Finite Group Databasebeta

← #10595415

Subgroups — #10595415

Order 576.

473 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 122 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#64155C211yesmaximal, characteristic, complemented
2C3×#946C212yesmaximal, complemented
3#63912C211yesmaximal, characteristic
4#63905C212yesmaximal, complemented
5#63904C212yesmaximal, complemented
6#63883C211yesmaximal, characteristic, complemented
7#63861C212yesmaximal, complemented
8#63854C212yesmaximal, complemented
9#63847C212yesmaximal, complemented
10#5638—32nomaximal
11C2×C32×Q8C2211yescharacteristic
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×C3×C4×D3C2212yes—
14C2×#626C2212yes—
15C2×#625C2211yescharacteristic
16C2×#624C2211yescharacteristic
17C2×#624C2211yescharacteristic
18#3752C2212yes—
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20#3750C2212yes—
21C3×(C2×D3)⋊C4C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26#3724C2211yescharacteristic
27#3724C2212yes—
28Dic3⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31Dic32C2212yes—
32C3×C22⋊Q8—32no—
33#946—32no—
34#946—61no—
35#946D312yescomplemented
36#929—32no—
37C4×#67—32no—
38#898—32no—
39#894—32no—
40#893—32no—
41D3×C4⋊C4—32no—
42#889—32no—
43#889—32no—
44#884—32no—
45#883—32no—
46#883—32no—
47#879—32no—
48C22×C3⋊D3C2311yescharacteristic
49C22×C3×D3C2312yes—
50C32×Q8—41no—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2312yes—
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2311yescharacteristic
55C4×C3⋊D3—41no—
56C2×C3×Dic3C2312yes—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C3×C4×D3—42no—
60#626—24no—
61#625—41no—
62#624—22no—
63#624—41no—
64#539—91noSylow
65C2×C3×Q8—21no—
66C2×C3×Q8C2×D312yes—
67C22×C3×C4—32no—
68C2×#67—32no—
69C2×#67—32no—
70C2×D12—32no—
71C2×C4×D3—32no—
72C2×C4×D3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—61no—
75C2×C4×D3C2×D312yes—
76C3×C4⋊C4—32no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—32no—
79C3×C22⋊C4—32no—
80C3×C22⋊C4—32no—
81#217—32no—
82#217—32no—
83(C2×D3)⋊C4—32no—
84(C2×D3)⋊C4—32no—
85(C2×D3)⋊C4—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—62no—
88(C2×D3)⋊C4C2×D312yes—
89(C2×D3)⋊C4C2×D312yes—
90C4⋊Dic3—32no—
91C4⋊Dic3—32no—
92C4⋊Dic3—61no—
93C4⋊Dic3C2×D312yes—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—62no—
101Dic3⋊C4C2×D312yes—
102Dic3⋊C4C2×D312yes—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C22×C32C2411yescharacteristic
106C2×C3⋊D3—21no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3#4112yes—
109C2×C3×D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—22no—
112C32×C4—21no—
113C32×C4—22no—
114C3⋊Dic3—21no—
115C3⋊Dic3—22no—
116C3×Dic3—22no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119#126—92no—
120#125—92no—
121#123—92no—
122#123—92no—
123C22⋊Q8—32no—
124C22⋊Q8—91no—
125#121—91no—
126C4×D4—92no—
127C2×C4⋊C4—91no—
128C23×C3—32no—
129C22×D3—32no—
130C22×D3—61no—
131C22×D3C22×D312yes—
132C3×Q8—42no—
133C3×Q8—42no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—122no—
140C2×C3×C4C22×D312yes—
141C2×C3×C4C22×D312yes—
142C2×C3×C4C22×D312yes—
143#67—64no—
144#67—122no—
145C2×Dic3—32no—
146C2×Dic3—32no—
147C2×Dic3—32no—
148C2×Dic3—61no—
149C2×Dic3—62no—
150C2×Dic3C22×D312yes—
151C2×Dic3C22×D312yes—
152C2×Dic3C22×D312yes—
153D12—64no—
154C4×D3—42no—
155C4×D3—64no—
156C4×D3—122no—
157C4×D3—122no—
158C4×D3—122no—
159C2×C32C2×#4111yescharacteristic
160C2×C32H(5,𝔽2)11yescharacteristic
161C2×C322−1+411yescharacteristic
162C3⋊D3—22no—
163C3×D3—42no—
164C2×Q8D3211yescharacteristic
165C2×D4—92no—
166C2×D4—91no—
167C22×C4—32no—
168C22×C4—91no—
169C22×C4—91no—
170C22×C4—91no—
171C4⋊C4—32no—
172C4⋊C4—32no—
173C4⋊C4—32no—
174C4⋊C4—92no—
175C4⋊C4—91no—
176C4⋊C4—92no—
177C4⋊C4—92no—
178C4⋊C4—91no—
179C22⋊C4—32no—
180C22⋊C4—32no—
181C22⋊C4—92no—
182C22⋊C4—92no—
183C22⋊C4—92no—
184C42—92no—
185C22×C3—21no—
186C22×C3—62no—
187C22×C3—62no—
188C22×C3—62no—
189C22×C3C23×D312yes—
190C2×D3—22no—
191C2×D3—22no—
192C2×D3—22no—
193C2×D3—34no—
194C2×D3—62no—
195C2×D3—62no—
196C2×D3—62no—
197C2×D3—121no—
198C2×D3—121no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—42no—
204C3×C4—62no—
205C3×C4—62no—
206C3×C4—62no—
207Dic3—22no—
208Dic3—22no—
209Dic3—22no—
210Dic3—62no—
211Dic3—62no—
212Dic3—62no—
213Dic3—121no—
214Dic3—122no—
215C32#53911yescharacteristic, complemented, Sylow
216C23—32no—
217C23—91no—
218Q8—41no—
219D4—184no—
220D4—361no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—92no—
225C2×C4—91no—
226C2×C4—122no—
227C2×C4—182no—
228C2×C4—361no—
229C2×C4—361no—
230C2×C4C2×D3211yescharacteristic
231C2×C4C2×D3212yes—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—122no—
236C2×C3#100412yes—
237C2×C3#100712yes—
238C2×C3D3×#4112yes—
239D3—42no—
240D3—64no—
241D3—122no—
242C22—62no—
243C22—62no—
244C22—62no—
245C22—92no—
246C22—181no—
247C22—181no—
248C22C22×D3211yescharacteristic
249C4—22no—
250C4—21no—
251C4—62no—
252C4—62no—
253C4—62no—
254C4—182no—
255C4—181no—
256C3—21no—
257C3#563812yescomplemented
258C2—122no—
259C2—182no—
260C2#6431611yescharacteristic
261C2#6431911yescharacteristic
262C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radical#63912C2
hypercenterC2×Q8D32
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64155C2semidirect
2. C3×#946C2semidirect
4. #63905C2semidirect
5. #63904C2semidirect
6. #63883C2semidirect
7. #63861C2semidirect
8. #63854C2semidirect
9. #63847C2semidirect
35. #946D3semidirect
215. C32#539semidirect
257. C3#5638semidirect

Maximal subgroups

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

subgroupindex
#641552
C3×#9462
#639122
#639052
#639042
#638832
#638612
#638542
#638472
#56383