Finite Group Databasebeta

← #80188

Subgroups — #80188

Order 384.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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×#946C211yesmaximal, characteristic, complemented
2#5627C211yesmaximal, characteristic
3C3×#451C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5188C212yesmaximal, complemented
6#5187C212yesmaximal, complemented
7#5043C211yesmaximal, characteristic, complemented
8#4814C212yesmaximal, complemented
9#4807C212yesmaximal, complemented
10#4736C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3190—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#239C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×Q16C2212yescomplemented
17C3×C22⋊Q8C2211yescharacteristic
18#946—21no—
19#946—21no—
20#946C2212yes—
21#946C2211yescharacteristic
22#946C2212yes—
23C2×#215C2212yescomplemented
24#930C2211yescharacteristic
25C2×(C2×D3)⋊C4—21no—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×Dic3⋊C4—21no—
28C2×D3×C8C2212yes—
29#895C2211yescharacteristic
30#894—21no—
31#893—21no—
32#892C2211yescharacteristic
33#882—21no—
34#67⋊C4—21no—
35C3×#103C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic
37#835C2212yescomplemented
38#835C2212yescomplemented
39#830C2211yescharacteristic
40#821C2212yescomplemented
41(C2×D3)⋊C8C2212yes—
42#818C2212yescomplemented
43#808C2212yescomplemented
44#525—31no—
45C2×C22⋊Q8—31no—
46#485—32no—
47#483—31no—
48#480—31no—
49#467—32no—
50#465—32no—
51#452—32no—
52#451D311yescharacteristic, complemented
53#448—31no—
54C2×C22⋊C8—31no—
55C23×D3D411yescharacteristic
56C2×C3×Q8—21no—
57C2×C3×Q8—42no—
58C2×C3×Q8C2311yescharacteristic
59C2×C3×Q8C2312yes—
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C2×#67—21no—
63C22×Dic3—21no—
64C22×Dic3D411yescharacteristic
65#239—42no—
66C2×D12—21no—
67C2×D12—21no—
68C2×D12C2311yescharacteristic
69C2×C4×D3—21no—
70C2×C4×D3—42no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C3×Q16—42no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80#215—42no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3C2312yes—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—22no—
92Dic3⋊C4—22no—
93C4×Dic3—21no—
94C2×C3⋊C8C2312yes—
95D3×C8—42no—
96C2×#41—31no—
97C22×Q8C2×D311yescharacteristic
98C23×C4—31no—
99C2×Q16C2×D312yescomplemented
100C2×SD16—32no—
101C22×C8—32no—
102#123—61no—
103C22⋊Q8—32no—
104C22⋊Q8—32no—
105C22⋊Q8—31no—
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—31no—
116C2×C4⋊C4—61no—
117C2×C22⋊C4—61no—
118#107—32no—
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—22no—
130C22×D3—21no—
131C22×D3—41no—
132C22×D3C2×D411yescharacteristic
133C22×D3C2×D411yescharacteristic
134C3×Q8—22no—
135C3×Q8—22no—
136C3×Q8—22no—
137C3×Q8—42no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4—42no—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2411yescharacteristic
145#67—42no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—41no—
149C2×Dic3—42no—
150C2×Dic3C2×D411yescharacteristic
151C2×Dic3C2×D411yescharacteristic
152D12—22no—
153D12—42no—
154D12—42no—
155C4×D3—22no—
156C4×D3—22no—
157C4×D3—41no—
158C4×D3—41no—
159C4×D3—42no—
160C3×C8—22no—
161C3⋊C8—22no—
162C24—31no—
163#41—122no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8C22×D311yescharacteristic
167C2×Q8C22×D312yes—
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—42no—
183SD16—122no—
184C2×C8—32no—
185C2×C8—122no—
186C2×C8C22×D312yes—
187C4⋊C4—21no—
188C4⋊C4—32no—
189C4⋊C4—31no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—61no—
193C4⋊C4—61no—
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#13512yes—
206C22×C3C22×D411yescharacteristic
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—21no—
212C2×D3—44no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3#13712yes—
217C3×C4—21no—
218C3×C4—41no—
219C3×C4—42no—
220C3×C4—41no—
221C3×C4H(5,𝔽2)12yes—
222Dic3—21no—
223Dic3—21no—
224Dic3—42no—
225Dic3—41no—
226C23—31no—
227C23—31no—
228C23—62no—
229C23—62no—
230C23—61no—
231C23—121no—
232C23D3×D411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—42no—
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×#13511yescharacteristic
264C2×C3C2×#13711yescharacteristic
265D3—22no—
266D3—41no—
267D3—81no—
268C22—21no—
269C22—21no—
270C22—32no—
271C22—61no—
272C22—61no—
273C22—61no—
274C22—61no—
275C22—61no—
276C22—124no—
277C22—121no—
278C22—121no—
279C22—121no—
280C22#91912yes—
281C22C2×D3×D411yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—61no—
287C4—61no—
288C4—122no—
289C4—121no—
290C4#100412yes—
291C3#319011yescharacteristic, complemented, Sylow
292C2—22no—
293C2—62no—
294C2—121no—
295C2—241no—
296C2#558911yescharacteristic
297C2C2×#91911yescharacteristic
298C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radical#5627C2
hypercenter#451D3
hypocenterC3#3190
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
3. C3×#451C2semidirect
4. #5227C2semidirect
5. #5188C2semidirect
6. #5187C2semidirect
7. #5043C2semidirect
8. #4814C2semidirect
9. #4807C2semidirect
10. #4736C2semidirect
11. D3×C22⋊C8C2semidirect
16. C2×C3×Q16C22semidirect
23. C2×#215C22semidirect
35. C3×#103C22semidirect
37. #835C22semidirect
38. #835C22semidirect
40. #821C22semidirect
42. #818C22semidirect
43. #808C22semidirect
52. #451D3semidirect
99. C2×Q16C2×D3semidirect
122. #103C2×D3semidirect
291. C3#3190semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
#56272
C3×#4512
#52272
#51882
#51872
#50432
#48142
#48072
#47362
D3×C22⋊C82
#31903