Finite Group Databasebeta

← #10591170

Subgroups — #10591170

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×#235C211yesmaximal, characteristic
2C3×#924C211yesmaximal, characteristic, complemented
3C2×#3718C211yesmaximal, characteristic, complemented
4C2×#3715C211yesmaximal, characteristic, complemented
5#63657C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#5239—31nomaximal
9#5047—31nomaximal
10C22×C32×C4C2211yescharacteristic
11C2×C3×#67—21no—
12C3×#235—22no—
13C3×#235—22no—
14C2×C3×D12C2211yescharacteristic
15C2×C3×C4×D3—21no—
16C2×C3×Dic6C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×#217—21no—
21C3×C4⋊Dic3C2211yescharacteristic
22C3×Dic3⋊C4—21no—
23#3718—22no—
24#3715—22no—
25C2×C3×#41—31no—
26C2×#235D311yescharacteristic
27C3×C22⋊Q8—31no—
28C2×#216—31no—
29C2×#216—31no—
30C2×#215—31no—
31C2×#214—31no—
32#924D311yescharacteristic, complemented
33#835—31no—
34#832—31no—
35#830—31no—
36#830—31no—
37#830—61no—
38#810—31no—
39#809—31no—
40C23×C32D411yescharacteristic
41C22×C3×D3—21no—
42C2×C32×C4—21no—
43C2×C32×C4—21no—
44C2×C32×C4D411yescharacteristic
45C2×C32×C4C2311yescharacteristic
46C3×#67—24no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C2×C3×Dic3—21no—
50C3×D12—21no—
51C3×D12D412yes—
52C3×C4×D3—24no—
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—31no—
65C2×C3×D4—61no—
66C22×C3×C4—21no—
67C22×C3×C4—61no—
68C22×C3×C4C2×D311yescharacteristic
69C22×C3×C4C2×D311yescharacteristic
70C2×#67—21no—
71#235—22no—
72#235—22no—
73C2×D12C2×D311yescharacteristic
74C2×C4×D3—21no—
75C2×Dic6C2×D311yescharacteristic
76C2×Dic6C2×D311yescharacteristic
77C3×C4⋊C4—31no—
78C3×C4⋊C4—61no—
79C3×C22⋊C4—61no—
80#217—21no—
81#216—62no—
82#216—62no—
83#215—62no—
84#214—62no—
85C4⋊Dic3C2×D311yescharacteristic
86Dic3⋊C4—21no—
87C2×C3⋊C8—31no—
88C2×C3⋊C8—31no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8—31no—
91C2×C3⋊C8—61no—
92C2×C3⋊C8—61no—
93C22×C32—21no—
94C22×C32—21no—
95C22×C32—21no—
96C22×C32C2×D411yescharacteristic
97C2×C3×D3—22no—
98C2×C3×D3—41no—
99C2×C3×D3—41no—
100C32×C4—21no—
101C32×C4C2×D411yescharacteristic
102C32×C4C2×D411yescharacteristic
103C3×Dic3—22no—
104C3×Dic3—41no—
105C3×Dic3—41no—
106C2×#41—31no—
107C2×Q16—91no—
108C2×SD16—91no—
109C22⋊Q8—31no—
110#103—91no—
111#102—91no—
112C22⋊C8—91no—
113C23×C3—21no—
114C23×C3—61no—
115C23×C3#6711yescharacteristic
116C23×C3#6711yescharacteristic
117C22×D3—21no—
118C3×Q8—32no—
119C3×Q8—61no—
120C3×Q8—61no—
121C3×Q8—61no—
122C3×D4—32no—
123C3×D4—64no—
124C3×D4—61no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—64no—
134C2×C3×C4—61no—
135C2×C3×C4—61no—
136C2×C3×C4—61no—
137C2×C3×C4#6711yescharacteristic
138C2×C3×C4#6711yescharacteristic
139C2×C3×C4C22×D311yescharacteristic
140C2×C3×C4C22×D311yescharacteristic
141#67—24no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145D12—21no—
146D12#6712yes—
147C4×D3—24no—
148Dic6—21no—
149Dic6—21no—
150Dic6—21no—
151Dic6#6712yes—
152C3⋊C8—61no—
153C3⋊C8—61no—
154C3⋊C8—61no—
155C3⋊C8—61no—
156C3⋊C8—121no—
157C3⋊C8—121no—
158C2×C32—41no—
159C2×C32#12011yescharacteristic
160C2×C32#13511yescharacteristic
161C2×C32#13711yescharacteristic
162C3×D3—42no—
163#41—62no—
164#41—62no—
165C2×Q8—31no—
166C2×Q8—31no—
167C2×D4—31no—
168C2×D4—61no—
169C22×C4—61no—
170C22×C4D3211yescharacteristic
171Q16—182no—
172SD16—182no—
173C2×C8—91no—
174C2×C8—91no—
175C4⋊C4—31no—
176C4⋊C4—61no—
177C22⋊C4—61no—
178C22×C3—22no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—62no—
189C22×C3—121no—
190C22×C3—121no—
191C22×C3C2×#6711yescharacteristic
192C22×C3C2×#6711yescharacteristic
193C2×D3—22no—
194C2×D3—41no—
195C2×D3—41no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—21no—
200C3×C4—41no—
201C3×C4—62no—
202C3×C4—121no—
203C3×C4—121no—
204C3×C4D3×D411yescharacteristic
205C3×C4D3×D411yescharacteristic
206C3×C4C2×#6711yescharacteristic
207C3×C4C2×#6711yescharacteristic
208Dic3—22no—
209Dic3—41no—
210Dic3—41no—
211C32#45211yescharacteristic, complemented, Sylow
212C23—61no—
213C23#62511yescharacteristic
214Q8—32no—
215Q8—61no—
216Q8—61no—
217Q8—61no—
218D4—32no—
219D4—64no—
220D4—61no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—64no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4#62511yescharacteristic
228C2×C4C2×D3211yescharacteristic
229C8—181no—
230C8—181no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—42no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—122no—
238C2×C3#93711yescharacteristic
239C2×C3#94211yescharacteristic
240C2×C3#95011yescharacteristic
241C2×C3#95011yescharacteristic
242C2×C3#95111yescharacteristic
243C2×C3#95311yescharacteristic
244D3—42no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—62no—
249C22—121no—
250C22—121no—
251C22C2×#62511yescharacteristic
252C4—21no—
253C4—62no—
254C4—121no—
255C4—121no—
256C4D3×#6711yescharacteristic
257C4D3×#6711yescharacteristic
258C3—21no—
259C3#504711yescharacteristic, complemented
260C3#523911yescharacteristic, complemented
261C2—41no—
262C2—122no—
263C2#6382211yescharacteristic
264C2#6382711yescharacteristic
265C2#6397611yescharacteristic
subgroupG/H
centerC22C2×#625
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC22×C32×C4C22
socleC22×C32C2×D4
prime radicalC2×C3×#235C2
hypercenterC22×C4D32
hypocenterC32#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#924C2semidirect
3. C2×#3718C2semidirect
4. C2×#3715C2semidirect
5. #63657C2semidirect
6. #63565C2semidirect
7. #63562C2semidirect
32. #924D3semidirect
211. C32#452semidirect
259. C3#5047semidirect
260. C3#5239semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
C3×#9242
C2×#37182
C2×#37152
#636572
#635652
#635622
#52393
#50473