Finite Group Databasebeta

← #69538

Subgroups — #69538

Order 384.

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

normal subgroups only

This group has 386 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
1C2×#929C211yesmaximal, characteristic
2#5486C211yesmaximal, characteristic
3C3×#381C211yesmaximal, characteristic, complemented
4#5211C211yesmaximal, characteristic, complemented
5C2×#806C211yesmaximal, characteristic
6#4527C211yesmaximal, characteristic, complemented
7#4526C211yesmaximal, characteristic, complemented
8#1890—31nomaximal, Sylow
9C23×C3×C4C2211yescharacteristic
10C22×#67C2211yescharacteristic
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4C2211yescharacteristic
13C2×#217C2211yescharacteristic
14C2×#217C2211yescharacteristic
15C2×#217C2211yescharacteristic
16#929—22no—
17#929C414yescomplemented
18C4×#67—22no—
19C2×(C2×D3)⋊C4—21no—
20C2×Dic3⋊C4—21no—
21#67⋊C4—22no—
22D3×C22⋊C4—21no—
23#877—21no—
24C3×#95—22no—
25C3⋊#95—22no—
26#806—22no—
27C2×#123—31no—
28#518—31no—
29C2×#99—31no—
30#381—31no—
31#381D311yescharacteristic, complemented
32#342—31no—
33#342—31no—
34C24×C3C2311yescharacteristic
35C23×D3D411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4—21no—
41C22×C3×C4—21no—
42C22×C3×C4—21no—
43C22×C3×C4C2×C412yescomplemented
44C22×C3×C4D412yes—
45C2×#67—21no—
46C2×#67—21no—
47C2×#67—21no—
48C2×#67—21no—
49C2×#67C2×C412yes—
50C2×#67D412yes—
51C22×Dic3—21no—
52C22×Dic3—21no—
53C22×Dic3D411yescharacteristic
54C22×Dic3D412yes—
55C2×C4×D3—22no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C22⋊C4—21no—
60#217—21no—
61#217—21no—
62#217—21no—
63#217—21no—
64#217—21no—
65#217C2×C412yescomplemented
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—24no—
68Dic3⋊C4—22no—
69Dic3⋊C4—24no—
70C4×Dic3—22no—
71C22×D4—31no—
72C23×C4C2×D311yescharacteristic
73#123—34no—
74#123—62no—
75C4×D4—62no—
76C4×D4—62no—
77#117—61no—
78C2×C4⋊C4—61no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—61no—
83C2×C22⋊C4—61no—
84C2×C22⋊C4C2×D311yescharacteristic
85C2×C22⋊C4C2×D311yescharacteristic
86#99—62no—
87#95—22no—
88#95—62no—
89C23×C3—42no—
90C23×C3C22×C411yescharacteristic
91C23×C3C2×D411yescharacteristic
92C23×C3C2×D411yescharacteristic
93C23×C3C2×D411yescharacteristic
94C23×C3C2×D411yescharacteristic
95C23×C3#4111yescharacteristic
96C23×C3#4111yescharacteristic
97C22×D3—21no—
98C22×D3—21no—
99C22×D3—42no—
100C22×D3C22⋊C412yes—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4—42no—
110C2×C3×C4—42no—
111C2×C3×C4—41no—
112#67—22no—
113#67—22no—
114#67—22no—
115#67—22no—
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—42no—
121C2×Dic3—42no—
122C2×Dic3—42no—
123C2×Dic3—41no—
124C2×Dic3C22⋊C412yes—
125C4×D3—82no—
126C24—31no—
127C24C22×D311yescharacteristic
128C2×D4—32no—
129C2×D4—32no—
130C2×D4—61no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—61no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—62no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4C4×D312yescomplemented
147C22×C4#6712yes—
148C4⋊C4—64no—
149C4⋊C4—62no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—32no—
155C22⋊C4—64no—
156C22⋊C4—62no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—61no—
160C22⋊C4—61no—
161C22⋊C4—61no—
162C42—62no—
163C22×C3—22no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3C2×C22⋊C411yescharacteristic
171C22×C3C4×D411yescharacteristic
172C22×C3C4×D411yescharacteristic
173C22×C3#12011yescharacteristic
174C22×C3#12111yescharacteristic
175C22×C3#12111yescharacteristic
176C22×C3#12311yescharacteristic
177C2×D3—22no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—81no—
181C3×C4—42no—
182C3×C4—42no—
183C3×C4—81no—
184Dic3—22no—
185Dic3—42no—
186Dic3—42no—
187Dic3—81no—
188C23—32no—
189C23—42no—
190C23—61no—
191C23—61no—
192C23—122no—
193C23C2×C4×D311yescharacteristic
194C23#23511yescharacteristic
195C23D3×D411yescharacteristic
196C23D3×D411yescharacteristic
197C23D3×D411yescharacteristic
198C23#23711yescharacteristic
199C23C2×#6711yescharacteristic
200D4—62no—
201D4—62no—
202D4—62no—
203D4—62no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—32no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4—121no—
224C2×C4—242no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—21no—
231C2×C3#38611yescharacteristic
232C2×C3#45812yes—
233D3—42no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—62no—
242C22—122no—
243C22—122no—
244C22—241no—
245C22D3×C22⋊C411yescharacteristic
246C22#67⋊C411yescharacteristic
247C22#88311yescharacteristic
248C22#88411yescharacteristic
249C22C4×#6711yescharacteristic
250C22#93711yescharacteristic
251C22#93811yescharacteristic
252C4—42no—
253C4—42no—
254C4—62no—
255C4—81no—
256C4—122no—
257C4—122no—
258C4—241no—
259C3#189011yescharacteristic, complemented, Sylow
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—122no—
267C2#495611yescharacteristic
268C2#505312yes—
subgroupG/H
centerC22#937
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#381C2
socleC22×C3#120
prime radicalC22×#67C22
cosocleC24×C3C23
hypercenter#381D3
hypocenterC3#1890
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
3. C3×#381C2semidirect
4. #5211C2semidirect
6. #4527C2semidirect
7. #4526C2semidirect
17. #929C4semidirect
31. #381D3semidirect
43. C22×C3×C4C2×C4semidirect
65. #217C2×C4semidirect
146. C22×C4C4×D3semidirect
259. C3#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#9292
#54862
C3×#3812
#52112
C2×#8062
#45272
#45262
#18903