Finite Group Databasebeta

← #10593923

Subgroups — #10593923

Order 576.

504 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64079C211yesmaximal, characteristic, complemented
2C3×C4×D12C211yesmaximal, characteristic, complemented
3C3×C4×D12C211yesmaximal, characteristic, complemented
4#63908C211yesmaximal, characteristic, complemented
5#63908C211yesmaximal, characteristic, complemented
6#63905C212yesmaximal, complemented
7D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
8#63885C212yesmaximal, complemented
9#63879C212yesmaximal, complemented
10#63858C212yesmaximal, complemented
11#63841C211yesmaximal, characteristic, complemented
12#5565—31nomaximal
13#5558—31nomaximal
14C2×C3×D12C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×#625C2212yes—
20C2×#625C2212yes—
21C2×D3×Dic3C2212yes—
22C32×C42C2211yescharacteristic
23C4⋊(C3⋊Dic3)C2211yescharacteristic
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25#3750C2212yes—
26#3750C2212yes—
27C3×(C2×D3)⋊C4C2212yes—
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×(C2×D3)⋊C4C2211yescharacteristic
30C3×C4⋊Dic3C2211yescharacteristic
31C3×C4⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×C4×D4—31no—
38C3×C4×D4—31no—
39#939—32no—
40#938—31no—
41#938—31no—
42#930—31no—
43#930—31no—
44#929—32no—
45C4×#67—32no—
46#926—31no—
47C2×C4⋊Dic3—31no—
48#898—32no—
49#892—31no—
50D3×C4⋊C4—31no—
51#884—32no—
52#883—32no—
53#67⋊C4—32no—
54#879—32no—
55C4×D12D311yescharacteristic, complemented
56C4×D12D311yescharacteristic, complemented
57#870—31no—
58#870—31no—
59#870—61no—
60C22×C3×D3C2312yes—
61C22×C3×D3C2311yescharacteristic
62C22×C3×D3C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2312yes—
67C2×C3⋊Dic3C2312yes—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C3×D12—22no—
72C3×D12—22no—
73C3×C4×D3—22no—
74C3×C4×D3—22no—
75C3×C4×D3—42no—
76#625—24no—
77#625—42no—
78D3×Dic3—42no—
79#553—91noSylow
80C2×C3×D4—31no—
81C2×C3×D4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C22×Dic3—32no—
90C2×D12C2×D311yescharacteristic
91C2×D12C2×D311yescharacteristic
92C2×C4×D3—32no—
93C2×C4×D3C2×D311yescharacteristic
94C2×C4×D3C2×D311yescharacteristic
95C2×C4×D3C2×D312yes—
96C3×C4⋊C4—31no—
97C3×C4⋊C4—31no—
98C3×C22⋊C4—32no—
99C3×C22⋊C4—31no—
100C3×C22⋊C4—31no—
101C3×C42—21no—
102C3×C42C2×D311yescharacteristic
103C3×C42C2×D311yescharacteristic
104#217—32no—
105#217—32no—
106#217—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4C2×D312yes—
111(C2×D3)⋊C4C2×D311yescharacteristic
112(C2×D3)⋊C4C2×D311yescharacteristic
113C4⋊Dic3—31no—
114C4⋊Dic3—31no—
115C4⋊Dic3—31no—
116C4⋊Dic3—31no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119C4⋊Dic3—61no—
120C4⋊Dic3C2×D311yescharacteristic
121C4⋊Dic3C2×D311yescharacteristic
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124Dic3⋊C4—32no—
125Dic3⋊C4—32no—
126Dic3⋊C4—32no—
127Dic3⋊C4—62no—
128Dic3⋊C4—62no—
129C4×Dic3—32no—
130C4×Dic3—32no—
131C22×C32C2411yescharacteristic
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3—22no—
135C2×C3×D3—21no—
136C2×C3×D3—22no—
137C2×C3×D3—21no—
138C2×C3×D3—21no—
139C2×C3×D3—22no—
140C2×C3×D3#4112yes—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4#4112yes—
144C3⋊Dic3—22no—
145C3⋊Dic3—22no—
146C3×Dic3—21no—
147C3×Dic3—21no—
148C3×Dic3—22no—
149#126—92no—
150#125—91no—
151#123—92no—
152#121—92no—
153#121—91no—
154#121—91no—
155C4×D4—31no—
156C4×D4—31no—
157C4×D4—92no—
158#117—91no—
159C2×C4⋊C4—91no—
160C23×C3—31no—
161C23×C3—32no—
162C23×C3—31no—
163C22×D3C22×D311yescharacteristic
164C22×D3C22×D312yes—
165C22×D3C22×D311yescharacteristic
166C3×D4—62no—
167C3×D4—62no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—31no—
172C2×C3×C4—32no—
173C2×C3×C4—31no—
174C2×C3×C4—62no—
175C2×C3×C4—62no—
176C2×C3×C4—122no—
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179C2×C3×C4C22×D311yescharacteristic
180C2×C3×C4C22×D311yescharacteristic
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D311yescharacteristic
183#67—64no—
184#67—64no—
185#67—122no—
186#67—122no—
187C2×Dic3—32no—
188C2×Dic3—32no—
189C2×Dic3—32no—
190C2×Dic3—32no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—122no—
194C2×Dic3C22×D311yescharacteristic
195C2×Dic3C22×D312yes—
196C2×Dic3C22×D311yescharacteristic
197D12—22no—
198D12—22no—
199C4×D3—22no—
200C4×D3—22no—
201C4×D3—42no—
202C4×D3—122no—
203C2×C32C2×#4111yescharacteristic
204C2×C32C2×#4111yescharacteristic
205C2×C32H(5,𝔽2)11yescharacteristic
206C3×D3—22no—
207C3×D3—42no—
208C3×D3—41no—
209C2×D4—31no—
210C2×D4—31no—
211C2×D4—92no—
212C2×D4—92no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—32no—
216C22×C4—92no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—92no—
220C4⋊C4—91no—
221C4⋊C4—91no—
222C4⋊C4—92no—
223C4⋊C4—92no—
224C22⋊C4—31no—
225C22⋊C4—32no—
226C22⋊C4—31no—
227C22⋊C4—92no—
228C22⋊C4—92no—
229C22⋊C4—92no—
230C42—92no—
231C42D3211yescharacteristic
232C22×C3—21no—
233C22×C3—32no—
234C22×C3—61no—
235C22×C3—61no—
236C22×C3—62no—
237C22×C3—61no—
238C22×C3—62no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—62no—
242C22×C3C23×D311yescharacteristic
243C22×C3C23×D311yescharacteristic
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—22no—
247C2×D3—21no—
248C2×D3—21no—
249C2×D3—22no—
250C2×D3—22no—
251C2×D3—21no—
252C2×D3#23512yes—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—22no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—62no—
263C3×C4#23512yes—
264C3×C4#23712yes—
265Dic3—21no—
266Dic3—21no—
267Dic3—22no—
268Dic3—62no—
269Dic3—62no—
270Dic3—62no—
271Dic3—62no—
272Dic3—122no—
273Dic3—122no—
274C32#55311yescharacteristic, complemented, Sylow
275C23—31no—
276C23—31no—
277C23—32no—
278D4—62no—
279D4—62no—
280D4—184no—
281D4—362no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—31no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—92no—
288C2×C4—92no—
289C2×C4—122no—
290C2×C4—362no—
291C2×C4C2×D3211yescharacteristic
292C2×C4C2×D3211yescharacteristic
293C2×C4C2×D3211yescharacteristic
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—62no—
298C2×C3—121no—
299C2×C3—122no—
300C2×C3C2×#23511yescharacteristic
301C2×C3C2×#23511yescharacteristic
302C2×C3C2×#23711yescharacteristic
303C2×C3#100411yescharacteristic
304C2×C3D3×#4111yescharacteristic
305C2×C3#100911yescharacteristic
306D3—22no—
307D3—42no—
308D3—41no—
309C22—32no—
310C22—61no—
311C22—61no—
312C22—61no—
313C22—62no—
314C22—61no—
315C22—62no—
316C22—62no—
317C22—61no—
318C22C22×D3211yescharacteristic
319C4—21no—
320C4—21no—
321C4—61no—
322C4—61no—
323C4—62no—
324C4—182no—
325C4—182no—
326C4#379512yes—
327C3—21no—
328C3#555811yescharacteristic, complemented
329C3#556511yescharacteristic, complemented
330C2—62no—
331C2—121no—
332C2—122no—
333C2C2×#379511yescharacteristic
334C2D3×#23511yescharacteristic
335C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64079C2semidirect
2. C3×C4×D12C2semidirect
3. C3×C4×D12C2semidirect
4. #63908C2semidirect
5. #63908C2semidirect
6. #63905C2semidirect
7. D3×C4⋊Dic3C2semidirect
8. #63885C2semidirect
9. #63879C2semidirect
10. #63858C2semidirect
11. #63841C2semidirect
55. C4×D12D3semidirect
56. C4×D12D3semidirect
274. C32#553semidirect
328. C3#5558semidirect
329. C3#5565semidirect

Maximal subgroups

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

subgroupindex
#640792
C3×C4×D122
#639082
#639052
D3×C4⋊Dic32
#638852
#638792
#638582
#638412
#55653
#55583