Finite Group Databasebeta

← #83312

Subgroups — #83312

Order 384.

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

normal subgroups only

This group has 392 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#5094C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5#5075C211yesmaximal, characteristic, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5042C212yesmaximal, complemented
8#5039C212yesmaximal, complemented
9#5016C212yesmaximal, complemented
10#5009C212yesmaximal, complemented
11#4924C211yesmaximal, characteristic, complemented
12#3251—31nomaximal, Sylow
13C2×C3×#41—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#124—21no—
17C3×#121—21no—
18C3×#121C2212yes—
19C3×C4×D4—21no—
20C3×C4×D4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×#214C2212yes—
23C2×#213C2212yes—
24#930C2211yescharacteristic
25#929—21no—
26C4×#67—21no—
27#924C2211yescharacteristic
28C2×#208C2212yes—
29#876—21no—
30#875C2211yescharacteristic
31C4×D12C2211yescharacteristic
32#872—21no—
33#870C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35#832C2212yes—
36#832C2212yes—
37#810C2212yes—
38#809C2212yes—
39#808C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2212yes—
42C4⋊(C3⋊C8)C2211yescharacteristic
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532D311yescharacteristic, complemented
45#529—31no—
46#478—32no—
47#475—31no—
48#474—31no—
49#469—32no—
50#468—32no—
51#463—32no—
52#460—31no—
53#459—31no—
54#423—31no—
55C3×#41—44no—
56C2×C3×Q8—21no—
57C2×C3×D4—22no—
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4C2312yes—
61C22×C3×C4—22no—
62C22×C3×C4D411yescharacteristic
63C22×C3×C4D411yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C2×#67—21no—
66C2×D12C2311yescharacteristic
67C2×C4×D3—21no—
68C2×Dic6C2311yescharacteristic
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2312yes—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—22no—
73C3×C42D411yescharacteristic
74C3×C42D411yescharacteristic
75C3×C42C2311yescharacteristic
76C3×C42C2311yescharacteristic
77#217—21no—
78#214—42no—
79#213—42no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84C4⋊Dic3C2311yescharacteristic
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90C4×Dic3—21no—
91#208—42no—
92C2×C3⋊C8C2312yes—
93C2×C3⋊C8C2312yes—
94C2×#41—21no—
95C2×SD16—32no—
96C2×D8—32no—
97C2×M16—32no—
98C4⋊Q8C2×D311yescharacteristic
99#127C2×D311yescharacteristic
100#126—61no—
101#125—31no—
102#124—21no—
103#124—31no—
104#123—61no—
105C22⋊Q8—31no—
106#121—21no—
107#121—31no—
108#121C2×D312yes—
109C4×Q8—31no—
110C4×D4—21no—
111C4×D4—31no—
112C4×D4—61no—
113C4×D4C2×D312yes—
114#117—61no—
115C2×C42C2×D311yescharacteristic
116#107—32no—
117#106—32no—
118C4⋊C8—31no—
119C4⋊C8—31no—
120C4⋊C8—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125C23×C3—22no—
126C23×C3C2×D411yescharacteristic
127C22×D3—21no—
128C3×Q8—42no—
129C3×D4—22no—
130C3×D4—22no—
131C3×D4—42no—
132C3×D4—44no—
133C3×D4—42no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—41no—
139C2×C3×C4—44no—
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—42no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150D12—21no—
151D12—21no—
152C4×D3—42no—
153Dic6—21no—
154Dic6—21no—
155C3⋊C8—22no—
156C3⋊C8—22no—
157#41—44no—
158C2×Q8—21no—
159C2×Q8—31no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—31no—
164C2×D4—61no—
165C2×D4C22×D312yes—
166C22×C4—22no—
167C22×C4—61no—
168C22×C4#6711yescharacteristic
169C22×C4#6711yescharacteristic
170C22×C4C22×D311yescharacteristic
171SD16—122no—
172D8—122no—
173M16—122no—
174C2×C8—32no—
175C2×C8—32no—
176C4⋊C4—21no—
177C4⋊C4—31no—
178C4⋊C4—31no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4C22×D312yes—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C42—61no—
192C42#6711yescharacteristic
193C42#6711yescharacteristic
194C42C22×D311yescharacteristic
195C42C22×D311yescharacteristic
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3C22×D411yescharacteristic
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—41no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—42no—
211C3×C4AGL(1,ℤ/8)12yes—
212C3×C4H(5,𝔽2)11yescharacteristic
213C3×C42−1+411yescharacteristic
214Dic3—41no—
215Dic3—41no—
216Dic3—41no—
217C23—22no—
218C23—61no—
219C23C2×#6711yescharacteristic
220Q8—42no—
221Q8—61no—
222Q8—61no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—44no—
227D4—42no—
228D4—61no—
229D4—61no—
230D4—122no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—44no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—122no—
241C2×C4C2×#6711yescharacteristic
242C2×C4C2×#6711yescharacteristic
243C2×C4C2×#6711yescharacteristic
244C2×C4C2×#6711yescharacteristic
245C2×C4C2×#6711yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—62no—
248C8—62no—
249C2×C3—41no—
250C2×C3—82no—
251C2×C3#53711yescharacteristic
252C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
253C2×C3#57511yescharacteristic
254D3—81no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—121no—
262C22—121no—
263C22—121no—
264C22C22×#6711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—121no—
271C4—121no—
272C4—121no—
273C4#93212yes—
274C4#100911yescharacteristic
275C4#101011yescharacteristic
276C3#325111yescharacteristic, complemented, Sylow
277C2—41no—
278C2—82no—
279C2—241no—
280C2C2×#93211yescharacteristic
281C2#582211yescharacteristic
282C2#583311yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#532C2
socleC22×C3C22×D4
hypercenter#532D3
hypocenterC3#3251
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. #5094C2semidirect
4. #5087C2semidirect
5. #5075C2semidirect
6. #5072C2semidirect
7. #5042C2semidirect
8. #5039C2semidirect
9. #5016C2semidirect
10. #5009C2semidirect
11. #4924C2semidirect
44. #532D3semidirect
276. C3#3251semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#54792
#50942
#50872
#50752
#50722
#50422
#50392
#50162
#50092
#49242
#32513