Finite Group Databasebeta

← #79474

Subgroups — #79474

Order 384.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5563C211yesmaximal, characteristic
2#5541C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5134C211yesmaximal, characteristic, complemented
5#5121C211yesmaximal, characteristic, complemented
6#5114C211yesmaximal, characteristic, complemented
7#5106C211yesmaximal, characteristic, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4847C211yesmaximal, characteristic, complemented
10#4737C211yesmaximal, characteristic, complemented
11#4735C211yesmaximal, characteristic, complemented
12#4723C211yesmaximal, characteristic, complemented
13#4723C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15#4702C211yesmaximal, characteristic, complemented
16#3348—31nomaximal, Sylow
17C2×C3×M16C2211yescharacteristic
18C22×C3×C8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20#938—21no—
21#935—21no—
22#930C2211yescharacteristic
23#930C2211yescharacteristic
24#926C2211yescharacteristic
25#926C2211yescharacteristic
26#926C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28#924C2211yescharacteristic
29#924C2211yescharacteristic
30C2×#204C2211yescharacteristic
31#892—21no—
32#889—21no—
33#885—21no—
34#885—21no—
35#879—21no—
36#877—21no—
37C4×D12C2211yescharacteristic
38#870C2211yescharacteristic
39#869C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×C4⋊C8C2211yescharacteristic
42C3×C22⋊C8C2211yescharacteristic
43C3×C22⋊C8C2211yescharacteristic
44C3×#97C2211yescharacteristic
45#821C2211yescharacteristic
46#821C2211yescharacteristic
47#821C2211yescharacteristic
48#818C2211yescharacteristic
49#818C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54#817C2211yescharacteristic
55#817C2211yescharacteristic
56#817C2211yescharacteristic
57#816C2211yescharacteristic
58#816C2211yescharacteristic
59#816C2211yescharacteristic
60#555—31no—
61#552—31no—
62#501—31no—
63#485—31no—
64#485—31no—
65#482—31no—
66#481—31no—
67#478—31no—
68#475—31no—
69#468—31no—
70#465—31no—
71#440—31no—
72#435D311yescharacteristic, complemented
73#428—31no—
74#427—31no—
75C2×C3×D4D411yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81C2×D12C2311yescharacteristic
82C2×C4×D3—21no—
83C2×Dic6—21no—
84C2×Dic6C2311yescharacteristic
85C3×M16—22no—
86C2×C3×C8—22no—
87C2×C3×C8C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4D411yescharacteristic
92C3×C22⋊C4D411yescharacteristic
93C3×C22⋊C4D411yescharacteristic
94C3×C42C2311yescharacteristic
95#217—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3C2311yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120#204—22no—
121C2×SD16—31no—
122C2×M16C2×D311yescharacteristic
123C22×C8C2×D311yescharacteristic
124C4⋊Q8—31no—
125#126—61no—
126#125—31no—
127#125—61no—
128#124—61no—
129#124—61no—
130#123—61no—
131C22⋊Q8—31no—
132C22⋊Q8—31no—
133#121—31no—
134#121—31no—
135#121—61no—
136C4×Q8—31no—
137C4×D4—31no—
138C4×D4C2×D311yescharacteristic
139#117—31no—
140#117—31no—
141#117—31no—
142#117—61no—
143#117—61no—
144C2×C4⋊C4—31no—
145#107—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#106—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106—31no—
154C4⋊C8C2×D311yescharacteristic
155#103—31no—
156#103—31no—
157#103—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161C22⋊C8C2×D311yescharacteristic
162C22⋊C8C2×D311yescharacteristic
163#97C2×D311yescharacteristic
164C23×C3C2×D411yescharacteristic
165C23×C3C2×D411yescharacteristic
166C22×D3—21no—
167C3×D4—41no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4C2×D411yescharacteristic
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2411yescharacteristic
177#67—81no—
178#67—81no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—42no—
187D12—21no—
188D12—21no—
189C4×D3—42no—
190Dic6—21no—
191Dic6—21no—
192Dic6—81no—
193C3×C8—21no—
194C3×C8—21no—
195C3×C8—21no—
196C3×C8#4112yes—
197C2×Q8—31no—
198C2×Q8—61no—
199C2×D4—31no—
200C2×D4—61no—
201C2×D4—61no—
202C2×D4D1211yescharacteristic
203C22×C4—61no—
204C22×C4—61no—
205C22×C4C22×D311yescharacteristic
206C22×C4C22×D311yescharacteristic
207SD16—62no—
208M16—22no—
209C2×C8—22no—
210C2×C8C22×D311yescharacteristic
211C2×C8C22×D311yescharacteristic
212C2×C8C22×D311yescharacteristic
213C2×C8C22×D311yescharacteristic
214C4⋊C4—31no—
215C4⋊C4—31no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4D1211yescharacteristic
228C22⋊C4—61no—
229C22⋊C4—61no—
230C22⋊C4—61no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4D1211yescharacteristic
237C22⋊C4D1211yescharacteristic
238C42—61no—
239C42—61no—
240C42—61no—
241C42—61no—
242C42C22×D311yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3C22×D411yescharacteristic
250C2×D3—41no—
251C2×D3—41no—
252C2×D3—41no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4C2×#4111yescharacteristic
258C3×C42−1+411yescharacteristic
259Dic3—41no—
260Dic3—41no—
261Dic3—41no—
262Dic3—41no—
263Dic3—41no—
264Dic3—41no—
265Dic3—41no—
266C23—61no—
267C23C2×D1211yescharacteristic
268C23C2×D1211yescharacteristic
269Q8—61no—
270Q8—61no—
271Q8—241no—
272D4—41no—
273D4—61no—
274D4—61no—
275D4—241no—
276D4—241no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4C2×D1211yescharacteristic
291C2×C4C2×D1211yescharacteristic
292C2×C4C2×D1211yescharacteristic
293C2×C4C2×D1211yescharacteristic
294C2×C4C23×D311yescharacteristic
295C8—21no—
296C8—21no—
297C8—21no—
298C8#23712yes—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3#54711yescharacteristic
302C2×C3#57511yescharacteristic
303C2×C3#57711yescharacteristic
304D3—81no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22C22×D1211yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—121no—
320C4—121no—
321C4—121no—
322C4—121no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4C2×#23711yescharacteristic
327C4#101011yescharacteristic
328C3#334811yescharacteristic, complemented, Sylow
329C2—41no—
330C2—41no—
331C2—241no—
332C2#555511yescharacteristic
333C2#574811yescharacteristic
334C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
prime radical#5563C2
hypercenter#435D3
hypocenterC3#3348
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #5541C2semidirect
3. C3×#435C2semidirect
4. #5134C2semidirect
5. #5121C2semidirect
6. #5114C2semidirect
7. #5106C2semidirect
8. #4851C2semidirect
9. #4847C2semidirect
10. #4737C2semidirect
11. #4735C2semidirect
12. #4723C2semidirect
13. #4723C2semidirect
14. #4710C2semidirect
15. #4702C2semidirect
72. #435D3semidirect
328. C3#3348semidirect

Maximal subgroups

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

subgroupindex
#55632
#55412
C3×#4352
#51342
#51212
#51142
#51062
#48512
#48472
#47372
#47352
#47232
#47102
#47022
#33483