Finite Group Databasebeta

← #83145

Subgroups — #83145

Order 384.

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

normal subgroups only

This group has 400 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
1C3×#529C211yesmaximal, characteristic, complemented
2#5477C211yesmaximal, characteristic, complemented
3#5068C212yesmaximal, complemented
4#5059C212yesmaximal, complemented
5#5047C212yesmaximal, complemented
6#5037C212yesmaximal, complemented
7#5031C211yesmaximal, characteristic, complemented
8#5027C211yesmaximal, characteristic, complemented
9#5019C211yesmaximal, characteristic, complemented
10#5015C211yesmaximal, characteristic, complemented
11#4925C211yesmaximal, characteristic, complemented
12#3212—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)C2211yescharacteristic
42C4⋊(C3⋊C8)C2211yescharacteristic
43C4×C3⋊C8C2212yes—
44#532—31no—
45#529D311yescharacteristic, complemented
46#495—32no—
47#488—32no—
48#462—31no—
49#461—31no—
50#460—31no—
51#459—31no—
52#452—32no—
53#449—32no—
54#431—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—31no—
120C4⋊C8—31no—
121#103—32no—
122#103—32no—
123#102—32no—
124#102—32no—
125C22⋊C8—32no—
126C4×C8—32no—
127C23×C3—21no—
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C3×Q8—21no—
131C3×Q8—21no—
132C3×D4—21no—
133C3×D4—21no—
134C3×D4—42no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—41no—
142C2×C3×C4—42no—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2411yescharacteristic
149#67—44no—
150C2×Dic3—22no—
151D12—22no—
152D12—22no—
153D12—42no—
154D12—42no—
155C4×D3—44no—
156Dic6—22no—
157Dic6—22no—
158C3⋊C8—22no—
159C3⋊C8—22no—
160#41—122no—
161#41—122no—
162C2×Q8—32no—
163C2×Q8C22×D311yescharacteristic
164C2×D4—21no—
165C2×D4—32no—
166C2×D4—62no—
167C2×D4—61no—
168C2×D4—61no—
169C2×D4C22×D311yescharacteristic
170C22×C4—21no—
171C22×C4—62no—
172C22×C4#6711yescharacteristic
173C22×C4#6711yescharacteristic
174C22×C4C22×D311yescharacteristic
175Q16—122no—
176SD16—122no—
177SD16—122no—
178D8—122no—
179C2×C8—32no—
180C2×C8—32no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4C22×D311yescharacteristic
188C4⋊C4C22×D311yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C42—21no—
197C42#6711yescharacteristic
198C42#6711yescharacteristic
199C42C22×D311yescharacteristic
200C42C22×D311yescharacteristic
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3C22×D411yescharacteristic
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4#13512yes—
219C3×C4H(5,𝔽2)11yescharacteristic
220C3×C4H(5,𝔽2)11yescharacteristic
221Dic3—42no—
222C23—21no—
223C23—62no—
224C23C2×#6711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—62no—
228Q8—62no—
229D4—21no—
230D4—21no—
231D4—42no—
232D4—62no—
233D4—62no—
234D4—122no—
235D4—124no—
236D4—122no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—62no—
246C2×C4—124no—
247C2×C4C2×#6711yescharacteristic
248C2×C4C2×#6711yescharacteristic
249C2×C4C2×#6711yescharacteristic
250C2×C4C2×#6711yescharacteristic
251C2×C4C2×#6711yescharacteristic
252C2×C4C23×D311yescharacteristic
253C8—62no—
254C8—62no—
255C2×C3—41no—
256C2×C3—81no—
257C2×C3#53511yescharacteristic
258C2×C3C2×#13511yescharacteristic
259C2×C3#57511yescharacteristic
260D3—82no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—122no—
268C22—122no—
269C22—122no—
270C22C22×#6711yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—122no—
279C4#95012yes—
280C4#100911yescharacteristic
281C4#100911yescharacteristic
282C3#321211yescharacteristic, complemented, Sylow
283C2—41no—
284C2—81no—
285C2—242no—
286C2C2×#95011yescharacteristic
287C2#582211yescharacteristic
288C2#583011yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#529C2
socleC22×C3C22×D4
hypercenter#529D3
hypocenterC3#3212
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. #5068C2semidirect
4. #5059C2semidirect
5. #5047C2semidirect
6. #5037C2semidirect
7. #5031C2semidirect
8. #5027C2semidirect
9. #5019C2semidirect
10. #5015C2semidirect
11. #4925C2semidirect
45. #529D3semidirect
282. C3#3212semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#54772
#50682
#50592
#50472
#50372
#50312
#50272
#50192
#50152
#49252
#32123