Finite Group Databasebeta

← #79770

Subgroups — #79770

Order 384.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5582C211yesmaximal, characteristic, complemented
2#5562C211yesmaximal, characteristic, complemented
3C3×#440C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic, complemented
5#5171C211yesmaximal, characteristic, complemented
6C4×#215C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4883C211yesmaximal, characteristic, complemented
9#4876C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4800C211yesmaximal, characteristic, complemented
12#4782C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4711C211yesmaximal, characteristic, complemented
15#4708C211yesmaximal, characteristic, complemented
16#3179—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C3×C4×Q8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20#947C2211yescharacteristic
21#946C2211yescharacteristic
22C2×#215C2211yescharacteristic
23#940C2211yescharacteristic
24#938C2211yescharacteristic
25C2×#213C2211yescharacteristic
26#930—21no—
27#929—21no—
28C2×D24C2211yescharacteristic
29C2×ΓL(1,𝔽25)C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#896—21no—
32#895C2211yescharacteristic
33#895C2211yescharacteristic
34#894—21no—
35#890C2211yescharacteristic
36#888C2211yescharacteristic
37#884—21no—
38#883—21no—
39#876—21no—
40#875—21no—
41C4×D12C2211yescharacteristic
42C4×D12C2211yescharacteristic
43#872C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47C3×#97C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#817C2211yescharacteristic
53#815C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#542—31no—
61#540—31no—
62#501—31no—
63#496—31no—
64#488—31no—
65#486—31no—
66#469—31no—
67#468—31no—
68#466—31no—
69#465—31no—
70#442—31no—
71#440D311yescharacteristic, complemented
72C4×SD16—31no—
73C4×D8—31no—
74#405—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12C2311yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3C2311yescharacteristic
88C2×Dic6—21no—
89C3×SD16—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96C3×C42—21no—
97C3×C42C2311yescharacteristic
98#217—21no—
99#215—22no—
100#213—22no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121D24—22no—
122ΓL(1,𝔽25)—22no—
123D3×C8—22no—
124C2×SD16—31no—
125C2×SD16C2×D311yescharacteristic
126C2×D8—31no—
127C2×D8—31no—
128C2×M16—31no—
129C22×C8—31no—
130C4⋊Q8—31no—
131#127—31no—
132#126—61no—
133#125—31no—
134#124—31no—
135#124—61no—
136#123—61no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—31no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146C4×Q8C2×D311yescharacteristic
147C4×D4—31no—
148C4×D4—31no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#107—31no—
152#107C2×D311yescharacteristic
153#106—31no—
154#106—31no—
155#103—31no—
156#103C2×D311yescharacteristic
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102C2×D311yescharacteristic
163#97—31no—
164#97C2×D311yescharacteristic
165C4×C8—31no—
166C4×C8—31no—
167C23×C3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3C2×D411yescharacteristic
171C3×Q8—21no—
172C3×Q8—21no—
173C3×D4—21no—
174C3×D4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—81no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2411yescharacteristic
182#67—81no—
183#67—81no—
184#67—81no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190D12—21no—
191D12—21no—
192D12—21no—
193D12—21no—
194D12—81no—
195D12—81no—
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—81no—
199C4×D3—81no—
200Dic6—81no—
201C3×C8—21no—
202C3×C8#4112yes—
203C3⋊C8—21no—
204C3⋊C8#4112yes—
205C2×Q8—61no—
206C2×Q8C22×D311yescharacteristic
207C2×D4—31no—
208C2×D4—31no—
209C2×D4—61no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4C22×D311yescharacteristic
215C22×C4—21no—
216C22×C4—31no—
217C22×C4—61no—
218C22×C4—61no—
219SD16—22no—
220SD16—62no—
221D8—62no—
222D8—62no—
223M16—62no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—62no—
227C2×C8C22×D311yescharacteristic
228C2×C8C22×D311yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—31no—
242C22⋊C4—31no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C42—21no—
252C42—31no—
253C42C22×D311yescharacteristic
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3C22×D411yescharacteristic
258C2×D3—21no—
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4C2×#4111yescharacteristic
273C3×C4C2×#4111yescharacteristic
274Dic3—21no—
275Dic3—21no—
276Dic3—21no—
277Dic3—41no—
278Dic3—41no—
279C23—21no—
280C23—31no—
281C23—61no—
282C23—61no—
283Q8—21no—
284Q8—21no—
285Q8—241no—
286D4—21no—
287D4—21no—
288D4—61no—
289D4—61no—
290D4—61no—
291D4—61no—
292D4—241no—
293D4—241no—
294D4—241no—
295D4—241no—
296D4—241no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—81no—
308C2×C4—241no—
309C2×C4—241no—
310C2×C4D3×D411yescharacteristic
311C2×C4D3×D411yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—21no—
314C8—32no—
315C8—61no—
316C8#23512yes—
317C2×C3—81no—
318C2×C3#53211yescharacteristic
319C2×C3#57611yescharacteristic
320C2×C3#57711yescharacteristic
321D3—41no—
322D3—81no—
323D3—81no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22—121no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—61no—
343C4—61no—
344C4—61no—
345C4—121no—
346C4—121no—
347C4C2×#23511yescharacteristic
348C4D3×#4111yescharacteristic
349C3#317911yescharacteristic, complemented, Sylow
350C2—81no—
351C2—121no—
352C2—241no—
353C2—241no—
354C2#554811yescharacteristic
355C2#577411yescharacteristic
356C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3179
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5582C2semidirect
2. #5562C2semidirect
3. C3×#440C2semidirect
4. #5176C2semidirect
5. #5171C2semidirect
6. C4×#215C2semidirect
7. C4×#213C2semidirect
8. #4883C2semidirect
9. #4876C2semidirect
10. #4812C2semidirect
11. #4800C2semidirect
12. #4782C2semidirect
13. #4760C2semidirect
14. #4711C2semidirect
15. #4708C2semidirect
71. #440D3semidirect
349. C3#3179semidirect

Maximal subgroups

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

subgroupindex
#55822
#55622
C3×#4402
#51762
#51712
C4×#2152
C4×#2132
#48832
#48762
#48122
#48002
#47822
#47602
#47112
#47082
#31793