Finite Group Databasebeta

← #10595037

Subgroups — #10595037

Order 576.

482 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 128 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×#3750C211yesmaximal, characteristic, complemented
2C3×C4×#67C212yesmaximal, complemented
3#63982C211yesmaximal, characteristic, complemented
4#63980C211yesmaximal, characteristic
5#63962C212yesmaximal, complemented
6#63905C212yesmaximal, complemented
7#63867C212yesmaximal, complemented
8#63859C212yesmaximal, complemented
9#63851C212yesmaximal, complemented
10#5564—32nomaximal
11C22×C32×C4C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C2×C3×#67C2212yes—
15C2×C3×C4×D3C2212yes—
16C2×#627C2211yescharacteristic
17C2×#625C2211yescharacteristic
18#3750C2212yes—
19#3750C2212yes—
20C3×#217C2212yes—
21C3×(C2×D3)⋊C4C2212yes—
22C3×Dic3⋊C4C2212yes—
23C3×C4×Dic3C2212yes—
24#3724C2211yescharacteristic
25#3724C2211yescharacteristic
26Dic3⋊Dic3C2212yes—
27Dic3⋊Dic3C2212yes—
28Dic3⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32C3×C4×D4—32no—
33#939—32no—
34#935—32no—
35#929—32no—
36C4×#67D312yescomplemented
37#924—32no—
38C2×Dic3⋊C4—32no—
39C2×Dic3⋊C4—61no—
40#896—32no—
41#889—32no—
42#886—32no—
43#883—32no—
44#879—32no—
45#878—32no—
46#876—32no—
47#870—32no—
48C23×C32C2311yescharacteristic
49C22×C3×D3C2312yes—
50C2×C32×C4—41no—
51C2×C32×C4C2312yes—
52C2×C3⋊Dic3—22no—
53C2×C3⋊Dic3—41no—
54C2×C3⋊Dic3C2312yes—
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2311yescharacteristic
57C3×#67—24no—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61C3×C4×D3—42no—
62#627—41no—
63#625—41no—
64#539—91noSylow
65C2×C3×D4—32no—
66C22×C3×C4—21no—
67C22×C3×C4—32no—
68C22×C3×C4C2×D312yes—
69C2×#67—32no—
70C2×#67C2×D312yes—
71C22×Dic3—32no—
72C22×Dic3—32no—
73C22×Dic3—61no—
74C22×Dic3—61no—
75C2×C4×D3C2×D312yes—
76C2×Dic6—32no—
77C3×C4⋊C4—32no—
78C3×C22⋊C4—32no—
79C3×C22⋊C4—32no—
80C3×C42—32no—
81#217—32no—
82#217—32no—
83#217—32no—
84#217C2×D312yes—
85(C2×D3)⋊C4—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4C2×D312yes—
89C4⋊Dic3—32no—
90C4⋊Dic3—32no—
91C4⋊Dic3—32no—
92Dic3⋊C4—32no—
93Dic3⋊C4—32no—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—62no—
102Dic3⋊C4—62no—
103Dic3⋊C4C2×D312yes—
104C4×Dic3C2×D312yes—
105C22×C32—21no—
106C22×C32—21no—
107C22×C32#4112yes—
108C22×C32C2411yescharacteristic
109C2×C3×D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—22no—
112C32×C4—22no—
113C3⋊Dic3—22no—
114C3⋊Dic3—21no—
115C3⋊Dic3—21no—
116C3×Dic3—22no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119#126—92no—
120#125—92no—
121#123—92no—
122#123—92no—
123C22⋊Q8—91no—
124C22⋊Q8—92no—
125#121—91no—
126C4×D4—32no—
127C2×C4⋊C4—91no—
128C23×C3—21no—
129C23×C3—32no—
130C23×C3C22×D312yes—
131C22×D3C22×D312yes—
132C3×D4—64no—
133C2×C3×C4—22no—
134C2×C3×C4—32no—
135C2×C3×C4—32no—
136C2×C3×C4—32no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—122no—
140C2×C3×C4C22×D312yes—
141C2×C3×C4C22×D312yes—
142#67—24no—
143#67—122no—
144C2×Dic3—32no—
145C2×Dic3—32no—
146C2×Dic3—32no—
147C2×Dic3—32no—
148C2×Dic3—62no—
149C2×Dic3—61no—
150C2×Dic3—61no—
151C2×Dic3—64no—
152C2×Dic3—64no—
153C2×Dic3—122no—
154C2×Dic3—122no—
155C2×Dic3C22×D312yes—
156C2×Dic3C22×D312yes—
157C2×Dic3C22×D312yes—
158C4×D3—42no—
159Dic6—122no—
160C2×C32—22no—
161C2×C32C2×#4111yescharacteristic
162C2×C32H(5,𝔽2)11yescharacteristic
163C2×C322−1+411yescharacteristic
164C3×D3—42no—
165C2×Q8—91no—
166C2×D4—32no—
167C2×D4—91no—
168C22×C4—32no—
169C22×C4—91no—
170C22×C4—91no—
171C22×C4D3211yescharacteristic
172C4⋊C4—32no—
173C4⋊C4—91no—
174C4⋊C4—92no—
175C4⋊C4—92no—
176C4⋊C4—91no—
177C4⋊C4—92no—
178C4⋊C4—92no—
179C4⋊C4—92no—
180C22⋊C4—32no—
181C22⋊C4—32no—
182C22⋊C4—92no—
183C22⋊C4—92no—
184C22⋊C4—92no—
185C42—32no—
186C22×C3—22no—
187C22×C3—22no—
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—21no—
191C22×C3—22no—
192C22×C3—62no—
193C22×C3—62no—
194C22×C3—62no—
195C22×C3#23514yes—
196C22×C3C23×D312yes—
197C2×D3—22no—
198C2×D3—22no—
199C2×D3—22no—
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—42no—
203C3×C4—62no—
204C3×C4—62no—
205C3×C4—62no—
206Dic3—22no—
207Dic3—22no—
208Dic3—22no—
209Dic3—64no—
210Dic3—62no—
211Dic3—62no—
212Dic3—62no—
213Dic3—62no—
214Dic3—121no—
215Dic3—121no—
216C32#53911yescharacteristic, complemented, Sylow
217C23—32no—
218C23C2×D3211yescharacteristic
219Q8—361no—
220D4—64no—
221D4—361no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—32no—
225C2×C4—41no—
226C2×C4—92no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—122no—
230C2×C4—182no—
231C2×C4—361no—
232C2×C4C2×D3212yes—
233C2×C3—24no—
234C2×C3—24no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—122no—
239C2×C3C2×#23512yes—
240C2×C3#100412yes—
241C2×C3#101012yes—
242D3—42no—
243C22—21no—
244C22—21no—
245C22—62no—
246C22—62no—
247C22—62no—
248C22#379812yes—
249C22C22×D3211yescharacteristic
250C4—22no—
251C4—62no—
252C4—62no—
253C4—62no—
254C4—182no—
255C4—181no—
256C4—181no—
257C3—21no—
258C3#556412yescomplemented
259C2—22no—
260C2—122no—
261C2#6429811yescharacteristic
262C2C2×#379811yescharacteristic
263C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#63980C2
hypercenterC22×C4D32
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3750C2semidirect
2. C3×C4×#67C2semidirect
3. #63982C2semidirect
5. #63962C2semidirect
6. #63905C2semidirect
7. #63867C2semidirect
8. #63859C2semidirect
9. #63851C2semidirect
36. C4×#67D3semidirect
216. C32#539semidirect
258. C3#5564semidirect

Maximal subgroups

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

subgroupindex
C2×#37502
C3×C4×#672
#639822
#639802
#639622
#639052
#638672
#638592
#638512
#55643