Finite Group Databasebeta

← #82935

Subgroups — #82935

Order 384.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#525C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3#5239C212yesmaximal, complemented
4#5207C211yesmaximal, characteristic, complemented
5#5052C212yesmaximal, complemented
6#5050C211yesmaximal, characteristic, complemented
7#5044C211yesmaximal, characteristic, complemented
8#5042C212yesmaximal, complemented
9#5040C211yesmaximal, characteristic, complemented
10#5034C211yesmaximal, characteristic, complemented
11#4996C212yesmaximal, complemented
12#3196—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C3×#123—21no—
17C3×C22⋊Q8C2211yescharacteristic
18C3×C22⋊Q8C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C3×#120—21no—
22C3×C4×D4—21no—
23C3×C4×D4—21no—
24C3×#117C2211yescharacteristic
25C2×#217—21no—
26C2×#216C2212yescomplemented
27C2×#214C2212yescomplemented
28C2×C4⋊Dic3C2211yescharacteristic
29#924—21no—
30#924—21no—
31#924C2212yes—
32#924C2212yes—
33C2×Dic3⋊C4—21no—
34C2×#208C2212yes—
35#835C2212yescomplemented
36#832C2212yescomplemented
37#830C2212yes—
38#830C2211yescharacteristic
39#830C2211yescharacteristic
40#810C2212yescomplemented
41#810C2212yescomplemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44#525D311yescharacteristic
45C2×C22⋊Q8—31no—
46#485—32no—
47#484—31no—
48#481—31no—
49#470—32no—
50#468—32no—
51#452—32no—
52#451—31no—
53#448—31no—
54#407—31no—
55C24×C3D411yescharacteristic
56C3×#41—42no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4—21no—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—21no—
63C22×C3×C4—42no—
64C22×C3×C4D411yescharacteristic
65C22×C3×C4D411yescharacteristic
66C22×C3×C4D411yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×Dic3—21no—
71C22×Dic3—21no—
72C2×Dic6—21no—
73C2×Dic6—42no—
74C2×Dic6C2312yes—
75C3×C4⋊C4—21no—
76C3×C4⋊C4—21no—
77C3×C4⋊C4C2311yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C22⋊C4—21no—
82C3×C22⋊C4—21no—
83C3×C22⋊C4—21no—
84C3×C42—21no—
85#217—22no—
86#217—22no—
87#216—42no—
88#214—42no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2312yes—
91Dic3⋊C4—22no—
92Dic3⋊C4—22no—
93#208—42no—
94C2×C3⋊C8C2312yes—
95C2×C3⋊C8C2312yes—
96C2×#41C2×D311yescharacteristic
97C22×Q8—31no—
98C23×C4C2×D311yescharacteristic
99C2×Q16—32no—
100C2×SD16—32no—
101C2×M16—32no—
102#123—21no—
103C22⋊Q8—32no—
104C22⋊Q8—32no—
105C22⋊Q8—61no—
106C22⋊Q8—61no—
107C22⋊Q8C2×D311yescharacteristic
108C22⋊Q8C2×D311yescharacteristic
109#121C2×D311yescharacteristic
110#121C2×D311yescharacteristic
111#120—21no—
112C4×D4—21no—
113C4×D4—21no—
114#117C2×D311yescharacteristic
115C2×C4⋊C4—31no—
116C2×C4⋊C4—61no—
117C2×C22⋊C4—61no—
118#107—32no—
119#106—32no—
120#103—32no—
121#103—32no—
122#103—32no—
123#102—32no—
124C22⋊C8—31no—
125C22⋊C8—31no—
126C22⋊C8—32no—
127C23×C3—22no—
128C23×C3—21no—
129C23×C3—41no—
130C23×C3—41no—
131C23×C3C2×D411yescharacteristic
132C23×C3C2×D411yescharacteristic
133C23×C3C2×D411yescharacteristic
134C3×Q8—22no—
135C3×D4—22no—
136C3×D4—42no—
137C3×D4—42no—
138C3×D4—42no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—41no—
146C2×C3×C4—41no—
147C2×C3×C4—41no—
148C2×C3×C4—42no—
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4C2×D411yescharacteristic
151C2×C3×C4C2×D411yescharacteristic
152C2×C3×C4C2411yescharacteristic
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157Dic6—22no—
158Dic6—22no—
159Dic6—42no—
160C3⋊C8—22no—
161C3⋊C8—22no—
162C24#6711yescharacteristic
163#41—42no—
164C2×Q8—32no—
165C2×Q8—61no—
166C2×Q8—122no—
167C2×Q8C22×D311yescharacteristic
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C22×D311yescharacteristic
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4#6711yescharacteristic
177C22×C4#6711yescharacteristic
178C22×C4#6711yescharacteristic
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182Q16—122no—
183SD16—122no—
184M16—122no—
185C2×C8—32no—
186C2×C8—32no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—32no—
190C4⋊C4—61no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4C22×D311yescharacteristic
194C4⋊C4C22×D311yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—62no—
201C22⋊C4—62no—
202C42—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—44no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3#13712yes—
216C22×C3C22×D411yescharacteristic
217C3×C4—21no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—41no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4H(5,𝔽2)12yes—
224Dic3—42no—
225Dic3—42no—
226C23—22no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×#6711yescharacteristic
231C23C2×#6711yescharacteristic
232C23C2×#6711yescharacteristic
233Q8—22no—
234Q8—62no—
235Q8—62no—
236Q8—122no—
237D4—22no—
238D4—42no—
239D4—42no—
240D4—42no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4C2×#6711yescharacteristic
256C2×C4C2×#6711yescharacteristic
257C2×C4C2×#6711yescharacteristic
258C2×C4C23×D311yescharacteristic
259C8—62no—
260C8—62no—
261C2×C3—22no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—81no—
265C2×C3#53411yescharacteristic
266C2×C3C2×#13711yescharacteristic
267C2×C3#57511yescharacteristic
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#95112yes—
281C22C22×#6711yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—122no—
289C4—122no—
290C4#100412yes—
291C3#319611yescharacteristic, complemented, Sylow
292C2—22no—
293C2—41no—
294C2—41no—
295C2—81no—
296C2#580411yescharacteristic
297C2#582211yescharacteristic
298C2C2×#95111yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#525C2
socleC22×C3C22×D4
hypercenter#525D3
hypocenterC3#3196
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. #5239C2semidirect
4. #5207C2semidirect
5. #5052C2semidirect
6. #5050C2semidirect
7. #5044C2semidirect
8. #5042C2semidirect
9. #5040C2semidirect
10. #5034C2semidirect
11. #4996C2semidirect
26. C2×#216C22semidirect
27. C2×#214C22semidirect
35. #835C22semidirect
36. #832C22semidirect
40. #810C22semidirect
41. #810C22semidirect
42. #808C22semidirect
43. #807C22semidirect
291. C3#3196semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
C2×#9242
#52392
#52072
#50522
#50502
#50442
#50422
#50402
#50342
#49962
#31963