Finite Group Databasebeta

← #77275

Subgroups — #77275

Order 384.

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

normal subgroups only

This group has 464 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
1C2×#947C211yesmaximal, characteristic
2C2×#889C211yesmaximal, characteristic, complemented
3C3×#395C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#4994C212yesmaximal, complemented
6#4993C212yesmaximal, complemented
7#4991C211yesmaximal, characteristic
8#4676C212yesmaximal, complemented
9#4675C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic
11#4650C211yesmaximal, characteristic, complemented
12#2820—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×D12C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17#947—24no—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#889—24no—
33C3×#95C2211yescharacteristic
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#124—31no—
40#399—32no—
41#397—32no—
42#396—32no—
43#396—32no—
44#395—31no—
45#395D311yescharacteristic, complemented
46#394—31no—
47#394—31no—
48#382—31no—
49C23×D3C2312yes—
50C2×C3×Q8—22no—
51C2×C3×Q8—42no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×D12—22no—
63C2×D12—42no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—42no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4—42no—
69(C2×D3)⋊C4—42no—
70(C2×D3)⋊C4—42no—
71(C2×D3)⋊C4—42no—
72C4⋊Dic3—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—42no—
78Dic3⋊C4—42no—
79C4×Dic3D414yes—
80C22×Q8C2×D311yescharacteristic
81C22×D4—31no—
82#125—64no—
83#124—64no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C4⋊C4C2×D312yes—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C42—31no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95C2×D311yescharacteristic
104#95C2×D312yes—
105C23×C3C2411yescharacteristic
106C22×D3—22no—
107C22×D3—22no—
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C3×Q8—44no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4C2×D412yes—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3C2×D412yes—
139C2×Dic3C2×D412yes—
140D12—44no—
141C24—32no—
142C2×Q8—22no—
143C2×Q8—42no—
144C2×D4—62no—
145C2×D4—122no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4C22×D312yes—
152C22×C4C22×D312yes—
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D311yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C22⋊C4—64no—
166C22⋊C4—64no—
167C22⋊C4—122no—
168C22⋊C4—122no—
169C22⋊C4—122no—
170C22⋊C4—122no—
171C42—34no—
172C22×C3C22×D411yescharacteristic
173C22×C3H(5,𝔽2)12yes—
174C22×C3H(5,𝔽2)11yescharacteristic
175C22×C32−1+412yes—
176C22×C32−1+411yescharacteristic
177C2×D3—42no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C3×C4—22no—
185C3×C4—41no—
186C3×C4—42no—
187C3×C4—41no—
188C3×C4—42no—
189Dic3—22no—
190Dic3—22no—
191Dic3—41no—
192Dic3—41no—
193Dic3—42no—
194C23—62no—
195C23—62no—
196C23—62no—
197C23—62no—
198C23—62no—
199C23—62no—
200C23—62no—
201C23C23×D311yescharacteristic
202Q8—44no—
203D4—124no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—32no—
217C2×C4—32no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4D3×D412yes—
230C2×C3#53511yescharacteristic
231C2×C3#53611yescharacteristic
232C2×C3#53711yescharacteristic
233C2×C3#56212yes—
234C2×C3#56312yes—
235D3—82no—
236C22—122no—
237C22—122no—
238C22—122no—
239C22—122no—
240C22—122no—
241C22—122no—
242C22—122no—
243C22C2×D3×D411yescharacteristic
244C22#100412yes—
245C22#100712yes—
246C22#100711yescharacteristic
247C22#100911yescharacteristic
248C4—22no—
249C4—42no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—62no—
254C4—62no—
255C4—122no—
256C4—121no—
257C4—121no—
258C3#282011yescharacteristic, complemented, Sylow
259C2—242no—
260C2#559211yescharacteristic
261C2#562911yescharacteristic
262C2#564712yes—
263C2#572411yescharacteristic
264C2#573512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2820
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#889C2semidirect
3. C3×#395C2semidirect
4. #5230C2semidirect
5. #4994C2semidirect
6. #4993C2semidirect
8. #4676C2semidirect
9. #4675C2semidirect
11. #4650C2semidirect
45. #395D3semidirect
258. C3#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#8892
C3×#3952
#52302
#49942
#49932
#49912
#46762
#46752
#46722
#46502
#28203