Finite Group Databasebeta

← #10595262

Subgroups — #10595262

Order 576.

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

normal subgroups only

This group has 585 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×#937C211yesmaximal, characteristic, complemented
3C3×#936C211yesmaximal, characteristic, complemented
4#63981C212yesmaximal, complemented
5#63977C212yesmaximal, complemented
6#63959C212yesmaximal, complemented
7#63953C212yesmaximal, complemented
8#63907C212yesmaximal, complemented
9#63852C212yesmaximal, complemented
10#5683—31nomaximal
11#5594—31nomaximal
12C23×C3⋊D3C2211yescharacteristic, complemented
13C23×C3×D3C2211yescharacteristic, complemented
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C3×#67C2212yes—
17C2×C3×Dic6C2211yescharacteristic, complemented
18C2×#627C2212yescomplemented
19C2×#626C2212yes—
20C2×#626C2212yes—
21#3757C2211yescharacteristic, complemented
22#3752C2212yescomplemented
23C3×#217C2212yescomplemented
24C3×#217C2212yescomplemented
25C3×#217C2211yescharacteristic, complemented
26C3×(C2×D3)⋊C4C2212yescomplemented
27C3×C4×Dic3C2211yescharacteristic, complemented
28#3722C2212yescomplemented
29#3722C2212yescomplemented
30(C2×D3)⋊Dic3C2212yescomplemented
31(C2×D3)⋊Dic3C2212yescomplemented
32Dic32C2212yescomplemented
33C3×#124—31no—
34C3×#120—31no—
35#953—32no—
36#947—32no—
37#937—31no—
38#937—31no—
39#937—32no—
40#937—61no—
41#937D311yescharacteristic, complemented
42#936—32no—
43#936D311yescharacteristic, complemented
44#885—32no—
45#885—32no—
46#885—32no—
47#885—32no—
48#885—32no—
49#882—32no—
50#882—32no—
51#875—32no—
52C23×C32C2311yescharacteristic
53C23×C32C2311yescharacteristic
54C22×C3⋊D3—42no—
55C22×C3⋊D3—41no—
56C22×C3⋊D3C2312yes—
57C22×C3×D3—41no—
58C22×C3×D3—42no—
59C22×C3×D3C2312yes—
60C32×D4—41no—
61C2×C32×C4C2311yescharacteristic
62#638—42no—
63C2×C3⋊Dic3C2312yes—
64C3×#67—42no—
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×Dic6—41no—
69#627—42no—
70#626—42no—
71#626—42no—
72H(3,𝔽4)—91noSylow
73C24×C3—31no—
74C23×D3—31no—
75C23×D3—31no—
76C23×D3—61no—
77C23×D3C2×D311yescharacteristic, complemented
78C2×C3×Q8—31no—
79C2×C3×D4—21no—
80C2×C3×D4—32no—
81C2×C3×D4C2×D311yescharacteristic
82C2×C3×D4C2×D311yescharacteristic
83C2×#67—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—62no—
88C2×#67C2×D312yes—
89C2×D12—32no—
90C2×D12—32no—
91C2×Dic6—32no—
92C2×Dic6—32no—
93C2×Dic6C2×D311yescharacteristic, complemented
94C3×C22⋊C4—32no—
95C3×C22⋊C4—32no—
96C3×C22⋊C4—31no—
97C3×C22⋊C4—32no—
98C3×C42—31no—
99#217—31no—
100#217—31no—
101#217—32no—
102#217—32no—
103#217—61no—
104#217C2×D311yescharacteristic, complemented
105#217C2×D312yescomplemented
106#217C2×D312yescomplemented
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—32no—
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—62no—
116(C2×D3)⋊C4C2×D312yescomplemented
117C4×Dic3—32no—
118C4×Dic3—32no—
119C4×Dic3C2×D311yescharacteristic, complemented
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—44no—
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C2×C3×D3—22no—
132C2×C3×D3—44no—
133C32×C4—21no—
134C3⋊Dic3—22no—
135C3×Dic3—22no—
136C3×Dic3—22no—
137C3×Dic3—22no—
138#124—31no—
139#124—92no—
140#124—92no—
141#124—92no—
142#124—92no—
143#120—31no—
144#120—92no—
145#120—92no—
146#120—91no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—32no—
150C23×C3—121no—
151C23×C3—122no—
152C23×C3C22×D311yescharacteristic
153C23×C3C22×D311yescharacteristic
154C23×C3C22×D311yescharacteristic
155C23×C3C22×D311yescharacteristic
156C22×D3—32no—
157C22×D3—32no—
158C22×D3—41no—
159C22×D3—42no—
160C22×D3—62no—
161C22×D3—121no—
162C22×D3—122no—
163C22×D3—121no—
164C22×D3—122no—
165C22×D3—124no—
166C22×D3—122no—
167C22×D3C22×D312yes—
168C3×Q8—121no—
169C3×D4—41no—
170C3×D4—41no—
171C3×D4—42no—
172C3×D4—122no—
173C2×C3×C4—21no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179#67—42no—
180#67—122no—
181#67—122no—
182#67—122no—
183#67—122no—
184#67—124no—
185C2×Dic3—32no—
186C2×Dic3—32no—
187C2×Dic3—62no—
188C2×Dic3C22×D312yes—
189C2×Dic3C22×D312yes—
190C2×Dic3C22×D312yes—
191D12—122no—
192D12—122no—
193Dic6—41no—
194Dic6—122no—
195Dic6—122no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32H(5,𝔽2)11yescharacteristic
199C2×C32H(5,𝔽2)12yes—
200C3⋊D3—42no—
201C3×D3—42no—
202C24—31no—
203C24—91no—
204C2×Q8—31no—
205C2×Q8—92no—
206C2×D4—32no—
207C2×D4—92no—
208C2×D4—92no—
209C2×D4—92no—
210C2×D4D3211yescharacteristic
211C22⋊C4—31no—
212C22⋊C4—32no—
213C22⋊C4—32no—
214C22⋊C4—32no—
215C22⋊C4—92no—
216C22⋊C4—92no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C22⋊C4—91no—
220C22⋊C4—92no—
221C42—31no—
222C42—92no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—41no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3—62no—
239C22×C3—124no—
240C22×C3C23×D311yescharacteristic
241C22×C3C23×D311yescharacteristic
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—44no—
246C2×D3—64no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—62no—
253C2×D3—122no—
254C2×D3—122no—
255C2×D3—128no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—62no—
262C3×C4—62no—
263C3×C4—62no—
264Dic3—22no—
265Dic3—22no—
266Dic3—22no—
267Dic3—62no—
268Dic3—62no—
269Dic3—122no—
270C32H(3,𝔽4)11yescharacteristic, complemented, Sylow
271C23—32no—
272C23—92no—
273C23—122no—
274C23—121no—
275C23—362no—
276C23—361no—
277C23C2×D3211yescharacteristic
278C23C2×D3211yescharacteristic
279Q8—121no—
280Q8—362no—
281D4—41no—
282D4—122no—
283D4—362no—
284D4—362no—
285D4—362no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—92no—
290C2×C4C2×D3211yescharacteristic
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—42no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—81no—
299C2×C3—122no—
300C2×C3#100412yes—
301C2×C3#100411yescharacteristic
302C2×C3#100411yescharacteristic
303C2×C3#100912yes—
304D3—42no—
305D3—124no—
306D3—122no—
307D3—122no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—62no—
313C22—62no—
314C22—62no—
315C22—124no—
316C22—182no—
317C22—182no—
318C22—182no—
319C22—364no—
320C22C22×D3211yescharacteristic
321C4—21no—
322C4—62no—
323C4—62no—
324C4—62no—
325C4—182no—
326C3—21no—
327C3#559411yescharacteristic, complemented
328C3#568311yescharacteristic, complemented
329C2—41no—
330C2—41no—
331C2—122no—
332C2—362no—
333C2#6431412yes—
334C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32H(3,𝔽4)
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×#937C2semidirect
3. C3×#936C2semidirect
4. #63981C2semidirect
5. #63977C2semidirect
6. #63959C2semidirect
7. #63953C2semidirect
8. #63907C2semidirect
9. #63852C2semidirect
12. C23×C3⋊D3C22semidirect
13. C23×C3×D3C22semidirect
17. C2×C3×Dic6C22semidirect
18. C2×#627C22semidirect
21. #3757C22semidirect
22. #3752C22semidirect
23. C3×#217C22semidirect
24. C3×#217C22semidirect
25. C3×#217C22semidirect
26. C3×(C2×D3)⋊C4C22semidirect
27. C3×C4×Dic3C22semidirect
28. #3722C22semidirect
29. #3722C22semidirect
30. (C2×D3)⋊Dic3C22semidirect
31. (C2×D3)⋊Dic3C22semidirect
32. Dic32C22semidirect
41. #937D3semidirect
43. #936D3semidirect
77. C23×D3C2×D3semidirect
93. C2×Dic6C2×D3semidirect
104. #217C2×D3semidirect
105. #217C2×D3semidirect
106. #217C2×D3semidirect
116. (C2×D3)⋊C4C2×D3semidirect
119. C4×Dic3C2×D3semidirect
270. C32H(3,𝔽4)semidirect
327. C3#5594semidirect
328. C3#5683semidirect

Maximal subgroups

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

subgroupindex
#641462
C3×#9372
C3×#9362
#639812
#639772
#639592
#639532
#639072
#638522
#56833
#55943