Finite Group Databasebeta

← #80315

Subgroups — #80315

Order 384.

430 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 119 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#5716C211yesmaximal, characteristic, complemented
2C4×#237C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5158C212yesmaximal, complemented
5#5152C212yesmaximal, complemented
6#5075C211yesmaximal, characteristic, complemented
7#5007C211yesmaximal, characteristic, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4835C211yesmaximal, characteristic, complemented
10#4779C212yesmaximal, complemented
11#4757C212yesmaximal, complemented
12#3115—31nomaximal, Sylow
13C2×#237—21no—
14C2×#237C2211yescharacteristic
15C2×C3×D8C2212yes—
16C3×#127C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#938—21no—
19#938C2212yes—
20#936—21no—
21D4×Dic3—21no—
22D4×Dic3C2211yescharacteristic
23D4×Dic3C2212yes—
24C2×#214C2212yes—
25C4×#67—21no—
26C2×C4×Dic3—21no—
27#892C2211yescharacteristic
28#887C2211yescharacteristic
29#67⋊C4—21no—
30#877—21no—
31C42×D3C2211yescharacteristic
32#869C2211yescharacteristic
33C4×Dic6C2211yescharacteristic
34C3×C4⋊C8C2211yescharacteristic
35C3×#102C2212yes—
36#832C2212yes—
37#832C2212yes—
38(C2×D3)⋊C8C2212yes—
39#817C2212yes—
40#816C2212yes—
41#815C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45C4×#41—31no—
46#482—32no—
47#475—31no—
48#474—31no—
49#460—31no—
50#459D311yescharacteristic, complemented
51#449—32no—
52#442—32no—
53#440—32no—
54#432—31no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C2×#67—21no—
61C2×#67—22no—
62C22×Dic3—21no—
63C22×Dic3—21no—
64C22×Dic3—22no—
65#237—22no—
66#237—22no—
67#237—44no—
68C2×C4×D3D411yescharacteristic
69C2×C4×D3D411yescharacteristic
70C2×C4×D3C2311yescharacteristic
71C2×Dic6—21no—
72C2×Dic6C2311yescharacteristic
73C3×D8—24no—
74C2×C3×C8C2312yes—
75C3×C4⋊C4C2311yescharacteristic
76C3×C22⋊C4—21no—
77C3×C42C2311yescharacteristic
78#217—21no—
79#217—22no—
80#217—22no—
81#214—24no—
82(C2×D3)⋊C4—21no—
83C4⋊Dic3—21no—
84C4⋊Dic3C2312yes—
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88C4×Dic3—21no—
89C4×Dic3—21no—
90C4×Dic3—21no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95C2×#41—31no—
96C2×#41—61no—
97C2×SD16—32no—
98C2×D8C2×D312yes—
99C4⋊Q8—31no—
100#127C2×D311yescharacteristic
101#124—61no—
102#121—32no—
103#121—61no—
104C4×Q8—31no—
105C4×Q8—31no—
106C4×D4—31no—
107C4×D4—32no—
108C4×D4—61no—
109C4×D4—61no—
110C4×D4—61no—
111C4×D4C2×D311yescharacteristic
112#117—31no—
113#117—61no—
114C2×C42—31no—
115C2×C42—61no—
116#107—32no—
117#106—32no—
118C4⋊C8—31no—
119C4⋊C8C2×D311yescharacteristic
120#103—32no—
121#102—32no—
122#102—32no—
123#102C2×D312yes—
124C22⋊C8—32no—
125#97—32no—
126C23×C3—21no—
127C23×C3—22no—
128C22×D3C2×D411yescharacteristic
129C3×D4—21no—
130C3×D4—22no—
131C3×D4—22no—
132C3×D4—42no—
133C3×D4—42no—
134C3×D4#4112yescomplemented
135C2×C3×C4—21no—
136C2×C3×C4—42no—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2411yescharacteristic
140#67—24no—
141#67—44no—
142C2×Dic3—24no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—44no—
147C2×Dic3—42no—
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151C4×D3—22no—
152C4×D3—21no—
153C4×D3—21no—
154C4×D3—41no—
155Dic6—21no—
156Dic6—42no—
157Dic6#4112yescomplemented
158C3×C8—22no—
159C3⋊C8—22no—
160#41—62no—
161#41—62no—
162#41—124no—
163C2×Q8—31no—
164C2×Q8—61no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—61no—
168C2×D4—62no—
169C2×D4C22×D312yes—
170C2×D4C22×D311yescharacteristic
171C22×C4—21no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—62no—
178SD16—64no—
179D8—24no—
180C2×C8—32no—
181C2×C8C22×D312yes—
182C4⋊C4—31no—
183C4⋊C4—32no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4C22×D311yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C42—31no—
194C42—31no—
195C42—31no—
196C42—61no—
197C42—61no—
198C42—61no—
199C42C22×D311yescharacteristic
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3C22×D411yescharacteristic
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C3×C4—21no—
211C3×C4—41no—
212C3×C4AGL(1,ℤ/8)12yes—
213C3×C4C2×#4111yescharacteristic
214C3×C4C2×#4111yescharacteristic
215Dic3—22no—
216Dic3—21no—
217Dic3—21no—
218Dic3—21no—
219Dic3—42no—
220Dic3—41no—
221C23—21no—
222C23—22no—
223C23—31no—
224Q8—32no—
225Q8—61no—
226Q8—122no—
227D4—22no—
228D4—21no—
229D4—22no—
230D4—42no—
231D4—42no—
232D4—64no—
233D4—124no—
234D4#23712yescomplemented
235C2×C4—21no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—42no—
240C2×C4—64no—
241C2×C4—61no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—121no—
248C2×C4—124no—
249C2×C4—122no—
250C2×C4D3×D411yescharacteristic
251C2×C4D3×D411yescharacteristic
252C2×C4C23×D311yescharacteristic
253C8—22no—
254C8—62no—
255C2×C3—42no—
256C2×C3—82no—
257C2×C3#52511yescharacteristic
258C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
259C2×C3#57511yescharacteristic
260D3—41no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22C2×D3×D411yescharacteristic
271C4—21no—
272C4—41no—
273C4—61no—
274C4—61no—
275C4—61no—
276C4—62no—
277C4—122no—
278C4—121no—
279C4#91112yes—
280C4C2×#23711yescharacteristic
281C4D3×#4111yescharacteristic
282C3#311511yescharacteristic, complemented, Sylow
283C2—42no—
284C2—82no—
285C2—121no—
286C2#560611yescharacteristic
287C2C2×#91111yescharacteristic
288C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3115
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. C4×#237C2semidirect
3. C3×#459C2semidirect
4. #5158C2semidirect
5. #5152C2semidirect
6. #5075C2semidirect
7. #5007C2semidirect
8. #4851C2semidirect
9. #4835C2semidirect
10. #4779C2semidirect
11. #4757C2semidirect
50. #459D3semidirect
134. C3×D4#41semidirect
157. Dic6#41semidirect
234. D4#237semidirect
282. C3#3115semidirect

Maximal subgroups

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

subgroupindex
#57162
C4×#2372
C3×#4592
#51582
#51522
#50752
#50072
#48512
#48352
#47792
#47572
#31153