Finite Group Databasebeta

← #10590618

Subgroups — #10590618

Order 576.

341 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 66 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×#3796C211yesmaximal, characteristic
2#64106C211yesmaximal, characteristic, complemented
3C2×#3719C211yesmaximal, characteristic, complemented
4C2×#3716C211yesmaximal, characteristic, complemented
5C3×#810C211yesmaximal, characteristic, complemented
6C3×#809C211yesmaximal, characteristic, complemented
7#63557C211yesmaximal, characteristic, complemented
8#4808—31nomaximal
9#4783—31nomaximal
10C2×C4×C3⋊D3C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×D12C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C2×#626—21no—
15C2×D3×Dic3—21no—
16#3796—22no—
17#3796—22no—
18C32×C4⋊C4C2211yescharacteristic
19#3752—21no—
20C4⋊(C3⋊Dic3)—21no—
21C2×C3×C3⋊C8C2211yescharacteristic
22C2×C3×C3⋊C8C2211yescharacteristic
23#3719—22no—
24#3716—22no—
25C2×#239—31no—
26C2×#237—31no—
27C2×#216—31no—
28C2×#214—31no—
29C2×Dic12—31no—
30C2×#204—31no—
31#897—31no—
32#897—31no—
33#897—61no—
34C3×#103—31no—
35C3×#102—31no—
36(C2×D3)⋊C8—31no—
37(C2×D3)⋊C8—31no—
38#810D311yescharacteristic, complemented
39#809D311yescharacteristic, complemented
40C22×C3⋊D3D411yescharacteristic
41C22×C3×D3—21no—
42C2×C32×C4—21no—
43C2×C32×C4C2311yescharacteristic
44C2×C3⋊Dic3—21no—
45C2×C3⋊Dic3D411yescharacteristic
46C4×C3⋊D3—21no—
47C4×C3⋊D3—21no—
48#634—21no—
49#634—21no—
50C2×C3×Dic3—21no—
51C3×D12—21no—
52C3×D12D412yes—
53C3×Dic6—21no—
54C3×Dic6D412yes—
55#626—24no—
56D3×Dic3—24no—
57C3×C3⋊C8—21no—
58C3×C3⋊C8—21no—
59#452—91noSylow
60C2×C3×Q8—31no—
61C2×C3×D4—31no—
62C2×#67—61no—
63C22×Dic3—61no—
64#239—62no—
65#239—62no—
66#237—62no—
67#237—62no—
68C2×D12—61no—
69C2×D12C2×D311yescharacteristic
70C2×C4×D3—31no—
71C2×C4×D3—31no—
72C2×C4×D3—61no—
73C2×C4×D3—61no—
74C2×Dic6—31no—
75C2×Dic6—31no—
76C2×Dic6—61no—
77C2×Dic6C2×D311yescharacteristic
78C2×C3×C8—31no—
79C2×C3×C8—31no—
80C3×C4⋊C4—21no—
81C3×C4⋊C4C2×D311yescharacteristic
82C3×C4⋊C4C2×D311yescharacteristic
83#216—62no—
84#214—62no—
85(C2×D3)⋊C4—61no—
86(C2×D3)⋊C4—61no—
87(C2×D3)⋊C4—62no—
88C4⋊Dic3—61no—
89C4⋊Dic3—61no—
90C4⋊Dic3—62no—
91C2×C3⋊C8C2×D311yescharacteristic
92C2×C3⋊C8C2×D311yescharacteristic
93Dic12—62no—
94#204—62no—
95C22×C32C2×D411yescharacteristic
96C2×C3⋊D3—21no—
97C2×C3⋊D3—21no—
98C2×C3⋊D3—21no—
99C2×C3×D3—22no—
100C2×C3×D3—41no—
101C2×C3×D3—41no—
102C32×C4—41no—
103C32×C4C2×D411yescharacteristic
104C32×C4C2×D411yescharacteristic
105C3⋊Dic3—21no—
106C3⋊Dic3—41no—
107C3×Dic3—22no—
108C2×#41—91no—
109C2×Q16—91no—
110C2×SD16—91no—
111C22⋊Q8—91no—
112#103—31no—
113#102—31no—
114C22⋊C8—91no—
115C23×C3—61no—
116C22×D3—21no—
117C22×D3—31no—
118C22×D3—31no—
119C22×D3—61no—
120C3×Q8—32no—
121C3×Q8—61no—
122C3×D4—32no—
123C3×D4—61no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—61no—
129C2×C3×C4C22×D311yescharacteristic
130C2×C3×C4C22×D311yescharacteristic
131#67—64no—
132C2×Dic3—21no—
133C2×Dic3—31no—
134C2×Dic3—31no—
135C2×Dic3—61no—
136C2×Dic3—64no—
137C2×Dic3—62no—
138C2×Dic3—61no—
139C2×Dic3—61no—
140D12—21no—
141D12—64no—
142D12D1212yes—
143C4×D3—64no—
144C4×D3—61no—
145C4×D3—61no—
146C4×D3—61no—
147C4×D3—61no—
148C4×D3—121no—
149C4×D3—121no—
150Dic6—21no—
151Dic6—61no—
152Dic6—61no—
153Dic6—61no—
154Dic6—61no—
155Dic6—121no—
156Dic6—121no—
157Dic6D1212yes—
158C3×C8—61no—
159C3×C8—61no—
160C3⋊C8—21no—
161C3⋊C8—21no—
162C2×C32#12011yescharacteristic
163C2×C32#13511yescharacteristic
164C2×C32#13711yescharacteristic
165C3⋊D3—41no—
166C3×D3—42no—
167#41—182no—
168#41—182no—
169C2×Q8—31no—
170C2×Q8—91no—
171C2×D4—31no—
172C2×D4—181no—
173C22×C4—91no—
174C22×C4—181no—
175Q16—182no—
176SD16—182no—
177C2×C8—31no—
178C2×C8—31no—
179C4⋊C4—181no—
180C4⋊C4D3211yescharacteristic
181C22⋊C4—181no—
182C22×C3—21no—
183C22×C3—62no—
184C22×C3—121no—
185C22×C3—121no—
186C22×C3D3×D411yescharacteristic
187C22×C3D3×D411yescharacteristic
188C2×D3—22no—
189C2×D3—41no—
190C2×D3—41no—
191C2×D3—62no—
192C2×D3—61no—
193C2×D3—61no—
194C2×D3—61no—
195C2×D3—61no—
196C2×D3—61no—
197C2×D3—61no—
198C2×D3—121no—
199C2×D3—121no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—42no—
203C3×C4—41no—
204C3×C4—41no—
205C3×C4—62no—
206C3×C4C2×D1211yescharacteristic
207C3×C4C2×D1211yescharacteristic
208C3×C4D3×D411yescharacteristic
209C3×C4D3×D411yescharacteristic
210Dic3—22no—
211Dic3—61no—
212Dic3—61no—
213Dic3—121no—
214Dic3—122no—
215Dic3—121no—
216Dic3—121no—
217C32#45211yescharacteristic, complemented, Sylow
218C23—61no—
219C23—91no—
220Q8—32no—
221Q8—61no—
222Q8—181no—
223Q8—181no—
224D4—32no—
225D4—61no—
226D4—184no—
227C2×C4—21no—
228C2×C4—61no—
229C2×C4—91no—
230C2×C4—184no—
231C2×C4—181no—
232C2×C4—181no—
233C2×C4—181no—
234C2×C4C2×D3211yescharacteristic
235C8—61no—
236C8—61no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—122no—
241C2×C3#88211yescharacteristic
242C2×C3#88211yescharacteristic
243C2×C3#91211yescharacteristic
244C2×C3#91511yescharacteristic
245C2×C3#91611yescharacteristic
246C2×C3#91811yescharacteristic
247D3—42no—
248D3—122no—
249D3—121no—
250D3—121no—
251C22—62no—
252C22—121no—
253C22—121no—
254C22—181no—
255C22—181no—
256C22—181no—
257C22#381111yescharacteristic
258C4—41no—
259C4—62no—
260C4—181no—
261C4—361no—
262C4D3×D1211yescharacteristic
263C4D3×D1211yescharacteristic
264C3—21no—
265C3#478311yescharacteristic, complemented
266C3#480811yescharacteristic, complemented
267C2—122no—
268C2—361no—
269C2#6392311yescharacteristic
270C2#6393611yescharacteristic
271C2#6395111yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC2×#3796C2
hypercenterC4⋊C4D32
hypocenterC32#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64106C2semidirect
3. C2×#3719C2semidirect
4. C2×#3716C2semidirect
5. C3×#810C2semidirect
6. C3×#809C2semidirect
7. #63557C2semidirect
38. #810D3semidirect
39. #809D3semidirect
217. C32#452semidirect
265. C3#4783semidirect
266. C3#4808semidirect

Maximal subgroups

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

subgroupindex
C2×#37962
#641062
C2×#37192
C2×#37162
C3×#8102
C3×#8092
#635572
#48083
#47833