Finite Group Databasebeta

← #83305

Subgroups — #83305

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×#532C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic, complemented
3#5239C212yesmaximal, complemented
4#5238C212yesmaximal, complemented
5#5090C211yesmaximal, characteristic, complemented
6#5083C211yesmaximal, characteristic, complemented
7#5075C211yesmaximal, characteristic, complemented
8#5072C211yesmaximal, characteristic, complemented
9#5059C212yesmaximal, complemented
10#5056C212yesmaximal, complemented
11#4925C211yesmaximal, characteristic, complemented
12#3212—31nomaximal, Sylow
13C2×C3×#41C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#124C2212yes—
17C3×#121—21no—
18C3×#121—21no—
19C3×C4×D4—21no—
20C3×C4×D4—21no—
21C2×C3×C42C2211yescharacteristic
22C2×#216C2212yes—
23C2×#215C2212yes—
24C2×#214C2212yes—
25C2×#213C2212yes—
26#930C2211yescharacteristic
27#929—21no—
28C4×#67—21no—
29#924C2211yescharacteristic
30#876—21no—
31#875C2211yescharacteristic
32C4×D12C2211yescharacteristic
33#872—21no—
34#870C2211yescharacteristic
35C4×Dic6C2211yescharacteristic
36#835C2212yes—
37#835C2212yes—
38#832C2212yes—
39#832C2212yes—
40#830C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42C4⋊(C3⋊C8)C2211yescharacteristic
43C4×C3⋊C8C2212yes—
44#532D311yescharacteristic, complemented
45#529—31no—
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—
55C3×#41—42no—
56C3×#41—42no—
57C2×C3×Q8C2312yes—
58C2×C3×D4—22no—
59C2×C3×D4—21no—
60C2×C3×D4—21no—
61C2×C3×D4C2312yes—
62C22×C3×C4—22no—
63C22×C3×C4D411yescharacteristic
64C22×C3×C4D411yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×#67—21no—
67C2×D12C2311yescharacteristic
68C2×C4×D3—21no—
69C2×Dic6C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C42D411yescharacteristic
75C3×C42D411yescharacteristic
76C3×C42C2311yescharacteristic
77C3×C42C2311yescharacteristic
78#217—21no—
79#216—42no—
80#215—42no—
81#214—42no—
82#213—42no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3C2311yescharacteristic
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3—21no—
94C2×C3⋊C8C2312yes—
95C2×C3⋊C8C2312yes—
96C2×#41C2×D312yes—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16—32no—
100C2×D8—32no—
101C4⋊Q8C2×D311yescharacteristic
102#127C2×D311yescharacteristic
103#126—61no—
104#125—31no—
105#124—31no—
106#124C2×D312yes—
107#123—61no—
108C22⋊Q8—31no—
109#121—21no—
110#121—21no—
111#121—31no—
112C4×Q8—31no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—31no—
116C4×D4—61no—
117#117—61no—
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—22no—
128C23×C3C2×D411yescharacteristic
129C22×D3—21no—
130C3×Q8—22no—
131C3×Q8—22no—
132C3×D4—22no—
133C3×D4—22no—
134C3×D4—42no—
135C3×D4—44no—
136C3×D4—42no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—41no—
142C2×C3×C4—44no—
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—42no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153D12—21no—
154D12—21no—
155C4×D3—42no—
156Dic6—21no—
157Dic6—21no—
158C3⋊C8—22no—
159C3⋊C8—22no—
160#41—42no—
161#41—42no—
162C2×Q8—31no—
163C2×Q8C22×D312yes—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—31no—
168C2×D4—61no—
169C2×D4C22×D312yes—
170C22×C4—22no—
171C22×C4—61no—
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—31no—
184C4⋊C4—31no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—61no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C22⋊C4—61no—
196C42—61no—
197C42#6711yescharacteristic
198C42#6711yescharacteristic
199C42C22×D311yescharacteristic
200C42C22×D311yescharacteristic
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3C22×D411yescharacteristic
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—41no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—42no—
216C3×C4#13512yes—
217C3×C4H(5,𝔽2)11yescharacteristic
218C3×C4H(5,𝔽2)11yescharacteristic
219Dic3—41no—
220Dic3—41no—
221Dic3—41no—
222C23—22no—
223C23—61no—
224C23C2×#6711yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—61no—
228Q8—61no—
229D4—22no—
230D4—22no—
231D4—42no—
232D4—44no—
233D4—42no—
234D4—61no—
235D4—61no—
236D4—122no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—122no—
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—82no—
257C2×C3#53511yescharacteristic
258C2×C3C2×#13511yescharacteristic
259C2×C3#57511yescharacteristic
260D3—81no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—121no—
268C22—121no—
269C22—121no—
270C22C22×#6711yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—121no—
277C4—121no—
278C4—121no—
279C4#95012yes—
280C4#100411yescharacteristic
281C4#100411yescharacteristic
282C3#321211yescharacteristic, complemented, Sylow
283C2—41no—
284C2—82no—
285C2—241no—
286C2#580511yescharacteristic
287C2C2×#95011yescharacteristic
288C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#532C2
socleC22×C3C22×D4
hypercenter#532D3
hypocenterC3#3212
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5479C2semidirect
3. #5239C2semidirect
4. #5238C2semidirect
5. #5090C2semidirect
6. #5083C2semidirect
7. #5075C2semidirect
8. #5072C2semidirect
9. #5059C2semidirect
10. #5056C2semidirect
11. #4925C2semidirect
44. #532D3semidirect
282. C3#3212semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#54792
#52392
#52382
#50902
#50832
#50752
#50722
#50592
#50562
#49252
#32123