Finite Group Databasebeta

← #81489

Subgroups — #81489

Order 384.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#5627C211yesmaximal, characteristic
2C2×#897C211yesmaximal, characteristic, complemented
3C3×#484C211yesmaximal, characteristic, complemented
4#5050C211yesmaximal, characteristic, complemented
5#4968C211yesmaximal, characteristic, complemented
6#4888C212yesmaximal, complemented
7#4886C212yesmaximal, complemented
8#4816C212yesmaximal, complemented
9#4808C212yesmaximal, complemented
10#4738C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3196—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C3×C22⋊Q8C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18#946C2211yescharacteristic
19C2×#216C2212yescomplemented
20#930C2211yescharacteristic
21C2×(C2×D3)⋊C4—21no—
22C2×C4⋊Dic3—21no—
23C2×#204C2212yescomplemented
24C2×ΓL(1,𝔽25)C2212yes—
25#897—21no—
26#897—21no—
27#897C2212yes—
28#897C2212yes—
29#895C2211yescharacteristic
30#894—21no—
31#893—21no—
32#892C2211yescharacteristic
33#882—21no—
34#67⋊C4—21no—
35C3×#107C2212yescomplemented
36C3×#103C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#830C2211yescharacteristic
39(C2×D3)⋊C8C2212yes—
40#816C2212yescomplemented
41#810C2212yescomplemented
42#809C2212yescomplemented
43#808C2212yescomplemented
44#525—31no—
45C2×C22⋊Q8—31no—
46#485—32no—
47#484D311yescharacteristic, complemented
48#481—31no—
49#470—32no—
50#468—32no—
51#452—32no—
52#451—31no—
53#448—31no—
54#407—31no—
55C23×D3D411yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4C2311yescharacteristic
59C2×#67—21no—
60C22×Dic3—21no—
61C22×Dic3D411yescharacteristic
62#239—42no—
63C2×D12—21no—
64C2×D12—21no—
65C2×D12C2311yescharacteristic
66C2×C4×D3—21no—
67C2×C4×D3—42no—
68C2×C4×D3D411yescharacteristic
69C2×C4×D3D411yescharacteristic
70C2×C4×D3C2311yescharacteristic
71C2×C4×D3C2311yescharacteristic
72C2×Dic6—21no—
73C2×Dic6—42no—
74C2×Dic6C2312yes—
75C2×C3×C8C2312yes—
76C3×C4⋊C4—21no—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2312yes—
79C3×C4⋊C4C2311yescharacteristic
80C3×C22⋊C4—21no—
81#216—42no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3C2311yescharacteristic
91Dic3⋊C4—21no—
92C4×Dic3—21no—
93C2×C3⋊C8C2312yes—
94#204—42no—
95ΓL(1,𝔽25)—42no—
96C2×#41—31no—
97C22×Q8—31no—
98C23×C4—31no—
99C2×Q16—32no—
100C2×SD16—32no—
101C2×M16—32no—
102#123—61no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—32no—
106C22⋊Q8—61no—
107C22⋊Q8—61no—
108C22⋊Q8C2×D311yescharacteristic
109#121—31no—
110#121—31no—
111#120—61no—
112C4×D4—61no—
113C4×D4—61no—
114#117—31no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4C2×D311yescharacteristic
117C2×C22⋊C4—61no—
118#107C2×D312yescomplemented
119#106—32no—
120#103—32no—
121#103—32no—
122#103C2×D312yescomplemented
123#102—32no—
124C22⋊C8—31no—
125C22⋊C8—32no—
126C22⋊C8C2×D311yescharacteristic
127C23×C3C2×D411yescharacteristic
128C22×D3—22no—
129C22×D3—21no—
130C22×D3—41no—
131C22×D3—41no—
132C22×D3C2×D411yescharacteristic
133C22×D3C2×D411yescharacteristic
134C3×Q8—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—41no—
146C2×Dic3—42no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149D12—22no—
150D12—42no—
151D12—42no—
152C4×D3—22no—
153C4×D3—22no—
154C4×D3—41no—
155C4×D3—41no—
156C4×D3—42no—
157Dic6—22no—
158Dic6—22no—
159Dic6—42no—
160C3×C8—22no—
161C3⋊C8—22no—
162C24—31no—
163#41—122no—
164C2×Q8—32no—
165C2×Q8—61no—
166C2×Q8—122no—
167C2×Q8C22×D311yescharacteristic
168C2×D4—31no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C22×C4—21no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—122no—
181C22×C4C22×D311yescharacteristic
182Q16—122no—
183SD16—122no—
184M16—122no—
185C2×C8—32no—
186C2×C8C22×D312yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—31no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—61no—
193C4⋊C4C22×D312yes—
194C4⋊C4C22×D311yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—61no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C42—61no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3#13712yes—
206C22×C3C22×D411yescharacteristic
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—21no—
212C2×D3—21no—
213C2×D3—44no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C3×C4—21no—
218C3×C4—41no—
219C3×C4—41no—
220C3×C4—42no—
221C3×C4H(5,𝔽2)12yes—
222Dic3—21no—
223Dic3—21no—
224Dic3—42no—
225Dic3—41no—
226C23—31no—
227C23—31no—
228C23—62no—
229C23—61no—
230C23—121no—
231C23—121no—
232C23D3×D411yescharacteristic
233Q8—22no—
234Q8—62no—
235Q8—62no—
236Q8—122no—
237D4—62no—
238D4—122no—
239D4—122no—
240D4—122no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—42no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4—122no—
257C2×C4D3×D411yescharacteristic
258C2×C4C23×D311yescharacteristic
259C8—22no—
260C8—62no—
261C2×C3—22no—
262C2×C3#53411yescharacteristic
263C2×C3C2×#13711yescharacteristic
264C2×C3#57511yescharacteristic
265D3—41no—
266D3—41no—
267D3—81no—
268C22—21no—
269C22—21no—
270C22—61no—
271C22—61no—
272C22—61no—
273C22—61no—
274C22—61no—
275C22—61no—
276C22—124no—
277C22—121no—
278C22—121no—
279C22—121no—
280C22#91812yes—
281C22C2×D3×D411yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—61no—
287C4—61no—
288C4—122no—
289C4—121no—
290C4#100912yes—
291C3#319611yescharacteristic, complemented, Sylow
292C2—22no—
293C2—121no—
294C2—121no—
295C2—241no—
296C2#565311yescharacteristic
297C2C2×#91811yescharacteristic
298C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#484C2
socleC22×C3C22×D4
prime radical#5627C2
hypercenter#484D3
hypocenterC3#3196
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#897C2semidirect
3. C3×#484C2semidirect
4. #5050C2semidirect
5. #4968C2semidirect
6. #4888C2semidirect
7. #4886C2semidirect
8. #4816C2semidirect
9. #4808C2semidirect
10. #4738C2semidirect
11. #4726C2semidirect
19. C2×#216C22semidirect
23. C2×#204C22semidirect
35. C3×#107C22semidirect
36. C3×#103C22semidirect
40. #816C22semidirect
41. #810C22semidirect
42. #809C22semidirect
43. #808C22semidirect
47. #484D3semidirect
118. #107C2×D3semidirect
122. #103C2×D3semidirect
291. C3#3196semidirect

Maximal subgroups

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

subgroupindex
#56272
C2×#8972
C3×#4842
#50502
#49682
#48882
#48862
#48162
#48082
#47382
#47262
#31963