Finite Group Databasebeta

← #10595418

Subgroups — #10595418

Order 576.

478 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 120 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
1#64156C211yesmaximal, characteristic, complemented
2C3×#946C212yesmaximal, complemented
3#63915C211yesmaximal, characteristic, complemented
4#63910C212yesmaximal, complemented
5#63906C212yesmaximal, complemented
6#63895C212yesmaximal, complemented
7#63870C211yesmaximal, characteristic, complemented
8#63854C212yesmaximal, complemented
9#63850C212yesmaximal, complemented
10#5647—32nomaximal
11C2×C32×Q8C2211yescharacteristic
12C2×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×D3C2212yes—
14C2×#627C2211yescharacteristic
15C2×#626C2212yes—
16C2×#626C2212yes—
17C2×#625C2211yescharacteristic
18C2×#624C2212yes—
19#3752C2212yes—
20#3752C2212yes—
21C4×C3⋊Dic3C2211yescharacteristic
22C3×(C2×D3)⋊C4C2212yes—
23C3×(C2×D3)⋊C4C2212yes—
24C3×C4⋊Dic3C2212yes—
25C3×Dic3⋊C4C2212yes—
26C3×Dic3⋊C4C2212yes—
27#3724C2211yescharacteristic
28#3724C2211yescharacteristic
29Dic3⋊Dic3C2212yes—
30#3722C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32C3×C22⋊Q8—32no—
33#947—32no—
34#947—61no—
35#946D312yescomplemented
36#930—32no—
37#929—32no—
38#897—32no—
39#896—32no—
40#895—32no—
41#894—32no—
42#894—32no—
43#889—32no—
44#885—32no—
45#884—32no—
46#884—32no—
47#883—32no—
48C22×C3⋊D3C2312yes—
49C22×C3×D3C2312yes—
50C32×Q8—41no—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2312yes—
53C2×C3⋊Dic3C2311yescharacteristic
54C2×C3⋊Dic3C2311yescharacteristic
55C4⋊(C3⋊D3)—41no—
56C2×C3×Dic3C2312yes—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C3×C4×D3—42no—
60#627—41no—
61#626—42no—
62#626—42no—
63#625—41no—
64#624—42no—
65#562—91noSylow
66C2×C3×Q8—21no—
67C2×C3×Q8C2×D312yes—
68C22×C3×C4—32no—
69C2×#67—32no—
70C2×#67—32no—
71C2×#67—32no—
72C2×D12—32no—
73C2×D12—32no—
74C2×D12—32no—
75C2×D12—61no—
76C2×C4×D3—32no—
77C2×C4×D3—32no—
78C2×C4×D3C2×D312yes—
79C2×Dic6—32no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—32no—
82C3×C4⋊C4—32no—
83C3×C22⋊C4—32no—
84C3×C22⋊C4—32no—
85#217—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—32no—
93(C2×D3)⋊C4—62no—
94(C2×D3)⋊C4—62no—
95(C2×D3)⋊C4C2×D312yes—
96(C2×D3)⋊C4C2×D312yes—
97C4⋊Dic3—32no—
98C4⋊Dic3C2×D312yes—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4C2×D312yes—
103Dic3⋊C4C2×D312yes—
104C4×Dic3—32no—
105C4×Dic3—61no—
106C22×C32C2411yescharacteristic
107C2×C3⋊D3—22no—
108C2×C3⋊D3—22no—
109C2×C3⋊D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C32×C4—21no—
114C32×C4—22no—
115C3⋊Dic3—21no—
116C3⋊Dic3—21no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119C3×Dic3—22no—
120#125—91no—
121#124—91no—
122#124—91no—
123#123—92no—
124#123—92no—
125C22⋊Q8—32no—
126C22⋊Q8—92no—
127#121—92no—
128#121—92no—
129C23×C3—32no—
130C22×D3—32no—
131C22×D3—32no—
132C22×D3—62no—
133C22×D3C22×D312yes—
134C3×Q8—42no—
135C3×Q8—42no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—122no—
142C2×C3×C4C22×D312yes—
143C2×C3×C4C22×D312yes—
144C2×C3×C4C22×D312yes—
145#67—122no—
146#67—122no—
147#67—122no—
148C2×Dic3—32no—
149C2×Dic3—32no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3C22×D312yes—
153C2×Dic3C22×D312yes—
154C2×Dic3C22×D312yes—
155D12—122no—
156D12—122no—
157D12—122no—
158D12—122no—
159C4×D3—42no—
160C4×D3—122no—
161C4×D3—122no—
162Dic6—122no—
163C2×C32H(5,𝔽2)11yescharacteristic
164C2×C32H(5,𝔽2)11yescharacteristic
165C2×C322−1+411yescharacteristic
166C3⋊D3—42no—
167C3×D3—42no—
168C2×Q8—91no—
169C2×Q8D3211yescharacteristic
170C2×D4—92no—
171C2×D4—92no—
172C2×D4—91no—
173C2×D4—91no—
174C22×C4—32no—
175C22×C4—92no—
176C4⋊C4—32no—
177C4⋊C4—32no—
178C4⋊C4—32no—
179C4⋊C4—92no—
180C4⋊C4—91no—
181C4⋊C4—91no—
182C22⋊C4—32no—
183C22⋊C4—32no—
184C22⋊C4—92no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4—92no—
188C42—91no—
189C22×C3—21no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—62no—
193C22×C3C23×D312yes—
194C2×D3—22no—
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—62no—
198C2×D3—62no—
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—122no—
204C2×D3—122no—
205C2×D3—122no—
206C3×C4—22no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—42no—
211C3×C4—62no—
212C3×C4—62no—
213C3×C4—62no—
214Dic3—22no—
215Dic3—22no—
216Dic3—22no—
217Dic3—62no—
218Dic3—62no—
219Dic3—121no—
220Dic3—121no—
221C32#56211yescharacteristic, complemented, Sylow
222C23—32no—
223C23—92no—
224Q8—41no—
225Q8—361no—
226D4—361no—
227D4—362no—
228D4—362no—
229D4—361no—
230C2×C4—32no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—122no—
236C2×C4—362no—
237C2×C4C2×D3211yescharacteristic
238C2×C4C2×D3212yes—
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3—122no—
243C2×C3#100412yes—
244C2×C3#100712yes—
245C2×C3#100912yes—
246D3—42no—
247D3—122no—
248D3—122no—
249D3—242no—
250C22—62no—
251C22—62no—
252C22—62no—
253C22—182no—
254C22—182no—
255C22—182no—
256C22C22×D3211yescharacteristic
257C4—22no—
258C4—21no—
259C4—62no—
260C4—62no—
261C4—62no—
262C4—181no—
263C4—181no—
264C3—21no—
265C3#564712yescomplemented
266C2—122no—
267C2—362no—
268C2#6431611yescharacteristic
269C2#6432011yescharacteristic
270C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
hypercenterC2×Q8D32
hypocenterC32#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64156C2semidirect
2. C3×#946C2semidirect
3. #63915C2semidirect
4. #63910C2semidirect
5. #63906C2semidirect
6. #63895C2semidirect
7. #63870C2semidirect
8. #63854C2semidirect
9. #63850C2semidirect
35. #946D3semidirect
221. C32#562semidirect
265. C3#5647semidirect

Maximal subgroups

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

subgroupindex
#641562
C3×#9462
#639152
#639102
#639062
#638952
#638702
#638542
#638502
#56473