Finite Group Databasebeta

← #80449

Subgroups — #80449

Order 384.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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×#916C212yesmaximal, complemented
2#5726C211yesmaximal, characteristic, complemented
3#5555C211yesmaximal, characteristic, complemented
4C3×#461C211yesmaximal, characteristic, complemented
5#5173C212yesmaximal, complemented
6#5092C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4813C212yesmaximal, complemented
11#4808C212yesmaximal, complemented
12#3283—31nomaximal, Sylow
13C2×#239C2212yes—
14C2×#237C2212yes—
15C2×C3×SD16C2212yes—
16C3×C4⋊Q8C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#946C2212yes—
19C2×#216C2212yes—
20C2×#216C2212yes—
21C2×#213C2211yescharacteristic
22#930—21no—
23C2×C4⋊Dic3—21no—
24#916—28no—
25C2×#204C2212yes—
26C2×D3×C8C2212yes—
27#897C2212yes—
28#895—21no—
29#894—22no—
30D3×C4⋊C4—21no—
31#886—22no—
32C4×D12C2211yescharacteristic
33C4×D12C2211yescharacteristic
34#869C2211yescharacteristic
35C3×C4⋊C8C2211yescharacteristic
36C3×#103C2212yes—
37#832C2211yescharacteristic
38(C2×D3)⋊C8C2212yes—
39#816C2212yes—
40#809C2211yescharacteristic
41#807C2211yescharacteristic
42C4×C3⋊C8C2211yescharacteristic
43C2×#135—32no—
44#547—31no—
45#547—31no—
46#494—31no—
47#467—32no—
48#461—31no—
49#461D311yescharacteristic, complemented
50#452—32no—
51#452—32no—
52C4×D8—31no—
53C8×D4—31no—
54C2×C3×Q8C2312yes—
55C2×C3×D4C2311yescharacteristic
56C22×C3×C4—21no—
57C2×#67—22no—
58C22×Dic3—22no—
59#239—24no—
60#239—42no—
61#237—24no—
62#237—42no—
63C2×D12—22no—
64C2×D12D411yescharacteristic
65C2×D12C2311yescharacteristic
66C2×C4×D3—21no—
67C2×C4×D3—22no—
68C2×C4×D3C2312yes—
69C2×Dic6C2312yes—
70C3×SD16—24no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4—21no—
73C3×C4⋊C4—21no—
74C3×C4⋊C4C2311yescharacteristic
75C3×C22⋊C4—21no—
76C3×C42C2311yescharacteristic
77#216—24no—
78#216—42no—
79#213D414yescomplemented
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4D412yes—
84C4⋊Dic3—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3—22no—
87C4⋊Dic3D411yescharacteristic
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—22no—
90C2×C3⋊C8C2311yescharacteristic
91C2×C3⋊C8C2311yescharacteristic
92#204—24no—
93D3×C8—24no—
94C2×#41—32no—
95C2×#41—32no—
96#135—68no—
97C2×Q16—32no—
98C2×Q16—32no—
99C2×SD16—32no—
100C2×SD16C2×D312yes—
101C2×D8—31no—
102C22×C8—32no—
103C4⋊Q8—31no—
104C4⋊Q8C2×D311yescharacteristic
105#123—62no—
106#123—62no—
107C22⋊Q8—32no—
108C22⋊Q8—32no—
109#121—61no—
110#121—61no—
111C4×D4—31no—
112C4×D4—31no—
113C4×D4C2×D311yescharacteristic
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116#107—31no—
117C4⋊C8C2×D311yescharacteristic
118#103—32no—
119#103C2×D312yes—
120#102—31no—
121#102—31no—
122C22⋊C8—32no—
123C4×C8—31no—
124C23×C3—21no—
125C22×D3—21no—
126C22×D3C2×D412yes—
127C3×Q8—22no—
128C3×Q8—22no—
129C3×D4—21no—
130C3×D4C2×D412yes—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137#67—44no—
138C2×Dic3—21no—
139C2×Dic3—22no—
140C2×Dic3—44no—
141C2×Dic3C2×D412yes—
142D12—22no—
143D12—21no—
144D12—44no—
145D12C2×D412yes—
146C4×D3—22no—
147C4×D3—22no—
148C4×D3—44no—
149C4×D3—42no—
150Dic6—22no—
151Dic6—22no—
152C3×C8—22no—
153C3⋊C8—21no—
154C3⋊C8C2×D412yes—
155#41—64no—
156#41—64no—
157#41—122no—
158#41—122no—
159C2×Q8—32no—
160C2×Q8C22×D312yes—
161C2×D4—31no—
162C2×D4—31no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4C22×D311yescharacteristic
166C22×C4—21no—
167C22×C4—32no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—61no—
171Q16—64no—
172Q16—122no—
173SD16—24no—
174SD16—64no—
175D8—34no—
176C2×C8—31no—
177C2×C8—31no—
178C2×C8—64no—
179C2×C8C22×D312yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—31no—
183C4⋊C4—31no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4C22×D311yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—32no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C22⋊C4—61no—
194C42C22×D311yescharacteristic
195C22×C3—22no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3C22×D411yescharacteristic
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—22no—
202C2×D3—22no—
203C2×D3—41no—
204C2×D3—41no—
205C3×C4—21no—
206C3×C4—42no—
207C3×C4—41no—
208C3×C4#13512yescomplemented
209C3×C4C22×D411yescharacteristic
210C3×C4H(5,𝔽2)11yescharacteristic
211Dic3—22no—
212Dic3—42no—
213Dic3—41no—
214C23—21no—
215C23—32no—
216C23—61no—
217Q8—22no—
218Q8—22no—
219Q8—62no—
220Q8—62no—
221D4—21no—
222D4—32no—
223D4—62no—
224D4—61no—
225D4—124no—
226D4—124no—
227D4D3×D412yes—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—32no—
231C2×C4—42no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—61no—
235C2×C4—62no—
236C2×C4—124no—
237C2×C4—124no—
238C2×C4—122no—
239C2×C4D3×D411yescharacteristic
240C2×C4D3×D411yescharacteristic
241C2×C4C23×D311yescharacteristic
242C8—22no—
243C8—32no—
244C8—61no—
245C2×C3—42no—
246C2×C3D4211yescharacteristic
247C2×C3C2×#13511yescharacteristic
248C2×C3#57811yescharacteristic
249D3—42no—
250D3—42no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—62no—
255C22—62no—
256C22—62no—
257C22—62no—
258C22—121no—
259C22—121no—
260C22C2×D3×D411yescharacteristic
261C4—21no—
262C4—41no—
263C4—42no—
264C4—62no—
265C4—121no—
266C4—122no—
267C4#91612yescomplemented
268C4C2×D3×D411yescharacteristic
269C4#100911yescharacteristic
270C3#328311yescharacteristic, complemented, Sylow
271C2—42no—
272C2—122no—
273C2—122no—
274C2#560911yescharacteristic
275C2C2×#91611yescharacteristic
276C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
hypercenter#461D3
hypocenterC3#3283
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#916C2semidirect
2. #5726C2semidirect
3. #5555C2semidirect
4. C3×#461C2semidirect
5. #5173C2semidirect
6. #5092C2semidirect
7. C4×#213C2semidirect
8. #4841C2semidirect
9. #4834C2semidirect
10. #4813C2semidirect
11. #4808C2semidirect
49. #461D3semidirect
79. #213D4semidirect
208. C3×C4#135semidirect
267. C4#916semidirect
270. C3#3283semidirect

Maximal subgroups

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

subgroupindex
C2×#9162
#57262
#55552
C3×#4612
#51732
#50922
C4×#2132
#48412
#48342
#48132
#48082
#32833