Finite Group Databasebeta

← #68174

Subgroups — #68174

Order 384.

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

normal subgroups only

This group has 266 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×#237C211yesmaximal, characteristic
2#4837C211yesmaximal, characteristic, complemented
3C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4456C211yesmaximal, characteristic, complemented
8#1489—31nomaximal, Sylow
9C2×#237C411yescharacteristic
10C3×C4×D4C2211yescharacteristic
11D4×Dic3C411yescharacteristic
12C4×#67—21no—
13C2×C4×Dic3—21no—
14C2×ΓL(1,𝔽25)—21no—
15C2×ΓL(1,𝔽25)—21no—
16#892C411yescharacteristic
17#887C411yescharacteristic
18#67⋊C4—21no—
19#877—21no—
20C42×D3C2211yescharacteristic
21C4×Dic6C2211yescharacteristic
22C3×C4⋊C8C2211yescharacteristic
23C3×C4×C8C2211yescharacteristic
24#815—21no—
25Dic3⋊C8—21no—
26C4⋊(C3⋊C8)C2211yescharacteristic
27C4×C3⋊C8C2211yescharacteristic
28C4×#41—31no—
29#423—31no—
30C4×M16—31no—
31Q8⋊C8—31no—
32Q8⋊C8—31no—
33D4⋊C8—31no—
34D4⋊C8D311yescharacteristic, complemented
35C2×C3×D4C2×C411yescharacteristic
36C22×C3×C4—21no—
37C2×#67—21no—
38C22×Dic3—21no—
39C22×Dic3—21no—
40#237—22no—
41#237—22no—
42C2×C4×D3C2×C411yescharacteristic
43C2×C4×D3D411yescharacteristic
44C2×C4×D3D411yescharacteristic
45C2×Dic6C2×C411yescharacteristic
46C2×C3×C8—21no—
47C2×C3×C8—21no—
48C3×C4⋊C4C2×C411yescharacteristic
49C3×C22⋊C4—21no—
50C3×C42C2311yescharacteristic
51#217—21no—
52(C2×D3)⋊C4—21no—
53C4⋊Dic3C2×C411yescharacteristic
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C4×Dic3—21no—
58C4×Dic3—21no—
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3D411yescharacteristic
61C4×Dic3D411yescharacteristic
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64ΓL(1,𝔽25)—24no—
65ΓL(1,𝔽25)—42no—
66C2×#41—31no—
67C2×M16—61no—
68C2×M16—61no—
69C4×Q8—31no—
70C4×Q8—31no—
71C4×D4—31no—
72C4×D4—61no—
73C4×D4—61no—
74C4×D4C2×D311yescharacteristic
75#117—31no—
76#117—61no—
77C2×C42—31no—
78C2×C42—61no—
79C4⋊C8—31no—
80C4⋊C8—61no—
81C4⋊C8C2×D311yescharacteristic
82#97—61no—
83C4×C8—31no—
84C4×C8C2×D311yescharacteristic
85C23×C3—21no—
86C22×D3C22⋊C411yescharacteristic
87C3×D4—21no—
88C3×D4M1612yescomplemented
89C2×C3×C4—21no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95#67—24no—
96C2×Dic3—24no—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—42no—
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C411yescharacteristic
102C2×Dic3C22⋊C411yescharacteristic
103C4×D3—21no—
104C4×D3—21no—
105C4×D3—21no—
106C4×D3—21no—
107C4×D3—22no—
108Dic6—21no—
109Dic6M1612yescomplemented
110C3×C8—22no—
111C3×C8—41no—
112C3⋊C8—22no—
113C3⋊C8—41no—
114#41—62no—
115#41—62no—
116C2×Q8—31no—
117C2×D4—61no—
118C2×D4C4×D311yescharacteristic
119C22×C4—21no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—31no—
123C22×C4—61no—
124C22×C4—61no—
125M16—64no—
126M16—122no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—61no—
130C2×C8—61no—
131C4⋊C4—31no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C4⋊C4C4×D311yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C42—31no—
139C42—31no—
140C42—31no—
141C42—61no—
142C42—61no—
143C42—61no—
144C42C22×D311yescharacteristic
145C22×C3—22no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3C2×C22⋊C411yescharacteristic
149C2×D3—21no—
150C2×D3—21no—
151C2×D3—21no—
152C3×C4—21no—
153C3×C4—41no—
154C3×C4C2×M1611yescharacteristic
155C3×C4C2×M1611yescharacteristic
156C3×C4AGL(1,ℤ/8)11yescharacteristic
157C3×C4#13711yescharacteristic
158Dic3—22no—
159Dic3—21no—
160Dic3—21no—
161Dic3—41no—
162Dic3#10412yes—
163C23—21no—
164C23—31no—
165Q8—32no—
166Q8—61no—
167D4—21no—
168D4—64no—
169D4ΓL(1,𝔽25)12yescomplemented
170C2×C4—21no—
171C2×C4—31no—
172C2×C4—31no—
173C2×C4—31no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—62no—
177C2×C4—64no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—122no—
185C2×C4C2×C4×D311yescharacteristic
186C2×C4D3×D411yescharacteristic
187C2×C4D3×D411yescharacteristic
188C8—22no—
189C8—41no—
190C8—62no—
191C8—121no—
192C2×C3—42no—
193C2×C3#40711yescharacteristic
194C2×C3#41711yescharacteristic
195C2×C3C2×#10411yescharacteristic
196D3—41no—
197C22—22no—
198C22—41no—
199C22—41no—
200C22—61no—
201C22—61no—
202C22—61no—
203C22D3×C22⋊C411yescharacteristic
204C4—21no—
205C4—32no—
206C4—41no—
207C4—61no—
208C4—62no—
209C4—61no—
210C4—121no—
211C4C2×ΓL(1,𝔽25)11yescharacteristic
212C4D3×M1611yescharacteristic
213C4#91111yescharacteristic
214C4#91511yescharacteristic
215C3#148911yescharacteristic, complemented, Sylow
216C2—42no—
217C2—121no—
218C2#472611yescharacteristic
219C2#477111yescharacteristic
220C2D3×#10411yescharacteristic
subgroupG/H
centerC2×C4D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×D4⋊C8C2
socleC22×C3C2×C22⋊C4
prime radicalC2×#237C4
cosocleC3×C42C23
hypercenterD4⋊C8D3
hypocenterC3#1489
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. #4837C2semidirect
3. C4×ΓL(1,𝔽25)C2semidirect
4. C3×D4⋊C8C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4484C2semidirect
7. #4456C2semidirect
34. D4⋊C8D3semidirect
88. C3×D4M16semidirect
109. Dic6M16semidirect
169. D4ΓL(1,𝔽25)semidirect
215. C3#1489semidirect

Maximal subgroups

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

subgroupindex
C4×#2372
#48372
C4×ΓL(1,𝔽25)2
C3×D4⋊C82
D4⋊(C3⋊C8)2
#44842
#44562
#14893