Finite Group Databasebeta

← #81467

Subgroups — #81467

Order 384.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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×#892C211yesmaximal, characteristic, complemented
3C3×#484C211yesmaximal, characteristic, complemented
4#5043C211yesmaximal, characteristic, complemented
5#4968C211yesmaximal, characteristic, complemented
6#4885C212yesmaximal, complemented
7#4880C212yesmaximal, complemented
8#4814C212yesmaximal, complemented
9#4804C212yesmaximal, complemented
10#4736C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3091—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C3×C22⋊Q8C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17#946C2211yescharacteristic
18#930C2211yescharacteristic
19C2×(C2×D3)⋊C4—21no—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4×Dic3—21no—
22C2×D3×C8C2212yes—
23#895C2211yescharacteristic
24#894—21no—
25#893—21no—
26#892—21no—
27#892—21no—
28#892C2211yescharacteristic
29#892C2212yes—
30#892C2212yes—
31#882—21no—
32#67⋊C4—21no—
33C3×#107C2212yescomplemented
34C3×#103C2212yescomplemented
35C3×C22⋊C8C2211yescharacteristic
36#835C2212yescomplemented
37#830C2211yescharacteristic
38#821C2212yescomplemented
39(C2×D3)⋊C8C2212yes—
40#818C2212yescomplemented
41#809C2212yescomplemented
42#808C2212yescomplemented
43#808C2212yescomplemented
44#525—31no—
45C2×#117—31no—
46#485—32no—
47#484D311yescharacteristic, complemented
48#483—31no—
49#482—32no—
50#481—31no—
51#480—31no—
52#427—32no—
53#416—32no—
54C2×C22⋊C8—31no—
55C23×D3D411yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4C2311yescharacteristic
59C2×#67—21no—
60C22×Dic3—21no—
61C22×Dic3D411yescharacteristic
62#239—24no—
63C2×D12—21no—
64C2×D12—21no—
65C2×D12C2311yescharacteristic
66C2×C4×D3—24no—
67C2×C4×D3—21no—
68C2×C4×D3D411yescharacteristic
69C2×C4×D3D411yescharacteristic
70C2×C4×D3C2311yescharacteristic
71C2×C4×D3C2311yescharacteristic
72C2×C3×C8C2312yes—
73C3×C4⋊C4—21no—
74C3×C4⋊C4—21no—
75C3×C4⋊C4C2311yescharacteristic
76C3×C4⋊C4C2312yes—
77C3×C22⋊C4—21no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2311yescharacteristic
86C4⋊Dic3C2312yes—
87Dic3⋊C4—21no—
88C4×Dic3—21no—
89C4×Dic3—22no—
90C4×Dic3—22no—
91C2×C3⋊C8C2312yes—
92D3×C8—24no—
93C2×#41—31no—
94C23×C4—31no—
95C22×C8—32no—
96#123—61no—
97C22⋊Q8—31no—
98C22⋊Q8C2×D311yescharacteristic
99#121—31no—
100#121—31no—
101#120—61no—
102C4×D4—61no—
103C4×D4—61no—
104#117—32no—
105#117—32no—
106#117—31no—
107#117—61no—
108#117—61no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4C2×D311yescharacteristic
111C2×C22⋊C4—61no—
112C2×C42—61no—
113#107—32no—
114#107C2×D312yescomplemented
115#106—32no—
116#106—32no—
117#103—32no—
118#103C2×D312yescomplemented
119#102—32no—
120#102—32no—
121C22⋊C8—31no—
122C22⋊C8—32no—
123C22⋊C8C2×D311yescharacteristic
124C23×C3C2×D411yescharacteristic
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—21no—
128C22×D3—41no—
129C22×D3C2×D411yescharacteristic
130C22×D3C2×D411yescharacteristic
131C3×Q8—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—42no—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—42no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—42no—
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146D12—22no—
147D12—42no—
148D12—42no—
149C4×D3—22no—
150C4×D3—22no—
151C4×D3—22no—
152C4×D3—42no—
153C4×D3#4114yes—
154C3×C8—22no—
155C3⋊C8—22no—
156C24—31no—
157#41—64no—
158C2×Q8C22×D311yescharacteristic
159C2×D4—31no—
160C2×D4—61no—
161C2×D4—61no—
162C2×D4—61no—
163C22×C4—21no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—64no—
170C22×C4—61no—
171C22×C4—61no—
172C22×C4C22×D311yescharacteristic
173C2×C8—32no—
174C2×C8—64no—
175C2×C8C22×D312yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—32no—
179C4⋊C4—31no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4C22×D312yes—
183C4⋊C4C22×D311yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—61no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C42—62no—
192C42—62no—
193C42—61no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3#13512yes—
197C22×C3C22×D411yescharacteristic
198C2×D3—21no—
199C2×D3—21no—
200C2×D3—21no—
201C2×D3—21no—
202C2×D3—21no—
203C2×D3—44no—
204C2×D3—41no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3#12312yes—
208C3×C4—21no—
209C3×C4—41no—
210C3×C4—41no—
211C3×C4—42no—
212C3×C4C2×#4112yes—
213Dic3—21no—
214Dic3—42no—
215Dic3—41no—
216Dic3#12312yes—
217C23—31no—
218C23—31no—
219C23—62no—
220C23—62no—
221C23—61no—
222C23—121no—
223C23D3×D411yescharacteristic
224Q8—22no—
225D4—62no—
226D4—122no—
227D4—122no—
228D4—122no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—31no—
234C2×C4—34no—
235C2×C4—31no—
236C2×C4—42no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—61no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4D3×D411yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—22no—
248C8—62no—
249C2×C3—22no—
250C2×C3C2×#12311yescharacteristic
251C2×C3C2×#13511yescharacteristic
252C2×C3#57511yescharacteristic
253D3—22no—
254D3—41no—
255D3—81no—
256C22—21no—
257C22—21no—
258C22—32no—
259C22—61no—
260C22—61no—
261C22—61no—
262C22—61no—
263C22—61no—
264C22—124no—
265C22—121no—
266C22—121no—
267C22—121no—
268C22#91912yes—
269C22C2×D3×D411yescharacteristic
270C4—21no—
271C4—32no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—61no—
276C4—122no—
277C4—121no—
278C4D3×#4112yes—
279C3#309111yescharacteristic, complemented, Sylow
280C2—22no—
281C2—62no—
282C2—121no—
283C2—241no—
284C2D3×#12311yescharacteristic
285C2C2×#91911yescharacteristic
286C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#484C2
socleC22×C3C22×D4
prime radical#5627C2
hypercenter#484D3
hypocenterC3#3091
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×#892C2semidirect
3. C3×#484C2semidirect
4. #5043C2semidirect
5. #4968C2semidirect
6. #4885C2semidirect
7. #4880C2semidirect
8. #4814C2semidirect
9. #4804C2semidirect
10. #4736C2semidirect
11. D3×C22⋊C8C2semidirect
33. C3×#107C22semidirect
34. C3×#103C22semidirect
36. #835C22semidirect
38. #821C22semidirect
40. #818C22semidirect
41. #809C22semidirect
42. #808C22semidirect
43. #808C22semidirect
47. #484D3semidirect
114. #107C2×D3semidirect
118. #103C2×D3semidirect
279. C3#3091semidirect

Maximal subgroups

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

subgroupindex
#56272
C2×#8922
C3×#4842
#50432
#49682
#48852
#48802
#48142
#48042
#47362
D3×C22⋊C82
#30913