Finite Group Databasebeta

← #78618

Subgroups — #78618

Order 384.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#5791C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
4#5116C212yesmaximal, complemented
5#5114C212yesmaximal, complemented
6#4739C211yesmaximal, characteristic, complemented
7#4738C211yesmaximal, characteristic, complemented
8#4736C211yesmaximal, characteristic, complemented
9#4733C212yesmaximal, complemented
10#4723C212yesmaximal, complemented
11#4721C211yesmaximal, characteristic, complemented
12#3190—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C22×C3×C8C2212yes—
17#953—21no—
18C2×#217—21no—
19#930C2211yescharacteristic
20#930C2211yescharacteristic
21#929—21no—
22C4×#67—21no—
23C4×#67—21no—
24#926C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26#924—21no—
27#924—21no—
28#924C2211yescharacteristic
29#924C2212yes—
30#924C2212yes—
31#924C2211yescharacteristic
32C2×Dic3⋊C4—21no—
33C2×Dic12C2212yescomplemented
34C2×#204C2212yescomplemented
35C3×C22⋊C8C2211yescharacteristic
36C3×C22⋊C8C2212yes—
37C3×C22⋊C8C2211yescharacteristic
38#821C2212yescomplemented
39#818C2212yescomplemented
40#817C2212yescomplemented
41#816C2212yescomplemented
42#816C2212yescomplemented
43#816C2212yescomplemented
44#525—31no—
45C2×C22⋊Q8—31no—
46#485—32no—
47#483—31no—
48#480—31no—
49#467—32no—
50#465—32no—
51#452—32no—
52#451—31no—
53#448—31no—
54C2×C22⋊C8D311yescharacteristic, complemented
55C24×C3D411yescharacteristic
56C22×C3×C4—42no—
57C22×C3×C4D411yescharacteristic
58C22×C3×C4D411yescharacteristic
59C22×C3×C4D411yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×#67—21no—
65C2×#67—21no—
66C22×Dic3—21no—
67C22×Dic3—21no—
68#235—42no—
69C2×D12C2311yescharacteristic
70C2×C4×D3—21no—
71C2×Dic6—21no—
72C2×Dic6—42no—
73C2×Dic6C2312yes—
74C2×Dic6C2311yescharacteristic
75C2×C3×C8—42no—
76C2×C3×C8C2312yes—
77C2×C3×C8C2312yes—
78#217—22no—
79#217—22no—
80#217—21no—
81#217—21no—
82#217—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3C2311yescharacteristic
87C4⋊Dic3C2311yescharacteristic
88C4⋊Dic3C2312yes—
89Dic3⋊C4—22no—
90Dic3⋊C4—22no—
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3—21no—
94Dic12—42no—
95#204—42no—
96C2×#41—31no—
97C22×Q8—31no—
98C23×C4C2×D311yescharacteristic
99C2×Q16—32no—
100C2×SD16—32no—
101C22×C8C2×D312yes—
102#123—61no—
103C22⋊Q8—32no—
104C22⋊Q8—32no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109#121—31no—
110#121—31no—
111#120—61no—
112C4×D4—61no—
113C4×D4—61no—
114#117—31no—
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⋊C8C2×D311yescharacteristic
125C22⋊C8C2×D311yescharacteristic
126C22⋊C8C2×D312yes—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—41no—
130C23×C3C2×D411yescharacteristic
131C23×C3C2×D411yescharacteristic
132C23×C3C2×D411yescharacteristic
133C22×D3—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2411yescharacteristic
144#67—42no—
145#67—42no—
146#67—42no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154D12—22no—
155C4×D3—42no—
156Dic6—22no—
157Dic6—22no—
158Dic6—22no—
159Dic6—42no—
160C3×C8—22no—
161C3×C8—22no—
162C24D1211yescharacteristic
163#41—122no—
164C2×Q8—32no—
165C2×Q8—31no—
166C2×Q8—61no—
167C2×Q8—122no—
168C2×D4—31no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C22×C4—42no—
173C22×C4—61no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4D1211yescharacteristic
177C22×C4D1211yescharacteristic
178C22×C4D1211yescharacteristic
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182Q16—122no—
183SD16—122no—
184C2×C8—42no—
185C2×C8C22×D312yes—
186C2×C8C22×D312yes—
187C4⋊C4—32no—
188C4⋊C4—31no—
189C4⋊C4—31no—
190C4⋊C4—61no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C22⋊C4—62no—
196C22⋊C4—62no—
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—44no—
211C22×C3#13512yes—
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×C4H(5,𝔽2)12yes—
221Dic3—42no—
222Dic3—41no—
223Dic3—42no—
224Dic3—41no—
225Dic3—41no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23—61no—
230C23C2×D1211yescharacteristic
231C23C2×D1211yescharacteristic
232C23C2×D1211yescharacteristic
233Q8—62no—
234Q8—62no—
235Q8—62no—
236Q8—122no—
237D4—62no—
238D4—122no—
239D4—122no—
240D4—122no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4C2×D1211yescharacteristic
256C2×C4C2×D1211yescharacteristic
257C2×C4C2×D1211yescharacteristic
258C2×C4C23×D311yescharacteristic
259C8—22no—
260C8—22no—
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—41no—
264C2×C3#53411yescharacteristic
265C2×C3C2×#13511yescharacteristic
266C2×C3C2×#13711yescharacteristic
267D3—81no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—121no—
277C22—121no—
278C22—121no—
279C22#90412yes—
280C22#90912yes—
281C22C22×D1211yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—122no—
286C4—121no—
287C4—121no—
288C4—121no—
289C4—122no—
290C4#100412yes—
291C3#319011yescharacteristic, complemented, Sylow
292C2—22no—
293C2—22no—
294C2—41no—
295C2—241no—
296C2#549311yescharacteristic
297C2C2×#90411yescharacteristic
298C2C2×#90911yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊C8C2
socleC22×C3C22×D4
prime radical#5791C2
hypercenterC2×C22⋊C8D3
hypocenterC3#3190
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C2×C3×C22⋊C8C2semidirect
4. #5116C2semidirect
5. #5114C2semidirect
6. #4739C2semidirect
7. #4738C2semidirect
8. #4736C2semidirect
9. #4733C2semidirect
10. #4723C2semidirect
11. #4721C2semidirect
33. C2×Dic12C22semidirect
34. C2×#204C22semidirect
38. #821C22semidirect
39. #818C22semidirect
40. #817C22semidirect
41. #816C22semidirect
42. #816C22semidirect
43. #816C22semidirect
54. C2×C22⋊C8D3semidirect
291. C3#3190semidirect

Maximal subgroups

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

subgroupindex
#57912
C2×#9242
C2×C3×C22⋊C82
#51162
#51142
#47392
#47382
#47362
#47332
#47232
#47212
#31903