Finite Group Databasebeta

← #10591461

Subgroups — #10591461

Order 576.

329 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 72 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×C3×#237C211yesmaximal, characteristic
2C2×C3×#214C211yesmaximal, characteristic, complemented
3#63856C211yesmaximal, characteristic, complemented
4C2×#3718C211yesmaximal, characteristic, complemented
5#63659C211yesmaximal, characteristic, complemented
6#63566C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#5239—31nomaximal
9#5173—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C3×#67—21no—
12C22×C3×Dic3—21no—
13C3×#237—22no—
14C3×#237—22no—
15C2×C3×C4×D3C2211yescharacteristic
16C2×C3×Dic6C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×#214—22no—
21C2×C3×C3⋊C8C2211yescharacteristic
22#3724—21no—
23(C2×D3)⋊Dic3—21no—
24#3718—22no—
25C2×C3×#41—31no—
26C2×#237D311yescharacteristic
27C2×C3×SD16—31no—
28#946—31no—
29C2×#216—31no—
30C2×#216—31no—
31C2×#214D311yescharacteristic, complemented
32#924—31no—
33#835—31no—
34#832—31no—
35#832—31no—
36#832—61no—
37#830—31no—
38(C2×D3)⋊C8—31no—
39#816—31no—
40C23×C32—21no—
41C22×C3×D3D411yescharacteristic
42C32×D4—21no—
43C32×D4D412yes—
44C2×C32×C4C2311yescharacteristic
45C2×C3⋊Dic3—21no—
46C3×#67—24no—
47C2×C3×Dic3—24no—
48C2×C3×Dic3—21no—
49C2×C3×Dic3—21no—
50C2×C3×Dic3D411yescharacteristic
51C3×C4×D3—21no—
52C3×C4×D3—21no—
53C3×Dic6—21no—
54C3×Dic6—21no—
55C3×Dic6—21no—
56C3×Dic6D412yes—
57C3⋊(C3⋊C8)—21no—
58C3×C3⋊C8—21no—
59#452—91noSylow
60C3×#41—62no—
61C3×#41—62no—
62C2×C3×Q8—31no—
63C2×C3×Q8—31no—
64C2×C3×D4—21no—
65C2×C3×D4—61no—
66C2×C3×D4C2×D311yescharacteristic
67C2×C3×D4C2×D311yescharacteristic
68C22×C3×C4—31no—
69C22×C3×C4—61no—
70C2×#67—21no—
71C22×Dic3—21no—
72#237—22no—
73#237—22no—
74C2×C4×D3C2×D311yescharacteristic
75C2×Dic6C2×D311yescharacteristic
76C2×Dic6C2×D311yescharacteristic
77C3×SD16—62no—
78C2×C3×C8—31no—
79#217—61no—
80#216—62no—
81#216—62no—
82#214—22no—
83(C2×D3)⋊C4—61no—
84C4⋊Dic3—31no—
85C4⋊Dic3—31no—
86C4⋊Dic3—61no—
87Dic3⋊C4—61no—
88Dic3⋊C4—61no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8—31no—
91C2×C3⋊C8—61no—
92C2×C3⋊C8C2×D311yescharacteristic
93C22×C32—22no—
94C22×C32—41no—
95C22×C32—41no—
96C22×C32C2×D411yescharacteristic
97C2×C3×D3—21no—
98C2×C3×D3—21no—
99C2×C3×D3—21no—
100C32×C4C2×D411yescharacteristic
101C32×C4C2×D411yescharacteristic
102C3⋊Dic3—41no—
103C3×Dic3—22no—
104C3×Dic3—21no—
105C3×Dic3—41no—
106C2×#41—31no—
107C2×Q16—91no—
108C2×SD16—31no—
109C22⋊Q8—91no—
110#103—91no—
111#102—91no—
112C22⋊C8—91no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—31no—
116C23×C3—41no—
117C22×D3#6711yescharacteristic
118C3×Q8—32no—
119C3×Q8—61no—
120C3×Q8—61no—
121C3×Q8—61no—
122C3×D4—21no—
123C3×D4—21no—
124C3×D4—22no—
125C3×D4—41no—
126C3×D4—64no—
127C3×D4#6712yes—
128C3×D4#6712yes—
129C2×C3×C4—21no—
130C2×C3×C4—31no—
131C2×C3×C4—64no—
132C2×C3×C4—61no—
133C2×C3×C4—61no—
134C2×C3×C4—61no—
135C2×C3×C4—61no—
136C2×C3×C4C22×D311yescharacteristic
137C2×C3×C4C22×D311yescharacteristic
138#67—24no—
139C2×Dic3—24no—
140C2×Dic3—21no—
141C2×Dic3—21no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—121no—
145C2×Dic3#6711yescharacteristic
146C4×D3—21no—
147C4×D3—21no—
148Dic6—21no—
149Dic6—21no—
150Dic6—21no—
151Dic6#6712yes—
152C3×C8—61no—
153C3⋊C8—21no—
154C3⋊C8—61no—
155C3⋊C8—61no—
156C3⋊C8—121no—
157C2×C32—42no—
158C2×C32#12011yescharacteristic
159C2×C32#13511yescharacteristic
160C2×C32#13711yescharacteristic
161C3×D3—41no—
162#41—62no—
163#41—62no—
164C2×Q8—31no—
165C2×Q8—31no—
166C2×D4—61no—
167C2×D4D3211yescharacteristic
168C22×C4—31no—
169C22×C4—61no—
170Q16—182no—
171SD16—62no—
172C2×C8—31no—
173C2×C8—91no—
174C4⋊C4—91no—
175C4⋊C4—181no—
176C22⋊C4—181no—
177C22×C3—22no—
178C22×C3—22no—
179C22×C3—21no—
180C22×C3—42no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—61no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—81no—
189C22×C3—81no—
190C22×C3D3×D411yescharacteristic
191C22×C3C2×#6711yescharacteristic
192C2×D3—21no—
193C2×D3—21no—
194C2×D3—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—61no—
198C3×C4—62no—
199C3×C4—121no—
200C3×C4D3×D411yescharacteristic
201C3×C4C2×#6711yescharacteristic
202C3×C4C2×#6711yescharacteristic
203C3×C4C2×#6711yescharacteristic
204Dic3—22no—
205Dic3—21no—
206Dic3—41no—
207Dic3—121no—
208Dic3—121no—
209Dic3—241no—
210C32#45211yescharacteristic, complemented, Sylow
211C23—21no—
212C23—31no—
213Q8—32no—
214Q8—61no—
215Q8—61no—
216Q8—61no—
217D4—21no—
218D4—64no—
219D4#62512yes—
220C2×C4—31no—
221C2×C4—64no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—181no—
227C2×C4C2×D3211yescharacteristic
228C8—61no—
229C8—181no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—82no—
236C2×C3—121no—
237C2×C3#91511yescharacteristic
238C2×C3#91611yescharacteristic
239C2×C3#93711yescharacteristic
240C2×C3#95011yescharacteristic
241C2×C3#95111yescharacteristic
242C2×C3#95311yescharacteristic
243D3—41no—
244C22—22no—
245C22—41no—
246C22—41no—
247C22—61no—
248C22—61no—
249C22—61no—
250C22D3×#6711yescharacteristic
251C4—61no—
252C4—62no—
253C4—121no—
254C4—361no—
255C4C2×#62511yescharacteristic
256C4D3×#6711yescharacteristic
257C3—21no—
258C3#517311yescharacteristic, complemented
259C3#523911yescharacteristic, complemented
260C2—42no—
261C2—121no—
262C2#6392911yescharacteristic
263C2#6393511yescharacteristic
264C2#6397611yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC2×C3×#237C2
hypercenterC2×D4D32
hypocenterC32#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#214C2semidirect
3. #63856C2semidirect
4. C2×#3718C2semidirect
5. #63659C2semidirect
6. #63566C2semidirect
7. (C2×D3)⋊(C3⋊C8)C2semidirect
31. C2×#214D3semidirect
210. C32#452semidirect
258. C3#5173semidirect
259. C3#5239semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2372
C2×C3×#2142
#638562
C2×#37182
#636592
#635662
(C2×D3)⋊(C3⋊C8)2
#52393
#51733