Finite Group Databasebeta

← #10594921

Subgroups — #10594921

Order 576.

618 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22×#634C211yesmaximal, characteristic, complemented
2C2×#3805C211yesmaximal, characteristic
3C3×#929C212yesmaximal, complemented
4#63962C212yesmaximal, complemented
5#63957C211yesmaximal, characteristic, complemented
6#63913C212yesmaximal, complemented
7#63896C214yesmaximal, complemented
8#63840C212yesmaximal, complemented
9#5648—32nomaximal
10C22×C32×C4C2211yescharacteristic
11C22×C3⋊Dic3C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C2×#634—24no—
14C2×#634C2212yes—
15C2×#634C2212yes—
16C2×C3×#67C2212yes—
17C2×#627C2211yescharacteristic
18C2×#625C2211yescharacteristic
19#3805—24no—
20C2×D3×Dic3C2212yes—
21C3×#217C2212yes—
22C3×(C2×D3)⋊C4C2214yes—
23C3×Dic3⋊C4C2214yes—
24#3724C2212yes—
25Dic3⋊Dic3C2214yes—
26(C2×D3)⋊Dic3C2214yes—
27Dic32C2212yes—
28C2×#237—32no—
29C22×Dic6—32no—
30C22×Dic6—61no—
31C3×#123—32no—
32#935—32no—
33#929D312yescomplemented
34#896—34no—
35#888—34no—
36#886—32no—
37#885—34no—
38#878—34no—
39#877—32no—
40C23×C32C2311yescharacteristic
41C22×C3×D3C2312yes—
42C2×C32×C4—22no—
43C2×C32×C4C2312yes—
44C2×C3⋊Dic3—22no—
45C2×C3⋊Dic3D414yes—
46C2×C3⋊Dic3C2312yes—
47C2×C3⋊Dic3C2311yescharacteristic
48C2×C3⋊Dic3C2311yescharacteristic
49#634—24no—
50#634—24no—
51C3×#67—24no—
52C2×C3×Dic3C2314yes—
53C2×C3×Dic3C2312yes—
54#627—22no—
55#625—22no—
56D3×Dic3—24no—
57#536—91noSylow
58C2×C3×D4—32no—
59C22×C3×C4—21no—
60C22×C3×C4C2×D312yes—
61C2×#67—32no—
62C2×#67C2×D312yes—
63C22×Dic3—32no—
64C22×Dic3—32no—
65C22×Dic3—32no—
66C22×Dic3—61no—
67C22×Dic3—61no—
68#237—68no—
69C2×C4×D3—32no—
70C2×Dic6—34no—
71C2×Dic6—34no—
72C2×Dic6—32no—
73C2×Dic6—62no—
74C2×Dic6—62no—
75C2×Dic6—68no—
76C2×Dic6—68no—
77C3×C4⋊C4—34no—
78C3×C22⋊C4—32no—
79C3×C22⋊C4—34no—
80#217—34no—
81#217C2×D312yes—
82(C2×D3)⋊C4—34no—
83(C2×D3)⋊C4C2×D314yes—
84C4⋊Dic3—34no—
85Dic3⋊C4—34no—
86Dic3⋊C4—34no—
87Dic3⋊C4C2×D314yes—
88C4×Dic3—34no—
89C22×C32—22no—
90C22×C32C2×D412yes—
91C22×C32C2411yescharacteristic
92C2×C3×D3—22no—
93C2×C3×D3—24no—
94C32×C4—22no—
95C3⋊Dic3—22no—
96C3⋊Dic3C2×D412yes—
97C3⋊Dic3C2×D412yes—
98C3×Dic3—22no—
99C3×Dic3—24no—
100C2×#41—91no—
101C22×Q8—91no—
102C4⋊Q8—92no—
103#124—92no—
104#123—32no—
105#123—92no—
106C22⋊Q8—94no—
107#117—91no—
108C23×C3—21no—
109C23×C3—32no—
110C23×C3C22×D312yes—
111C22×D3C22×D312yes—
112C3×D4—64no—
113C2×C3×C4—24no—
114C2×C3×C4—24no—
115C2×C3×C4—22no—
116C2×C3×C4—34no—
117C2×C3×C4—32no—
118C2×C3×C4C22×D314yes—
119#67—24no—
120#67—64no—
121C2×Dic3—38no—
122C2×Dic3—32no—
123C2×Dic3—32no—
124C2×Dic3—34no—
125C2×Dic3—64no—
126C2×Dic3—61no—
127C2×Dic3—61no—
128C2×Dic3—62no—
129C2×Dic3—64no—
130C2×Dic3—64no—
131C2×Dic3—64no—
132C2×Dic3C22×D314yes—
133C2×Dic3C22×D312yes—
134C4×D3—64no—
135Dic6—64no—
136Dic6—68no—
137Dic6—68no—
138Dic6—68no—
139Dic6—68no—
140C2×C32—22no—
141C2×C32C22×D411yescharacteristic
142C2×C322−1+412yes—
143C3×D3—42no—
144#41—184no—
145C2×Q8—91no—
146C2×Q8—92no—
147C2×Q8—92no—
148C2×Q8—184no—
149C2×D4—32no—
150C2×D4—91no—
151C22×C4—92no—
152C22×C4—91no—
153C22×C4—91no—
154C22×C4D3211yescharacteristic
155C4⋊C4—34no—
156C4⋊C4—92no—
157C4⋊C4—94no—
158C22⋊C4—32no—
159C22⋊C4—34no—
160C22⋊C4—94no—
161C42—92no—
162C22×C3—24no—
163C22×C3—24no—
164C22×C3—21no—
165C22×C3—22no—
166C22×C3—62no—
167C22×C3—64no—
168C22×C3D3×D414yes—
169C22×C3C23×D312yes—
170C2×D3—22no—
171C2×D3—24no—
172C3×C4—24no—
173C3×C4—24no—
174C3×C4—62no—
175C3×C4—64no—
176Dic3—22no—
177Dic3—24no—
178Dic3—34no—
179Dic3—34no—
180Dic3—62no—
181Dic3—62no—
182Dic3—64no—
183Dic3—64no—
184C32#53611yescharacteristic, complemented, Sylow
185C23—32no—
186C23C2×D3211yescharacteristic
187Q8—184no—
188Q8—184no—
189Q8—182no—
190D4—64no—
191D4—182no—
192C2×C4—22no—
193C2×C4—34no—
194C2×C4—32no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—94no—
198C2×C4—92no—
199C2×C4—184no—
200C2×C4—182no—
201C2×C4C2×D3212yes—
202C2×C3—24no—
203C2×C3—24no—
204C2×C3—22no—
205C2×C3—21no—
206C2×C3—122no—
207C2×C3C2×D3×D412yes—
208C2×C3#101014yes—
209D3—42no—
210C22—22no—
211C22—64no—
212C22—62no—
213C22#381112yes—
214C22C22×D3211yescharacteristic
215C4—22no—
216C4—64no—
217C4—62no—
218C4—92no—
219C4—92no—
220C4—182no—
221C3—21no—
222C3#564812yescomplemented
223C2—22no—
224C2—122no—
225C2#6429812yes—
226C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×#3805C2
hypercenterC22×C4D32
hypocenterC32#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634C2semidirect
3. C3×#929C2semidirect
4. #63962C2semidirect
5. #63957C2semidirect
6. #63913C2semidirect
7. #63896C2semidirect
8. #63840C2semidirect
33. #929D3semidirect
184. C32#536semidirect
222. C3#5648semidirect

Maximal subgroups

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

subgroupindex
C22×#6342
C2×#38052
C3×#9292
#639622
#639572
#639132
#638962
#638402
#56483