Finite Group Databasebeta

← #10591656

Subgroups — #10591656

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#64155C211yesmaximal, characteristic, complemented
3C2×C3×#216C211yesmaximal, characteristic, complemented
4C2×C3×#215C211yesmaximal, characteristic, complemented
5#63566C211yesmaximal, characteristic, complemented
6#63563C211yesmaximal, characteristic, complemented
7#63557C211yesmaximal, characteristic, complemented
8#5187—31nomaximal
9#5173—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C4×C3⋊D3C2211yescharacteristic
12C2×C3×D12C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C2×#626—21no—
15C2×D3×Dic3—21no—
16#3796—22no—
17#3796—22no—
18#3752—21no—
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20#3750—21no—
21C3×#216—22no—
22C3×#215—22no—
23C2×C3×C3⋊C8C2211yescharacteristic
24C2×C3×C3⋊C8C2211yescharacteristic
25C2×#239—31no—
26C2×#237—31no—
27C2×C3×Q16—31no—
28C2×C3×SD16—31no—
29#946—31no—
30#946—31no—
31#946—61no—
32C2×#216D311yescharacteristic, complemented
33C2×#215D311yescharacteristic, complemented
34#835—31no—
35#832—31no—
36#821—31no—
37(C2×D3)⋊C8—31no—
38(C2×D3)⋊C8—31no—
39#816—31no—
40C22×C3⋊D3D411yescharacteristic
41C22×C3×D3—21no—
42C32×Q8—21no—
43C32×Q8—21no—
44C2×C32×C4—21no—
45C2×C32×C4C2311yescharacteristic
46C2×C3⋊Dic3—21no—
47C2×C3⋊Dic3D411yescharacteristic
48C4×C3⋊D3—21no—
49C4×C3⋊D3—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—21no—
61C2×C3×Q8—31no—
62C2×C3×Q8C2×D311yescharacteristic
63C2×C3×Q8C2×D311yescharacteristic
64C2×C3×D4—31no—
65C2×#67—61no—
66C22×Dic3—61no—
67#239—62no—
68#239—62no—
69#237—62no—
70#237—62no—
71C2×D12—61no—
72C2×D12C2×D311yescharacteristic
73C2×C4×D3—31no—
74C2×C4×D3—31no—
75C2×C4×D3—61no—
76C2×C4×D3—61no—
77C2×Dic6C2×D311yescharacteristic
78C3×Q16—62no—
79C3×SD16—62no—
80C2×C3×C8—31no—
81C2×C3×C8—31no—
82#216—22no—
83#215—22no—
84(C2×D3)⋊C4—61no—
85(C2×D3)⋊C4—61no—
86(C2×D3)⋊C4—62no—
87C4⋊Dic3—31no—
88C4⋊Dic3—31no—
89C4⋊Dic3—61no—
90Dic3⋊C4—61no—
91Dic3⋊C4—61no—
92Dic3⋊C4—62no—
93C2×C3⋊C8C2×D311yescharacteristic
94C2×C3⋊C8C2×D311yescharacteristic
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—31no—
110C2×SD16—31no—
111C22⋊Q8—91no—
112#103—91no—
113#102—91no—
114C22⋊C8—91no—
115C23×C3—61no—
116C22×D3—21no—
117C22×D3—31no—
118C22×D3—31no—
119C22×D3—61no—
120C3×Q8—21no—
121C3×Q8—21no—
122C3×Q8—21no—
123C3×Q8—21no—
124C3×Q8—32no—
125C3×Q8—41no—
126C3×Q8—41no—
127C3×Q8—61no—
128C3×D4—32no—
129C3×D4—61no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—61no—
135C2×C3×C4C22×D311yescharacteristic
136C2×C3×C4C22×D311yescharacteristic
137#67—64no—
138C2×Dic3—21no—
139C2×Dic3—31no—
140C2×Dic3—31no—
141C2×Dic3—61no—
142C2×Dic3—64no—
143C2×Dic3—62no—
144C2×Dic3—61no—
145C2×Dic3—61no—
146D12—21no—
147D12—64no—
148D12#6712yes—
149C4×D3—64no—
150C4×D3—61no—
151C4×D3—61no—
152C4×D3—61no—
153C4×D3—61no—
154C4×D3—121no—
155C4×D3—121no—
156Dic6—21no—
157Dic6#6712yes—
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×Q8D3211yescharacteristic
171C2×D4—31no—
172C2×D4—181no—
173C22×C4—91no—
174C22×C4—181no—
175Q16—62no—
176SD16—62no—
177C2×C8—31no—
178C2×C8—31no—
179C4⋊C4—91no—
180C4⋊C4—181no—
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×C4D3×D411yescharacteristic
207C3×C4D3×D411yescharacteristic
208C3×C4C2×#6711yescharacteristic
209C3×C4C2×#6711yescharacteristic
210Dic3—22no—
211Dic3—61no—
212Dic3—61no—
213Dic3—121no—
214Dic3—122no—
215Dic3—121no—
216Dic3—121no—
217C32#45211yescharacteristic, complemented, Sylow
218C23—61no—
219C23—91no—
220Q8—21no—
221Q8—21no—
222Q8—32no—
223Q8—61no—
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#91511yescharacteristic
242C2×C3#91611yescharacteristic
243C2×C3#91811yescharacteristic
244C2×C3#91911yescharacteristic
245C2×C3#93711yescharacteristic
246C2×C3#93711yescharacteristic
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×#6711yescharacteristic
263C4D3×#6711yescharacteristic
264C3—21no—
265C3#517311yescharacteristic, complemented
266C3#518711yescharacteristic, complemented
267C2—122no—
268C2—361no—
269C2#6395011yescharacteristic
270C2#6395111yescharacteristic
271C2#6398311yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC2×#3796C2
hypercenterC2×Q8D32
hypocenterC32#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #64155C2semidirect
3. C2×C3×#216C2semidirect
4. C2×C3×#215C2semidirect
5. #63566C2semidirect
6. #63563C2semidirect
7. #63557C2semidirect
32. C2×#216D3semidirect
33. C2×#215D3semidirect
217. C32#452semidirect
265. C3#5173semidirect
266. C3#5187semidirect

Maximal subgroups

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

subgroupindex
C2×#37962
#641552
C2×C3×#2162
C2×C3×#2152
#635662
#635632
#635572
#51873
#51733