Finite Group Databasebeta

← #82739

Subgroups — #82739

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
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2#5791C211yesmaximal, characteristic
3#5228C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5207C211yesmaximal, characteristic, complemented
6#5052C212yesmaximal, complemented
7#5049C212yesmaximal, complemented
8#5047C212yesmaximal, complemented
9#5045C212yesmaximal, complemented
10#4970C211yesmaximal, characteristic, complemented
11#4968C211yesmaximal, characteristic, complemented
12#3196—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C2×#235C2211yescharacteristic
16C3×C22⋊Q8—21no—
17C3×C22⋊Q8—21no—
18C3×C22⋊Q8C2212yes—
19C3×C22⋊Q8C2212yes—
20C2×C3×C4⋊C4—21no—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4—21no—
23#953—21no—
24C2×#216C2212yescomplemented
25C2×#215C2212yescomplemented
26#930C2211yescharacteristic
27#930C2211yescharacteristic
28#929—21no—
29C4×#67—21no—
30C4×#67—21no—
31#926C2211yescharacteristic
32#924C2211yescharacteristic
33#924C2211yescharacteristic
34C2×#208C2212yes—
35#835C2212yescomplemented
36#835C2212yescomplemented
37#830C2211yescharacteristic
38#830C2211yescharacteristic
39#830C2212yes—
40#810C2212yescomplemented
41#809C2212yescomplemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44#525—31no—
45C2×C22⋊Q8D311yescharacteristic, complemented
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
56C2×C3×Q8—21no—
57C2×C3×Q8—42no—
58C2×C3×Q8C2312yes—
59C22×C3×C4—21no—
60C22×C3×C4—21no—
61C22×C3×C4—42no—
62C22×C3×C4D411yescharacteristic
63C22×C3×C4D411yescharacteristic
64C22×C3×C4D411yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C2×#67—21no—
69C2×#67—21no—
70C2×#67—21no—
71#235—42no—
72C2×D12C2311yescharacteristic
73C2×C4×D3—21no—
74C2×Dic6C2311yescharacteristic
75C3×C4⋊C4—22no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2312yes—
79C3×C22⋊C4—22no—
80C3×C22⋊C4—22no—
81#217—21no—
82#217—21no—
83#217—21no—
84#216—42no—
85#215—42no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3C2311yescharacteristic
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92C4×Dic3—21no—
93#208—42no—
94C2×C3⋊C8C2312yes—
95C2×C3⋊C8C2312yes—
96C2×#41—31no—
97C22×Q8C2×D311yescharacteristic
98C23×C4C2×D311yescharacteristic
99C2×Q16—32no—
100C2×SD16—32no—
101C2×M16—32no—
102#123—61no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107C22⋊Q8C2×D312yes—
108C22⋊Q8C2×D312yes—
109#121—31no—
110#121—31no—
111#120—61no—
112C4×D4—61no—
113C4×D4—61no—
114#117—31no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×D311yescharacteristic
117C2×C22⋊C4—21no—
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—41no—
129C23×C3—41no—
130C23×C3C2×D411yescharacteristic
131C23×C3C2×D411yescharacteristic
132C23×C3C2×D411yescharacteristic
133C22×D3—21no—
134C3×Q8—22no—
135C3×Q8—22no—
136C3×Q8—42no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2×D411yescharacteristic
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4C2411yescharacteristic
151#67—42no—
152#67—42no—
153#67—42no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157D12—22no—
158C4×D3—42no—
159Dic6—22no—
160C3⋊C8—22no—
161C3⋊C8—22no—
162C24#6711yescharacteristic
163#41—122no—
164C2×Q8—21no—
165C2×Q8—31no—
166C2×Q8—42no—
167C2×Q8C22×D312yes—
168C2×D4—31no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
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—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—31no—
191C4⋊C4—31no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4C22×D312yes—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C42—61no—
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#13712yes—
213C22×C3C22×D411yescharacteristic
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C3×C4—21no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—42no—
221C3×C4—42no—
222C3×C4H(5,𝔽2)12yes—
223Dic3—41no—
224Dic3—41no—
225Dic3—41no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—61no—
230C23C2×#6711yescharacteristic
231C23C2×#6711yescharacteristic
232C23C2×#6711yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—62no—
237D4—62no—
238D4—122no—
239D4—122no—
240D4—122no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—61no—
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#53411yescharacteristic
265C2×C3C2×#13711yescharacteristic
266C2×C3#57511yescharacteristic
267D3—81no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—121no—
278C22—121no—
279C22—121no—
280C22#94212yes—
281C22C22×#6711yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—121no—
288C4—121no—
289C4—121no—
290C4#100412yes—
291C3#319611yescharacteristic, complemented, Sylow
292C2—22no—
293C2—41no—
294C2—41no—
295C2—241no—
296C2#580411yescharacteristic
297C2C2×#94211yescharacteristic
298C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊Q8C2
socleC22×C3C22×D4
prime radical#5791C2
hypercenterC2×C22⋊Q8D3
hypocenterC3#3196
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
3. #5228C2semidirect
4. #5227C2semidirect
5. #5207C2semidirect
6. #5052C2semidirect
7. #5049C2semidirect
8. #5047C2semidirect
9. #5045C2semidirect
10. #4970C2semidirect
11. #4968C2semidirect
24. C2×#216C22semidirect
25. C2×#215C22semidirect
35. #835C22semidirect
36. #835C22semidirect
40. #810C22semidirect
41. #809C22semidirect
42. #808C22semidirect
43. #807C22semidirect
45. C2×C22⋊Q8D3semidirect
291. C3#3196semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
#57912
#52282
#52272
#52072
#50522
#50492
#50472
#50452
#49702
#49682
#31963