Finite Group Databasebeta

← #82825

Subgroups — #82825

Order 384.

498 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#929C211yesmaximal, characteristic
3#5231C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5C4×#217C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic, complemented
7#4990C212yesmaximal, complemented
8#4985C211yesmaximal, characteristic, complemented
9#4976C212yesmaximal, complemented
10#4959C212yesmaximal
11#4952C212yesmaximal, complemented
12#2663—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C3×C22⋊Q8—24no—
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22#929—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#123—31no—
39C2×C22⋊Q8D311yescharacteristic, complemented
40#400—31no—
41#399—31no—
42#397—32no—
43#395—31no—
44#394—32no—
45#393—31no—
46#389—32no—
47#388—32no—
48C4×C22⋊C4—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C22×C3×C4—42no—
54C22×C3×C4D414yes—
55C22×C3×C4C2312yes—
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—22no—
60C2×#67—42no—
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—24no—
68C3×C22⋊C4—24no—
69#217—22no—
70#217—24no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—42no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—24no—
78Dic3⋊C4—42no—
79C4×Dic3—42no—
80C22×Q8C2×D311yescharacteristic
81C22×D4—31no—
82C23×C4C2×D311yescharacteristic
83#123—64no—
84C22⋊Q8—24no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4C2×D312yes—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4C2×D312yes—
98C2×C42—32no—
99#95—31no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95—32no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3C2×D412yes—
110C23×C3C2411yescharacteristic
111C22×D3—21no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—21no—
115C22×D3—21no—
116C3×Q8—44no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—44no—
126C2×C3×C4C2×D412yes—
127C2×C3×C4C2×D412yes—
128#67—44no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3#4114yes—
142C24—31no—
143C24C22×D311yescharacteristic
144C2×Q8—22no—
145C2×Q8—42no—
146C2×D4—62no—
147C2×D4—122no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—42no—
154C22×C4#6714yes—
155C22×C4C22×D312yes—
156C22×C4C22×D312yes—
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—24no—
161C4⋊C4—64no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—122no—
165C22⋊C4—24no—
166C22⋊C4—64no—
167C22⋊C4—64no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22⋊C4—122no—
172C42—122no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3C22×D411yescharacteristic
179C22×C3C2×#4112yes—
180C22×C3H(5,𝔽2)12yes—
181C22×C32−1+411yescharacteristic
182C22×C32−1+411yescharacteristic
183C2×D3—41no—
184C2×D3—41no—
185C2×D3—41no—
186C2×D3—42no—
187C2×D3—42no—
188C3×C4—22no—
189C3×C4—22no—
190C3×C4—42no—
191C3×C4—42no—
192Dic3—24no—
193Dic3—41no—
194Dic3—41no—
195Dic3—41no—
196Dic3—42no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—62no—
202C23—61no—
203C23—61no—
204C23—62no—
205C23—61no—
206C23C2×#6712yes—
207C23C23×D311yescharacteristic
208Q8—44no—
209D4—124no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—34no—
219C2×C4—44no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4C2×#6712yes—
233C2×C4C2×#6712yes—
234C2×C3—42no—
235C2×C3#53211yescharacteristic
236C2×C3#53411yescharacteristic
237C2×C3#53611yescharacteristic
238C2×C3#53912yes—
239C2×C3#54212yes—
240D3—81no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—42no—
245C22—42no—
246C22—122no—
247C22—122no—
248C22—121no—
249C22—121no—
250C22—121no—
251C22#100412yes—
252C22#100711yescharacteristic
253C22#100711yescharacteristic
254C22D3×#4112yes—
255C22C22×#6711yescharacteristic
256C4—22no—
257C4—22no—
258C4—42no—
259C4—42no—
260C4—64no—
261C4—122no—
262C4—121no—
263C4—121no—
264C4—121no—
265C3#266311yescharacteristic, complemented, Sylow
266C2—42no—
267C2—241no—
268C2#563812yes—
269C2#564012yes—
270C2#580411yescharacteristic
271C2#581611yescharacteristic
272C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C22⋊Q8C2
prime radicalC22×#67C22
hypercenterC2×C22⋊Q8D3
hypocenterC3#2663
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
3. #5231C2semidirect
4. #5230C2semidirect
5. C4×#217C2semidirect
6. #4993C2semidirect
7. #4990C2semidirect
8. #4985C2semidirect
9. #4976C2semidirect
11. #4952C2semidirect
39. C2×C22⋊Q8D3semidirect
265. C3#2663semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#9292
#52312
#52302
C4×#2172
#49932
#49902
#49852
#49762
#49592
#49522
#26633