Finite Group Databasebeta

← #69592

Subgroups — #69592

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#5504C211yesmaximal, characteristic
2C3×#384C211yesmaximal, characteristic, complemented
3#4966C211yesmaximal, characteristic
4#4964C211yesmaximal, characteristic, complemented
5C2×#805C211yesmaximal, characteristic
6#4530C211yesmaximal, characteristic, complemented
7#4523C211yesmaximal, characteristic, complemented
8#1859—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C2×C3×C4⋊C4—22no—
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C42C2211yescharacteristic
14C4×#67—22no—
15#926C2211yescharacteristic
16C2×#208C2211yescharacteristic
17C2×#208C2211yescharacteristic
18#893—21no—
19#892—21no—
20D3×C4⋊C4—21no—
21#887—21no—
22#810—21no—
23#809—21no—
24#808—21no—
25#807—21no—
26#805C414yescomplemented
27#520—31no—
28#428—31no—
29C2×#104—31no—
30#417—31no—
31#384D311yescharacteristic, complemented
32#343—31no—
33#339—31no—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4D412yes—
37C22×C3×C4C2311yescharacteristic
38C2×#67—21no—
39#235—21no—
40#235—21no—
41#235C2×C412yes—
42#235Q812yes—
43C2×D12D411yescharacteristic
44C2×C4×D3—21no—
45C2×C4×D3—22no—
46C2×Dic6D411yescharacteristic
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—44no—
52C3×C42—21no—
53C3×C42C2×C412yescomplemented
54#217—21no—
55(C2×D3)⋊C4—22no—
56C4⋊Dic3D411yescharacteristic
57C4⋊Dic3D411yescharacteristic
58Dic3⋊C4—22no—
59C4×Dic3—22no—
60C4×Dic3—21no—
61#208—21no—
62#208—21no—
63#208—21no—
64#208C2×C412yescomplemented
65C2×C3⋊C8—21no—
66C2×C3⋊C8—21no—
67C2×#41—31no—
68C2×M16—31no—
69C2×M16—31no—
70C4×Q8—61no—
71C4×D4—61no—
72C4×D4—62no—
73#117—31no—
74#117—61no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—61no—
77C2×C4⋊C4C2×D311yescharacteristic
78C2×C4⋊C4C2×D311yescharacteristic
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—22no—
91C2×C3×C4—42no—
92C2×C3×C4—42no—
93C2×C3×C4—41no—
94C2×C3×C4C22×C411yescharacteristic
95C2×C3×C4C2×D411yescharacteristic
96C2×C3×C4C2×D411yescharacteristic
97C2×C3×C4C2×Q811yescharacteristic
98C2×C3×C4#4111yescharacteristic
99C2×C3×C4#4111yescharacteristic
100#67—22no—
101#67—22no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105D12—21no—
106D12C4⋊C412yescomplemented
107C4×D3—22no—
108C4×D3—22no—
109C4×D3—82no—
110Dic6—21no—
111Dic6C4⋊C412yescomplemented
112C3⋊C8—22no—
113C3⋊C8—41no—
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—21no—
123C22×C4—62no—
124C22×C4—61no—
125C22×C4#6712yes—
126C22×C4C22×D311yescharacteristic
127M16—32no—
128M16—61no—
129M16—61no—
130M16—61no—
131C2×C8—61no—
132C2×C8—61no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—31no—
138C4⋊C4—31no—
139C4⋊C4—44no—
140C4⋊C4—62no—
141C22⋊C4—62no—
142C22⋊C4—61no—
143C42—21no—
144C42—62no—
145C42—61no—
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—42no—
157C3×C4—81no—
158C3×C4C2×C4⋊C411yescharacteristic
159C3×C4C4×D411yescharacteristic
160C3×C4C22⋊Q811yescharacteristic
161C3×C4#12311yescharacteristic
162Dic3—22no—
163Dic3—41no—
164Dic3—41no—
165C23—61no—
166C23C2×#6711yescharacteristic
167Q8—32no—
168Q8—61no—
169D4—32no—
170D4—62no—
171D4—62no—
172D4—61no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—61no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—242no—
186C2×C4C2×C4×D311yescharacteristic
187C2×C4#23511yescharacteristic
188C2×C4D3×D411yescharacteristic
189C2×C4D3×D411yescharacteristic
190C2×C4D3×Q811yescharacteristic
191C2×C4#23911yescharacteristic
192C8—62no—
193C8—121no—
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—
205C22C4×#6711yescharacteristic
206C22#93711yescharacteristic
207C22#94611yescharacteristic
208C4—24no—
209C4—42no—
210C4—62no—
211C4—81no—
212C4—121no—
213C4—121no—
214C4D3×C4⋊C411yescharacteristic
215C4#89311yescharacteristic
216C4#89411yescharacteristic
217C4#89611yescharacteristic
218C3#185911yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2—122no—
222C2#498911yescharacteristic
223C2#507711yescharacteristic
224C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#384C2
socleC22×C3#120
prime radicalC2×#235C22
cosocleC22×C3×C4C23
hypercenter#384D3
hypocenterC3#1859
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C3×#384C2semidirect
4. #4964C2semidirect
6. #4530C2semidirect
7. #4523C2semidirect
26. #805C4semidirect
31. #384D3semidirect
53. C3×C42C2×C4semidirect
64. #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
#55042
C3×#3842
#49662
#49642
C2×#8052
#45302
#45232
#18593