Finite Group Databasebeta

← #79239

Subgroups — #79239

Order 384.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#5517C211yesmaximal, characteristic, complemented
2C2×#893C211yesmaximal, characteristic
3C3×#428C211yesmaximal, characteristic, complemented
4#5131C211yesmaximal, characteristic
5#5118C211yesmaximal, characteristic, complemented
6#5001C211yesmaximal, characteristic
7#4964C211yesmaximal, characteristic, complemented
8#4891C212yesmaximal, complemented
9#4890C212yesmaximal, complemented
10#4871C212yesmaximal, complemented
11#4870C212yesmaximal, complemented
12#3077—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3—21no—
15C2×C3×M16C2211yescharacteristic
16C3×#117C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4—21no—
19#926C2211yescharacteristic
20C2×C4×Dic3C2211yescharacteristic
21C2×#208C2211yescharacteristic
22#893—21no—
23#893—21no—
24#893C2212yes—
25#893C2212yes—
26#890C2211yescharacteristic, complemented
27#890C2211yescharacteristic, complemented
28#888C2211yescharacteristic, complemented
29#888C2211yescharacteristic, complemented
30#878—21no—
31#878—21no—
32C4×Dic6—21no—
33C4×Dic6—21no—
34C3×#107C2212yescomplemented
35C3×#106C2212yescomplemented
36#821C2212yes—
37#821C2212yes—
38Dic3⋊C8C2212yescomplemented
39Dic3⋊C8C2212yescomplemented
40#809C2212yes—
41#809C2212yes—
42#808C2212yescomplemented
43#807C2212yescomplemented
44#533—31no—
45C2×C4×D4—31no—
46#478—32no—
47#478—32no—
48#476—32no—
49#474—32no—
50#428—31no—
51#428D311yescharacteristic, complemented
52#423—31no—
53#418—31no—
54#418—31no—
55C23×D3—21no—
56C22×C3×C4—21no—
57C22×C3×C4C2311yescharacteristic
58C22×Dic3D411yescharacteristic
59C22×Dic3D411yescharacteristic
60C2×D12—21no—
61C2×D12—22no—
62C2×D12Q814yescomplemented
63C2×D12C2312yes—
64C2×C4×D3—22no—
65C2×C4×D3—22no—
66C2×C4×D3—44no—
67C2×Dic6—21no—
68C2×Dic6—21no—
69C3×M16—22no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2312yes—
73C3×C4⋊C4C2311yescharacteristic
74C3×C4⋊C4C2311yescharacteristic
75C3×C22⋊C4—21no—
76C3×C42—21no—
77#217—21no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80C4⋊Dic3—21no—
81C4⋊Dic3—21no—
82C4⋊Dic3C2311yescharacteristic
83C4⋊Dic3C2311yescharacteristic
84Dic3⋊C4—21no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88C4×Dic3—21no—
89C4×Dic3D411yescharacteristic
90C4×Dic3D411yescharacteristic
91C4×Dic3C2311yescharacteristic
92C4×Dic3C2311yescharacteristic
93#208—22no—
94C2×C3⋊C8C2312yes—
95C22×D4—31no—
96C23×C4—61no—
97C2×M16—31no—
98C2×M16C2×D311yescharacteristic
99C4⋊Q8—31no—
100C4⋊Q8—31no—
101#125—31no—
102#125—31no—
103C22⋊Q8—61no—
104C22⋊Q8—61no—
105C4×Q8—61no—
106C4×Q8—61no—
107C4×D4—32no—
108C4×D4—32no—
109C4×D4—61no—
110C4×D4—61no—
111#117—31no—
112#117C2×D311yescharacteristic
113C2×C4⋊C4C2×D311yescharacteristic
114C2×C22⋊C4—61no—
115C2×C42—31no—
116#107—32no—
117#107C2×D312yescomplemented
118#106—32no—
119#106C2×D312yescomplemented
120C4⋊C8—32no—
121C4⋊C8—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125#102—32no—
126C23×C3C2×D411yescharacteristic
127C22×D3—24no—
128C22×D3—22no—
129C22×D3—42no—
130C22×D3—42no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—42no—
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×Q812yes—
137C2×C3×C4#4112yes—
138C2×C3×C4C2411yescharacteristic
139C2×Dic3—22no—
140C2×Dic3—21no—
141C2×Dic3—21no—
142C2×Dic3—41no—
143C2×Dic3C2×D411yescharacteristic
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147D12—24no—
148D12—22no—
149D12C2×Q814yes—
150C4×D3—44no—
151C4×D3—44no—
152Dic6—42no—
153Dic6—42no—
154C3×C8—22no—
155C3⋊C8—22no—
156C24—61no—
157C2×Q8—61no—
158C2×Q8—61no—
159C2×D4—34no—
160C2×D4—32no—
161C2×D4—62no—
162C2×D4—61no—
163C22×C4—21no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—62no—
167C22×C4—62no—
168C22×C4—124no—
169C22×C4C22×D311yescharacteristic
170M16—22no—
171M16—62no—
172C2×C8—32no—
173C2×C8C22×D312yes—
174C4⋊C4—21no—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4C22×D311yescharacteristic
184C4⋊C4C22×D311yescharacteristic
185C4⋊C4C22×D312yes—
186C22⋊C4—21no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—61no—
190C42—21no—
191C42—31no—
192C42—31no—
193C42—31no—
194C42—31no—
195C42—61no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3C22⋊Q812yes—
199C22×C3C22×D411yescharacteristic
200C2×D3—24no—
201C2×D3—48no—
202C2×D3—42no—
203C2×D3—42no—
204C3×C4—41no—
205C3×C4—42no—
206C3×C4—41no—
207C3×C4C22⋊Q812yes—
208C3×C4C22×Q811yescharacteristic
209C3×C4C2×#4111yescharacteristic
210Dic3—21no—
211Dic3—21no—
212Dic3—21no—
213Dic3—41no—
214Dic3—41no—
215Dic3AGL(1,ℤ/8)12yes—
216C23—64no—
217C23—62no—
218C23—122no—
219C23—122no—
220C23D3×D411yescharacteristic
221Q8—122no—
222Q8—122no—
223D4—34no—
224D4—64no—
225D4—62no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—42no—
234C2×C4—62no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—124no—
238C2×C4—124no—
239C2×C4—121no—
240C2×C4#23512yes—
241C2×C4D3×D411yescharacteristic
242C2×C4D3×Q812yes—
243C2×C4C23×D311yescharacteristic
244C8—22no—
245C8—62no—
246C2×C3—22no—
247C2×C3C2×C22⋊Q811yescharacteristic
248C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
249C2×C3#57511yescharacteristic
250D3—44no—
251C22—21no—
252C22—21no—
253C22—64no—
254C22—128no—
255C22—122no—
256C22—122no—
257C22#89612yes—
258C22C2×D3×D411yescharacteristic
259C4—32no—
260C4—42no—
261C4—41no—
262C4—41no—
263C4—61no—
264C4—61no—
265C4—61no—
266C4—121no—
267C4—121no—
268C4#89612yes—
269C4C2×#23511yescharacteristic
270C4C2×D3×Q811yescharacteristic
271C3#307711yescharacteristic, complemented, Sylow
272C2—22no—
273C2—124no—
274C2C2×#89611yescharacteristic
275C2D3×AGL(1,ℤ/8)11yescharacteristic
276C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC22×D12C22
hypercenter#428D3
hypocenterC3#3077
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #5517C2semidirect
3. C3×#428C2semidirect
5. #5118C2semidirect
7. #4964C2semidirect
8. #4891C2semidirect
9. #4890C2semidirect
10. #4871C2semidirect
11. #4870C2semidirect
26. #890C22semidirect
27. #890C22semidirect
28. #888C22semidirect
29. #888C22semidirect
34. C3×#107C22semidirect
35. C3×#106C22semidirect
38. Dic3⋊C8C22semidirect
39. Dic3⋊C8C22semidirect
42. #808C22semidirect
43. #807C22semidirect
51. #428D3semidirect
62. C2×D12Q8semidirect
117. #107C2×D3semidirect
119. #106C2×D3semidirect
271. C3#3077semidirect

Maximal subgroups

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

subgroupindex
#55172
C2×#8932
C3×#4282
#51312
#51182
#50012
#49642
#48912
#48902
#48712
#48702
#30773