Finite Group Databasebeta

← #78676

Subgroups — #78676

Order 384.

402 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 117 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#5833C212yesmaximal, complemented
2C3×#408C211yesmaximal, characteristic, complemented
3#5134C212yesmaximal, complemented
4#5116C211yesmaximal, characteristic, complemented
5#5115C211yesmaximal, characteristic, complemented
6#4736C212yesmaximal, complemented
7#4734C212yesmaximal, complemented
8#4733C212yesmaximal, complemented
9#4722C212yesmaximal, complemented
10#3203—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C2×#235C2212yes—
13C2×C3×M16C2211yescharacteristic
14C22×C3×C8C2211yescharacteristic
15#946—22no—
16#939—22no—
17#939—22no—
18#938—22no—
19#930C2212yes—
20#930C2212yes—
21C2×C4⋊Dic3C2212yes—
22#924C2212yes—
23#924C2212yes—
24C2×Dic12C2211yescharacteristic
25C2×D24C2211yescharacteristic
26C2×#204C2212yes—
27C3×C22⋊C8C2212yes—
28C3×C22⋊C8C2212yes—
29#821C2212yes—
30#821C2212yes—
31#818C2212yes—
32#818C2211yescharacteristic
33#818C2211yescharacteristic
34#816C2212yes—
35#816C2212yes—
36#537—32no—
37#484—32no—
38#480—32no—
39#468—32no—
40#467—31no—
41#466—31no—
42#452—32no—
43#449—32no—
44#408D311yescharacteristic, complemented
45C3×#41—41no—
46C3×#41—41no—
47C2×C3×Q8D411yescharacteristic
48C2×C3×D4D412yes—
49C2×C3×D4D411yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C2×#67—22no—
53C2×#67—22no—
54C2×#67—22no—
55C22×Dic3—22no—
56#235—42no—
57#235—42no—
58C2×D12C2312yes—
59C2×C4×D3—22no—
60C2×Dic6C2312yes—
61C3×M16—41no—
62C2×C3×C8—41no—
63C2×C3×C8C2311yescharacteristic
64C2×C3×C8C2312yes—
65C2×C3×C8C2311yescharacteristic
66#217—22no—
67#217—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3C2312yes—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74Dic3⋊C4—22no—
75Dic12—41no—
76D24—41no—
77#204—42no—
78C2×#41—32no—
79C2×#41C2×D311yescharacteristic
80C2×Q16—31no—
81C2×SD16—32no—
82C2×D8—31no—
83C2×M16C2×D311yescharacteristic
84C22×C8C2×D311yescharacteristic
85C22⋊Q8—32no—
86C22⋊Q8—32no—
87C22⋊Q8—62no—
88#121—32no—
89#121—32no—
90#121—62no—
91#121—62no—
92#121—62no—
93C2×C4⋊C4—32no—
94#107—31no—
95#107—31no—
96#107—32no—
97#103—32no—
98#103—32no—
99#102—32no—
100#102—32no—
101C22⋊C8C2×D312yes—
102C22⋊C8C2×D312yes—
103C23×C3C2×D411yescharacteristic
104C23×C3C2×D412yes—
105C22×D3—22no—
106C3×Q8—41no—
107C3×D4—42no—
108C3×D4—41no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2411yescharacteristic
116#67—44no—
117#67—82no—
118#67—82no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—82no—
123D12—22no—
124D12—22no—
125C4×D3—44no—
126Dic6—22no—
127Dic6—22no—
128C3×C8—22no—
129C3×C8—21no—
130C3×C8—21no—
131#41—41no—
132#41—41no—
133#41—122no—
134#41—122no—
135C2×Q8—32no—
136C2×Q8D1211yescharacteristic
137C2×D4—32no—
138C2×D4—62no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4D1211yescharacteristic
142C2×D4D1212yes—
143C22×C4—62no—
144C22×C4—62no—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147Q16—121no—
148SD16—122no—
149D8—121no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D311yescharacteristic
153C2×C8C22×D312yes—
154C2×C8C22×D311yescharacteristic
155C4⋊C4—32no—
156C4⋊C4—32no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22×C3—22no—
165C22×C3—21no—
166C22×C3—22no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—22no—
170C22×C3C22×D411yescharacteristic
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—22no—
175C3×C4—21no—
176C3×C4H(5,𝔽2)11yescharacteristic
177C3×C4H(5,𝔽2)11yescharacteristic
178Dic3—42no—
179Dic3—42no—
180Dic3—42no—
181C23—62no—
182C23C2×D1211yescharacteristic
183C23C2×D1212yes—
184Q8—41no—
185Q8—62no—
186Q8—62no—
187D4—42no—
188D4—41no—
189D4—62no—
190D4—62no—
191D4—124no—
192D4—242no—
193D4—242no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—124no—
202C2×C4—242no—
203C2×C4C2×D1211yescharacteristic
204C2×C4C2×D1212yes—
205C2×C4C23×D311yescharacteristic
206C8—21no—
207C8—21no—
208C8—22no—
209C2×C3—42no—
210C2×C3—41no—
211C2×C3#53411yescharacteristic
212C2×C3#57611yescharacteristic
213C2×C3#57811yescharacteristic
214D3—82no—
215C22—22no—
216C22—21no—
217C22—22no—
218C22—21no—
219C22—21no—
220C22—22no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22C22×D1211yescharacteristic
225C4—22no—
226C4—21no—
227C4—122no—
228C4—122no—
229C4—122no—
230C4#100411yescharacteristic
231C4#100411yescharacteristic
232C3#320311yescharacteristic, complemented, Sylow
233C2—42no—
234C2—41no—
235C2—242no—
236C2#549311yescharacteristic
237C2#575211yescharacteristic
238C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#408C2
socleC22×C3C22×D4
hypercenter#408D3
hypocenterC3#3203
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #5833C2semidirect
2. C3×#408C2semidirect
3. #5134C2semidirect
4. #5116C2semidirect
5. #5115C2semidirect
6. #4736C2semidirect
7. #4734C2semidirect
8. #4733C2semidirect
9. #4722C2semidirect
44. #408D3semidirect
232. C3#3203semidirect

Maximal subgroups

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

subgroupindex
#58332
C3×#4082
#51342
#51162
#51152
#47362
#47342
#47332
#47222
#32033