Finite Group Databasebeta

← #83250

Subgroups — #83250

Order 384.

456 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 157 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×#944C211yesmaximal, characteristic
3#5230C212yesmaximal
4#5229C211yesmaximal, characteristic
5#4982C212yesmaximal
6#4979C212yesmaximal
7#4978C212yesmaximal
8#4976C212yesmaximal
9#4933C212yesmaximal
10#2734—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C3×C4⋊Q8—24no—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18#944—24no—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4C2212yes—
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×C4×Dic3C2211yescharacteristic
26C2×C4×Dic3C2211yescharacteristic
27C3⋊#95C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×C4⋊Q8—31no—
35C2×C4⋊Q8D311yescharacteristic
36#398—32no—
37#395—32no—
38#395—32no—
39#391—31no—
40#389—32no—
41#387—32no—
42#387—32no—
43C2×C3×Q8—22no—
44C2×C3×Q8—41no—
45C2×C3×Q8—41no—
46C2×C3×Q8—41no—
47C2×C3×Q8—41no—
48C2×C3×Q8D414yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C2×Dic6—22no—
59C2×Dic6—41no—
60C2×Dic6—41no—
61C3×C4⋊C4—24no—
62C3×C4⋊C4—24no—
63C3×C42—22no—
64C4⋊Dic3—41no—
65Dic3⋊C4—24no—
66Dic3⋊C4—24no—
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69Dic3⋊C4Q818yescomplemented
70C4×Dic3—22no—
71C4×Dic3—41no—
72C22×Q8—31no—
73C22×Q8C2×D311yescharacteristic
74C22×Q8C2×D311yescharacteristic
75C4⋊Q8—24no—
76C4⋊Q8—64no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4C2×D312yes—
84C2×C4⋊C4C2×D312yes—
85C2×C42—31no—
86C2×C42—31no—
87C2×C42C2×D311yescharacteristic
88#95—32no—
89#95—32no—
90#95—32no—
91#95—32no—
92#95—32no—
93#95—32no—
94#95—32no—
95C23×C3C2411yescharacteristic
96C3×Q8—44no—
97C3×Q8—44no—
98C2×C3×C4—22no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D412yes—
110C2×C3×C4C2×D412yes—
111C2×C3×C4C2×D412yes—
112C2×C3×C4C2×Q814yes—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—24no—
122C2×Dic3C2×Q814yes—
123C2×Dic3C2×Q814yes—
124Dic6—44no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—62no—
131C2×Q8—121no—
132C2×Q8—121no—
133C2×Q8#6714yes—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—32no—
137C22×C4—32no—
138C22×C4C22×D312yes—
139C22×C4C22×D312yes—
140C22×C4C22×D311yescharacteristic
141C22×C4C22×D311yescharacteristic
142C22×C4C22×D311yescharacteristic
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—38no—
146C4⋊C4—64no—
147C4⋊C4—64no—
148C4⋊C4—122no—
149C4⋊C4—122no—
150C4⋊C4—121no—
151C42—22no—
152C42—62no—
153C42—121no—
154C22×C3C22×D411yescharacteristic
155C22×C3C22×Q812yes—
156C22×C3H(5,𝔽2)11yescharacteristic
157C22×C3H(5,𝔽2)11yescharacteristic
158C22×C32−1+412yes—
159C3×C4—22no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—24no—
163C3×C4—42no—
164Dic3—24no—
165Dic3—24no—
166Dic3—42no—
167Dic3—42no—
168C23C23×D311yescharacteristic
169Q8—44no—
170Q8—44no—
171Q8—124no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—34no—
184C2×C4—34no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—64no—
193C2×C4—62no—
194C2×C4D3×Q814yes—
195C2×C4C2×#6712yes—
196C2×C4C2×#6712yes—
197C2×C4C2×#6712yes—
198C2×C3#53411yescharacteristic
199C2×C3#54412yes—
200C2×C3D4×Q812yes—
201C2×C3Q8211yescharacteristic
202C2×C3#56311yescharacteristic
203C22#100411yescharacteristic
204C22#100411yescharacteristic
205C22C2×D3×Q812yes—
206C22#100712yes—
207C22C22×#6711yescharacteristic
208C4—22no—
209C4—22no—
210C4—22no—
211C4—24no—
212C4—42no—
213C4—64no—
214C4—64no—
215C4—122no—
216C4—122no—
217C3#273411yescharacteristic, complemented, Sylow
218C2#572211yescharacteristic
219C2#572912yes—
220C2#573511yescharacteristic
221C2#580411yescharacteristic
222C2Q8×#6712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, prime radical, cosocleC23×C3C24
FittingC2×C3×C4⋊Q8C2
hypercenterC2×C4⋊Q8D3
hypocenterC3#2734
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
69. Dic3⋊C4Q8semidirect
217. C3#2734semidirect

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×#9442
#52302
#52292
#49822
#49792
#49782
#49762
#49332
#27343