Finite Group Databasebeta

← #10594974

Subgroups — #10594974

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×#930C211yesmaximal, characteristic
3C3×#924C211yesmaximal, characteristic, complemented
4#67⋊Dic3C212yesmaximal, complemented
5#63962C212yesmaximal, complemented
6#63887C212yesmaximal, complemented
7#63884C212yesmaximal, complemented
8#63858C212yesmaximal, complemented
9#63846C212yesmaximal, complemented
10#5645—31nomaximal
11#5602—31nomaximal
12C22×C32×C4C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic
15C2×C3×#67C2212yes—
16C2×C3×D12C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×D3×Dic3C2212yes—
19#3750C2212yes—
20#3750C2212yes—
21C3×#217C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2211yescharacteristic
24C3×C4⋊Dic3C2211yescharacteristic
25C3×Dic3⋊C4C2212yes—
26Dic3⋊Dic3C2212yes—
27Dic3⋊Dic3C2212yes—
28Dic3⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32Dic32C2212yes—
33C3×C22⋊Q8—31no—
34C3×#121—31no—
35#946—32no—
36#935—32no—
37#935—32no—
38D4×Dic3—32no—
39#930D311yescharacteristic
40#924D311yescharacteristic, complemented
41C2×Dic3⋊C4—31no—
42C2×Dic3⋊C4—31no—
43C2×Dic3⋊C4—61no—
44#898—32no—
45#894—32no—
46#890—32no—
47#889—32no—
48#886—32no—
49#67⋊C4—32no—
50#879—32no—
51#878—32no—
52C23×C32C2311yescharacteristic
53C22×C3×D3C2312yes—
54C2×C32×C4—41no—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3—22no—
58C2×C3⋊Dic3—41no—
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2312yes—
61C3×#67—24no—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64C2×C3×Dic3C2312yes—
65C3×D12—41no—
66C3×Dic6—41no—
67D3×Dic3—42no—
68#539—91noSylow
69C2×C3×Q8—31no—
70C2×C3×D4—32no—
71C2×C3×D4—31no—
72C22×C3×C4—21no—
73C22×C3×C4C2×D311yescharacteristic
74C22×C3×C4C2×D311yescharacteristic
75C2×#67C2×D312yes—
76C22×Dic3—31no—
77C22×Dic3—31no—
78C22×Dic3—31no—
79C22×Dic3—31no—
80C22×Dic3—32no—
81C22×Dic3—61no—
82C22×Dic3—61no—
83C2×D12C2×D311yescharacteristic
84C2×C4×D3—32no—
85C2×Dic6C2×D311yescharacteristic
86C3×C4⋊C4—31no—
87C3×C4⋊C4—32no—
88C3×C4⋊C4—31no—
89C3×C22⋊C4—32no—
90C3×C22⋊C4—32no—
91#217—32no—
92#217—32no—
93#217—32no—
94#217C2×D312yes—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4C2×D312yes—
99C4⋊Dic3—32no—
100C4⋊Dic3—32no—
101C4⋊Dic3—32no—
102C4⋊Dic3C2×D311yescharacteristic
103C4⋊Dic3C2×D311yescharacteristic
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—32no—
110Dic3⋊C4—32no—
111Dic3⋊C4—62no—
112Dic3⋊C4—62no—
113Dic3⋊C4C2×D312yes—
114C4×Dic3—32no—
115C4×Dic3—32no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32#4112yes—
119C22×C32C2411yescharacteristic
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C32×C4—21no—
124C32×C4—21no—
125C3⋊Dic3—22no—
126C3⋊Dic3—22no—
127C3×Dic3—22no—
128C3×Dic3—22no—
129C3×Dic3—22no—
130#126—92no—
131#125—92no—
132#123—92no—
133#123—92no—
134C22⋊Q8—31no—
135C22⋊Q8—92no—
136#121—31no—
137C4×D4—92no—
138C2×C4⋊C4—91no—
139C23×C3—21no—
140C23×C3—32no—
141C23×C3C22×D311yescharacteristic
142C23×C3C22×D311yescharacteristic
143C22×D3C22×D312yes—
144C3×Q8—121no—
145C3×D4—64no—
146C3×D4—121no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—32no—
150C2×C3×C4—32no—
151C2×C3×C4—32no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—42no—
155C2×C3×C4C22×D311yescharacteristic
156C2×C3×C4C22×D311yescharacteristic
157C2×C3×C4C22×D311yescharacteristic
158C2×C3×C4C22×D311yescharacteristic
159#67—24no—
160C2×Dic3—32no—
161C2×Dic3—32no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—62no—
165C2×Dic3—62no—
166C2×Dic3—64no—
167C2×Dic3—62no—
168C2×Dic3—62no—
169C2×Dic3—122no—
170C2×Dic3—121no—
171C2×Dic3—121no—
172C2×Dic3—122no—
173C2×Dic3C22×D312yes—
174C2×Dic3C22×D312yes—
175C2×Dic3C22×D312yes—
176D12—41no—
177C4×D3—122no—
178Dic6—41no—
179C2×C32—22no—
180C2×C32C2×#4111yescharacteristic
181C2×C32H(5,𝔽2)11yescharacteristic
182C2×C322−1+411yescharacteristic
183C3×D3—42no—
184C2×Q8—31no—
185C2×D4—32no—
186C2×D4—31no—
187C22×C4—92no—
188C22×C4—91no—
189C22×C4—91no—
190C22×C4D3211yescharacteristic
191C4⋊C4—31no—
192C4⋊C4—32no—
193C4⋊C4—31no—
194C4⋊C4—92no—
195C4⋊C4—92no—
196C4⋊C4—92no—
197C4⋊C4—92no—
198C4⋊C4—92no—
199C22⋊C4—32no—
200C22⋊C4—32no—
201C22⋊C4—92no—
202C22⋊C4—92no—
203C22⋊C4—92no—
204C42—92no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—22no—
213C22×C3—62no—
214C22×C3—62no—
215C22×C3—62no—
216C22×C3#23712yes—
217C22×C3#23912yes—
218C22×C3C23×D311yescharacteristic
219C22×C3C23×D311yescharacteristic
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4—41no—
229C3×C4—62no—
230C3×C4—62no—
231C3×C4—62no—
232Dic3—22no—
233Dic3—22no—
234Dic3—22no—
235Dic3—64no—
236Dic3—62no—
237Dic3—62no—
238Dic3—62no—
239Dic3—62no—
240Dic3—122no—
241C32#53911yescharacteristic, complemented, Sylow
242C23—32no—
243C23C2×D3211yescharacteristic
244Q8—121no—
245D4—64no—
246D4—121no—
247C2×C4—32no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—41no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4—182no—
254C2×C4—362no—
255C2×C4—361no—
256C2×C4C2×D3211yescharacteristic
257C2×C4C2×D3211yescharacteristic
258C2×C3—24no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—122no—
265C2×C3C2×#23711yescharacteristic
266C2×C3#100411yescharacteristic
267C2×C3#100411yescharacteristic
268C2×C3C2×#23911yescharacteristic
269C2×C3#101011yescharacteristic
270C2×C3#101011yescharacteristic
271D3—42no—
272C22—21no—
273C22—21no—
274C22—62no—
275C22—62no—
276C22—62no—
277C22#379612yes—
278C22C22×D3211yescharacteristic
279C4—21no—
280C4—21no—
281C4—62no—
282C4—62no—
283C4—62no—
284C4—182no—
285C4—182no—
286C3—21no—
287C3#560211yescharacteristic, complemented
288C3#564511yescharacteristic, complemented
289C2—22no—
290C2—122no—
291C2C2×#379611yescharacteristic
292C2#6429811yescharacteristic
293C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC3×#930C2
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
3. C3×#924C2semidirect
4. #67⋊Dic3C2semidirect
5. #63962C2semidirect
6. #63887C2semidirect
7. #63884C2semidirect
8. #63858C2semidirect
9. #63846C2semidirect
40. #924D3semidirect
241. C32#539semidirect
287. C3#5602semidirect
288. C3#5645semidirect

Maximal subgroups

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

subgroupindex
C2×#37502
C3×#9302
C3×#9242
#67⋊Dic32
#639622
#638872
#638842
#638582
#638462
#56453
#56023