Finite Group Databasebeta

← #70025

Subgroups — #70025

Order 384.

278 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 65 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#5483C211yesmaximal, characteristic, complemented
2C2×C3×#99C211yesmaximal, characteristic
3C2×#834C211yesmaximal, characteristic
4C2×#834C211yesmaximal, characteristic
5#4527C211yesmaximal, characteristic, complemented
6#4527C211yesmaximal, characteristic, complemented
7#4526C211yesmaximal, characteristic, complemented
8#2032—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×#217C2211yescharacteristic
13C2×#217C2211yescharacteristic
14C2×#217C2211yescharacteristic
15C2×#217C2211yescharacteristic
16#924—22no—
17#878—22no—
18#878—22no—
19C3×#99—22no—
20#834—22no—
21#834—22no—
22#543—31no—
23C2×#99—31no—
24C2×#99—31no—
25C2×#99D311yescharacteristic
26#342—31no—
27#342—31no—
28#342—31no—
29C24×C3Q811yescharacteristic, complemented
30C24×C3C2311yescharacteristic
31C2×C3×D4—21no—
32C2×C3×D4—21no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4Q811yescharacteristic, complemented
40C22×Dic3—21no—
41C22×Dic3—21no—
42C22×Dic3—21no—
43C22×Dic3—21no—
44C2×Dic6—22no—
45C3×C22⋊C4—21no—
46C3×C22⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4D412yes—
49#217—21no—
50#217—21no—
51#217—21no—
52#217—21no—
53#217—21no—
54#217—21no—
55#217D412yes—
56#217D412yes—
57C4⋊Dic3—22no—
58Dic3⋊C4—24no—
59C22×D4C2×D311yescharacteristic
60C22⋊Q8—62no—
61C22⋊Q8—62no—
62C22⋊Q8—62no—
63C2×C22⋊C4—31no—
64C2×C22⋊C4—31no—
65C2×C22⋊C4—31no—
66C2×C22⋊C4—31no—
67C2×C22⋊C4C2×D311yescharacteristic
68C2×C22⋊C4C2×D311yescharacteristic
69#99—22no—
70#99—62no—
71#99—62no—
72C23×C3—21no—
73C23×C3—21no—
74C23×C3—21no—
75C23×C3—81no—
76C23×C3—81no—
77C23×C3C2×D411yescharacteristic
78C23×C3C2×D411yescharacteristic
79C23×C3C2×D411yescharacteristic
80C23×C3C2×Q811yescharacteristic
81C23×C3#4111yescharacteristic
82C23×C3#4111yescharacteristic
83C23×C3#4111yescharacteristic
84C3×D4—41no—
85C3×D4—41no—
86C3×D4—41no—
87C3×D4—41no—
88C2×C3×C4—22no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—41no—
93C2×C3×C4—81no—
94C2×C3×C4—81no—
95C2×Dic3—22no—
96C2×Dic3—22no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3—81no—
100C2×Dic3—81no—
101C2×Dic3—81no—
102C2×Dic3—81no—
103Dic6—82no—
104C24Dic611yescharacteristic, complemented
105C24C22×D311yescharacteristic
106C2×Q8—62no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—61no—
116C22×C4—61no—
117C22×C4—61no—
118C22×C4—61no—
119C22×C4Dic611yescharacteristic, complemented
120C4⋊C4—64no—
121C4⋊C4—62no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—32no—
126C22⋊C4—32no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4#6712yes—
134C22×C3—41no—
135C22×C3—41no—
136C22×C3—41no—
137C22×C3—41no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3—81no—
143C22×C3—81no—
144C22×C3—81no—
145C22×C3#12011yescharacteristic
146C22×C3C22⋊Q811yescharacteristic
147C22×C3C22⋊Q811yescharacteristic
148C22×C3C22⋊Q811yescharacteristic
149C22×C3#12411yescharacteristic
150C22×C3#12411yescharacteristic
151C22×C3#12411yescharacteristic
152C3×C4—42no—
153C3×C4—41no—
154C3×C4—81no—
155Dic3—42no—
156Dic3—42no—
157Dic3—81no—
158Dic3—81no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—81no—
163C23—81no—
164C23C2×Dic611yescharacteristic
165C23#23511yescharacteristic
166C23D3×D411yescharacteristic
167C23D3×D411yescharacteristic
168C23#23711yescharacteristic
169C23#23711yescharacteristic
170C23C2×#6711yescharacteristic
171Q8—242no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—241no—
188C2×C4—241no—
189C2×C4—241no—
190C2×C4—241no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—81no—
199C2×C3#39311yescharacteristic
200C22—41no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22#87811yescharacteristic
212C22#87811yescharacteristic
213C22#88511yescharacteristic
214C22#88511yescharacteristic
215C22#92411yescharacteristic
216C22#93611yescharacteristic
217C22#93711yescharacteristic
218C4—42no—
219C4—41no—
220C4—81no—
221C4—122no—
222C4—122no—
223C4—241no—
224C4—241no—
225C3#203211yescharacteristic, complemented, Sylow
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—81no—
234C2#494711yescharacteristic
subgroupG/H
centerC2#4947
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC2×C3×#99C2
socleC2×C3#393
prime radicalC22×C3×D4C22
hypercenterC2×#99D3
hypocenterC3#2032
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. #5483C2semidirect
5. #4527C2semidirect
6. #4527C2semidirect
7. #4526C2semidirect
29. C24×C3Q8semidirect
39. C22×C3×C4Q8semidirect
104. C24Dic6semidirect
119. C22×C4Dic6semidirect
225. C3#2032semidirect

Maximal subgroups

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

subgroupindex
#54832
C2×C3×#992
C2×#8342
#45272
#45262
#20323