Finite Group Databasebeta

← #70313

Subgroups — #70313

Order 384.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#5519C211yesmaximal, characteristic, complemented
2C3×#418C211yesmaximal, characteristic
3#5221C211yesmaximal, characteristic
4#5215C211yesmaximal, characteristic
5#4555C211yesmaximal, characteristic, complemented
6#4554C211yesmaximal, characteristic, complemented
7#4532C211yesmaximal, characteristic, complemented
8#2027—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C3×#117C2211yescharacteristic
12C2×#217—22no—
13#926C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15C2×C4×Dic3C2211yescharacteristic
16C2×#208C2211yescharacteristic
17#890—21no—
18#888—21no—
19#878—22no—
20#870—21no—
21#869—21no—
22C3×#102—21no—
23C3×#102—21no—
24#832—21no—
25#832—21no—
26#544—31no—
27#418—31no—
28#418D311yescharacteristic
29#390—31no—
30#343—31no—
31#339—31no—
32#339—31no—
33C24×C3—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4Q812yescomplemented
40C22×C3×C4C2311yescharacteristic
41C22×Dic3—21no—
42C22×Dic3D412yes—
43C2×Dic6—22no—
44C3×M16—21no—
45C3×M16—21no—
46C2×C3×C8—21no—
47C3×C4⋊C4D411yescharacteristic
48C3×C4⋊C4D411yescharacteristic
49C3×C22⋊C4—21no—
50C3×C42—21no—
51#217—21no—
52#217—44no—
53C4⋊Dic3—22no—
54C4⋊Dic3—21no—
55C4⋊Dic3—21no—
56C4⋊Dic3D411yescharacteristic
57C4⋊Dic3D411yescharacteristic
58Dic3⋊C4—22no—
59Dic3⋊C4—22no—
60C4×Dic3—21no—
61C4×Dic3—21no—
62C4×Dic3—21no—
63#208—21no—
64#208—21no—
65C2×C3⋊C8—21no—
66C22×D4C2×D311yescharacteristic
67C2×M16—31no—
68C2×M16C2×D311yescharacteristic
69C4⋊Q8—61no—
70C4⋊Q8—61no—
71#125—61no—
72#125—61no—
73C22⋊Q8—62no—
74#117—31no—
75#117C2×D311yescharacteristic
76C2×C4⋊C4—31no—
77C2×C22⋊C4—62no—
78C2×C42—31no—
79#102—21no—
80#102—21no—
81#102—61no—
82#102—61no—
83C23×C3—22no—
84C23×C3—42no—
85C23×C3—42no—
86C23×C3—41no—
87C23×C3—41no—
88C23×C3C2×D411yescharacteristic
89C3×D4—22no—
90C3×D4—22no—
91C3×D4—41no—
92C3×D4—41no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4C2×D411yescharacteristic
96C2×C3×C4C2×D411yescharacteristic
97C2×C3×C4C2×Q811yescharacteristic
98C2×C3×C4#4111yescharacteristic
99C2×C3×C4#4111yescharacteristic
100C2×C3×C4#4111yescharacteristic
101C2×Dic3—22no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—22no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—42no—
108Dic6—82no—
109C3×C8—41no—
110C3⋊C8—41no—
111C24—21no—
112C2×Q8—62no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4Dic612yescomplemented
119C22×C4—32no—
120C22×C4—61no—
121C22×C4C22×D311yescharacteristic
122M16—21no—
123M16—21no—
124M16—61no—
125M16—61no—
126C2×C8—21no—
127C2×C8—61no—
128C4⋊C4—31no—
129C4⋊C4—31no—
130C4⋊C4—62no—
131C4⋊C4—62no—
132C4⋊C4—62no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4#6711yescharacteristic
136C4⋊C4#6711yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—61no—
139C22⋊C4—124no—
140C42—21no—
141C42—61no—
142C42—61no—
143C42—61no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3—44no—
147C22×C3—42no—
148C22×C3—82no—
149C22×C3—81no—
150C22×C3—81no—
151C22×C3#12011yescharacteristic
152C22×C3C22⋊Q811yescharacteristic
153C22×C3#12411yescharacteristic
154C3×C4—41no—
155C3×C4—41no—
156C3×C4C22⋊Q811yescharacteristic
157C3×C4C22⋊Q811yescharacteristic
158C3×C4#12411yescharacteristic
159C3×C4#12411yescharacteristic
160Dic3—42no—
161Dic3—41no—
162Dic3—42no—
163Dic3—41no—
164C23—22no—
165C23—42no—
166C23—42no—
167C23—41no—
168C23—41no—
169C23D3×D411yescharacteristic
170Q8—242no—
171D4—22no—
172D4—22no—
173D4—41no—
174D4—41no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—62no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—62no—
181C2×C4—121no—
182C2×C4—121no—
183C2×C4—122no—
184C2×C4C2×Dic611yescharacteristic
185C2×C4#23511yescharacteristic
186C2×C4D3×D411yescharacteristic
187C2×C4#23711yescharacteristic
188C2×C4#23711yescharacteristic
189C2×C4C2×#6711yescharacteristic
190C8—41no—
191C8—121no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—82no—
195C2×C3#39311yescharacteristic
196C2×C3#45311yescharacteristic
197C2×C3#45411yescharacteristic
198C22—21no—
199C22—21no—
200C22—44no—
201C22—42no—
202C22—82no—
203C22—81no—
204C22—81no—
205C22#87811yescharacteristic
206C22#88511yescharacteristic
207C22#93711yescharacteristic
208C4—41no—
209C4—41no—
210C4—122no—
211C4—122no—
212C4—121no—
213C4—121no—
214C4#87811yescharacteristic
215C4#88511yescharacteristic
216C4#92411yescharacteristic
217C4#93611yescharacteristic
218C3#202711yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2—82no—
222C2#494711yescharacteristic
223C2#519811yescharacteristic
224C2#520111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#418C2
socleC22×C3#120
prime radicalC22×C3×D4C22
hypercenter#418D3
hypocenterC3#2027
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #5519C2semidirect
5. #4555C2semidirect
6. #4554C2semidirect
7. #4532C2semidirect
39. C2×C3×D4Q8semidirect
118. C2×D4Dic6semidirect
218. C3#2027semidirect

Maximal subgroups

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

subgroupindex
#55192
C3×#4182
#52212
#52152
#45552
#45542
#45322
#20273