Finite Group Databasebeta

← #79951

Subgroups — #79951

Order 384.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#5584C211yesmaximal, characteristic
2#5570C211yesmaximal, characteristic, complemented
3C3×#446C211yesmaximal, characteristic, complemented
4#4849C212yesmaximal, complemented
5#4849C211yesmaximal, characteristic, complemented
6#4845C212yesmaximal
7#4841C211yesmaximal, characteristic
8#4716C212yesmaximal, complemented
9#4715C212yesmaximal
10#4699C211yesmaximal, characteristic, complemented
11C4×#204C211yesmaximal, characteristic
12#3398—31nomaximal, Sylow
13C3×C4×Q8C2211yescharacteristic
14#946—21no—
15Q8×Dic3—21no—
16C2×Dic12C2212yes—
17C2×Dic12C2212yes—
18C2×Dic12C2212yes—
19C2×#204C2211yescharacteristic
20C2×#204C2212yes—
21#898—22no—
22#892—21no—
23#890—21no—
24#890—22no—
25#875C2212yes—
26C4×D12C2211yescharacteristic
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29#869C2212yes—
30C4×Dic6C2211yescharacteristic
31C4×Dic6C2212yes—
32C4×Dic6C2212yes—
33C3×C4⋊C8C2211yescharacteristic
34C3×C4⋊C8C2212yes—
35C3×#97C2212yes—
36C3×C4×C8C2211yescharacteristic
37#821C2211yescharacteristic
38#817C2211yescharacteristic
39#816C2211yescharacteristic
40#816C2212yes—
41#816C2212yes—
42#557—31no—
43#556—31no—
44#497—32no—
45#494—31no—
46#464—32no—
47#462—32no—
48#462—31no—
49#461—31no—
50#446D311yescharacteristic, complemented
51#441—32no—
52C4×SD16—31no—
53C2×C3×Q8D411yescharacteristic
54C2×D12C2311yescharacteristic
55C2×C4×D3—21no—
56C2×Dic6C2311yescharacteristic
57C2×Dic6C2312yes—
58C2×Dic6C2312yes—
59C2×C3×C8C2312yes—
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4D411yescharacteristic
63C3×C4⋊C4D412yes—
64C3×C42C2311yescharacteristic
65C3×C42C2312yes—
66(C2×D3)⋊C4—21no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69C4⋊Dic3—22no—
70C4⋊Dic3—21no—
71C4⋊Dic3—22no—
72C4⋊Dic3—21no—
73C4⋊Dic3—21no—
74C4⋊Dic3—21no—
75C4⋊Dic3C2311yescharacteristic
76C4⋊Dic3C2311yescharacteristic
77Dic3⋊C4—22no—
78Dic3⋊C4—21no—
79Dic3⋊C4—22no—
80C4×Dic3—22no—
81C4×Dic3—21no—
82C4×Dic3—22no—
83Dic12—24no—
84Dic12—42no—
85Dic12—42no—
86#204—22no—
87#204—42no—
88C2×Q16—32no—
89C2×Q16—32no—
90C2×Q16—32no—
91C2×SD16—31no—
92C2×SD16—32no—
93C4⋊Q8—31no—
94C4⋊Q8—31no—
95C4⋊Q8—32no—
96#126—62no—
97#125—61no—
98#125—62no—
99#124—32no—
100C22⋊Q8—61no—
101C4×Q8—32no—
102C4×Q8—31no—
103C4×Q8—32no—
104C4×Q8—61no—
105C4×Q8C2×D311yescharacteristic
106C4×D4—31no—
107#117—61no—
108#106—31no—
109C4⋊C8C2×D311yescharacteristic
110C4⋊C8C2×D312yes—
111#103—31no—
112#103—32no—
113#103—32no—
114#102—31no—
115#97C2×D312yes—
116C4×C8C2×D311yescharacteristic
117C22×D3—21no—
118C3×Q8—22no—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2411yescharacteristic
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129D12—21no—
130D12—21no—
131C4×D3—42no—
132Dic6—21no—
133Dic6—22no—
134Dic6—22no—
135Dic6—22no—
136Dic6—21no—
137Dic6—22no—
138C3×C8—22no—
139C3×C8—21no—
140C3×C8#4112yes—
141C2×Q8—32no—
142C2×Q8—31no—
143C2×Q8—32no—
144C2×Q8D1211yescharacteristic
145C2×D4—31no—
146C22×C4—61no—
147Q16—64no—
148Q16—122no—
149Q16—122no—
150SD16—62no—
151SD16—122no—
152C2×C8C22×D311yescharacteristic
153C2×C8C22×D312yes—
154C2×C8C22×D311yescharacteristic
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—61no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—61no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4D1211yescharacteristic
167C4⋊C4D1212yes—
168C22⋊C4—62no—
169C22⋊C4—61no—
170C22⋊C4—62no—
171C42—62no—
172C42—62no—
173C42—61no—
174C42C22×D311yescharacteristic
175C42C22×D312yes—
176C22×C3C22×D411yescharacteristic
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—21no—
183C3×C4#13712yes—
184C3×C4C2×#4111yescharacteristic
185C3×C4H(5,𝔽2)11yescharacteristic
186Dic3—41no—
187Dic3—42no—
188Dic3—42no—
189Dic3—41no—
190Dic3—41no—
191C23—61no—
192Q8—22no—
193Q8—61no—
194Q8—62no—
195Q8—62no—
196Q8—62no—
197Q8—62no—
198Q8—61no—
199D4—61no—
200D4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—62no—
204C2×C4—61no—
205C2×C4—62no—
206C2×C4—122no—
207C2×C4C2×D1212yes—
208C2×C4C2×D1212yes—
209C2×C4C2×D1211yescharacteristic
210C2×C4C2×D1211yescharacteristic
211C2×C4C23×D311yescharacteristic
212C8—22no—
213C8—21no—
214C8#23912yes—
215C2×C3#55011yescharacteristic
216C2×C3C2×#13711yescharacteristic
217C2×C3#57811yescharacteristic
218D3—81no—
219C22—121no—
220C22—121no—
221C22—121no—
222C22C22×D1211yescharacteristic
223C4—22no—
224C4—22no—
225C4—21no—
226C4—121no—
227C4—121no—
228C4—122no—
229C4—121no—
230C4—122no—
231C4#90912yes—
232C4C2×#23911yescharacteristic
233C4#100911yescharacteristic
234C3#339811yescharacteristic, complemented, Sylow
235C2—241no—
236C2#557711yescharacteristic
237C2C2×#90911yescharacteristic
238C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#446C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenter#446D3
hypocenterC3#3398
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. #5570C2semidirect
3. C3×#446C2semidirect
4. #4849C2semidirect
5. #4849C2semidirect
8. #4716C2semidirect
10. #4699C2semidirect
50. #446D3semidirect
234. C3#3398semidirect

Maximal subgroups

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

subgroupindex
#55842
#55702
C3×#4462
#48492
#48452
#48412
#47162
#47152
#46992
C4×#2042
#33983