#217D411yescharacteristic
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4D411yescharacteristic
106C4⋊Dic3—21no—
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4D411yescharacteristic
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120#204—22no—
121D3×C8—22no—
122C2×#41—61no—
123C2×SD16—31no—
124C22×C8—31no—
125C22×C8C2×D311yescharacteristic
126C4⋊Q8—31no—
127C4⋊Q8—31no—
128#125—61no—
129#123—61no—
130#123—61no—
131C22⋊Q8—31no—
132C22⋊Q8—31no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136#121—31no—
137#121—31no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—61no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4C2×D311yescharacteristic
145C2×C4⋊C4C2×D311yescharacteristic
146#106—31no—
147#106—31no—
148#106—31no—
149#106—31no—
150#106—31no—
151#106C2×D311yescharacteristic
152#106C2×D311yescharacteristic
153#106C2×D311yescharacteristic
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162C22⋊C8—31no—
163C22⋊C8—31no—
164C4×C8—31no—
165C23×C3C2×D411yescharacteristic
166C22×D3—21no—
167C22×D3C2×D411yescharacteristic
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—42no—
175C2×C3×C4—81no—
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178#67—42no—
179#67SD1614yescomplemented
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185C2×Dic3C2×D411yescharacteristic
186D12—21no—
187D12—21no—
188D12—42no—
189C4×D3—21no—
190C4×D3—21no—
191C4×D3—42no—
192C4×D3—42no—
193C4×D3—81no—
194Dic6—21no—
195Dic6—21no—
196Dic6—42no—
197Dic6—81no—
198C3×C8—21no—
199C3×C8#4112yescomplemented
200C3⋊C8—21no—
201C3⋊C8—21no—
202#41—124no—
203C2×Q8—31no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×D4—31no—
207C2×D4—31no—
208C2×D4—61no—
209C2×D4—61no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—31no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4C22×D311yescharacteristic
217SD16—62no—
218C2×C8—22no—
219C2×C8—31no—
220C2×C8—31no—
221C2×C8—62no—
222C2×C8C22×D311yescharacteristic
223C2×C8C22×D311yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4C22×D311yescharacteristic
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—31no—
241C22⋊C4—31no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C42—31no—
248C42—61no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3C2×SD1612yes—
252C22×C3C22×D411yescharacteristic
253C2×D3—21no—
254C2×D3—21no—
255C2×D3—41no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3C2×SD1612yes—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4C2×#4111yescharacteristic
265C3×C42−1+411yescharacteristic
266Dic3—21no—
267Dic3—21no—
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271Dic3C2×SD1612yes—
272C23—31no—
273C23—61no—
274C23D3×D411yescharacteristic
275Q8—61no—
276Q8—61no—
277Q8—122no—
278Q8—241no—
279D4—34no—
280D4—61no—
281D4—61no—
282D4—122no—
283D4—122no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—42no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—81no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4—241no—
303C2×C4D3×D411yescharacteristic
304C2×C4C23×D311yescharacteristic
305C8—21no—
306C8—61no—
307C8—61no—
308C8#23512yescomplemented
309C2×C3—22no—
310C2×C3#54711yescharacteristic
311C2×C3C22×SD1611yescharacteristic
312C2×C3#57711yescharacteristic
313D3—22no—
314D3—81no—
315C22—21no—
316C22—21no—
317C22—32no—
318C22—61no—
319C22—61no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22D3×SD1612yes—
324C22C2×D3×D411yescharacteristic
325C4—21no—
326C4—32no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—61no—
332C4—61no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4C2×#23511yescharacteristic
337C4#100711yescharacteristic
338C3#333511yescharacteristic, complemented, Sylow
339C2—22no—
340C2—62no—
341C2—241no—
342C2#550711yescharacteristic
343C2C2×D3×SD1611yescharacteristic
344C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#106C2
socleC22×C3C22×D4
prime radical#5507C2
hypercenterC2×#106D3
hypocenterC3#3335
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. #5511C2semidirect
3. C2×C3×#106C2semidirect
4. #5114C2semidirect
5. C8×#67C2semidirect
6. #4971C2semidirect
7. #4968C2semidirect
8. #4870C2semidirect
9. #4864C2semidirect
10. #4863C2semidirect
11. #4862C2semidirect
12. D3×#106C2semidirect
13. #4855C2semidirect
14. #4854C2semidirect
15. #4852C2semidirect
74. C2×#106D3semidirect
179. #67SD16semidirect
199. C3×C8#41semidirect
308. C8#235semidirect
338. C3#3335semidirect

Maximal subgroups

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

subgroupindex
#55112
#55072
C2×C3×#1062
#51142
C8×#672
#49712
#49682
#48702
#48642
#48632
#48622
D3×#1062
#48552
#48542
#48522
#33353