Finite Group Databasebeta

← #10595404

Subgroups — #10595404

Order 576.

670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 670 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×#3832C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3C3×#946C212yesmaximal, complemented
4#63912C212yesmaximal, complemented
5#63910C214yesmaximal, complemented
6#63909C212yesmaximal, complemented
7#63900C211yesmaximal, characteristic, complemented
8#63865C212yesmaximal, complemented
9#5629—32nomaximal
10C2×C32×Q8C2211yescharacteristic
11#3832—24no—
12C2×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C4⋊(C3⋊D3)C2212yes—
14C2×C4×C3⋊D3C2212yes—
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×D3C2212yes—
17C2×#627C2211yescharacteristic
18C2×#626C2212yes—
19C2×#626C2214yes—
20C2×#625C2211yescharacteristic
21C2×#624C2212yes—
22#3798—24no—
23C3×(C2×D3)⋊C4C2214yes—
24C3×C4⋊Dic3C2212yes—
25C3×Dic3⋊C4C2214yes—
26#3724C2212yes—
27#3722C2214yes—
28C2×#239—32no—
29C2×#239—61no—
30C2×#235—32no—
31C3×C22⋊Q8—32no—
32#946D312yescomplemented
33#930—32no—
34#896—34no—
35#895—32no—
36#895—34no—
37D3×C4⋊C4—32no—
38#884—34no—
39#884—34no—
40C22×C3⋊D3C2312yes—
41C22×C3⋊D3C2311yescharacteristic
42C22×C3×D3C2312yes—
43C32×Q8—22no—
44C2×C32×C4C2312yes—
45C2×C32×C4C2311yescharacteristic
46C2×C3⋊Dic3C2311yescharacteristic
47C4⋊(C3⋊D3)—24no—
48C4⋊(C3⋊D3)—22no—
49C4×C3⋊D3—24no—
50C4×C3⋊D3D414yescomplemented
51C2×C3×Dic3C2312yes—
52C2×C3×Dic3C2314yes—
53C3×C4×D3—24no—
54#627—22no—
55#626—24no—
56#626—44no—
57#625—22no—
58#624—42no—
59#537—91noSylow
60C2×C3×Q8—21no—
61C2×C3×Q8C2×D312yes—
62C22×C3×C4—32no—
63C2×#67—32no—
64C2×#67—32no—
65C2×#67—34no—
66#239—68no—
67#239—68no—
68#235—68no—
69C2×D12—32no—
70C2×D12—34no—
71C2×D12—32no—
72C2×D12—34no—
73C2×D12—61no—
74C2×D12—62no—
75C2×C4×D3—32no—
76C2×C4×D3—34no—
77C2×C4×D3—34no—
78C2×C4×D3—61no—
79C2×C4×D3—62no—
80C2×C4×D3C2×D312yes—
81C2×Dic6—32no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4—34no—
84C3×C22⋊C4—34no—
85(C2×D3)⋊C4—34no—
86(C2×D3)⋊C4—34no—
87(C2×D3)⋊C4C2×D314yes—
88C4⋊Dic3C2×D312yes—
89Dic3⋊C4—34no—
90Dic3⋊C4C2×D314yes—
91C22×C32C2411yescharacteristic
92C2×C3⋊D3—22no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—21no—
95C2×C3⋊D3—22no—
96C2×C3⋊D3—22no—
97C2×C3⋊D3C2×D412yes—
98C2×C3×D3—22no—
99C2×C3×D3—22no—
100C2×C3×D3—22no—
101C32×C4—22no—
102C32×C4C2×D412yes—
103C3⋊Dic3C2×D412yes—
104C3×Dic3—22no—
105C3×Dic3—24no—
106C2×#41—91no—
107C2×#41—91no—
108C22⋊Q8—32no—
109C22⋊Q8—92no—
110#121—92no—
111#121—94no—
112#121—92no—
113C2×C4⋊C4—91no—
114C23×C3—32no—
115C22×D3—32no—
116C22×D3—34no—
117C22×D3—61no—
118C22×D3—62no—
119C22×D3C22×D312yes—
120C3×Q8—24no—
121C3×Q8—24no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—34no—
125C2×C3×C4—32no—
126C2×C3×C4—64no—
127C2×C3×C4C22×D314yes—
128C2×C3×C4C22×D312yes—
129#67—64no—
130#67—64no—
131#67—124no—
132C2×Dic3—32no—
133C2×Dic3—61no—
134C2×Dic3C22×D314yes—
135C2×Dic3C22×D312yes—
136D12—68no—
137D12—68no—
138D12—64no—
139D12—64no—
140D12—64no—
141D12—124no—
142C4×D3—24no—
143C4×D3—38no—
144C4×D3—64no—
145C4×D3—68no—
146C4×D3—68no—
147C4×D3—124no—
148Dic6—64no—
149C2×C32C22×D411yescharacteristic
150C2×C32H(5,𝔽2)11yescharacteristic
151C2×C322−1+411yescharacteristic
152C3⋊D3—22no—
153C3⋊D3—42no—
154C3×D3—42no—
155#41—184no—
156#41—184no—
157C2×Q8—91no—
158C2×Q8D3211yescharacteristic
159C2×D4—92no—
160C2×D4—94no—
161C2×D4—91no—
162C2×D4—92no—
163C2×D4—91no—
164C22×C4—32no—
165C22×C4—92no—
166C22×C4—91no—
167C22×C4—92no—
168C4⋊C4—32no—
169C4⋊C4—34no—
170C4⋊C4—92no—
171C22⋊C4—34no—
172C22⋊C4—94no—
173C22×C3—21no—
174C22×C3—62no—
175C22×C3—62no—
176C22×C3—62no—
177C22×C3C23×D312yes—
178C2×D3—22no—
179C2×D3—22no—
180C2×D3—22no—
181C2×D3—34no—
182C2×D3—62no—
183C2×D3—62no—
184C2×D3—62no—
185C2×D3—64no—
186C2×D3—64no—
187C2×D3—64no—
188C2×D3—64no—
189C2×D3—121no—
190C2×D3—121no—
191C2×D3—122no—
192C2×D3—122no—
193C3×C4—24no—
194C3×C4—24no—
195C3×C4—22no—
196C3×C4—62no—
197C3×C4—64no—
198C3×C4D3×D414yes—
199Dic3—22no—
200Dic3—24no—
201Dic3—34no—
202Dic3—62no—
203C32#53711yescharacteristic, complemented, Sylow
204C23—32no—
205C23—91no—
206C23—92no—
207Q8—22no—
208Q8—182no—
209D4—184no—
210D4—184no—
211D4—182no—
212D4—182no—
213D4—364no—
214C2×C4—34no—
215C2×C4—32no—
216C2×C4—64no—
217C2×C4—91no—
218C2×C4—94no—
219C2×C4—184no—
220C2×C4—362no—
221C2×C4C2×D3212yes—
222C2×C4C2×D3211yescharacteristic
223C2×C3—21no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—122no—
227C2×C3C2×D3×D412yes—
228C2×C3#100712yes—
229C2×C3#100912yes—
230D3—42no—
231D3—64no—
232D3—122no—
233D3—124no—
234D3—124no—
235C22—62no—
236C22—62no—
237C22—62no—
238C22—92no—
239C22—181no—
240C22—181no—
241C22—182no—
242C22—182no—
243C22—182no—
244C22C22×D3211yescharacteristic
245C4—22no—
246C4—64no—
247C4—62no—
248C4—92no—
249C4#381112yes—
250C3—21no—
251C3#562912yescomplemented
252C2—122no—
253C2—182no—
254C2—362no—
255C2#6431611yescharacteristic
256C2#6432011yescharacteristic
257C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
hypercenterC2×Q8D32
hypocenterC32#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#3832C2semidirect
2. C2×#3798C2semidirect
3. C3×#946C2semidirect
4. #63912C2semidirect
5. #63910C2semidirect
6. #63909C2semidirect
7. #63900C2semidirect
8. #63865C2semidirect
32. #946D3semidirect
50. C4×C3⋊D3D4semidirect
203. C32#537semidirect
251. C3#5629semidirect

Maximal subgroups

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

subgroupindex
C2×#38322
C2×#37982
C3×#9462
#639122
#639102
#639092
#639002
#638652
#56293