Finite Group Databasebeta

← #10595261

Subgroups — #10595261

Order 576.

558 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 120 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#64149C211yesmaximal, characteristic, complemented
2C3×#937C211yesmaximal, characteristic, complemented
3C3×#935C211yesmaximal, characteristic, complemented
4#63978C212yesmaximal, complemented
5#63977C212yesmaximal, complemented
6#63975C212yesmaximal, complemented
7#63958C212yesmaximal, complemented
8#63910C212yesmaximal, complemented
9#63862C212yesmaximal, complemented
10#5666—31nomaximal
11#5595—31nomaximal
12C23×C3×D3C2211yescharacteristic
13C2×C32×D4C2211yescharacteristic
14C2×#638C2212yes—
15C2×#638C2212yescomplemented
16C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2211yescharacteristic, complemented
19C2×#626C2212yes—
20C2×#626C2212yes—
21C2×#624C2212yescomplemented
22C4×C3⋊Dic3C2211yescharacteristic, complemented
23C3×#217C2212yes—
24C3×#217C2211yescharacteristic
25C3×#217C2211yescharacteristic, complemented
26C3×(C2×D3)⋊C4C2212yes—
27C3×Dic3⋊C4C2212yescomplemented
28Dic3⋊Dic3C2212yescomplemented
29Dic3⋊Dic3C2212yescomplemented
30#3722C2212yescomplemented
31(C2×D3)⋊Dic3C2212yes—
32(C2×D3)⋊Dic3C2212yes—
33C3×#123—31no—
34C3×#120—31no—
35#953—32no—
36#940—31no—
37#940—31no—
38#940—61no—
39#939—32no—
40#938—32no—
41#937D311yescharacteristic, complemented
42#935D311yescharacteristic, complemented
43#930—32no—
44#898—32no—
45#895—32no—
46#884—32no—
47#884—32no—
48#883—32no—
49#883—32no—
50#882—32no—
51#879—32no—
52C23×C32C2311yescharacteristic
53C23×C32C2311yescharacteristic
54C22×C3⋊D3C2312yes—
55C22×C3×D3—41no—
56C22×C3×D3—42no—
57C22×C3×D3C2312yes—
58C32×D4—41no—
59C2×C32×C4C2311yescharacteristic
60#638—42no—
61#638—42no—
62C2×C3⋊Dic3C2312yes—
63C4⋊(C3⋊D3)—41no—
64C3×#67—42no—
65C2×C3×Dic3—41no—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2312yes—
69#626—42no—
70#626—42no—
71#624—42no—
72#560—91noSylow
73C24×C3—31no—
74C23×D3C2×D311yescharacteristic
75C2×C3×D4—21no—
76C2×C3×D4—32no—
77C2×C3×D4C2×D311yescharacteristic
78C2×C3×D4C2×D311yescharacteristic
79C22×C3×C4—31no—
80C2×#67—32no—
81C2×#67—32no—
82C2×#67—32no—
83C2×#67—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—62no—
87C2×#67—62no—
88C2×#67C2×D312yes—
89C22×Dic3C2×D311yescharacteristic, complemented
90C2×D12—32no—
91C2×D12—32no—
92C2×D12—31no—
93C2×D12—31no—
94C2×D12—61no—
95C2×C4×D3—32no—
96C2×C4×D3—32no—
97C3×C4⋊C4—32no—
98C3×C22⋊C4—32no—
99C3×C22⋊C4—31no—
100C3×C22⋊C4—31no—
101C3×C22⋊C4—32no—
102#217—32no—
103#217—32no—
104#217C2×D311yescharacteristic
105#217C2×D312yes—
106#217C2×D311yescharacteristic, complemented
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4C2×D312yes—
112C4⋊Dic3—32no—
113C4⋊Dic3—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4C2×D312yescomplemented
117C4×Dic3—31no—
118C4×Dic3—31no—
119C4×Dic3—61no—
120C22×C32—21no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—21no—
124C22×C32C2411yescharacteristic
125C2×C3⋊D3—22no—
126C2×C3⋊D3—22no—
127C2×C3⋊D3—22no—
128C2×C3×D3—22no—
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C2×C3×D3—44no—
132C32×C4—21no—
133C3⋊Dic3—22no—
134C3×Dic3—22no—
135C3×Dic3—22no—
136C3×Dic3—22no—
137#127—91no—
138#126—92no—
139#123—31no—
140#123—92no—
141#121—92no—
142#121—92no—
143#121—92no—
144#120—31no—
145#120—92no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—32no—
149C23×C3—121no—
150C23×C3—122no—
151C23×C3C22×D311yescharacteristic
152C23×C3C22×D311yescharacteristic
153C23×C3C22×D311yescharacteristic
154C23×C3C22×D311yescharacteristic
155C22×D3—32no—
156C22×D3—32no—
157C22×D3—41no—
158C22×D3—42no—
159C22×D3—62no—
160C22×D3C22×D312yes—
161C3×D4—41no—
162C3×D4—41no—
163C3×D4—42no—
164C3×D4—122no—
165C2×C3×C4—21no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—121no—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172#67—42no—
173#67—122no—
174#67—122no—
175#67—122no—
176#67—122no—
177#67—122no—
178#67—122no—
179#67—124no—
180#67—124no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—41no—
184C2×Dic3—62no—
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D312yes—
188D12—121no—
189D12—121no—
190D12—122no—
191D12—122no—
192D12—122no—
193C4×D3—122no—
194C4×D3—122no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32H(5,𝔽2)11yescharacteristic
198C2×C32H(5,𝔽2)12yes—
199C3⋊D3—42no—
200C3×D3—42no—
201C24—31no—
202C2×D4—32no—
203C2×D4—92no—
204C2×D4—92no—
205C2×D4—92no—
206C2×D4—92no—
207C2×D4—91no—
208C2×D4D3211yescharacteristic
209C22×C4—31no—
210C22×C4—92no—
211C4⋊C4—32no—
212C4⋊C4—92no—
213C4⋊C4—92no—
214C22⋊C4—31no—
215C22⋊C4—32no—
216C22⋊C4—31no—
217C22⋊C4—32no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C42—91no—
222C22×C3—21no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3—124no—
239C22×C3C23×D311yescharacteristic
240C22×C3C23×D311yescharacteristic
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—44no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—122no—
252C2×D3—122no—
253C2×D3—122no—
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4—62no—
260Dic3—22no—
261Dic3—22no—
262Dic3—22no—
263Dic3—64no—
264Dic3—62no—
265Dic3—62no—
266C32#56011yescharacteristic, complemented, Sylow
267C23—32no—
268C23—92no—
269C23—122no—
270C23—121no—
271C23C2×D3211yescharacteristic
272C23C2×D3211yescharacteristic
273D4—41no—
274D4—122no—
275D4—361no—
276D4—362no—
277D4—362no—
278D4—362no—
279D4—362no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—92no—
284C2×C4—121no—
285C2×C4—362no—
286C2×C4C2×D3211yescharacteristic
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—122no—
296C2×C3#100412yes—
297C2×C3#100411yescharacteristic
298C2×C3#100411yescharacteristic
299C2×C3#100912yes—
300D3—42no—
301D3—122no—
302D3—122no—
303D3—242no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—62no—
309C22—62no—
310C22—62no—
311C22—124no—
312C22—182no—
313C22—182no—
314C22—182no—
315C22C22×D3211yescharacteristic
316C4—21no—
317C4—62no—
318C4—62no—
319C4—62no—
320C4—182no—
321C3—21no—
322C3#559511yescharacteristic, complemented
323C3#566611yescharacteristic, complemented
324C2—41no—
325C2—41no—
326C2—122no—
327C2—362no—
328C2#6431412yes—
329C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64149C2semidirect
2. C3×#937C2semidirect
3. C3×#935C2semidirect
4. #63978C2semidirect
5. #63977C2semidirect
6. #63975C2semidirect
7. #63958C2semidirect
8. #63910C2semidirect
9. #63862C2semidirect
15. C2×#638C22semidirect
16. C2×C4⋊(C3⋊D3)C22semidirect
18. C22×C3×Dic3C22semidirect
21. C2×#624C22semidirect
22. C4×C3⋊Dic3C22semidirect
25. C3×#217C22semidirect
27. C3×Dic3⋊C4C22semidirect
28. Dic3⋊Dic3C22semidirect
29. Dic3⋊Dic3C22semidirect
30. #3722C22semidirect
41. #937D3semidirect
42. #935D3semidirect
89. C22×Dic3C2×D3semidirect
106. #217C2×D3semidirect
116. Dic3⋊C4C2×D3semidirect
266. C32#560semidirect
322. C3#5595semidirect
323. C3#5666semidirect

Maximal subgroups

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

subgroupindex
#641492
C3×#9372
C3×#9352
#639782
#639772
#639752
#639582
#639102
#638622
#56663
#55953