Finite Group Databasebeta

← #10594514

Subgroups — #10594514

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
1C2×#3795C212yesmaximal, complemented
2#64097C211yesmaximal, characteristic, complemented
3C3×#894C212yesmaximal, complemented
4#63902C211yesmaximal, characteristic, complemented
5#63902C211yesmaximal, characteristic, complemented
6#63896C212yesmaximal, complemented
7#63891C212yesmaximal, complemented
8#63885C212yesmaximal, complemented
9D3×Dic3⋊C4C212yesmaximal, complemented
10#5655—32nomaximal
11C2×#634C2212yes—
12C2×C3×D12C2212yes—
13C2×C3×C4×D3C2212yes—
14C2×#625C2212yes—
15C2×#625C2211yescharacteristic
16C2×#625C2211yescharacteristic
17C2×D3×Dic3C2212yes—
18C2×D3×Dic3C2212yes—
19C2×D3×Dic3C2212yes—
20#3795—28no—
21C32×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23#3750C2212yes—
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×(C2×D3)⋊C4C2212yes—
27C3×(C2×D3)⋊C4C2212yes—
28C3×Dic3⋊C4C2212yes—
29#3724C2211yescharacteristic
30#3724C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32(C2×D3)⋊Dic3C2212yes—
33C2×#237—32no—
34C2×#235—32no—
35C3×#123—32no—
36#939—32no—
37C2×Dic3⋊C4—32no—
38#896—32no—
39#894D312yescomplemented
40D3×C4⋊C4—32no—
41#888—32no—
42#888—61no—
43#886—32no—
44#884—32no—
45#883—32no—
46#67⋊C4—32no—
47#67⋊C4—32no—
48#878—32no—
49C22×C3×D3C2312yes—
50C22×C3×D3C2312yes—
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56#634—24no—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C3×D12—24no—
60C3×C4×D3—24no—
61#625—24no—
62#625—22no—
63#625D414yescomplemented
64D3×Dic3—24no—
65D3×Dic3—24no—
66D3×Dic3—24no—
67#547—91noSylow
68C2×C3×D4—32no—
69C22×C3×C4—32no—
70C2×#67—32no—
71C2×#67—32no—
72C2×#67—32no—
73C2×#67—32no—
74C22×Dic3—32no—
75C22×Dic3—32no—
76C22×Dic3—32no—
77#237—68no—
78#235—68no—
79C2×D12C2×D312yes—
80C2×C4×D3—32no—
81C2×C4×D3—32no—
82C2×C4×D3—32no—
83C2×C4×D3C2×D312yes—
84C2×Dic6—32no—
85C2×Dic6—32no—
86C2×Dic6—62no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4C2×D312yes—
90C3×C22⋊C4—32no—
91C3×C22⋊C4—32no—
92C3×C22⋊C4—32no—
93#217—32no—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4C2×D312yes—
96(C2×D3)⋊C4C2×D312yes—
97(C2×D3)⋊C4C2×D312yes—
98C4⋊Dic3—32no—
99C4⋊Dic3—32no—
100C4⋊Dic3—61no—
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—62no—
107Dic3⋊C4C2×D312yes—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C22×C32C2411yescharacteristic
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C2×C3×D3—22no—
114C2×C3×D3—22no—
115C2×C3×D3—22no—
116C2×C3×D3C2×D414yes—
117C32×C4—21no—
118C32×C4—22no—
119C3⋊Dic3—22no—
120C3⋊Dic3C2×D412yes—
121C3⋊Dic3#4112yescomplemented
122C3×Dic3—22no—
123C3×Dic3—22no—
124C2×#41—92no—
125C4⋊Q8—91no—
126#123—32no—
127#123—92no—
128C22⋊Q8—92no—
129#121—92no—
130C4×D4—91no—
131C4×D4—91no—
132C2×C4⋊C4—92no—
133C23×C3—32no—
134C23×C3—32no—
135C22×D3C22×D312yes—
136C22×D3C22×D312yes—
137C3×D4—64no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—32no—
141C2×C3×C4—32no—
142C2×C3×C4—64no—
143C2×C3×C4C22×D312yes—
144C2×C3×C4C22×D312yes—
145C2×C3×C4C22×D312yes—
146#67—38no—
147#67—64no—
148#67—64no—
149#67—64no—
150C2×Dic3—32no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—32no—
154C2×Dic3—61no—
155C2×Dic3—61no—
156C2×Dic3—62no—
157C2×Dic3—64no—
158C2×Dic3—64no—
159C2×Dic3—64no—
160C2×Dic3C22×D312yes—
161C2×Dic3C22×D312yes—
162D12—24no—
163C4×D3—24no—
164C4×D3—64no—
165C4×D3—64no—
166C4×D3—64no—
167Dic6—64no—
168Dic6—64no—
169Dic6—124no—
170C2×C32C22×D411yescharacteristic
171C2×C32C2×#4111yescharacteristic
172C2×C322−1+411yescharacteristic
173C3×D3—24no—
174C3×D3—42no—
175#41—188no—
176C2×Q8—92no—
177C2×D4—32no—
178C2×D4—91no—
179C2×D4—91no—
180C2×D4—92no—
181C22×C4—32no—
182C22×C4—92no—
183C22×C4—92no—
184C22×C4—92no—
185C4⋊C4—32no—
186C4⋊C4—91no—
187C4⋊C4—92no—
188C4⋊C4—91no—
189C4⋊C4—92no—
190C4⋊C4—91no—
191C4⋊C4D3211yescharacteristic
192C22⋊C4—32no—
193C22⋊C4—32no—
194C22⋊C4—32no—
195C22⋊C4—92no—
196C42—91no—
197C22×C3—21no—
198C22×C3—34no—
199C22×C3—62no—
200C22×C3—62no—
201C22×C3—62no—
202C22×C3—62no—
203C22×C3—62no—
204C22×C3C23×D312yes—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—22no—
208C2×D3—22no—
209C2×D3—22no—
210C2×D3D3×D414yes—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—42no—
216C3×C4—62no—
217C3×C4—62no—
218Dic3—22no—
219Dic3—22no—
220Dic3—34no—
221Dic3—34no—
222Dic3—62no—
223Dic3—62no—
224Dic3—62no—
225Dic3—62no—
226Dic3—122no—
227C32#54711yescharacteristic, complemented, Sylow
228C23—32no—
229C23—32no—
230Q8—184no—
231D4—64no—
232D4—94no—
233D4—182no—
234D4—184no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—64no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—92no—
241C2×C4—184no—
242C2×C4—184no—
243C2×C4—184no—
244C2×C4C2×D3211yescharacteristic
245C2×C4C2×D3212yes—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—21no—
249C2×C3—64no—
250C2×C3—122no—
251C2×C3C2×D3×D412yes—
252C2×C3D3×#4112yes—
253C2×C3#101012yes—
254D3—24no—
255D3—42no—
256C22—34no—
257C22—62no—
258C22—62no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22C22×D3211yescharacteristic
263C4—21no—
264C4—22no—
265C4—62no—
266C4—62no—
267C4—92no—
268C4—92no—
269C4—182no—
270C3—21no—
271C3#565512yescomplemented
272C2—64no—
273C2—122no—
274C2#6429811yescharacteristic
275C2D32×D411yescharacteristic
276C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3795C2semidirect
2. #64097C2semidirect
3. C3×#894C2semidirect
4. #63902C2semidirect
5. #63902C2semidirect
6. #63896C2semidirect
7. #63891C2semidirect
8. #63885C2semidirect
9. D3×Dic3⋊C4C2semidirect
39. #894D3semidirect
63. #625D4semidirect
121. C3⋊Dic3#41semidirect
227. C32#547semidirect
271. C3#5655semidirect

Maximal subgroups

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

subgroupindex
C2×#37952
#640972
C3×#8942
#639022
#638962
#638912
#638852
D3×Dic3⋊C42
#56553