Finite Group Databasebeta

← #10595276

Subgroups — #10595276

Order 576.

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

normal subgroups only

This group has 431 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#64144C211yesmaximal, characteristic
2C3×#936C211yesmaximal, characteristic
3C3×#935C211yesmaximal, characteristic
4#63982C212yesmaximal, complemented
5#63964C212yesmaximal, complemented
6#63956C212yesmaximal, complemented
7#63956C212yesmaximal, complemented
8#63867C212yesmaximal, complemented
9#63839C212yesmaximal, complemented
10#5672—31nomaximal
11#5651—31nomaximal
12C2×C32×D4C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C2×C3×Dic6C2211yescharacteristic
16C2×#627C2212yes—
17#3757C2211yescharacteristic
18#3757C2212yes—
19#3750C2212yes—
20C3×#217C2211yescharacteristic
21C3×#217C2212yes—
22C3×#217C2212yes—
23C3×#217C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×C4×Dic3C2211yescharacteristic
26#3724C2212yes—
27#3724C2212yes—
28Dic3⋊Dic3C2212yes—
29Dic3⋊Dic3C2212yes—
30Dic3⋊Dic3C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic32C2212yes—
33C3×#124—31no—
34C3×#123—31no—
35#944—32no—
36#936D311yescharacteristic
37#935—31no—
38#935—31no—
39#935—61no—
40#935D311yescharacteristic
41#924—32no—
42#889—32no—
43#888—32no—
44#879—32no—
45#879—32no—
46#879—32no—
47#879—32no—
48#878—32no—
49#878—32no—
50#878—32no—
51#870—32no—
52C23×C32C2311yescharacteristic
53C23×C32C2311yescharacteristic
54C32×D4—41no—
55C2×C32×C4C2311yescharacteristic
56C2×C3⋊Dic3—41no—
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2312yes—
59C2×C3×Dic3—41no—
60C2×C3×Dic3C2312yes—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64C3×Dic6—41no—
65#627—42no—
66#563—91noSylow
67C2×C3×Q8—31no—
68C2×C3×D4—21no—
69C2×C3×D4C2×D311yescharacteristic
70C2×C3×D4C2×D311yescharacteristic
71C22×C3×C4—31no—
72C22×Dic3—31no—
73C22×Dic3—31no—
74C22×Dic3—61no—
75C22×Dic3C2×D311yescharacteristic
76C2×Dic6—32no—
77C2×Dic6—32no—
78C2×Dic6C2×D311yescharacteristic
79C3×C4⋊C4—32no—
80C3×C22⋊C4—32no—
81C3×C22⋊C4—32no—
82C3×C22⋊C4—32no—
83C3×C22⋊C4—31no—
84C3×C42—31no—
85#217—32no—
86#217—31no—
87#217—31no—
88#217—32no—
89#217—62no—
90#217—61no—
91#217C2×D312yes—
92#217C2×D312yes—
93#217C2×D311yescharacteristic
94#217C2×D312yes—
95C4⋊Dic3—32no—
96C4⋊Dic3—32no—
97C4⋊Dic3—32no—
98C4⋊Dic3—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—62no—
110Dic3⋊C4C2×D312yes—
111C4×Dic3—32no—
112C4×Dic3—32no—
113C4×Dic3C2×D311yescharacteristic
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—22no—
117C22×C32—22no—
118C22×C32C2411yescharacteristic
119C32×C4—21no—
120C3⋊Dic3—22no—
121C3⋊Dic3—22no—
122C3×Dic3—22no—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C3×Dic3—22no—
126C4⋊Q8—92no—
127#126—92no—
128#126—92no—
129#125—92no—
130#124—31no—
131#123—31no—
132#123—91no—
133C22⋊Q8—92no—
134C22⋊Q8—92no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3C22×D311yescharacteristic
138C23×C3C22×D311yescharacteristic
139C23×C3C22×D311yescharacteristic
140C23×C3C22×D311yescharacteristic
141C3×Q8—121no—
142C3×D4—41no—
143C3×D4—41no—
144C3×D4—42no—
145C2×C3×C4—21no—
146C2×C3×C4—32no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—121no—
151C2×C3×C4C22×D311yescharacteristic
152C2×C3×C4C22×D311yescharacteristic
153C2×Dic3—32no—
154C2×Dic3—32no—
155C2×Dic3—32no—
156C2×Dic3—32no—
157C2×Dic3—41no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—121no—
161C2×Dic3—121no—
162C2×Dic3—122no—
163C2×Dic3C22×D312yes—
164C2×Dic3C22×D312yes—
165C2×Dic3C22×D312yes—
166C2×Dic3C22×D312yes—
167Dic6—41no—
168Dic6—122no—
169Dic6—122no—
170C2×C32—41no—
171C2×C32—41no—
172C2×C32H(5,𝔽2)11yescharacteristic
173C2×C322−1+412yes—
174C2×Q8—31no—
175C2×Q8—92no—
176C2×D4D3211yescharacteristic
177C22×C4—31no—
178C22×C4—91no—
179C4⋊C4—32no—
180C4⋊C4—92no—
181C4⋊C4—92no—
182C4⋊C4—92no—
183C4⋊C4—92no—
184C4⋊C4—92no—
185C4⋊C4—92no—
186C4⋊C4—92no—
187C22⋊C4—32no—
188C22⋊C4—32no—
189C22⋊C4—32no—
190C22⋊C4—31no—
191C22⋊C4—92no—
192C22⋊C4—91no—
193C42—31no—
194C42—92no—
195C22×C3—22no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—21no—
205C22×C3—42no—
206C22×C3—41no—
207C22×C3—42no—
208C22×C3C23×D311yescharacteristic
209C22×C3C23×D311yescharacteristic
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—41no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—62no—
216C3×C4—62no—
217Dic3—22no—
218Dic3—22no—
219Dic3—22no—
220Dic3—22no—
221Dic3—62no—
222Dic3—62no—
223Dic3—62no—
224Dic3—62no—
225Dic3—122no—
226Dic3—122no—
227C32#56311yescharacteristic, complemented, Sylow
228C23C2×D3211yescharacteristic
229C23C2×D3211yescharacteristic
230Q8—121no—
231Q8—362no—
232D4—41no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—92no—
238C2×C4—92no—
239C2×C4—121no—
240C2×C4—361no—
241C2×C4C2×D3211yescharacteristic
242C2×C3—22no—
243C2×C3—21no—
244C2×C3—42no—
245C2×C3—41no—
246C2×C3—41no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—81no—
250C2×C3#100411yescharacteristic
251C2×C3#100411yescharacteristic
252C2×C3#101012yes—
253C2×C3#101012yes—
254C22—22no—
255C22—22no—
256C22—21no—
257C22—21no—
258C22C22×D3211yescharacteristic
259C4—21no—
260C4—62no—
261C4—62no—
262C4—62no—
263C4—62no—
264C4—182no—
265C4—182no—
266C3—21no—
267C3#565111yescharacteristic, complemented
268C3#567211yescharacteristic, complemented
269C2—41no—
270C2—41no—
271C2#6430912yes—
272C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
Fitting, prime radicalC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #63982C2semidirect
5. #63964C2semidirect
6. #63956C2semidirect
7. #63956C2semidirect
8. #63867C2semidirect
9. #63839C2semidirect
227. C32#563semidirect
267. C3#5651semidirect
268. C3#5672semidirect

Maximal subgroups

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

subgroupindex
#641442
C3×#9362
C3×#9352
#639822
#639642
#639562
#638672
#638392
#56723
#56513