Finite Group Databasebeta

← #79398

Subgroups — #79398

Order 384.

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

normal subgroups only

This group has 364 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#5526C211yesmaximal, characteristic
2#5518C211yesmaximal, characteristic, complemented
3C3×#433C211yesmaximal, characteristic, complemented
4#4849C212yesmaximal, complemented
5#4845C212yesmaximal, complemented
6#4739C212yesmaximal, complemented
7#4733C212yesmaximal
8#4715C211yesmaximal, characteristic, complemented
9#4703C211yesmaximal, characteristic, complemented
10#4699C211yesmaximal, characteristic, complemented
11#4697C211yesmaximal, characteristic, complemented
12#3223—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C3×#117C2211yescharacteristic
16#924C2212yes—
17C2×Dic12C2212yes—
18C2×Dic12C2212yes—
19C2×Dic12C2212yes—
20C2×#204C2212yes—
21#897—22no—
22#889—22no—
23#886—22no—
24#879—22no—
25#875C2211yescharacteristic
26#875C2211yescharacteristic
27#870C2211yescharacteristic
28#869C2211yescharacteristic
29#869C2211yescharacteristic
30#869C2211yescharacteristic
31C4×Dic6C2212yes—
32C3×C4⋊C8C2212yes—
33C3×C22⋊C8C2212yes—
34C3×#97C2211yescharacteristic
35C3×C4×C8C2211yescharacteristic
36#821C2212yes—
37#816C2212yes—
38#816C2212yes—
39#816C2212yes—
40#541—31no—
41#536—31no—
42#497—31no—
43#494—31no—
44#491—31no—
45#488—31no—
46#464—32no—
47#462—32no—
48#452—32no—
49#451—32no—
50#433D311yescharacteristic, complemented
51C22×C3×C4C2311yescharacteristic
52C2×#67—21no—
53C22×Dic3—21no—
54#235—42no—
55C2×D12C2311yescharacteristic
56C2×C4×D3—21no—
57C2×Dic6—21no—
58C2×Dic6—42no—
59C2×Dic6C2311yescharacteristic
60C2×Dic6C2311yescharacteristic
61C2×Dic6C2311yescharacteristic
62C2×Dic6C2312yes—
63C2×C3×C8C2312yes—
64C2×C3×C8C2312yes—
65C3×C4⋊C4D412yes—
66C3×C22⋊C4D412yes—
67C3×C42C2311yescharacteristic
68C3×C42C2311yescharacteristic
69#217—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74C4⋊Dic3—22no—
75C4⋊Dic3C2312yes—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79C4×Dic3—22no—
80Dic12—42no—
81Dic12—42no—
82Dic12—42no—
83#204—42no—
84C2×#41—31no—
85C22×Q8—31no—
86C2×Q16—32no—
87C2×Q16—32no—
88C2×Q16—32no—
89C2×SD16—32no—
90C4⋊Q8—31no—
91C4⋊Q8—31no—
92C4⋊Q8—31no—
93#126—62no—
94#125—31no—
95#125—62no—
96#124—31no—
97#124—31no—
98#123—62no—
99C22⋊Q8—32no—
100C22⋊Q8—62no—
101C4×Q8—32no—
102#117C2×D311yescharacteristic
103C4⋊C8C2×D312yes—
104#103—32no—
105#103—32no—
106#103—32no—
107#102—32no—
108C22⋊C8C2×D312yes—
109#97C2×D311yescharacteristic
110C4×C8C2×D311yescharacteristic
111C23×C3C2×D411yescharacteristic
112C22×D3—21no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2411yescharacteristic
118#67—42no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—42no—
125D12—22no—
126C4×D3—42no—
127Dic6—22no—
128Dic6—22no—
129Dic6—22no—
130Dic6—22no—
131Dic6—22no—
132Dic6—42no—
133C3×C8—22no—
134C3×C8—22no—
135#41—122no—
136C2×Q8—31no—
137C2×Q8—31no—
138C2×Q8—31no—
139C2×Q8—32no—
140C2×Q8—61no—
141C2×Q8—122no—
142C2×D4—31no—
143C2×D4—61no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4C22×D311yescharacteristic
147Q16—122no—
148Q16—122no—
149Q16—122no—
150SD16—122no—
151C2×C8C22×D312yes—
152C2×C8C22×D312yes—
153C4⋊C4—32no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4D1212yes—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4D1212yes—
165C42—62no—
166C42C22×D311yescharacteristic
167C42C22×D311yescharacteristic
168C22×C3—21no—
169C22×C3—22no—
170C22×C3C22×D411yescharacteristic
171C2×D3—41no—
172C2×D3—42no—
173C3×C4—21no—
174C3×C4—22no—
175C3×C4—22no—
176C3×C4H(5,𝔽2)12yes—
177Dic3—41no—
178Dic3—41no—
179Dic3—41no—
180Dic3—42no—
181Dic3—42no—
182C23—61no—
183C23C2×D1211yescharacteristic
184Q8—62no—
185Q8—62no—
186Q8—62no—
187Q8—62no—
188Q8—62no—
189Q8—122no—
190D4—62no—
191D4—122no—
192C2×C4—22no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4C2×D1212yes—
201C2×C4C2×D1212yes—
202C2×C4C2×D1211yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—22no—
205C8—22no—
206C2×C3—41no—
207C2×C3#53511yescharacteristic
208C2×C3#57812yes—
209D3—81no—
210C22—22no—
211C22—21no—
212C22—122no—
213C22—121no—
214C22C22×D1211yescharacteristic
215C4—22no—
216C4—22no—
217C4—21no—
218C4—122no—
219C4—121no—
220C4—121no—
221C4—121no—
222C4—122no—
223C4#100912yes—
224C3#322311yescharacteristic, complemented, Sylow
225C2—41no—
226C2—241no—
227C2#552511yescharacteristic
228C2#575312yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#433C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenter#433D3
hypocenterC3#3223
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #5518C2semidirect
3. C3×#433C2semidirect
4. #4849C2semidirect
5. #4845C2semidirect
6. #4739C2semidirect
8. #4715C2semidirect
9. #4703C2semidirect
10. #4699C2semidirect
11. #4697C2semidirect
50. #433D3semidirect
224. C3#3223semidirect

Maximal subgroups

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

subgroupindex
#55262
#55182
C3×#4332
#48492
#48452
#47392
#47332
#47152
#47032
#46992
#46972
#32233