Finite Group Databasebeta

← #68175

Subgroups — #68175

Order 384.

316 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×D3×D4C211yesmaximal, characteristic
2#4835C211yesmaximal, characteristic, complemented
3C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4483C211yesmaximal, characteristic, complemented
7#4459C211yesmaximal, characteristic, complemented
8#1490—31nomaximal, Sylow
9C2×D3×D4C411yescharacteristic
10C22×C4×D3—21no—
11C3×C4×D4C2211yescharacteristic
12D4×Dic3C411yescharacteristic
13C4×#67—21no—
14C2×ΓL(1,𝔽25)—21no—
15#893C411yescharacteristic
16D3×C4⋊C4C411yescharacteristic
17#67⋊C4—21no—
18D3×C22⋊C4—21no—
19C4×D12C2211yescharacteristic
20C42×D3C2211yescharacteristic
21C3×C4⋊C8C2211yescharacteristic, complemented
22C3×C4×C8C2211yescharacteristic, complemented
23(C2×D3)⋊C8—21no—
24#815—21no—
25#815—21no—
26C4⋊(C3⋊C8)C2211yescharacteristic, complemented
27C4×C3⋊C8C2211yescharacteristic, complemented
28C2×C4×D4—31no—
29#432—31no—
30C4×M16—31no—
31D4⋊C8—31no—
32D4⋊C8—31no—
33D4⋊C8—31no—
34D4⋊C8D311yescharacteristic, complemented
35C23×D3—21no—
36C2×C3×D4C2×C411yescharacteristic
37C22×C3×C4—21no—
38C2×#67—21no—
39C22×Dic3—21no—
40D3×D4—22no—
41D3×D4—22no—
42C2×D12C2×C411yescharacteristic
43C2×C4×D3—21no—
44C2×C4×D3—21no—
45C2×C4×D3—21no—
46C2×C4×D3—42no—
47C2×C4×D3—42no—
48C2×C4×D3C2×C411yescharacteristic
49C2×C4×D3D411yescharacteristic
50C2×C4×D3D411yescharacteristic
51C2×C3×C8—21no—
52C2×C3×C8—21no—
53C3×C4⋊C4C2×C411yescharacteristic
54C3×C22⋊C4—21no—
55C3×C42C2311yescharacteristic
56#217—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59C4⋊Dic3C2×C411yescharacteristic
60Dic3⋊C4—21no—
61C4×Dic3C2×C411yescharacteristic
62C4×Dic3D411yescharacteristic
63C4×Dic3D411yescharacteristic
64C2×C3⋊C8—21no—
65C2×C3⋊C8—21no—
66ΓL(1,𝔽25)—24no—
67C22×D4—31no—
68C23×C4—61no—
69C2×M16—61no—
70C4×D4—31no—
71C4×D4—31no—
72C4×D4—31no—
73C4×D4—61no—
74C4×D4—61no—
75C4×D4C2×D311yescharacteristic
76C2×C4⋊C4—31no—
77C2×C22⋊C4—61no—
78C2×C42—31no—
79C4⋊C8—31no—
80C4⋊C8C2×D311yescharacteristic, complemented
81C22⋊C8—61no—
82#97—61no—
83#97—61no—
84C4×C8—31no—
85C4×C8C2×D311yescharacteristic, complemented
86C23×C3—21no—
87C22×D3—24no—
88C22×D3—21no—
89C22×D3—42no—
90C22×D3—42no—
91C22×D3C22⋊C411yescharacteristic
92C3×D4—21no—
93C3×D4M1612yescomplemented
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4C22×C411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D411yescharacteristic
100#67—24no—
101C2×Dic3—21no—
102C2×Dic3—42no—
103C2×Dic3C22⋊C411yescharacteristic
104C2×Dic3C22⋊C411yescharacteristic
105C2×Dic3C22⋊C411yescharacteristic
106D12—21no—
107D12M1612yescomplemented
108C4×D3—22no—
109C4×D3—21no—
110C4×D3—21no—
111C4×D3—21no—
112C4×D3—21no—
113C4×D3—42no—
114C4×D3—42no—
115C4×D3—41no—
116C4×D3—41no—
117C3×C8—22no—
118C3×C8—41no—
119C3⋊C8—22no—
120C3⋊C8—41no—
121C24—61no—
122C2×D4—31no—
123C2×D4—61no—
124C2×D4—62no—
125C2×D4—62no—
126C2×D4C4×D311yescharacteristic
127C22×C4—21no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C22×C4—61no—
132C22×C4—61no—
133C22×C4—61no—
134C22×C4—61no—
135C22×C4—122no—
136C22×C4—122no—
137M16—64no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—61no—
141C2×C8—61no—
142C4⋊C4—31no—
143C4⋊C4—61no—
144C4⋊C4C4×D311yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C42—31no—
150C42—31no—
151C42—31no—
152C42C22×D311yescharacteristic
153C22×C3—22no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3C2×C22⋊C411yescharacteristic
157C2×D3—21no—
158C2×D3—21no—
159C2×D3—22no—
160C2×D3—48no—
161C2×D3—41no—
162C2×D3—41no—
163C2×D3#10412yes—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4C2×M1611yescharacteristic
167C3×C4C2×M1611yescharacteristic
168C3×C4AGL(1,ℤ/8)11yescharacteristic
169C3×C4AGL(1,ℤ/8)11yescharacteristic
170Dic3—21no—
171Dic3—21no—
172Dic3—21no—
173Dic3—41no—
174C23—21no—
175C23—31no—
176C23—64no—
177C23—61no—
178C23—122no—
179C23—122no—
180D4—21no—
181D4—32no—
182D4—64no—
183D4—61no—
184D4ΓL(1,𝔽25)12yescomplemented
185C2×C4—21no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—62no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4C2×C4×D311yescharacteristic
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D411yescharacteristic
205C8—22no—
206C8—41no—
207C8—62no—
208C8—121no—
209C2×C3—42no—
210C2×C3#40711yescharacteristic
211C2×C3#41811yescharacteristic
212C2×C3C2×#10411yescharacteristic
213D3—22no—
214D3—42no—
215C22—22no—
216C22—32no—
217C22—41no—
218C22—41no—
219C22—61no—
220C22—61no—
221C22—62no—
222C22—128no—
223C22—121no—
224C22—121no—
225C22D3×C22⋊C411yescharacteristic
226C4—21no—
227C4—41no—
228C4—61no—
229C4—61no—
230C4—61no—
231C4—121no—
232C4C2×ΓL(1,𝔽25)11yescharacteristic
233C4D3×M1611yescharacteristic
234C4#91111yescharacteristic
235C4#91411yescharacteristic
236C3#149011yescharacteristic, complemented, Sylow
237C2—42no—
238C2—62no—
239C2—122no—
240C2#472611yescharacteristic
241C2#477011yescharacteristic
242C2D3×#10411yescharacteristic
subgroupG/H
centerC2×C4D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×D4⋊C8C2
socleC22×C3C2×C22⋊C4
prime radicalC2×D3×D4C4
cosocleC3×C42C23
hypercenterD4⋊C8D3
hypocenterC3#1490
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #4835C2semidirect
3. C4×ΓL(1,𝔽25)C2semidirect
4. C3×D4⋊C8C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4483C2semidirect
7. #4459C2semidirect
21. C3×C4⋊C8C22semidirect
22. C3×C4×C8C22semidirect
26. C4⋊(C3⋊C8)C22semidirect
27. C4×C3⋊C8C22semidirect
34. D4⋊C8D3semidirect
80. C4⋊C8C2×D3semidirect
85. C4×C8C2×D3semidirect
93. C3×D4M16semidirect
107. D12M16semidirect
184. D4ΓL(1,𝔽25)semidirect
236. C3#1490semidirect

Maximal subgroups

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

subgroupindex
C4×D3×D42
#48352
C4×ΓL(1,𝔽25)2
C3×D4⋊C82
D4⋊(C3⋊C8)2
#44832
#44592
#14903