Finite Group Databasebeta

← #10593986

Subgroups — #10593986

Order 576.

527 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×D12C211yesmaximal, characteristic, complemented
3C3×#872C211yesmaximal, characteristic, complemented
4#63912C212yesmaximal, complemented
5#63911C211yesmaximal, characteristic, complemented
6#63909C211yesmaximal, characteristic, complemented
7#63908C211yesmaximal, characteristic, complemented
8#63906C212yesmaximal, complemented
9#63869C211yesmaximal, characteristic, complemented
10#63853C212yesmaximal, complemented
11#63850C212yesmaximal, complemented
12#5562—31nomaximal
13#5535—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×#626C2212yes—
20C2×#626C2212yes—
21C2×#626C2212yes—
22C2×#625C2212yes—
23C32×C42C2211yescharacteristic
24#3752C2212yes—
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×(C2×D3)⋊C4C2212yes—
27C3×(C2×D3)⋊C4C2211yescharacteristic
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×C4⋊Dic3C2211yescharacteristic
30C3×Dic3⋊C4C2211yescharacteristic
31C3×Dic3⋊C4C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35#3722C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37C3×C4×D4—31no—
38C3×#117—31no—
39#940—31no—
40#938—31no—
41#930—31no—
42#930—32no—
43#930—31no—
44#929—32no—
45#895—31no—
46#894—32no—
47#894—32no—
48#890—31no—
49#884—32no—
50#884—32no—
51#884—32no—
52#883—32no—
53#874—31no—
54C4×D12—31no—
55C4×D12—31no—
56C4×D12—61no—
57C4×D12D311yescharacteristic, complemented
58#872D311yescharacteristic, complemented
59#870—31no—
60C22×C3⋊D3C2312yes—
61C22×C3×D3C2312yes—
62C22×C3×D3C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2312yes—
67C4⋊(C3⋊D3)—22no—
68C4×C3⋊D3—42no—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×D12—22no—
74C3×C4×D3—22no—
75C3×C4×D3—42no—
76#626—42no—
77#626—42no—
78#626—42no—
79#625—42no—
80#540—91noSylow
81C2×C3×D4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×D12—31no—
90C2×D12—31no—
91C2×D12—32no—
92C2×D12—32no—
93C2×D12—32no—
94C2×D12—61no—
95C2×D12C2×D311yescharacteristic
96C2×C4×D3—32no—
97C2×C4×D3—32no—
98C2×C4×D3—62no—
99C2×C4×D3C2×D311yescharacteristic
100C2×C4×D3C2×D312yes—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—31no—
104C3×C22⋊C4—32no—
105C3×C22⋊C4—31no—
106C3×C22⋊C4—31no—
107C3×C42—21no—
108C3×C42—31no—
109C3×C42C2×D311yescharacteristic
110C3×C42C2×D311yescharacteristic
111#217—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—32no—
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—62no—
118(C2×D3)⋊C4C2×D311yescharacteristic
119(C2×D3)⋊C4C2×D311yescharacteristic
120(C2×D3)⋊C4C2×D312yes—
121C4⋊Dic3—31no—
122C4⋊Dic3—31no—
123C4⋊Dic3—32no—
124C4⋊Dic3—61no—
125C4⋊Dic3C2×D311yescharacteristic
126Dic3⋊C4—32no—
127Dic3⋊C4—32no—
128Dic3⋊C4—32no—
129Dic3⋊C4C2×D311yescharacteristic
130Dic3⋊C4C2×D311yescharacteristic
131C4×Dic3C2×D311yescharacteristic
132C22×C32C2411yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3—22no—
135C2×C3⋊D3—22no—
136C2×C3×D3—21no—
137C2×C3×D3—22no—
138C2×C3×D3—21no—
139C2×C3×D3—22no—
140C2×C3×D3—21no—
141C2×C3×D3—22no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4#4112yes—
145C3⋊Dic3—22no—
146C3×Dic3—21no—
147C3×Dic3—21no—
148C3×Dic3—21no—
149C3×Dic3—22no—
150#127—91no—
151#125—91no—
152#123—92no—
153#123—92no—
154#121—92no—
155#121—92no—
156#121—91no—
157#121—91no—
158C4×D4—31no—
159C4×D4—91no—
160#117—31no—
161C23×C3—32no—
162C23×C3—31no—
163C22×D3—32no—
164C22×D3—32no—
165C22×D3—62no—
166C22×D3C22×D312yes—
167C22×D3C22×D311yescharacteristic
168C3×D4—62no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—32no—
175C2×C3×C4—31no—
176C2×C3×C4—62no—
177C2×C3×C4—122no—
178C2×C3×C4C22×D311yescharacteristic
179C2×C3×C4C22×D311yescharacteristic
180C2×C3×C4C22×D311yescharacteristic
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D311yescharacteristic
183C2×C3×C4C22×D311yescharacteristic
184#67—122no—
185#67—122no—
186#67—122no—
187#67—122no—
188#67—122no—
189C2×Dic3—32no—
190C2×Dic3—32no—
191C2×Dic3—62no—
192C2×Dic3C22×D311yescharacteristic
193C2×Dic3C22×D311yescharacteristic
194C2×Dic3C22×D312yes—
195C2×Dic3C22×D311yescharacteristic
196D12—22no—
197D12—64no—
198D12—62no—
199D12—62no—
200D12—122no—
201D12—122no—
202D12—122no—
203C4×D3—22no—
204C4×D3—42no—
205C4×D3—122no—
206C4×D3—122no—
207C4×D3—124no—
208C2×C32C2×#4111yescharacteristic
209C2×C32H(5,𝔽2)11yescharacteristic
210C2×C32H(5,𝔽2)11yescharacteristic
211C3⋊D3—42no—
212C3×D3—41no—
213C3×D3—42no—
214C2×D4—31no—
215C2×D4—92no—
216C2×D4—92no—
217C2×D4—92no—
218C2×D4—92no—
219C2×D4—91no—
220C22×C4—31no—
221C22×C4—32no—
222C22×C4—92no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—92no—
227C4⋊C4—92no—
228C4⋊C4—91no—
229C22⋊C4—31no—
230C22⋊C4—31no—
231C22⋊C4—32no—
232C22⋊C4—92no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C42—31no—
236C42D3211yescharacteristic
237C22×C3—21no—
238C22×C3—61no—
239C22×C3—62no—
240C22×C3—61no—
241C22×C3—62no—
242C22×C3—61no—
243C22×C3—62no—
244C22×C3C23×D311yescharacteristic
245C22×C3C23×D311yescharacteristic
246C2×D3—21no—
247C2×D3—22no—
248C2×D3—22no—
249C2×D3—21no—
250C2×D3—22no—
251C2×D3—21no—
252C2×D3—64no—
253C2×D3—62no—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—122no—
260C2×D3—122no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—22no—
267C3×C4—41no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—62no—
272C3×C4#23512yes—
273C3×C4#23512yes—
274Dic3—21no—
275Dic3—22no—
276Dic3—21no—
277Dic3—21no—
278Dic3—62no—
279Dic3—62no—
280Dic3—122no—
281C32#54011yescharacteristic, complemented, Sylow
282C23—31no—
283C23—32no—
284C23—92no—
285D4—62no—
286D4—182no—
287D4—362no—
288D4—362no—
289D4—362no—
290D4—362no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—31no—
295C2×C4—62no—
296C2×C4—92no—
297C2×C4—122no—
298C2×C4—362no—
299C2×C4C2×D3211yescharacteristic
300C2×C4C2×D3211yescharacteristic
301C2×C4C2×D3211yescharacteristic
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—122no—
306C2×C3—121no—
307C2×C3C2×#23511yescharacteristic
308C2×C3C2×#23511yescharacteristic
309C2×C3#100411yescharacteristic
310C2×C3#100911yescharacteristic
311C2×C3#100911yescharacteristic
312C2×C3#100911yescharacteristic
313D3—41no—
314D3—42no—
315D3—124no—
316D3—122no—
317D3—122no—
318C22—61no—
319C22—62no—
320C22—62no—
321C22—61no—
322C22—62no—
323C22—61no—
324C22—182no—
325C22—182no—
326C22—182no—
327C22C22×D3211yescharacteristic
328C4—21no—
329C4—21no—
330C4—61no—
331C4—62no—
332C4—61no—
333C4—61no—
334C4—182no—
335C4#379812yes—
336C3—21no—
337C3#553511yescharacteristic, complemented
338C3#556211yescharacteristic, complemented
339C2—121no—
340C2—122no—
341C2—362no—
342C2C2×#379811yescharacteristic
343C2#6430711yescharacteristic
344C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D3)C2semidirect
2. C3×C4×D12C2semidirect
3. C3×#872C2semidirect
4. #63912C2semidirect
5. #63911C2semidirect
6. #63909C2semidirect
7. #63908C2semidirect
8. #63906C2semidirect
9. #63869C2semidirect
10. #63853C2semidirect
11. #63850C2semidirect
57. C4×D12D3semidirect
58. #872D3semidirect
281. C32#540semidirect
337. C3#5535semidirect
338. C3#5562semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D3)2
C3×C4×D122
C3×#8722
#639122
#639112
#639092
#639082
#639062
#638692
#638532
#638502
#55623
#55353