Finite Group Databasebeta

← #83238

Subgroups — #83238

Order 384.

540 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 185 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×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×Q8×Dic3C211yesmaximal, characteristic
3C2×#944C211yesmaximal, characteristic
4C2×#944C211yesmaximal, characteristic
5#5229C211yesmaximal, characteristic
6#4982C214yesmaximal
7#4976C214yesmaximal
8C4×Dic3⋊C4C211yesmaximal, characteristic
9#4929C211yesmaximal, characteristic
10#2546—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C3×C4⋊Q8—24no—
15C2×C3×C4⋊C4C2214yes—
16C2×C3×C42C2211yescharacteristic
17Q8×Dic3—24no—
18#944—24no—
19#944—24no—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊Dic3C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2214yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27C2×C4×Dic3C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C3⋊#95C2214yes—
30C3⋊#95C2212yes—
31C3⋊#95C2214yes—
32C2×C4⋊Q8—31no—
33C2×C4⋊Q8—31no—
34C2×C4⋊Q8D311yescharacteristic
35C2×C4×Q8—31no—
36#398—34no—
37#391—31no—
38#391—31no—
39#389—34no—
40C4×C4⋊C4—31no—
41C2×C3×Q8—22no—
42C2×C3×Q8—22no—
43C2×C3×Q8—22no—
44C2×C3×Q8—22no—
45C2×C3×Q8—22no—
46C2×C3×Q8D414yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C2×Dic6—22no—
57C2×Dic6—22no—
58C2×Dic6—22no—
59C3×C4⋊C4—28no—
60C3×C42—22no—
61C4⋊Dic3—22no—
62C4⋊Dic3—24no—
63Dic3⋊C4—28no—
64Dic3⋊C4—24no—
65Dic3⋊C4—24no—
66Dic3⋊C4Q818yescomplemented
67C4×Dic3—22no—
68C4×Dic3—24no—
69C4×Dic3D414yes—
70C22×Q8—31no—
71C22×Q8C2×D311yescharacteristic
72C22×Q8C2×D311yescharacteristic
73C4⋊Q8—24no—
74C4⋊Q8—64no—
75C4⋊Q8—64no—
76C4×Q8—64no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—34no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4C2×D314yes—
84C2×C42—31no—
85C2×C42—32no—
86C2×C42—31no—
87C2×C42C2×D311yescharacteristic
88#95—32no—
89#95—34no—
90#95—34no—
91C23×C3C2411yescharacteristic
92C3×Q8—28no—
93C3×Q8—44no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4C2×D412yes—
102C2×C3×C4C2×D414yes—
103C2×C3×C4C2×D412yes—
104C2×C3×C4C2×Q814yes—
105C2×C3×C4#4112yes—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—24no—
114C2×Dic3C2×D412yes—
115C2×Dic3C2×D412yes—
116C2×Dic3C2×Q814yes—
117C2×Dic3C2×Q814yes—
118Dic6—44no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—62no—
125C2×Q8—62no—
126C2×Q8—62no—
127C2×Q8#6714yes—
128C22×C4—31no—
129C22×C4—32no—
130C22×C4—31no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4C22×D312yes—
134C22×C4C22×D312yes—
135C22×C4C22×D312yes—
136C22×C4C22×D311yescharacteristic
137C4⋊C4—28no—
138C4⋊C4—38no—
139C4⋊C4—64no—
140C4⋊C4—64no—
141C4⋊C4—64no—
142C4⋊C4—62no—
143C4⋊C4—68no—
144C42—22no—
145C42—34no—
146C42—64no—
147C42—62no—
148C22×C3C22×D411yescharacteristic
149C22×C3C22×D411yescharacteristic
150C22×C3C22×Q812yes—
151C22×C3C2×#4111yescharacteristic
152C22×C32−1+412yes—
153C3×C4—24no—
154C3×C4—24no—
155C3×C4—42no—
156C3×C4#12114yes—
157Dic3—22no—
158Dic3—24no—
159Dic3—42no—
160Dic3—42no—
161Dic3C4⋊Q814yescomplemented
162C23C23×D311yescharacteristic
163Q8—28no—
164Q8—44no—
165Q8—124no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—32no—
174C2×C4—34no—
175C2×C4—32no—
176C2×C4—34no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—64no—
184C2×C4—62no—
185C2×C4D3×D412yes—
186C2×C4#23712yes—
187C2×C4D3×Q814yes—
188C2×C4C2×#6714yes—
189C2×C4C2×#6712yes—
190C2×C3C2×#12111yescharacteristic
191C2×C3C2×C4⋊Q811yescharacteristic
192C2×C3#53611yescharacteristic
193C2×C3D4×Q812yes—
194C2×C3#55712yes—
195C22C2×D3×D411yescharacteristic
196C22C2×#23711yescharacteristic
197C22C2×D3×Q812yes—
198C22#100712yes—
199C22C22×#6711yescharacteristic
200C4—24no—
201C4—24no—
202C4—34no—
203C4—42no—
204C4—62no—
205C4—64no—
206C4—122no—
207C4—122no—
208C4#93914yes—
209C3#254611yescharacteristic, complemented, Sylow
210C2#572112yes—
211C2D3×C4⋊Q811yescharacteristic
212C2#572411yescharacteristic
213C2C2×#93911yescharacteristic
214C2Q8×#6712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC2×C3×C4⋊Q8C2
hypercenterC2×C4⋊Q8D3
hypocenterC3#2546
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
66. Dic3⋊C4Q8semidirect
161. Dic3C4⋊Q8semidirect
209. C3#2546semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×Q8×Dic32
C2×#9442
#52292
#49822
#49762
C4×Dic3⋊C42
#49292
#25463