Finite Group Databasebeta

← #83530

Subgroups — #83530

Order 384.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×#547C211yesmaximal, characteristic, complemented
2C2×#950C212yesmaximal, complemented
3#5553C211yesmaximal, characteristic, complemented
4#5238C212yesmaximal, complemented
5#5090C211yesmaximal, characteristic, complemented
6#5085C211yesmaximal, characteristic, complemented
7#5048C212yesmaximal, complemented
8#5047C212yesmaximal, complemented
9C4×#214C211yesmaximal, characteristic, complemented
10#5015C211yesmaximal, characteristic, complemented
11D4×C3⋊C8C211yesmaximal, characteristic, complemented
12#3284—31nomaximal, Sylow
13C2×C3×#41C2212yes—
14C2×#235C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×#123—22no—
17C3×C22⋊Q8C2212yes—
18C3×#121—21no—
19C3×C4×D4C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21C2×C3×C4⋊C4—21no—
22#950—28no—
23C2×#216C2212yes—
24C2×#215C2212yes—
25C2×#215C2212yes—
26#940—21no—
27C2×#214C2211yescharacteristic
28C2×#213C2212yes—
29#930C2212yes—
30C22×C3⋊C8C2212yes—
31#893—21no—
32#884—22no—
33#874C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35#835C2212yes—
36#832C2211yescharacteristic
37#830C2212yes—
38#810C2211yescharacteristic
39#809C2212yes—
40#808C2211yescharacteristic
41C4⋊(C3⋊C8)C2211yescharacteristic
42C4×C3⋊C8C2211yescharacteristic
43C2×#135—32no—
44#550—31no—
45#547D311yescharacteristic, complemented
46#492—31no—
47#465—32no—
48#462—31no—
49#459—31no—
50#452—32no—
51#449—32no—
52C4×SD16—31no—
53C8×D4—31no—
54C3×#41—24no—
55C3×#41—42no—
56C2×C3×Q8C2312yes—
57C2×C3×D4—22no—
58C2×C3×D4D411yescharacteristic
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4—22no—
62C22×C3×C4C2312yes—
63C2×#67—22no—
64C2×#67—22no—
65#235—24no—
66#235—42no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12C2312yes—
70C2×C4×D3—22no—
71C2×Dic6C2311yescharacteristic
72C3×C4⋊C4—21no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—21no—
75C3×C4⋊C4D411yescharacteristic
76C3×C4⋊C4C2311yescharacteristic
77C3×C22⋊C4—22no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4D412yes—
80C3×C42C2311yescharacteristic
81#216—24no—
82#215—24no—
83#215—42no—
84#214D414yescomplemented
85#213—24no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3C2311yescharacteristic
88Dic3⋊C4—21no—
89C4×Dic3—21no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8C2311yescharacteristic
92C2×C3⋊C8C2312yes—
93C2×C3⋊C8C2311yescharacteristic
94C2×#41—32no—
95C2×#41C2×D312yes—
96#135—68no—
97C2×Q16—32no—
98C2×SD16—31no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×D8—32no—
102C22×C8—32no—
103C4⋊Q8C2×D311yescharacteristic
104#127—31no—
105#127—61no—
106#123—22no—
107C22⋊Q8C2×D312yes—
108#121—21no—
109#121—32no—
110#121—62no—
111C4×Q8—31no—
112C4×D4—61no—
113C4×D4C2×D311yescharacteristic
114C4×D4C2×D311yescharacteristic
115C2×C4⋊C4—21no—
116#106—31no—
117C4⋊C8—31no—
118#103—32no—
119#103—31no—
120#102—31no—
121#102—32no—
122C22⋊C8—32no—
123C4×C8—31no—
124C23×C3—21no—
125C23×C3C2×D412yes—
126C22×D3—22no—
127C3×Q8—22no—
128C3×Q8—22no—
129C3×D4—22no—
130C3×D4—21no—
131C3×D4—44no—
132C3×D4C2×D412yes—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—44no—
138C2×C3×C4—42no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—44no—
144#67—82no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147D12—22no—
148D12—22no—
149D12—42no—
150D12—81no—
151C4×D3—44no—
152Dic6—21no—
153Dic6C2×D412yes—
154C3⋊C8—22no—
155C3⋊C8—21no—
156C3⋊C8C2×D412yes—
157#41—24no—
158#41—42no—
159#41—64no—
160#41—122no—
161C2×Q8—31no—
162C2×Q8C22×D312yes—
163C2×D4—22no—
164C2×D4—32no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—61no—
168C2×D4—61no—
169C2×D4#6711yescharacteristic
170C2×D4C22×D311yescharacteristic
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—62no—
174C22×C4C22×D312yes—
175Q16—64no—
176SD16—34no—
177SD16—64no—
178SD16—122no—
179D8—64no—
180C2×C8—31no—
181C2×C8—31no—
182C2×C8—32no—
183C2×C8—64no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—31no—
188C4⋊C4—61no—
189C4⋊C4#6711yescharacteristic
190C4⋊C4C22×D311yescharacteristic
191C22⋊C4—22no—
192C22⋊C4—21no—
193C22⋊C4—62no—
194C22⋊C4#6712yes—
195C42—61no—
196C42C22×D311yescharacteristic
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3C22×D411yescharacteristic
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C3×C4—22no—
208C3×C4—21no—
209C3×C4—42no—
210C3×C4—41no—
211C3×C4#13512yescomplemented
212C3×C4C22×D411yescharacteristic
213C3×C4H(5,𝔽2)11yescharacteristic
214Dic3—22no—
215Dic3—41no—
216C23—21no—
217C23—62no—
218C23C2×#6712yes—
219Q8—22no—
220Q8—22no—
221Q8—32no—
222Q8—61no—
223D4—22no—
224D4—21no—
225D4—44no—
226D4—62no—
227D4—62no—
228D4—124no—
229D4—122no—
230D4—241no—
231D4—242no—
232D4D3×D412yes—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—124no—
242C2×C4C2×#6711yescharacteristic
243C2×C4C2×#6712yes—
244C2×C4C2×#6711yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—32no—
247C8—62no—
248C8—61no—
249C2×C3—42no—
250C2×C3—42no—
251C2×C3D4211yescharacteristic
252C2×C3C2×#13511yescharacteristic
253C2×C3#57711yescharacteristic
254D3—82no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—122no—
262C22—122no—
263C22—122no—
264C22C22×#6711yescharacteristic
265C4—22no—
266C4—21no—
267C4—42no—
268C4—41no—
269C4—62no—
270C4—121no—
271C4#95012yescomplemented
272C4C2×D3×D411yescharacteristic
273C4#100411yescharacteristic
274C3#328411yescharacteristic, complemented, Sylow
275C2—42no—
276C2—42no—
277C2—242no—
278C2D4×#6711yescharacteristic
279C2C2×#95011yescharacteristic
280C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#547C2
socleC22×C3C22×D4
hypercenter#547D3
hypocenterC3#3284
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. C2×#950C2semidirect
3. #5553C2semidirect
4. #5238C2semidirect
5. #5090C2semidirect
6. #5085C2semidirect
7. #5048C2semidirect
8. #5047C2semidirect
9. C4×#214C2semidirect
10. #5015C2semidirect
11. D4×C3⋊C8C2semidirect
45. #547D3semidirect
84. #214D4semidirect
211. C3×C4#135semidirect
271. C4#950semidirect
274. C3#3284semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
C2×#9502
#55532
#52382
#50902
#50852
#50482
#50472
C4×#2142
#50152
D4×C3⋊C82
#32843