Finite Group Databasebeta

← #10595266

Subgroups — #10595266

Order 576.

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

normal subgroups only

This group has 569 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#64146C211yesmaximal, characteristic, complemented
2C3×#940C211yesmaximal, characteristic, complemented
3C3×#935C211yesmaximal, characteristic, complemented
4#63981C212yesmaximal, complemented
5#63978C212yesmaximal, complemented
6#63971C212yesmaximal, complemented
7#63969C212yesmaximal, complemented
8#63886C212yesmaximal, complemented
9#63868C212yesmaximal, complemented
10#5720—31nomaximal
11#5666—31nomaximal
12C23×C3⋊D3C2211yescharacteristic
13C2×C32×D4C2211yescharacteristic
14C2×#638C2212yes—
15C2×C3×#67C2212yes—
16C2×C3×#67C2212yescomplemented
17C22×C3×Dic3C2211yescharacteristic, complemented
18C2×C3×D12C2211yescharacteristic, complemented
19C2×#626C2212yes—
20C2×#626C2212yes—
21C2×D3×Dic3C2212yescomplemented
22#3757C2211yescharacteristic
23#3752C2212yes—
24C3×#217C2212yes—
25C3×#217C2211yescharacteristic, complemented
26C3×Dic3⋊C4C2212yescomplemented
27C3×C4×Dic3C2211yescharacteristic, complemented
28#3724C2212yescomplemented
29Dic3⋊Dic3C2212yescomplemented
30#3722C2212yes—
31#3722C2212yes—
32(C2×D3)⋊Dic3C2212yescomplemented
33C3×#127—31no—
34C3×#123—31no—
35#940D311yescharacteristic, complemented
36#939—32no—
37#939—32no—
38#939—32no—
39#937—31no—
40#937—31no—
41#937—32no—
42#937—61no—
43#935—32no—
44#935D311yescharacteristic, complemented
45#930—32no—
46#898—32no—
47#895—32no—
48#884—32no—
49#883—32no—
50#882—32no—
51#876—32no—
52C23×C32C2311yescharacteristic
53C23×C32C2311yescharacteristic
54C22×C3⋊D3—42no—
55C22×C3⋊D3—41no—
56C22×C3⋊D3C2312yes—
57C22×C3×D3C2312yes—
58C32×D4—41no—
59C2×C32×C4C2311yescharacteristic
60#638—42no—
61C2×C3⋊Dic3C2312yes—
62C3×#67—42no—
63C3×#67—42no—
64C2×C3×Dic3—41no—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×D12—41no—
69#626—42no—
70#626—42no—
71D3×Dic3—42no—
72#560—91noSylow
73C23×D3—31no—
74C23×D3—31no—
75C23×D3—61no—
76C2×C3×D4—21no—
77C2×C3×D4—32no—
78C2×C3×D4—32no—
79C2×C3×D4—31no—
80C2×C3×D4C2×D311yescharacteristic
81C2×C3×D4C2×D311yescharacteristic
82C22×C3×C4—31no—
83C2×#67—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—62no—
88C2×#67C2×D312yes—
89C2×#67C2×D312yescomplemented
90C22×Dic3—32no—
91C22×Dic3C2×D311yescharacteristic, complemented
92C2×D12—32no—
93C2×D12—32no—
94C2×D12C2×D311yescharacteristic, complemented
95C2×C4×D3—32no—
96C3×C4⋊C4—32no—
97C3×C22⋊C4—32no—
98C3×C22⋊C4—31no—
99C3×C42—31no—
100#217—31no—
101#217—31no—
102#217—32no—
103#217—61no—
104#217C2×D312yes—
105#217C2×D311yescharacteristic, complemented
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—62no—
114C4⋊Dic3—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4C2×D312yescomplemented
119C4×Dic3C2×D311yescharacteristic, complemented
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—22no—
123C22×C32—22no—
124C22×C32C2411yescharacteristic
125C2×C3⋊D3—22no—
126C2×C3⋊D3—22no—
127C2×C3⋊D3—22no—
128C2×C3⋊D3—44no—
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C2×C3×D3—22no—
132C32×C4—21no—
133C3⋊Dic3—22no—
134C3×Dic3—22no—
135C3×Dic3—22no—
136C3×Dic3—22no—
137#127—31no—
138#126—92no—
139#123—31no—
140#123—92no—
141#121—92no—
142#121—92no—
143#121—92no—
144#120—92no—
145#120—91no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—32no—
149C23×C3C22×D311yescharacteristic
150C23×C3C22×D311yescharacteristic
151C23×C3C22×D311yescharacteristic
152C23×C3C22×D311yescharacteristic
153C22×D3—32no—
154C22×D3—32no—
155C22×D3—62no—
156C22×D3—121no—
157C22×D3—122no—
158C22×D3—121no—
159C22×D3—122no—
160C22×D3—124no—
161C22×D3—122no—
162C22×D3C22×D312yes—
163C3×D4—41no—
164C3×D4—41no—
165C3×D4—42no—
166C3×D4—121no—
167C3×D4—122no—
168C3×D4—122no—
169C2×C3×C4—21no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—32no—
173C2×C3×C4—121no—
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176#67—42no—
177#67—42no—
178#67—122no—
179#67—122no—
180#67—122no—
181#67—122no—
182#67—124no—
183C2×Dic3—32no—
184C2×Dic3—32no—
185C2×Dic3—41no—
186C2×Dic3—62no—
187C2×Dic3—122no—
188C2×Dic3C22×D312yes—
189C2×Dic3C22×D312yes—
190C2×Dic3C22×D312yes—
191D12—41no—
192D12—122no—
193D12—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—91no—
202C2×D4—32no—
203C2×D4—32no—
204C2×D4—31no—
205C2×D4—92no—
206C2×D4—92no—
207C2×D4—92no—
208C2×D4D3211yescharacteristic
209C22×C4—31no—
210C22×C4—92no—
211C4⋊C4—32no—
212C4⋊C4—92no—
213C4⋊C4—92no—
214C22⋊C4—32no—
215C22⋊C4—31no—
216C22⋊C4—92no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C22⋊C4—91no—
220C22⋊C4—92no—
221C42—31no—
222C22×C3—22no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—41no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3C23×D311yescharacteristic
239C22×C3C23×D311yescharacteristic
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—64no—
244C2×D3—62no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—122no—
251C2×D3—122no—
252C2×D3—128no—
253C2×D3—124no—
254C2×D3—124no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—62no—
259C3×C4—62no—
260C3×C4—62no—
261Dic3—22no—
262Dic3—22no—
263Dic3—22no—
264Dic3—62no—
265Dic3—62no—
266Dic3—122no—
267C32#56011yescharacteristic, complemented, Sylow
268C23—32no—
269C23—92no—
270C23—362no—
271C23—361no—
272C23C2×D3211yescharacteristic
273C23C2×D3211yescharacteristic
274D4—41no—
275D4—121no—
276D4—122no—
277D4—122no—
278D4—362no—
279D4—362no—
280D4—362no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—92no—
285C2×C4—121no—
286C2×C4—362no—
287C2×C4C2×D3211yescharacteristic
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—42no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—81no—
296C2×C3—122no—
297C2×C3#100412yes—
298C2×C3#100411yescharacteristic
299C2×C3#100411yescharacteristic
300C2×C3#100912yes—
301D3—42no—
302D3—124no—
303D3—122no—
304D3—122no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—62no—
310C22—62no—
311C22—62no—
312C22—182no—
313C22—182no—
314C22—182no—
315C22—364no—
316C22C22×D3211yescharacteristic
317C4—21no—
318C4—62no—
319C4—62no—
320C4—62no—
321C4—182no—
322C3—21no—
323C3#566611yescharacteristic, complemented
324C3#572011yescharacteristic, complemented
325C2—41no—
326C2—41no—
327C2—122no—
328C2—362no—
329C2#6431412yes—
330C2#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. #64146C2semidirect
2. C3×#940C2semidirect
3. C3×#935C2semidirect
4. #63981C2semidirect
5. #63978C2semidirect
6. #63971C2semidirect
7. #63969C2semidirect
8. #63886C2semidirect
9. #63868C2semidirect
16. C2×C3×#67C22semidirect
17. C22×C3×Dic3C22semidirect
18. C2×C3×D12C22semidirect
21. C2×D3×Dic3C22semidirect
25. C3×#217C22semidirect
26. C3×Dic3⋊C4C22semidirect
27. C3×C4×Dic3C22semidirect
28. #3724C22semidirect
29. Dic3⋊Dic3C22semidirect
32. (C2×D3)⋊Dic3C22semidirect
35. #940D3semidirect
44. #935D3semidirect
89. C2×#67C2×D3semidirect
91. C22×Dic3C2×D3semidirect
94. C2×D12C2×D3semidirect
105. #217C2×D3semidirect
118. Dic3⋊C4C2×D3semidirect
119. C4×Dic3C2×D3semidirect
267. C32#560semidirect
323. C3#5666semidirect
324. C3#5720semidirect

Maximal subgroups

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

subgroupindex
#641462
C3×#9402
C3×#9352
#639812
#639782
#639712
#639692
#638862
#638682
#57203
#56663