Finite Group Databasebeta

← #10594036

Subgroups — #10594036

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#64095C211yesmaximal, characteristic
2C3×#879C212yesmaximal, complemented
3#63982C212yesmaximal, complemented
4#63975C211yesmaximal, characteristic
5#63953C211yesmaximal, characteristic
6#63918C212yesmaximal, complemented
7#63895C212yesmaximal, complemented
8#63868C212yesmaximal, complemented
9#63845C212yesmaximal, complemented
10#5712—32nomaximal
11C2×#638C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C2×#627C2212yes—
14C2×#627C2211yescharacteristic
15C2×#624C2211yescharacteristic
16C32×C22⋊C4C2211yescharacteristic
17#3752C2212yes—
18C4⋊(C3⋊Dic3)C2212yes—
19C3×#217C2212yes—
20C3×#217C2212yes—
21C3×C4⋊Dic3C2212yes—
22C3×Dic3⋊C4C2212yes—
23C3×Dic3⋊C4C2212yes—
24C3×C4×Dic3C2212yes—
25#3724C2212yes—
26Dic3⋊Dic3C2212yes—
27Dic3⋊Dic3C2212yes—
28Dic3⋊Dic3C2212yes—
29#3722C2211yescharacteristic
30#3722C2212yes—
31Dic32C2211yescharacteristic
32C3×#126—32no—
33#898—32no—
34#897—32no—
35#896—32no—
36#890—32no—
37#889—32no—
38#888—32no—
39#886—32no—
40#886—61no—
41#885—32no—
42#883—32no—
43#879D312yescomplemented
44#878—32no—
45#878—32no—
46#876—32no—
47#869—32no—
48C23×C32C2311yescharacteristic
49C22×C3⋊D3C2311yescharacteristic
50C2×C32×C4C2312yes—
51#638—41no—
52C2×C3⋊Dic3—41no—
53C2×C3⋊Dic3C2311yescharacteristic
54C2×C3⋊Dic3C2312yes—
55C2×C3×Dic3C2312yes—
56C2×C3×Dic3C2312yes—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59#627—42no—
60#627—41no—
61#624—41no—
62#563—91noSylow
63C2×#67—32no—
64C2×#67—61no—
65C22×Dic3—32no—
66C22×Dic3—61no—
67C2×C4×D3—32no—
68C2×Dic6—32no—
69C2×Dic6—32no—
70C2×Dic6—32no—
71C3×C4⋊C4—32no—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—32no—
74C3×C22⋊C4—21no—
75C3×C22⋊C4—32no—
76C3×C22⋊C4—32no—
77C3×C22⋊C4C2×D312yes—
78C3×C42—32no—
79#217C2×D312yes—
80#217C2×D312yes—
81(C2×D3)⋊C4—32no—
82(C2×D3)⋊C4—32no—
83(C2×D3)⋊C4—32no—
84(C2×D3)⋊C4—32no—
85(C2×D3)⋊C4—32no—
86(C2×D3)⋊C4—62no—
87C4⋊Dic3—32no—
88C4⋊Dic3—32no—
89C4⋊Dic3—32no—
90C4⋊Dic3—32no—
91C4⋊Dic3—32no—
92C4⋊Dic3—62no—
93C4⋊Dic3C2×D312yes—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4C2×D312yes—
100Dic3⋊C4C2×D312yes—
101C4×Dic3—32no—
102C4×Dic3C2×D312yes—
103C22×C32—21no—
104C22×C32—22no—
105C22×C32C2411yescharacteristic
106C2×C3⋊D3—21no—
107C2×C3⋊D3—22no—
108C32×C4—22no—
109C3⋊Dic3—22no—
110C3⋊Dic3—21no—
111C3×Dic3—22no—
112C3×Dic3—22no—
113C3×Dic3—22no—
114C3×Dic3—22no—
115C4⋊Q8—92no—
116#126—32no—
117#126—92no—
118#125—92no—
119#124—91no—
120#123—91no—
121#123—91no—
122C22⋊Q8—92no—
123C22⋊Q8—92no—
124C23×C3—21no—
125C23×C3C22×D312yes—
126C22×D3—32no—
127C22×D3—61no—
128C2×C3×C4—22no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—32no—
132C2×C3×C4—32no—
133C2×C3×C4C22×D312yes—
134C2×C3×C4C22×D312yes—
135#67—122no—
136#67—122no—
137C2×Dic3—32no—
138C2×Dic3—32no—
139C2×Dic3—32no—
140C2×Dic3—61no—
141C2×Dic3—62no—
142C2×Dic3—122no—
143C2×Dic3—122no—
144C2×Dic3C22×D312yes—
145C2×Dic3C22×D312yes—
146C2×Dic3C22×D312yes—
147C2×Dic3C22×D312yes—
148C4×D3—122no—
149Dic6—122no—
150Dic6—122no—
151Dic6—122no—
152C2×C32—41no—
153C2×C32H(5,𝔽2)11yescharacteristic
154C2×C322−1+412yes—
155C3⋊D3—41no—
156C2×Q8—92no—
157C2×Q8—91no—
158C2×D4—91no—
159C22×C4—91no—
160C22×C4—91no—
161C4⋊C4—32no—
162C4⋊C4—32no—
163C4⋊C4—32no—
164C4⋊C4—92no—
165C4⋊C4—92no—
166C4⋊C4—92no—
167C4⋊C4—92no—
168C4⋊C4—92no—
169C22⋊C4—32no—
170C22⋊C4—32no—
171C22⋊C4—92no—
172C22⋊C4—91no—
173C22⋊C4—92no—
174C22⋊C4D3211yescharacteristic
175C42—32no—
176C42—91no—
177C22×C3—22no—
178C22×C3—22no—
179C22×C3—22no—
180C22×C3—22no—
181C22×C3—21no—
182C22×C3—42no—
183C22×C3C23×D312yes—
184C2×D3—62no—
185C2×D3—62no—
186C2×D3—62no—
187C2×D3—121no—
188C2×D3—122no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—62no—
193C3×C4—62no—
194C3×C4—62no—
195C3×C4—62no—
196Dic3—22no—
197Dic3—22no—
198Dic3—22no—
199Dic3—22no—
200Dic3—62no—
201Dic3—62no—
202Dic3—62no—
203Dic3—122no—
204Dic3—121no—
205C32#56311yescharacteristic, complemented, Sylow
206C23—91no—
207C23C2×D3211yescharacteristic
208Q8—362no—
209Q8—361no—
210D4—361no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—91no—
216C2×C4—92no—
217C2×C4—361no—
218C2×C4—361no—
219C2×C4C2×D3212yes—
220C2×C3—21no—
221C2×C3—22no—
222C2×C3—42no—
223C2×C3—42no—
224C2×C3#100912yes—
225C2×C3#101012yes—
226C2×C3#101012yes—
227D3—122no—
228D3—241no—
229C22—21no—
230C22—22no—
231C22—181no—
232C22—182no—
233C22C22×D3211yescharacteristic
234C4—22no—
235C4—62no—
236C4—62no—
237C4—62no—
238C4—62no—
239C4—182no—
240C4—181no—
241C3—21no—
242C3#571212yescomplemented
243C2—41no—
244C2—361no—
245C2#6430811yescharacteristic
246C2#6430912yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×#638C22
hypercenterC22⋊C4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#879C2semidirect
3. #63982C2semidirect
6. #63918C2semidirect
7. #63895C2semidirect
8. #63868C2semidirect
9. #63845C2semidirect
43. #879D3semidirect
205. C32#563semidirect
242. C3#5712semidirect

Maximal subgroups

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

subgroupindex
#640952
C3×#8792
#639822
#639752
#639532
#639182
#638952
#638682
#638452
#57123