Finite Group Databasebeta

← #79976

Subgroups — #79976

Order 384.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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#5579C211yesmaximal, characteristic
2Q8×Dic6C211yesmaximal, characteristic, complemented
3C3×#446C211yesmaximal, characteristic, complemented
4#4850C211yesmaximal, characteristic, complemented
5#4846C212yesmaximal
6#4842C211yesmaximal, characteristic
7#4829C212yesmaximal, complemented
8#4710C212yesmaximal
9#4702C212yesmaximal, complemented
10C4×#204C211yesmaximal, characteristic
11#4682C211yesmaximal, characteristic, complemented
12#3385—31nomaximal, Sylow
13C3×C4×Q8C2211yescharacteristic
14#946—21no—
15Q8×Dic3—21no—
16C2×#204C2211yescharacteristic
17#896—22no—
18D3×C4⋊C4—21no—
19#888—21no—
20#888—22no—
21C4×D12C2211yescharacteristic
22C4×D12C2212yes—
23#870C2212yes—
24#869C2211yescharacteristic
25#869C2211yescharacteristic
26#869C2212yes—
27C4×Dic6C2211yescharacteristic
28C4×Dic6C2212yes—
29C3×C4⋊C8C2212yes—
30C3×C4⋊C8C2211yescharacteristic
31C3×#97C2212yes—
32C3×C4×C8C2211yescharacteristic
33#821C2211yescharacteristic
34#821C2212yes—
35#818C2212yes—
36#818C2212yes—
37#818C2212yes—
38#817C2211yescharacteristic
39#817C2212yes—
40#816C2211yescharacteristic
41#816C2212yes—
42Q82—31no—
43#554—31no—
44#501—32no—
45#500—31no—
46#479—32no—
47#477—31no—
48#476—31no—
49#474—32no—
50#446D311yescharacteristic, complemented
51#440—32no—
52C4×SD16—31no—
53C2×C3×Q8D411yescharacteristic
54C2×D12C2311yescharacteristic
55C2×C4×D3—21no—
56C2×C4×D3—22no—
57C2×Dic6—22no—
58C2×Dic6—21no—
59C2×Dic6—21no—
60C2×Dic6C2311yescharacteristic
61C2×C3×C8C2312yes—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4D411yescharacteristic
65C3×C4⋊C4D412yes—
66C3×C42C2311yescharacteristic
67C3×C42C2312yes—
68(C2×D3)⋊C4—21no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3—21no—
72C4⋊Dic3—21no—
73C4⋊Dic3C2311yescharacteristic
74C4⋊Dic3C2312yes—
75C4⋊Dic3C2311yescharacteristic
76C4⋊Dic3C2312yes—
77Dic3⋊C4—22no—
78Dic3⋊C4—21no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C4×Dic3—22no—
82C4×Dic3—21no—
83#204Q814yescomplemented
84C2×SD16—31no—
85C4⋊Q8—31no—
86C4⋊Q8—32no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8—62no—
90#125—32no—
91C22⋊Q8—61no—
92C22⋊Q8—62no—
93C4×Q8—31no—
94C4×Q8—32no—
95C4×Q8—61no—
96C4×Q8C2×D311yescharacteristic
97C4×D4—31no—
98C4×D4—32no—
99C2×C4⋊C4—61no—
100#107—32no—
101#107—32no—
102#107—32no—
103#106—31no—
104#106—32no—
105C4⋊C8C2×D312yes—
106C4⋊C8C2×D311yescharacteristic
107#103—31no—
108#103—32no—
109#102—31no—
110#102—32no—
111#97C2×D312yes—
112C4×C8C2×D311yescharacteristic
113C22×D3—21no—
114C3×Q8—22no—
115C2×C3×C4C2×D412yes—
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120C2×Dic3—21no—
121C2×Dic3—22no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125D12—21no—
126D12C2×Q812yes—
127C4×D3—42no—
128C4×D3—82no—
129Dic6—21no—
130Dic6—81no—
131Dic6—82no—
132Dic6—81no—
133Dic6C2×Q812yes—
134C3×C8—22no—
135C3×C8—21no—
136C3×C8C2×Q812yes—
137C2×Q8—31no—
138C2×Q8—61no—
139C2×Q8—61no—
140C2×Q8—62no—
141C2×Q8D1211yescharacteristic
142C2×D4—31no—
143C22×C4—61no—
144C22×C4—62no—
145SD16—34no—
146C2×C8C22×D311yescharacteristic
147C2×C8C22×D312yes—
148C2×C8C22×D311yescharacteristic
149C4⋊C4—31no—
150C4⋊C4—32no—
151C4⋊C4—31no—
152C4⋊C4—32no—
153C4⋊C4—62no—
154C4⋊C4—61no—
155C4⋊C4—61no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4—62no—
160C4⋊C4D1211yescharacteristic
161C4⋊C4D1212yes—
162C22⋊C4—61no—
163C22⋊C4—62no—
164C42—62no—
165C42—61no—
166C42C22×D311yescharacteristic
167C42C22×D312yes—
168C22×C3C22×D411yescharacteristic
169C2×D3—22no—
170C2×D3—41no—
171C2×D3—41no—
172C3×C4—22no—
173C3×C4—21no—
174C3×C4—22no—
175C3×C4#13712yes—
176C3×C4C22×Q811yescharacteristic
177C3×C42−1+411yescharacteristic
178Dic3—22no—
179Dic3—41no—
180Dic3—42no—
181Dic3—41no—
182Dic3—42no—
183C23—61no—
184Q8—22no—
185Q8—32no—
186Q8—61no—
187Q8—242no—
188Q8—241no—
189Q8—241no—
190D4—32no—
191D4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—62no—
197C2×C4—122no—
198C2×C4—242no—
199C2×C4C2×D1212yes—
200C2×C4C2×D1211yescharacteristic
201C2×C4C2×D1212yes—
202C2×C4C2×D1211yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—21no—
205C8—22no—
206C8D3×Q812yes—
207C2×C3D4×Q811yescharacteristic
208C2×C3C2×#13711yescharacteristic
209C2×C3#57611yescharacteristic
210D3—42no—
211C22—62no—
212C22—121no—
213C22—121no—
214C22C22×D1211yescharacteristic
215C4—22no—
216C4—22no—
217C4—21no—
218C4—62no—
219C4—121no—
220C4—122no—
221C4—121no—
222C4—122no—
223C4#90912yes—
224C4C2×D3×Q811yescharacteristic
225C4#101011yescharacteristic
226C3#338511yescharacteristic, complemented, Sylow
227C2—122no—
228C2Q8×D1211yescharacteristic
229C2C2×#90911yescharacteristic
230C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#446C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenter#446D3
hypocenterC3#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
2. Q8×Dic6C2semidirect
3. C3×#446C2semidirect
4. #4850C2semidirect
7. #4829C2semidirect
9. #4702C2semidirect
11. #4682C2semidirect
50. #446D3semidirect
83. #204Q8semidirect
226. C3#3385semidirect

Maximal subgroups

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

subgroupindex
#55792
Q8×Dic62
C3×#4462
#48502
#48462
#48422
#48292
#47102
#47022
C4×#2042
#46822
#33853