Finite Group Databasebeta

← #83147

Subgroups — #83147

Order 384.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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×#529C211yesmaximal, characteristic, complemented
2#5477C211yesmaximal, characteristic, complemented
3#5069C211yesmaximal, characteristic, complemented
4#5067C211yesmaximal, characteristic, complemented
5#5060C211yesmaximal, characteristic, complemented
6#5058C211yesmaximal, characteristic, complemented
7#5047C212yesmaximal, complemented
8#5037C212yesmaximal, complemented
9#5028C212yesmaximal, complemented
10#5016C212yesmaximal, complemented
11#4926C211yesmaximal, characteristic, complemented
12#3219—31nomaximal, Sylow
13C2×#235C2212yes—
14C3×#126—21no—
15C3×#125C2211yescharacteristic
16C3×#124C2211yescharacteristic
17C3×#123—21no—
18C3×C22⋊Q8C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×D4—21no—
22C3×C4×D4C2211yescharacteristic
23C3×#117—21no—
24C2×C3×C42C2211yescharacteristic
25C2×#216C2212yes—
26C2×#215C2212yes—
27C2×#214C2212yes—
28C2×#213C2212yes—
29#930—21no—
30#930—21no—
31#875C2212yes—
32#874C2211yescharacteristic
33C4×D12—21no—
34C4×D12—21no—
35#869C2211yescharacteristic
36#830C2212yes—
37#810C2212yes—
38#810C2212yes—
39#809C2212yes—
40#809C2212yes—
41C4⋊(C3⋊C8)C2212yes—
42#803C2211yescharacteristic
43#803C2211yescharacteristic
44#532—31no—
45#529D311yescharacteristic, complemented
46#497—31no—
47#496—31no—
48#491—31no—
49#490—31no—
50#464—32no—
51#463—32no—
52#452—32no—
53#449—32no—
54#432—31no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—21no—
57C2×C3×D4C2311yescharacteristic
58C22×C3×C4—21no—
59C22×C3×C4D411yescharacteristic
60C22×C3×C4D411yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63#235—42no—
64#235—42no—
65C2×D12—21no—
66C2×D12—21no—
67C2×D12C2312yes—
68C2×C4×D3—22no—
69C2×Dic6C2312yes—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—21no—
73C3×C4⋊C4—21no—
74C3×C4⋊C4C2311yescharacteristic
75C3×C4⋊C4C2311yescharacteristic
76C3×C22⋊C4—21no—
77C3×C22⋊C4—21no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C42—21no—
82C3×C42D411yescharacteristic
83C3×C42D411yescharacteristic
84C3×C42C2311yescharacteristic
85C3×C42C2311yescharacteristic
86#216—42no—
87#215—42no—
88#214—42no—
89#213—42no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—22no—
92C4⋊Dic3—21no—
93C4⋊Dic3—21no—
94C2×C3⋊C8C2312yes—
95C2×C3⋊C8C2312yes—
96C2×#41—32no—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16—32no—
100C2×D8—32no—
101C4⋊Q8—31no—
102#127—31no—
103#126—21no—
104#125C2×D311yescharacteristic
105#124—32no—
106#124C2×D311yescharacteristic
107#123—21no—
108C22⋊Q8C2×D311yescharacteristic
109#121—61no—
110#121—61no—
111#121C2×D311yescharacteristic
112C4×Q8C2×D311yescharacteristic
113C4×D4—21no—
114C4×D4—61no—
115C4×D4—61no—
116C4×D4C2×D311yescharacteristic
117#117—21no—
118C2×C42C2×D311yescharacteristic
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C22⋊C8—32no—
125#97—31no—
126#97—31no—
127C23×C3—21no—
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C3×Q8—22no—
131C3×D4—22no—
132C3×D4—42no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2411yescharacteristic
146#67—44no—
147C2×Dic3—22no—
148D12—22no—
149D12—22no—
150D12—42no—
151D12—42no—
152C4×D3—44no—
153Dic6—22no—
154Dic6—22no—
155C3⋊C8—22no—
156C3⋊C8—22no—
157#41—122no—
158#41—122no—
159C2×Q8—32no—
160C2×Q8C22×D311yescharacteristic
161C2×D4—21no—
162C2×D4—32no—
163C2×D4—62no—
164C2×D4—61no—
165C2×D4—61no—
166C2×D4C22×D311yescharacteristic
167C22×C4—21no—
168C22×C4—62no—
169C22×C4#6711yescharacteristic
170C22×C4#6711yescharacteristic
171C22×C4C22×D311yescharacteristic
172Q16—122no—
173SD16—122no—
174SD16—122no—
175D8—122no—
176C2×C8—32no—
177C2×C8—32no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4C22×D311yescharacteristic
185C4⋊C4C22×D311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C42—21no—
194C42#6711yescharacteristic
195C42#6711yescharacteristic
196C42C22×D311yescharacteristic
197C42C22×D311yescharacteristic
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3C22×D411yescharacteristic
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—42no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—41no—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4H(5,𝔽2)12yes—
217Dic3—42no—
218C23—21no—
219C23—62no—
220C23C2×#6711yescharacteristic
221Q8—22no—
222Q8—62no—
223Q8—62no—
224D4—22no—
225D4—42no—
226D4—62no—
227D4—62no—
228D4—122no—
229D4—122no—
230D4—124no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—62no—
239C2×C4—124no—
240C2×C4C2×#6711yescharacteristic
241C2×C4C2×#6711yescharacteristic
242C2×C4C2×#6711yescharacteristic
243C2×C4C2×#6711yescharacteristic
244C2×C4C2×#6711yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—62no—
247C8—62no—
248C2×C3—41no—
249C2×C3—81no—
250C2×C3#53511yescharacteristic
251C2×C3#57511yescharacteristic
252C2×C3#57511yescharacteristic
253D3—82no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—122no—
261C22—122no—
262C22—122no—
263C22C22×#6711yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—122no—
273C4#100912yes—
274C3#321911yescharacteristic, complemented, Sylow
275C2—41no—
276C2—81no—
277C2—242no—
278C2#582211yescharacteristic
279C2#582211yescharacteristic
280C2#583011yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#529C2
socleC22×C3C22×D4
hypercenter#529D3
hypocenterC3#3219
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5477C2semidirect
3. #5069C2semidirect
4. #5067C2semidirect
5. #5060C2semidirect
6. #5058C2semidirect
7. #5047C2semidirect
8. #5037C2semidirect
9. #5028C2semidirect
10. #5016C2semidirect
11. #4926C2semidirect
45. #529D3semidirect
274. C3#3219semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#54772
#50692
#50672
#50602
#50582
#50472
#50372
#50282
#50162
#49262
#32193