Finite Group Databasebeta

← #68758

Subgroups — #68758

Order 384.

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

normal subgroups only

This group has 324 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#5521C211yesmaximal, characteristic
2C2×#823C211yesmaximal, characteristic
3#5121C211yesmaximal, characteristic, complemented
4#4995C211yesmaximal, characteristic, complemented
5#4939C211yesmaximal, characteristic
6C3×#339C211yesmaximal, characteristic, complemented
7#4523C211yesmaximal, characteristic, complemented
8#1856—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C3×#117C2211yescharacteristic
12C2×C3×C42C2211yescharacteristic
13C2×(C2×D3)⋊C4—22no—
14#926C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×#208C2211yescharacteristic
17#893—21no—
18#893—21no—
19D3×C22⋊C4—22no—
20C4×D12—21no—
21C4×D12—21no—
22#823C414yescomplemented
23#818—21no—
24#817—21no—
25#808—21no—
26#807—21no—
27#518—31no—
28#428—31no—
29#428—31no—
30C2×#100—31no—
31#390—31no—
32#339—31no—
33#339D311yescharacteristic, complemented
34C23×D3D412yes—
35C22×C3×C4—21no—
36C22×C3×C4C2311yescharacteristic
37C22×Dic3—21no—
38C2×D12—21no—
39C2×D12—21no—
40C2×D12—21no—
41C2×D12—21no—
42C2×D12C2×C412yes—
43C2×D12Q812yes—
44C2×C4×D3—22no—
45C2×C4×D3—22no—
46C3×M16—21no—
47C3×M16C2×C412yescomplemented
48C2×C3×C8—21no—
49C3×C4⋊C4D411yescharacteristic
50C3×C4⋊C4D411yescharacteristic
51C3×C22⋊C4—21no—
52C3×C42—21no—
53C3×C42—21no—
54C3×C42—21no—
55#217—21no—
56(C2×D3)⋊C4—22no—
57(C2×D3)⋊C4—22no—
58(C2×D3)⋊C4—44no—
59C4⋊Dic3—21no—
60C4⋊Dic3—21no—
61C4⋊Dic3D411yescharacteristic
62C4⋊Dic3D411yescharacteristic
63C4×Dic3—21no—
64#208—21no—
65#208C2×C412yescomplemented
66C2×C3⋊C8—21no—
67C22×D4—31no—
68C2×M16—31no—
69C2×M16C2×D311yescharacteristic
70C4×D4—61no—
71C4×D4—61no—
72C4×D4—61no—
73C4×D4—61no—
74#117—31no—
75#117C2×D311yescharacteristic
76C2×C4⋊C4—31no—
77C2×C22⋊C4—62no—
78C2×C22⋊C4—62no—
79C2×C42C2×D311yescharacteristic
80#107—61no—
81#107—61no—
82#106—61no—
83#106—61no—
84#100—34no—
85C23×C3C2×D411yescharacteristic
86C22×D3—22no—
87C22×D3—22no—
88C22×D3—42no—
89C22×D3—42no—
90C22×D3C4⋊C414yes—
91C2×C3×C4—22no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4C22×C411yescharacteristic
97C2×C3×C4C2×D411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×Q811yescharacteristic
100C2×C3×C4#4111yescharacteristic
101C2×C3×C4#4111yescharacteristic
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—42no—
105C2×Dic3—41no—
106D12—22no—
107D12—22no—
108D12—22no—
109D12—22no—
110C4×D3—82no—
111C4×D3—82no—
112C3×C8—22no—
113C3⋊C8—22no—
114C24—32no—
115C2×D4—32no—
116C2×D4—32no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4—61no—
121C22×C4—21no—
122C22×C4—62no—
123C22×C4—62no—
124C22×C4—61no—
125C22×C4C22×D311yescharacteristic
126M16—21no—
127M16—32no—
128M16—61no—
129M16C4×D312yescomplemented
130C2×C8—21no—
131C2×C8—61no—
132C4⋊C4—31no—
133C4⋊C4—31no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4D1211yescharacteristic
137C4⋊C4D1211yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—62no—
140C22⋊C4—62no—
141C22⋊C4—61no—
142C22⋊C4—124no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—61no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3C2×C4⋊C411yescharacteristic
150C22×C3#12011yescharacteristic
151C22×C3#12311yescharacteristic
152C2×D3—24no—
153C2×D3—42no—
154C2×D3—42no—
155C2×D3—44no—
156C2×D3—82no—
157C3×C4—42no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4C4×D411yescharacteristic
161C3×C4C4×D411yescharacteristic
162C3×C4C22⋊Q811yescharacteristic
163C3×C4C22⋊Q811yescharacteristic
164Dic3—41no—
165Dic3—41no—
166Dic3—81no—
167C23—34no—
168C23—62no—
169C23—62no—
170C23—122no—
171C23—122no—
172C23D3×D411yescharacteristic
173D4—62no—
174D4—62no—
175D4—62no—
176D4—62no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—122no—
185C2×C4—121no—
186C2×C4—242no—
187C2×C4—242no—
188C2×C4C2×C4×D311yescharacteristic
189C2×C4C2×D1211yescharacteristic
190C2×C4#23511yescharacteristic
191C2×C4D3×D411yescharacteristic
192C2×C4#23711yescharacteristic
193C2×C4D3×Q811yescharacteristic
194C8—22no—
195C8—62no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3#38511yescharacteristic
199C2×C3#45311yescharacteristic
200C2×C3#45511yescharacteristic
201D3—44no—
202C22—21no—
203C22—21no—
204C22—64no—
205C22—124no—
206C22—122no—
207C22—122no—
208C22—242no—
209C22#88211yescharacteristic
210C22#88311yescharacteristic
211C22D3×C4⋊C411yescharacteristic
212C4—42no—
213C4—41no—
214C4—41no—
215C4—121no—
216C4—121no—
217C4—241no—
218C4C4×D1211yescharacteristic
219C4#67⋊C411yescharacteristic
220C4#89611yescharacteristic
221C4#89711yescharacteristic
222C3#185611yescharacteristic, complemented, Sylow
223C2—21no—
224C2—21no—
225C2—124no—
226C2#466811yescharacteristic
227C2#482211yescharacteristic
228C2#482511yescharacteristic
subgroupG/H
centerC22#882
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
prime radicalC22×D12C22
cosocleC22×C3×C4C23
hypercenter#339D3
hypocenterC3#1856
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. #5121C2semidirect
4. #4995C2semidirect
6. C3×#339C2semidirect
7. #4523C2semidirect
22. #823C4semidirect
33. #339D3semidirect
47. C3×M16C2×C4semidirect
65. #208C2×C4semidirect
129. M16C4×D3semidirect
222. C3#1856semidirect

Maximal subgroups

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

subgroupindex
#55212
C2×#8232
#51212
#49952
#49392
C3×#3392
#45232
#18563