Finite Group Databasebeta

← #68771

Subgroups — #68771

Order 384.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#5523C211yesmaximal, characteristic
2#5133C211yesmaximal, characteristic
3#4995C211yesmaximal, characteristic, complemented
4#4935C211yesmaximal, characteristic, complemented
5C2×#805C211yesmaximal, characteristic
6C3×#339C211yesmaximal, characteristic, complemented
7#4554C211yesmaximal, characteristic, complemented
8#1859—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C3×#117C2211yescharacteristic
12C2×C3×C42C2211yescharacteristic
13C2×C4⋊Dic3—22no—
14C2×C4⋊Dic3C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×#208C2211yescharacteristic
17#892—21no—
18D3×C4⋊C4—21no—
19#67⋊C4—22no—
20C4×D12—21no—
21C4×Dic6—21no—
22#821—21no—
23#816—21no—
24#808—21no—
25#807—21no—
26#805C414yescomplemented
27#520—31no—
28#428—31no—
29C2×#104—31no—
30#417—31no—
31#384—31no—
32#343—31no—
33#339D311yescharacteristic, complemented
34C22×C3×C4—21no—
35C22×C3×C4C2311yescharacteristic
36C2×#67—21no—
37C22×Dic3—21no—
38C22×Dic3D412yes—
39#235—21no—
40#235—21no—
41#235C2×C412yes—
42#235Q812yes—
43C2×D12D411yescharacteristic
44C2×C4×D3—21no—
45C2×C4×D3—22no—
46C2×Dic6D411yescharacteristic
47C3×M16—21no—
48C3×M16—21no—
49C2×C3×C8—21no—
50C3×C4⋊C4D411yescharacteristic
51C3×C4⋊C4D411yescharacteristic
52C3×C22⋊C4—21no—
53C3×C42—21no—
54C3×C42—21no—
55C3×C42C2×C412yescomplemented
56(C2×D3)⋊C4—22no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—21no—
60C4⋊Dic3—21no—
61C4⋊Dic3—44no—
62Dic3⋊C4—22no—
63C4×Dic3—22no—
64#208—21no—
65#208C2×C412yescomplemented
66C2×C3⋊C8—21no—
67C2×#41—31no—
68C2×M16—31no—
69C2×M16C2×D311yescharacteristic
70C4×Q8—61no—
71C4×D4—61no—
72C4×D4—62no—
73#117—61no—
74#117C2×D311yescharacteristic
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#104—34no—
83#103—61no—
84#102—61no—
85C23×C3C2×D411yescharacteristic
86C22×D3—21no—
87C2×C3×C4—22no—
88C2×C3×C4—22no—
89C2×C3×C4—22no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95C2×C3×C4C2×Q811yescharacteristic
96C2×C3×C4#4111yescharacteristic
97C2×C3×C4#4111yescharacteristic
98#67—22no—
99#67—22no—
100C2×Dic3—21no—
101C2×Dic3—22no—
102C2×Dic3—42no—
103C2×Dic3—42no—
104C2×Dic3—41no—
105D12—21no—
106D12C4⋊C412yescomplemented
107C4×D3—22no—
108C4×D3—22no—
109C4×D3—82no—
110Dic6—21no—
111Dic6C4⋊C412yescomplemented
112C3×C8—41no—
113C3⋊C8—22no—
114#41—32no—
115#41—32no—
116#41—61no—
117#41—61no—
118C2×Q8—31no—
119C2×D4—31no—
120C2×D4—61no—
121C22×C4—21no—
122C22×C4—32no—
123C22×C4—62no—
124C22×C4—61no—
125C22×C4—61no—
126C22×C4C22×D311yescharacteristic
127M16—21no—
128M16—21no—
129M16—32no—
130M16—61no—
131C2×C8—21no—
132C2×C8—61no—
133C4⋊C4—62no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—124no—
139C4⋊C4D1211yescharacteristic
140C4⋊C4D1211yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—62no—
143C42—21no—
144C42—21no—
145C42—62no—
146C42C4×D312yescomplemented
147C22×C3—21no—
148C22×C3—21no—
149C22×C3C4×D411yescharacteristic
150C22×C3#12011yescharacteristic
151C22×C3C22⋊Q811yescharacteristic
152C2×D3—22no—
153C2×D3—41no—
154C2×D3—41no—
155C3×C4—24no—
156C3×C4—41no—
157C3×C4—41no—
158C3×C4C2×C4⋊C411yescharacteristic
159C3×C4C4×D411yescharacteristic
160C3×C4C22⋊Q811yescharacteristic
161C3×C4#12311yescharacteristic
162Dic3—22no—
163Dic3—42no—
164Dic3—81no—
165C23—61no—
166C23D3×D411yescharacteristic
167Q8—32no—
168Q8—61no—
169D4—32no—
170D4—62no—
171D4—62no—
172D4—61no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—61no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—122no—
183C2×C4—122no—
184C2×C4—121no—
185C2×C4—242no—
186C2×C4C2×C4×D311yescharacteristic
187C2×C4C2×D1211yescharacteristic
188C2×C4#23511yescharacteristic
189C2×C4D3×D411yescharacteristic
190C2×C4#23711yescharacteristic
191C2×C4D3×Q811yescharacteristic
192C8—41no—
193C8—62no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3#38511yescharacteristic
197C2×C3#45311yescharacteristic
198C2×C3#45611yescharacteristic
199D3—42no—
200C22—21no—
201C22—21no—
202C22—62no—
203C22—121no—
204C22—121no—
205C22#67⋊C411yescharacteristic
206C22#88211yescharacteristic
207C22#89611yescharacteristic
208C4—24no—
209C4—41no—
210C4—41no—
211C4—62no—
212C4—122no—
213C4—241no—
214C4C4×D1211yescharacteristic
215C4#88311yescharacteristic
216C4D3×C4⋊C411yescharacteristic
217C4#89711yescharacteristic
218C3#185911yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2—122no—
222C2#466811yescharacteristic
223C2#482211yescharacteristic
224C2#482611yescharacteristic
subgroupG/H
centerC22#882
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
prime radicalC2×#235C22
cosocleC22×C3×C4C23
hypercenter#339D3
hypocenterC3#1859
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
3. #4995C2semidirect
4. #4935C2semidirect
6. C3×#339C2semidirect
7. #4554C2semidirect
26. #805C4semidirect
33. #339D3semidirect
55. C3×C42C2×C4semidirect
65. #208C2×C4semidirect
106. D12C4⋊C4semidirect
111. Dic6C4⋊C4semidirect
146. C42C4×D3semidirect
218. C3#1859semidirect

Maximal subgroups

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

subgroupindex
#55232
#51332
#49952
#49352
C2×#8052
C3×#3392
#45542
#18593