Finite Group Databasebeta

← #10593939

Subgroups — #10593939

Order 576.

519 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#875C211yesmaximal, characteristic, complemented
3C3×C4×Dic6C211yesmaximal, characteristic, complemented
4#63911C211yesmaximal, characteristic
5#63907C212yesmaximal, complemented
6Dic3×D12C211yesmaximal, characteristic, complemented
7#63890C212yesmaximal, complemented
8#63881C212yesmaximal, complemented
9#63864C211yesmaximal, characteristic, complemented
10#63853C212yesmaximal, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5681—31nomaximal
13#5585—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×C3×D12C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic
19C2×#626C2212yes—
20C2×#626C2212yes—
21C2×D3×Dic3C2212yes—
22C32×C42C2211yescharacteristic
23#3752C2212yes—
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×(C2×D3)⋊C4C2212yes—
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2211yescharacteristic
29C3×Dic3⋊C4C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31C3×C4×Dic3C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×#124—31no—
38C3×C4×Q8—31no—
39#947—32no—
40#947—31no—
41Q8×Dic3—31no—
42#940—31no—
43D4×Dic3—31no—
44#894—32no—
45#894—32no—
46#893—32no—
47#886—32no—
48#885—32no—
49#883—32no—
50#67⋊C4—32no—
51#875—31no—
52#875D311yescharacteristic, complemented
53#874—31no—
54C4×D12—31no—
55C4×D12—31no—
56C4×D12—31no—
57C4×D12—61no—
58C4×Dic6—31no—
59C4×Dic6D311yescharacteristic, complemented
60C22×C3⋊D3C2312yes—
61C22×C3×D3C2312yes—
62C2×C32×C4C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C4⋊(C3⋊D3)—22no—
67C4×C3⋊D3—42no—
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×D12—22no—
74C3×Dic6—22no—
75C3×Dic6—22no—
76#626—24no—
77#626—42no—
78D3×Dic3—42no—
79#559—91noSylow
80C2×C3×Q8—31no—
81C2×C3×Q8—31no—
82C2×C3×D4—31no—
83C2×#67—32no—
84C2×#67—32no—
85C22×Dic3—32no—
86C2×D12—32no—
87C2×D12—31no—
88C2×D12—31no—
89C2×D12—32no—
90C2×D12—61no—
91C2×D12C2×D311yescharacteristic
92C2×C4×D3—32no—
93C2×C4×D3—32no—
94C2×C4×D3—32no—
95C2×C4×D3—62no—
96C2×Dic6C2×D311yescharacteristic
97C2×Dic6C2×D311yescharacteristic
98C3×C4⋊C4—31no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C22⋊C4—32no—
102C3×C22⋊C4—32no—
103C3×C42—21no—
104C3×C42—31no—
105C3×C42—31no—
106C3×C42C2×D311yescharacteristic
107C3×C42C2×D311yescharacteristic
108#217—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—32no—
116(C2×D3)⋊C4—62no—
117(C2×D3)⋊C4C2×D312yes—
118(C2×D3)⋊C4C2×D312yes—
119C4⋊Dic3—31no—
120C4⋊Dic3—31no—
121C4⋊Dic3—32no—
122C4⋊Dic3—61no—
123C4⋊Dic3C2×D311yescharacteristic
124Dic3⋊C4—32no—
125Dic3⋊C4C2×D311yescharacteristic
126Dic3⋊C4C2×D311yescharacteristic
127C4×Dic3—32no—
128C4×Dic3—32no—
129C4×Dic3C2×D311yescharacteristic
130C4×Dic3C2×D311yescharacteristic
131C22×C32C2411yescharacteristic
132C2×C3⋊D3—22no—
133C2×C3⋊D3—22no—
134C2×C3⋊D3—22no—
135C2×C3×D3—22no—
136C2×C3×D3—22no—
137C2×C3×D3—22no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4#4112yes—
141C3⋊Dic3—22no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3—21no—
146C3×Dic3#4112yes—
147#127—91no—
148#124—31no—
149#124—91no—
150#124—92no—
151#123—92no—
152#123—92no—
153C4×Q8—31no—
154C4×Q8—91no—
155C4×D4—91no—
156C4×D4—92no—
157C4×D4—91no—
158C23×C3—32no—
159C22×D3—32no—
160C22×D3—32no—
161C22×D3—62no—
162C22×D3C22×D312yes—
163C3×Q8—62no—
164C3×Q8—62no—
165C3×D4—62no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179C2×C3×C4C22×D311yescharacteristic
180#67—64no—
181#67—122no—
182C2×Dic3—32no—
183C2×Dic3—32no—
184C2×Dic3—62no—
185C2×Dic3—122no—
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D311yescharacteristic
190C2×Dic3C22×D311yescharacteristic
191D12—22no—
192D12—64no—
193D12—64no—
194D12—62no—
195D12—62no—
196D12—122no—
197C4×D3—122no—
198C4×D3—122no—
199C4×D3—122no—
200C4×D3—124no—
201Dic6—22no—
202Dic6—22no—
203C2×C32C2×#4111yescharacteristic
204C2×C32C2×#4111yescharacteristic
205C2×C32H(5,𝔽2)11yescharacteristic
206C3⋊D3—42no—
207C3×D3—42no—
208C2×Q8—31no—
209C2×Q8—31no—
210C2×D4—31no—
211C2×D4—92no—
212C2×D4—92no—
213C2×D4—91no—
214C22×C4—92no—
215C22×C4—92no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—92no—
220C4⋊C4—91no—
221C22⋊C4—32no—
222C22⋊C4—32no—
223C22⋊C4—92no—
224C22⋊C4—92no—
225C22⋊C4—92no—
226C22⋊C4—92no—
227C42—31no—
228C42—31no—
229C42—92no—
230C42D3211yescharacteristic
231C22×C3—21no—
232C22×C3—62no—
233C22×C3—62no—
234C22×C3—62no—
235C22×C3C23×D311yescharacteristic
236C22×C3C23×D311yescharacteristic
237C2×D3—22no—
238C2×D3—22no—
239C2×D3—22no—
240C2×D3—64no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—62no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—122no—
248C2×D3—122no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—32no—
256C3×C4—41no—
257C3×C4—62no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4#23712yes—
262C3×C4#23912yes—
263Dic3—21no—
264Dic3—22no—
265Dic3—21no—
266Dic3—21no—
267Dic3—62no—
268Dic3—62no—
269Dic3—122no—
270Dic3#23512yes—
271C32#55911yescharacteristic, complemented, Sylow
272C23—32no—
273C23—92no—
274Q8—62no—
275Q8—62no—
276D4—62no—
277D4—184no—
278D4—182no—
279D4—362no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—32no—
285C2×C4—92no—
286C2×C4—362no—
287C2×C4—362no—
288C2×C4C2×D3211yescharacteristic
289C2×C4C2×D3211yescharacteristic
290C2×C4C2×D3211yescharacteristic
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—122no—
295C2×C3C2×#23511yescharacteristic
296C2×C3C2×#23711yescharacteristic
297C2×C3C2×#23911yescharacteristic
298C2×C3D3×#4111yescharacteristic
299C2×C3#100911yescharacteristic
300C2×C3#100911yescharacteristic
301D3—42no—
302D3—124no—
303D3—122no—
304D3—122no—
305C22—62no—
306C22—62no—
307C22—62no—
308C22—182no—
309C22—182no—
310C22—182no—
311C22C22×D3211yescharacteristic
312C4—21no—
313C4—21no—
314C4—32no—
315C4—62no—
316C4—61no—
317C4—61no—
318C4—61no—
319C4—182no—
320C4#379612yes—
321C3—21no—
322C3#558511yescharacteristic, complemented
323C3#568111yescharacteristic, complemented
324C2—122no—
325C2—362no—
326C2C2×#379611yescharacteristic
327C2D3×#23511yescharacteristic
328C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radical#63911C2
hypercenterC42D32
hypocenterC32#559
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×#875C2semidirect
3. C3×C4×Dic6C2semidirect
5. #63907C2semidirect
6. Dic3×D12C2semidirect
7. #63890C2semidirect
8. #63881C2semidirect
9. #63864C2semidirect
10. #63853C2semidirect
11. Dic3×Dic6C2semidirect
52. #875D3semidirect
59. C4×Dic6D3semidirect
271. C32#559semidirect
322. C3#5585semidirect
323. C3#5681semidirect

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×#8752
C3×C4×Dic62
#639112
#639072
Dic3×D122
#638902
#638812
#638642
#638532
Dic3×Dic62
#56813
#55853