Finite Group Databasebeta

← #10595038

Subgroups — #10595038

Order 576.

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

normal subgroups only

This group has 547 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
1#64139C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
3C3×#924C211yesmaximal, characteristic, complemented
4#63977C212yesmaximal, complemented
5#63952C212yesmaximal, complemented
6#63910C212yesmaximal, complemented
7#63907C212yesmaximal, complemented
8#63861C212yesmaximal, complemented
9#63854C212yesmaximal, complemented
10#5644—31nomaximal
11#5494—31nomaximal
12C23×C3×D3C2211yescharacteristic
13C22×C32×C4C2211yescharacteristic
14C2×#638C2212yes—
15C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C22×C3×Dic3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic, complemented
18C2×#626C2212yes—
19C2×#626C2212yes—
20C2×#624C2212yescomplemented
21#3752C2212yescomplemented
22C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
23C3×#217C2212yes—
24C3×(C2×D3)⋊C4C2212yes—
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic, complemented
27C3×Dic3⋊C4C2212yescomplemented
28#3724C2212yescomplemented
29#3722C2212yescomplemented
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32Dic32C2212yescomplemented
33C3×C22⋊Q8—31no—
34C2×C3×C22⋊C4—31no—
35#953—32no—
36#947—32no—
37#930—31no—
38#930—31no—
39#930—61no—
40C4×#67—32no—
41C2×(C2×D3)⋊C4D311yescharacteristic, complemented
42#924D311yescharacteristic, complemented
43#898—32no—
44#895—32no—
45#894—32no—
46#885—32no—
47#884—32no—
48#883—32no—
49#882—32no—
50#67⋊C4—32no—
51#879—32no—
52C23×C32C2311yescharacteristic
53C22×C3⋊D3C2312yes—
54C22×C3×D3—22no—
55C22×C3×D3—41no—
56C22×C3×D3—41no—
57C22×C3×D3C2312yes—
58C2×C32×C4—41no—
59C2×C32×C4C2311yescharacteristic
60C2×C32×C4C2311yescharacteristic
61#638—24no—
62C2×C3⋊Dic3C2312yes—
63C4⋊(C3⋊D3)—41no—
64C2×C3×Dic3—41no—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×Dic6—41no—
69#626—42no—
70#626—42no—
71#624—42no—
72#538—91noSylow
73C24×C3—31no—
74C23×D3C2×D311yescharacteristic
75C2×C3×Q8—31no—
76C22×C3×C4—21no—
77C22×C3×C4—31no—
78C22×C3×C4C2×D311yescharacteristic
79C22×C3×C4C2×D311yescharacteristic
80C2×#67—32no—
81C2×#67—32no—
82C2×#67—32no—
83C2×#67—32no—
84C2×#67—62no—
85C22×Dic3C2×D311yescharacteristic
86C2×D12—32no—
87C2×D12—32no—
88C2×D12—31no—
89C2×D12—31no—
90C2×D12—61no—
91C2×C4×D3—32no—
92C2×C4×D3—32no—
93C2×Dic6C2×D311yescharacteristic, complemented
94C3×C4⋊C4—31no—
95C3×C4⋊C4—32no—
96C3×C22⋊C4—32no—
97C3×C22⋊C4—32no—
98C3×C22⋊C4—32no—
99#217—32no—
100#217—32no—
101#217C2×D312yes—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—62no—
109(C2×D3)⋊C4C2×D312yes—
110(C2×D3)⋊C4C2×D312yes—
111C4⋊Dic3—31no—
112C4⋊Dic3—31no—
113C4⋊Dic3—61no—
114C4⋊Dic3C2×D311yescharacteristic, complemented
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4C2×D312yescomplemented
118C4×Dic3—32no—
119C4×Dic3—32no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32#4112yes—
123C22×C32C2411yescharacteristic
124C2×C3⋊D3—22no—
125C2×C3⋊D3—22no—
126C2×C3⋊D3—22no—
127C2×C3×D3—22no—
128C2×C3×D3—22no—
129C2×C3×D3—22no—
130C2×C3×D3—44no—
131C32×C4—21no—
132C32×C4—21no—
133C3⋊Dic3—22no—
134C3×Dic3—22no—
135C3×Dic3—22no—
136C3×Dic3—22no—
137#126—92no—
138#124—92no—
139#123—92no—
140C22⋊Q8—31no—
141#121—92no—
142#121—91no—
143#120—92no—
144C4×D4—92no—
145C2×C22⋊C4—31no—
146C23×C3—21no—
147C23×C3—32no—
148C23×C3—62no—
149C23×C3—121no—
150C23×C3—121no—
151C23×C3C22×D311yescharacteristic
152C23×C3C22×D311yescharacteristic
153C22×D3—22no—
154C22×D3—32no—
155C22×D3—32no—
156C22×D3—41no—
157C22×D3—41no—
158C22×D3—62no—
159C22×D3C22×D312yes—
160C3×Q8—121no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—32no—
164C2×C3×C4—32no—
165C2×C3×C4—32no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—121no—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174#67—64no—
175#67—64no—
176#67—124no—
177#67—122no—
178#67—122no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—41no—
182C2×Dic3—62no—
183C2×Dic3C22×D312yes—
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D312yes—
186D12—121no—
187D12—121no—
188D12—122no—
189D12—122no—
190D12—122no—
191C4×D3—122no—
192C4×D3—122no—
193Dic6—41no—
194C2×C32—22no—
195C2×C32C2×#4111yescharacteristic
196C2×C32H(5,𝔽2)11yescharacteristic
197C2×C32H(5,𝔽2)11yescharacteristic
198C3⋊D3—42no—
199C3×D3—42no—
200C24—31no—
201C2×Q8—31no—
202C2×D4—92no—
203C2×D4—92no—
204C2×D4—92no—
205C2×D4—91no—
206C22×C4—31no—
207C22×C4—92no—
208C22×C4D3211yescharacteristic
209C4⋊C4—32no—
210C4⋊C4—31no—
211C4⋊C4—92no—
212C4⋊C4—91no—
213C22⋊C4—32no—
214C22⋊C4—32no—
215C22⋊C4—32no—
216C22⋊C4—92no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C42—92no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—22no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—62no—
230C22×C3—62no—
231C22×C3—62no—
232C22×C3—124no—
233C22×C3#23512yes—
234C22×C3#23912yes—
235C22×C3C23×D311yescharacteristic
236C22×C3C23×D311yescharacteristic
237C2×D3—22no—
238C2×D3—22no—
239C2×D3—22no—
240C2×D3—44no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—62no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—122no—
250C3×C4—22no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—62no—
259Dic3—22no—
260Dic3—22no—
261Dic3—22no—
262Dic3—62no—
263Dic3—62no—
264Dic3—122no—
265C32#53811yescharacteristic, complemented, Sylow
266C23—32no—
267C23—62no—
268C23—92no—
269C23—121no—
270C23—121no—
271C23C2×D3211yescharacteristic
272Q8—121no—
273D4—184no—
274D4—361no—
275D4—362no—
276D4—362no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—41no—
281C2×C4—92no—
282C2×C4—121no—
283C2×C4—362no—
284C2×C4C2×D3211yescharacteristic
285C2×C4C2×D3211yescharacteristic
286C2×C3—22no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—42no—
292C2×C3—122no—
293C2×C3C2×#23511yescharacteristic
294C2×C3#100411yescharacteristic
295C2×C3#100411yescharacteristic
296C2×C3#100411yescharacteristic
297C2×C3C2×#23911yescharacteristic
298C2×C3#100911yescharacteristic
299D3—42no—
300D3—122no—
301D3—122no—
302D3—242no—
303C22—21no—
304C22—21no—
305C22—62no—
306C22—62no—
307C22—62no—
308C22—124no—
309C22—182no—
310C22—182no—
311C22—182no—
312C22#379912yes—
313C22C22×D3211yescharacteristic
314C4—21no—
315C4—21no—
316C4—62no—
317C4—62no—
318C4—62no—
319C4—182no—
320C3—21no—
321C3#549411yescharacteristic, complemented
322C3#564411yescharacteristic, complemented
323C2—22no—
324C2—122no—
325C2—362no—
326C2C2×#379911yescharacteristic
327C2#6430711yescharacteristic
328C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64139C2semidirect
2. C2×C3×(C2×D3)⋊C4C2semidirect
3. C3×#924C2semidirect
4. #63977C2semidirect
5. #63952C2semidirect
6. #63910C2semidirect
7. #63907C2semidirect
8. #63861C2semidirect
9. #63854C2semidirect
15. C2×C4⋊(C3⋊D3)C22semidirect
17. C2×C3×Dic6C22semidirect
20. C2×#624C22semidirect
21. #3752C22semidirect
22. C4⋊(C3⋊Dic3)C22semidirect
26. C3×C4⋊Dic3C22semidirect
27. C3×Dic3⋊C4C22semidirect
28. #3724C22semidirect
29. #3722C22semidirect
32. Dic32C22semidirect
41. C2×(C2×D3)⋊C4D3semidirect
42. #924D3semidirect
93. C2×Dic6C2×D3semidirect
114. C4⋊Dic3C2×D3semidirect
117. Dic3⋊C4C2×D3semidirect
265. C32#538semidirect
321. C3#5494semidirect
322. C3#5644semidirect

Maximal subgroups

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

subgroupindex
#641392
C2×C3×(C2×D3)⋊C42
C3×#9242
#639772
#639522
#639102
#639072
#638612
#638542
#56443
#54943