Finite Group Databasebeta

← #10595259

Subgroups — #10595259

Order 576.

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

normal subgroups only

This group has 543 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#64144C211yesmaximal, characteristic, complemented
2C3×#940C211yesmaximal, characteristic, complemented
3C3×#937C211yesmaximal, characteristic, complemented
4#63980C212yesmaximal, complemented
5#63976C212yesmaximal, complemented
6#63970C212yesmaximal, complemented
7#63969C212yesmaximal, complemented
8#63885C212yesmaximal, complemented
9#63859C212yesmaximal, complemented
10#5720—31nomaximal
11#5595—31nomaximal
12C23×C3×D3C2211yescharacteristic
13C2×C32×D4C2211yescharacteristic
14C22×C3⋊Dic3C2211yescharacteristic, complemented
15C2×C3×#67C2212yes—
16C2×C3×#67C2212yes—
17C2×C3×#67C2212yescomplemented
18C2×C3×D12C2211yescharacteristic, complemented
19C2×#625C2212yes—
20C2×#625C2212yes—
21C2×D3×Dic3C2212yescomplemented
22#3757C2212yes—
23#3757C2211yescharacteristic, complemented
24#3750C2212yescomplemented
25C3×#217C2211yescharacteristic
26C3×(C2×D3)⋊C4C2212yes—
27C3×C4×Dic3C2211yescharacteristic, complemented
28#3724C2212yescomplemented
29Dic3⋊Dic3C2212yescomplemented
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32(C2×D3)⋊Dic3C2212yescomplemented
33C3×#127—31no—
34C3×#120—31no—
35#953—32no—
36#940D311yescharacteristic, complemented
37#939—32no—
38#939—32no—
39#939—32no—
40#939—32no—
41#938—32no—
42#937—32no—
43#937D311yescharacteristic, complemented
44#935—31no—
45#935—31no—
46#935—32no—
47#935—61no—
48#884—32no—
49#883—32no—
50#879—32no—
51#876—32no—
52C23×C32C2311yescharacteristic
53C23×C32C2311yescharacteristic
54C22×C3×D3—41no—
55C22×C3×D3—42no—
56C22×C3×D3C2312yes—
57C22×C3×D3C2312yes—
58C32×D4—41no—
59C2×C32×C4C2311yescharacteristic
60C2×C3⋊Dic3—41no—
61C2×C3⋊Dic3C2312yes—
62C2×C3⋊Dic3C2312yes—
63C3×#67—42no—
64C3×#67—42no—
65C3×#67—42no—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×D12—41no—
69#625—42no—
70#625—42no—
71D3×Dic3—42no—
72#560—91noSylow
73C24×C3—31no—
74C23×D3C2×D311yescharacteristic
75C2×C3×D4—21no—
76C2×C3×D4—32no—
77C2×C3×D4—32no—
78C2×C3×D4—32no—
79C2×C3×D4—31no—
80C2×C3×D4C2×D311yescharacteristic
81C2×C3×D4C2×D311yescharacteristic
82C2×#67—32no—
83C2×#67—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67C2×D312yes—
87C2×#67C2×D312yes—
88C2×#67C2×D312yescomplemented
89C22×Dic3—31no—
90C22×Dic3—31no—
91C22×Dic3—32no—
92C22×Dic3—61no—
93C2×D12C2×D311yescharacteristic, complemented
94C2×C4×D3—32no—
95C3×C22⋊C4—31no—
96C3×C22⋊C4—32no—
97C3×C42—31no—
98#217—32no—
99#217—32no—
100#217—31no—
101#217—31no—
102#217—32no—
103#217—32no—
104#217—32no—
105#217—62no—
106#217—61no—
107#217C2×D311yescharacteristic
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4C2×D312yes—
112C4⋊Dic3—32no—
113Dic3⋊C4—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—62no—
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—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—31no—
138#126—92no—
139#123—92no—
140#123—91no—
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—32no—
150C23×C3—122no—
151C23×C3—121no—
152C23×C3C22×D311yescharacteristic
153C23×C3C22×D311yescharacteristic
154C23×C3C22×D311yescharacteristic
155C23×C3C22×D311yescharacteristic
156C22×D3—42no—
157C22×D3—41no—
158C22×D3C22×D312yes—
159C22×D3C22×D312yes—
160C3×D4—41no—
161C3×D4—41no—
162C3×D4—42no—
163C3×D4—121no—
164C3×D4—122no—
165C3×D4—122no—
166C3×D4—122no—
167C2×C3×C4—21no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172#67—42no—
173#67—42no—
174#67—42no—
175#67—122no—
176#67—122no—
177#67—122no—
178#67—122no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—62no—
184C2×Dic3—62no—
185C2×Dic3—122no—
186C2×Dic3—121no—
187C2×Dic3—121no—
188C2×Dic3—122no—
189C2×Dic3C22×D312yes—
190C2×Dic3C22×D312yes—
191D12—41no—
192C4×D3—122no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32H(5,𝔽2)12yes—
196C2×C32H(5,𝔽2)11yescharacteristic
197C3×D3—42no—
198C3×D3—42no—
199C24—31no—
200C2×D4—32no—
201C2×D4—32no—
202C2×D4—32no—
203C2×D4—31no—
204C2×D4—92no—
205C2×D4—92no—
206C2×D4D3211yescharacteristic
207C22×C4—92no—
208C22×C4—91no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C22⋊C4—31no—
213C22⋊C4—32no—
214C22⋊C4—92no—
215C22⋊C4—92no—
216C22⋊C4—92no—
217C22⋊C4—92no—
218C22⋊C4—91no—
219C42—31no—
220C22×C3—22no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—22no—
229C22×C3—21no—
230C22×C3—42no—
231C22×C3—41no—
232C22×C3—42no—
233C22×C3—62no—
234C22×C3—62no—
235C22×C3—62no—
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—22no—
246C2×D3—22no—
247C2×D3—22no—
248C2×D3—44no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—62no—
253C3×C4—62no—
254Dic3—22no—
255Dic3—22no—
256Dic3—62no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—122no—
261Dic3—122no—
262C32#56011yescharacteristic, complemented, Sylow
263C23—32no—
264C23—32no—
265C23—122no—
266C23—121no—
267C23C2×D3211yescharacteristic
268C23C2×D3211yescharacteristic
269D4—41no—
270D4—121no—
271D4—122no—
272D4—122no—
273D4—122no—
274D4—362no—
275D4—362no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—362no—
281C2×C4—361no—
282C2×C4C2×D3211yescharacteristic
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—42no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—81no—
291C2×C3—122no—
292C2×C3—122no—
293C2×C3#100412yes—
294C2×C3#100412yes—
295C2×C3#100411yescharacteristic
296C2×C3#100411yescharacteristic
297D3—42no—
298D3—42no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—62no—
304C22—62no—
305C22—62no—
306C22—62no—
307C22—62no—
308C22—62no—
309C22—124no—
310C22C22×D3211yescharacteristic
311C4—21no—
312C4—62no—
313C4—62no—
314C4—182no—
315C4—182no—
316C3—21no—
317C3#559511yescharacteristic, complemented
318C3#572011yescharacteristic, complemented
319C2—41no—
320C2—41no—
321C2—122no—
322C2—122no—
323C2#6431312yes—
324C2#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. #64144C2semidirect
2. C3×#940C2semidirect
3. C3×#937C2semidirect
4. #63980C2semidirect
5. #63976C2semidirect
6. #63970C2semidirect
7. #63969C2semidirect
8. #63885C2semidirect
9. #63859C2semidirect
14. C22×C3⋊Dic3C22semidirect
17. C2×C3×#67C22semidirect
18. C2×C3×D12C22semidirect
21. C2×D3×Dic3C22semidirect
23. #3757C22semidirect
24. #3750C22semidirect
27. C3×C4×Dic3C22semidirect
28. #3724C22semidirect
29. Dic3⋊Dic3C22semidirect
32. (C2×D3)⋊Dic3C22semidirect
36. #940D3semidirect
43. #937D3semidirect
88. C2×#67C2×D3semidirect
93. C2×D12C2×D3semidirect
119. C4×Dic3C2×D3semidirect
262. C32#560semidirect
317. C3#5595semidirect
318. C3#5720semidirect

Maximal subgroups

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

subgroupindex
#641442
C3×#9402
C3×#9372
#639802
#639762
#639702
#639692
#638852
#638592
#57203
#55953