Finite Group Databasebeta

← #10595271

Subgroups — #10595271

Order 576.

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

normal subgroups only

This group has 622 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
1C2×#3830C211yesmaximal, characteristic
2C22×#627C211yesmaximal, characteristic, complemented
3C3×#935C212yesmaximal, complemented
4#63982C212yesmaximal, complemented
5#63966C212yesmaximal, complemented
6#63957C211yesmaximal, characteristic, complemented
7#63953C212yesmaximal, complemented
8#63865C212yesmaximal, complemented
9#63840C212yesmaximal, complemented
10#5648—32nomaximal
11C2×C32×D4C2211yescharacteristic
12C2×#638C2211yescharacteristic
13C2×#638C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C22×C3⋊Dic3C2211yescharacteristic
16#3830—24no—
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×#634C2211yescharacteristic
19C22×C3×Dic3C2212yes—
20C2×#627—24no—
21C2×#627C2212yes—
22C2×#627C2212yes—
23C3×#217C2214yes—
24C3×#217C2212yes—
25C3×Dic3⋊C4C2214yes—
26#3724C2212yes—
27Dic3⋊Dic3C2214yes—
28#3722C2214yes—
29Dic32C2212yes—
30C2×#237—32no—
31C2×#237—61no—
32C22×Dic6—32no—
33C3×#123—32no—
34#935D312yescomplemented
35#929—32no—
36#896—34no—
37#888—34no—
38#886—32no—
39#885—34no—
40#878—34no—
41#877—32no—
42C23×C32C2311yescharacteristic
43C23×C32C2311yescharacteristic
44C22×C3⋊D3C2311yescharacteristic
45C32×D4—22no—
46C2×C32×C4C2311yescharacteristic
47#638—22no—
48#638—22no—
49C2×C3⋊Dic3—22no—
50C2×C3⋊Dic3D414yes—
51C2×C3⋊Dic3C2311yescharacteristic
52C2×C3⋊Dic3C2311yescharacteristic
53C2×C3⋊Dic3C2311yescharacteristic
54C4×C3⋊D3—22no—
55#634—22no—
56C2×C3×Dic3—24no—
57C2×C3×Dic3C2314yes—
58C2×C3×Dic3C2314yes—
59#627—24no—
60#627—24no—
61#536—91noSylow
62C2×C3×D4—21no—
63C2×C3×D4C2×D312yes—
64C22×C3×C4—32no—
65C2×#67—32no—
66C2×#67—32no—
67C2×#67—61no—
68C2×#67—61no—
69C22×Dic3—32no—
70C22×Dic3—32no—
71C22×Dic3—61no—
72C22×Dic3—61no—
73C22×Dic3C2×D312yes—
74#237—68no—
75#237—68no—
76C2×C4×D3—32no—
77C2×C4×D3—61no—
78C2×Dic6—32no—
79C2×Dic6—34no—
80C2×Dic6—34no—
81C2×Dic6—61no—
82C2×Dic6—68no—
83C3×C4⋊C4—34no—
84C3×C22⋊C4—32no—
85C3×C22⋊C4—34no—
86#217C2×D312yes—
87#217C2×D314yes—
88(C2×D3)⋊C4—34no—
89(C2×D3)⋊C4—34no—
90C4⋊Dic3—34no—
91Dic3⋊C4—34no—
92Dic3⋊C4—34no—
93Dic3⋊C4C2×D314yes—
94C4×Dic3—34no—
95C22×C32—21no—
96C22×C32—22no—
97C22×C32—22no—
98C22×C32C2×D412yes—
99C22×C32C2411yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—22no—
102C32×C4—21no—
103C3⋊Dic3—21no—
104C3⋊Dic3C2×D412yes—
105C3⋊Dic3C2×D412yes—
106C3×Dic3—24no—
107C3×Dic3—24no—
108C2×#41—91no—
109C22×Q8—91no—
110C4⋊Q8—92no—
111#124—92no—
112#123—32no—
113#123—92no—
114C22⋊Q8—92no—
115C22⋊Q8—92no—
116#117—91no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3C22×D312yes—
120C23×C3C22×D312yes—
121C22×D3—32no—
122C22×D3—61no—
123C3×D4—24no—
124C3×D4—24no—
125C2×C3×C4—21no—
126C2×C3×C4—34no—
127C2×C3×C4—34no—
128C2×C3×C4—64no—
129C2×C3×C4C22×D312yes—
130#67—64no—
131#67—64no—
132#67—64no—
133#67—64no—
134C2×Dic3—24no—
135C2×Dic3—38no—
136C2×Dic3—32no—
137C2×Dic3—32no—
138C2×Dic3—32no—
139C2×Dic3—64no—
140C2×Dic3—61no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—64no—
144C2×Dic3—64no—
145C2×Dic3C22×D314yes—
146C2×Dic3C22×D314yes—
147C4×D3—64no—
148C4×D3—64no—
149Dic6—68no—
150Dic6—68no—
151Dic6—64no—
152Dic6—64no—
153C2×C32—22no—
154C2×C32—41no—
155C2×C32C22×D411yescharacteristic
156C2×C322−1+412yes—
157C3⋊D3—41no—
158#41—184no—
159C2×Q8—92no—
160C2×Q8—92no—
161C2×Q8—91no—
162C2×Q8—184no—
163C2×D4—91no—
164C2×D4—91no—
165C2×D4D3211yescharacteristic
166C22×C4—32no—
167C22×C4—91no—
168C22×C4—91no—
169C22×C4—91no—
170C4⋊C4—34no—
171C4⋊C4—92no—
172C4⋊C4—94no—
173C22⋊C4—32no—
174C22⋊C4—34no—
175C22⋊C4—94no—
176C42—92no—
177C22×C3—24no—
178C22×C3—22no—
179C22×C3—22no—
180C22×C3—24no—
181C22×C3—21no—
182C22×C3—22no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3D3×D414yes—
186C22×C3C23×D312yes—
187C2×D3—62no—
188C2×D3—62no—
189C2×D3—64no—
190C2×D3—122no—
191C3×C4—22no—
192C3×C4—22no—
193C3×C4—64no—
194C3×C4—64no—
195Dic3—24no—
196Dic3—24no—
197Dic3—34no—
198Dic3—34no—
199Dic3—62no—
200Dic3—62no—
201Dic3—62no—
202Dic3—62no—
203C32#53611yescharacteristic, complemented, Sylow
204C23—91no—
205C23C2×D3211yescharacteristic
206C23C2×D3211yescharacteristic
207Q8—182no—
208Q8—184no—
209Q8—184no—
210D4—22no—
211D4—182no—
212D4—182no—
213C2×C4—34no—
214C2×C4—34no—
215C2×C4—64no—
216C2×C4—91no—
217C2×C4—91no—
218C2×C4—94no—
219C2×C4—91no—
220C2×C4—182no—
221C2×C4—182no—
222C2×C4C2×D3211yescharacteristic
223C2×C3—24no—
224C2×C3—22no—
225C2×C3—21no—
226C2×C3—42no—
227C2×C3—42no—
228C2×C3—42no—
229C2×C3C2×D3×D412yes—
230C2×C3#101014yes—
231D3—122no—
232D3—122no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—182no—
237C22—181no—
238C22#381112yes—
239C22C22×D3211yescharacteristic
240C4—21no—
241C4—64no—
242C4—64no—
243C4—92no—
244C4—92no—
245C4—181no—
246C3—21no—
247C3#564812yescomplemented
248C2—22no—
249C2—41no—
250C2—361no—
251C2#6430912yes—
252C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
prime radicalC2×#3830C2
hypercenterC2×D4D32
hypocenterC32#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#627C2semidirect
3. C3×#935C2semidirect
4. #63982C2semidirect
5. #63966C2semidirect
6. #63957C2semidirect
7. #63953C2semidirect
8. #63865C2semidirect
9. #63840C2semidirect
34. #935D3semidirect
203. C32#536semidirect
247. C3#5648semidirect

Maximal subgroups

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

subgroupindex
C2×#38302
C22×#6272
C3×#9352
#639822
#639662
#639572
#639532
#638652
#638402
#56483