Finite Group Databasebeta

← #70363

Subgroups — #70363

Order 384.

302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 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#5827C211yesmaximal, characteristic
2C2×C3×#104C211yesmaximal, characteristic
3#5232C211yesmaximal, characteristic
4#5121C211yesmaximal, characteristic, complemented
5#4935C211yesmaximal, characteristic, complemented
6#4554C211yesmaximal, characteristic, complemented
7#4523C211yesmaximal, characteristic, complemented
8#1859—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C2×C3×C42C2211yescharacteristic
12Q8×Dic3—21no—
13D4×Dic3—21no—
14D4×Dic3—22no—
15#926—21no—
16#926C2211yescharacteristic
17C2×C4⋊Dic3—21no—
18C2×C4⋊Dic3—22no—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2211yescharacteristic
21C2×#208C2211yescharacteristic
22C3×#104C414yescomplemented
23#835—21no—
24#832—21no—
25#818—21no—
26#817—21no—
27#520—31no—
28#428—31no—
29C2×#104D311yescharacteristic
30#417—31no—
31#384—31no—
32#343—31no—
33#339—31no—
34C3×#41—21no—
35C3×#41—21no—
36C3×#41C2×C412yes—
37C3×#41Q812yes—
38C2×C3×Q8D411yescharacteristic
39C2×C3×D4—21no—
40C2×C3×D4D411yescharacteristic
41C22×C3×C4—21no—
42C22×C3×C4—21no—
43C22×C3×C4C2311yescharacteristic
44C22×Dic3—22no—
45C22×Dic3—21no—
46C22×Dic3D412yes—
47C3×M16—21no—
48C3×M16C2×C412yescomplemented
49C2×C3×C8—21no—
50C3×C42—21no—
51C3×C42C2×C412yescomplemented
52#217—22no—
53#217—21no—
54C4⋊Dic3—22no—
55C4⋊Dic3—21no—
56C4⋊Dic3—21no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—44no—
60C4⋊Dic3D411yescharacteristic
61C4⋊Dic3D411yescharacteristic
62C4×Dic3—22no—
63C4×Dic3—21no—
64#208—21no—
65#208—21no—
66C2×C3⋊C8—21no—
67C2×#41C2×D311yescharacteristic
68C2×M16—31no—
69C2×M16C2×D311yescharacteristic
70C4×Q8—61no—
71C4×D4—62no—
72C4×D4—61no—
73#117—31no—
74#117—61no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—61no—
78C2×C4⋊C4—62no—
79C2×C42C2×D311yescharacteristic
80#107—61no—
81#106—61no—
82#104Dic314yescomplemented
83#103—61no—
84#102—61no—
85C23×C3—21no—
86C23×C3C2×D411yescharacteristic
87C3×Q8—21no—
88C3×Q8C4⋊C412yescomplemented
89C3×D4—22no—
90C3×D4—22no—
91C3×D4—21no—
92C3×D4C4⋊C412yescomplemented
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—21no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4C22×C411yescharacteristic
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×Q811yescharacteristic
103C2×C3×C4#4111yescharacteristic
104C2×C3×C4#4111yescharacteristic
105C2×Dic3—21no—
106C2×Dic3—22no—
107C2×Dic3—21no—
108C2×Dic3—42no—
109C2×Dic3—42no—
110C2×Dic3—41no—
111C2×Dic3—82no—
112C3×C8—22no—
113C3⋊C8—41no—
114#41—21no—
115#41—21no—
116#41Dic612yes—
117#41C2×Dic312yes—
118C2×Q8D1211yescharacteristic
119C2×D4—21no—
120C2×D4D1211yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—32no—
124C22×C4—62no—
125C22×C4—61no—
126C22×C4C22×D311yescharacteristic
127M16—21no—
128M16—61no—
129M16—61no—
130M16C2×Dic312yescomplemented
131C2×C8—21no—
132C2×C8—61no—
133C4⋊C4—31no—
134C4⋊C4—31no—
135C4⋊C4—62no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—124no—
141C22⋊C4—62no—
142C22⋊C4—61no—
143C42—21no—
144C42—62no—
145C42—61no—
146C42C2×Dic312yescomplemented
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—22no—
150C22×C3—41no—
151C22×C3—41no—
152C22×C3C4×D411yescharacteristic
153C22×C3#12011yescharacteristic
154C22×C3C22⋊Q811yescharacteristic
155C3×C4—24no—
156C3×C4—22no—
157C3×C4C2×C4⋊C411yescharacteristic
158C3×C4C4×D411yescharacteristic
159C3×C4C22⋊Q811yescharacteristic
160C3×C4#12311yescharacteristic
161Dic3—42no—
162Dic3—41no—
163Dic3—41no—
164Dic3—81no—
165C23—21no—
166C23D3×D411yescharacteristic
167Q8—21no—
168Q8C4⋊Dic312yescomplemented
169D4—22no—
170D4—22no—
171D4—21no—
172D4C4⋊Dic312yescomplemented
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—62no—
182C2×C4—122no—
183C2×C4—122no—
184C2×C4—121no—
185C2×C4—242no—
186C2×C4C2×Dic611yescharacteristic
187C2×C4C2×D1211yescharacteristic
188C2×C4D3×D411yescharacteristic
189C2×C4#23711yescharacteristic
190C2×C4#23711yescharacteristic
191C2×C4C22×Dic311yescharacteristic
192C8—22no—
193C8—121no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—42no—
197C2×C3#38511yescharacteristic
198C2×C3#45311yescharacteristic
199C2×C3#45611yescharacteristic
200C22—21no—
201C22—21no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22#87811yescharacteristic
206C22#88211yescharacteristic
207C22D4×Dic311yescharacteristic
208C4—24no—
209C4—22no—
210C4—122no—
211C4—121no—
212C4—121no—
213C4—241no—
214C4#87811yescharacteristic
215C4#88611yescharacteristic
216C4C2×C4⋊Dic311yescharacteristic
217C4D4×Dic311yescharacteristic
218C3#185911yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2—42no—
222C2#482211yescharacteristic
223C2#482611yescharacteristic
224C2#495011yescharacteristic
subgroupG/H
centerC22#882
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#104C2
socleC22×C3#120
prime radicalC2×C3×#41C22
cosocleC22×C3×C4C23
hypercenterC2×#104D3
hypocenterC3#1859
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
4. #5121C2semidirect
5. #4935C2semidirect
6. #4554C2semidirect
7. #4523C2semidirect
22. C3×#104C4semidirect
48. C3×M16C2×C4semidirect
51. C3×C42C2×C4semidirect
82. #104Dic3semidirect
88. C3×Q8C4⋊C4semidirect
92. C3×D4C4⋊C4semidirect
130. M16C2×Dic3semidirect
146. C42C2×Dic3semidirect
168. Q8C4⋊Dic3semidirect
172. D4C4⋊Dic3semidirect
218. C3#1859semidirect

Maximal subgroups

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

subgroupindex
#58272
C2×C3×#1042
#52322
#51212
#49352
#45542
#45232
#18593