Finite Group Databasebeta

← #83232

Subgroups — #83232

Order 384.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×D3×Q8C211yesmaximal, characteristic
2C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#946C211yesmaximal, characteristic
4C2×#944C211yesmaximal, characteristic, complemented
5#5229C211yesmaximal, characteristic, complemented
6#4989C214yesmaximal
7#4979C214yesmaximal, complemented
8C4×(C2×D3)⋊C4C211yesmaximal, characteristic
9#4929C211yesmaximal, characteristic, complemented
10#2434—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic
13C2×D3×Q8—24no—
14C2×D3×Q8—24no—
15C2×D3×Q8—24no—
16C22×C4×D3C2212yes—
17C22×C4×D3C2211yescharacteristic
18C22×Dic6C2212yes—
19C22×Dic6C2211yescharacteristic
20C3×C4⋊Q8—24no—
21C2×C3×C4⋊C4C2214yes—
22C2×C3×C42C2211yescharacteristic
23#946—24no—
24#944—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2214yes—
30C2×C4×Dic3C2211yescharacteristic
31C3⋊#95C2214yes—
32C3⋊#95C2212yes—
33C3⋊#95C2214yes—
34C23×Q8—31no—
35C2×C4⋊Q8—31no—
36C2×C4⋊Q8D311yescharacteristic, complemented
37C2×C22⋊Q8—31no—
38#395—34no—
39#391—31no—
40#391—31no—
41#385—34no—
42C4×C22⋊C4—31no—
43C23×D3C2311yescharacteristic
44C2×C3×Q8—22no—
45C2×C3×Q8—22no—
46C2×C3×Q8—22no—
47C2×C3×Q8—22no—
48C2×C3×Q8—22no—
49C2×C3×Q8D414yes—
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58D3×Q8—232no—
59C2×C4×D3—24no—
60C2×C4×D3—22no—
61C2×C4×D3—24no—
62C2×C4×D3—22no—
63C2×C4×D3—24no—
64C2×C4×D3D414yes—
65C2×Dic6—22no—
66C2×Dic6—24no—
67C2×Dic6—22no—
68C2×Dic6—24no—
69C2×Dic6—24no—
70C2×Dic6D414yes—
71C3×C4⋊C4—28no—
72C3×C42—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4Q818yescomplemented
75C4⋊Dic3—22no—
76Dic3⋊C4—28no—
77Dic3⋊C4—24no—
78C4×Dic3—22no—
79C22×Q8—31no—
80C22×Q8—32no—
81C22×Q8—64no—
82C22×Q8—64no—
83C22×Q8—64no—
84C22×Q8C2×D311yescharacteristic
85C22×Q8C2×D311yescharacteristic
86C23×C4—31no—
87C23×C4—32no—
88C4⋊Q8—24no—
89C4⋊Q8—64no—
90C22⋊Q8—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—34no—
94C2×C4⋊C4C2×D314yes—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C42—31no—
98C2×C42C2×D311yescharacteristic
99#95—32no—
100#95—34no—
101#95—34no—
102C23×C3C2411yescharacteristic
103C22×D3—21no—
104C22×D3—21no—
105C22×D3—22no—
106C22×D3C2×D412yes—
107C22×D3C2×Q814yes—
108C3×Q8—28no—
109C3×Q8—44no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4C2×D412yes—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2×D414yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×Q814yescomplemented
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—24no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D414yes—
132C2×Dic3C2×Q814yescomplemented
133C4×D3—216no—
134C4×D3—28no—
135Dic6—216no—
136Dic6—28no—
137C24—31no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—34no—
144C2×Q8—64no—
145C2×Q8—64no—
146C2×Q8—632no—
147C2×Q8—62no—
148C2×Q8—64no—
149C2×Q8—62no—
150C2×Q8#6714yes—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—32no—
154C22×C4—34no—
155C22×C4—32no—
156C22×C4—64no—
157C22×C4—64no—
158C22×C4—62no—
159C22×C4—64no—
160C22×C4—62no—
161C22×C4C22×D312yes—
162C22×C4C22×D312yes—
163C22×C4C22×D312yes—
164C22×C4C22×D311yescharacteristic
165C4⋊C4—28no—
166C4⋊C4—64no—
167C4⋊C4—68no—
168C4⋊C4—62no—
169C22⋊C4—38no—
170C22⋊C4—64no—
171C42—22no—
172C42—62no—
173C22×C3C22×D411yescharacteristic
174C22×C3C22×D411yescharacteristic
175C22×C3C22×D411yescharacteristic
176C22×C3C22×Q812yes—
177C22×C32−1+412yes—
178C2×D3—22no—
179C2×D3—22no—
180C2×D3—24no—
181C2×D3—24no—
182C2×D3C4⋊Q814yescomplemented
183C3×C4—24no—
184C3×C4—24no—
185C3×C4—42no—
186C3×C4#12014yes—
187Dic3—22no—
188Dic3—24no—
189Dic3—24no—
190Dic3—42no—
191C23—32no—
192C23—34no—
193C23—61no—
194C23—62no—
195C23—61no—
196C23C23×D311yescharacteristic
197Q8—28no—
198Q8—44no—
199Q8—616no—
200Q8—68no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—32no—
209C2×C4—34no—
210C2×C4—34no—
211C2×C4—616no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—62no—
216C2×C4—61no—
217C2×C4—62no—
218C2×C4—68no—
219C2×C4—62no—
220C2×C4—64no—
221C2×C4D3×D412yes—
222C2×C4D3×D412yes—
223C2×C4D3×Q814yescomplemented
224C2×C4C2×#6714yes—
225C2×C4C2×#6712yes—
226C2×C3C2×#12011yescharacteristic
227C2×C3C2×C4⋊Q811yescharacteristic
228C2×C3#53611yescharacteristic
229C2×C3D4×Q812yes—
230C2×C3D4×Q812yes—
231D3—24no—
232C22—34no—
233C22—62no—
234C22—64no—
235C22—62no—
236C22—64no—
237C22C2×D3×D411yescharacteristic
238C22C2×D3×D411yescharacteristic
239C22C2×D3×Q812yes—
240C22#100712yes—
241C22C22×#6711yescharacteristic
242C4—24no—
243C4—24no—
244C4—42no—
245C4—64no—
246C4—62no—
247C4—64no—
248C4—122no—
249C4#93714yes—
250C3#243411yescharacteristic, complemented, Sylow
251C2—64no—
252C2D3×C4⋊Q811yescharacteristic
253C2#572411yescharacteristic
254C2#572712yes—
255C2C2×#93711yescharacteristic
256C2Q8×#6712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4⋊Q8C2
prime radicalC23×D3C23
hypercenterC2×C4⋊Q8D3
hypocenterC3#2434
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊Q8C2semidirect
4. C2×#944C2semidirect
5. #5229C2semidirect
7. #4979C2semidirect
9. #4929C2semidirect
36. C2×C4⋊Q8D3semidirect
74. (C2×D3)⋊C4Q8semidirect
121. C2×C3×C4C2×Q8semidirect
132. C2×Dic3C2×Q8semidirect
182. C2×D3C4⋊Q8semidirect
223. C2×C4D3×Q8semidirect
250. C3#2434semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×C3×C4⋊Q82
C2×#9462
C2×#9442
#52292
#49892
#49792
C4×(C2×D3)⋊C42
#49292
#24343