Finite Group Databasebeta

← #10596107

Subgroups — #10596107

Order 576.

684 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155C211yesmaximal, characteristic, complemented
3#64153C211yesmaximal, characteristic, complemented
4C4×#638C211yesmaximal, characteristic
5C2×#3750C211yesmaximal, characteristic, complemented
6#64105C211yesmaximal, characteristic, complemented
7#64105C211yesmaximal, characteristic, complemented
8C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
9#64098C211yesmaximal, characteristic, complemented
10#64098C211yesmaximal, characteristic, complemented
11#64096C211yesmaximal, characteristic, complemented
12#64090C211yesmaximal, characteristic
13#64090C211yesmaximal, characteristic
14#64087C211yesmaximal, characteristic, complemented
15#64087C211yesmaximal, characteristic, complemented
16#5636—34nomaximal
17C2×C32×Q8C2211yescharacteristic
18C22×C32×C4C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C22×C3⋊Dic3C2211yescharacteristic
21C22×C3⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic
23C2×C4×C3⋊D3C2211yescharacteristic
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×#634C2211yescharacteristic
26C2×#634C2211yescharacteristic
27C32×C4⋊C4C2211yescharacteristic
28C32×C4⋊C4C2211yescharacteristic
29C32×C4⋊C4C2211yescharacteristic
30C32×C22⋊C4C2211yescharacteristic
31C32×C22⋊C4C2211yescharacteristic
32#3757C2211yescharacteristic
33#3752C2211yescharacteristic
34#3752C2211yescharacteristic
35#3752C2211yescharacteristic
36C4⋊(C3⋊Dic3)C2211yescharacteristic
37C4⋊(C3⋊Dic3)C2211yescharacteristic
38#3750C2211yescharacteristic
39#3750C2211yescharacteristic
40#3750C2211yescharacteristic
41#3750C2211yescharacteristic
42#3750C2211yescharacteristic
43#3750C2211yescharacteristic
44#3750C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47#3750C2211yescharacteristic
48#3750C2211yescharacteristic
49C4×C3⋊Dic3C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×C22⋊Q8D314yescomplemented
53#946—34no—
54#944—34no—
55C4×#67—34no—
56C2×Dic3⋊C4—34no—
57#896—34no—
58#896—34no—
59D3×C4⋊C4—34no—
60#889—34no—
61#889—34no—
62#887—34no—
63#67⋊C4—34no—
64#67⋊C4—34no—
65#878—34no—
66#878—34no—
67C23×C32C2311yescharacteristic
68C22×C3⋊D3C2311yescharacteristic
69C32×Q8—41no—
70C2×C32×C4—41no—
71C2×C32×C4C2311yescharacteristic
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C32×C4C2311yescharacteristic
77#638Q814yescomplemented
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3—41no—
80C2×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊Dic3C2311yescharacteristic
86C2×C3⋊Dic3C2311yescharacteristic
87C4×C3⋊D3—22no—
88C4×C3⋊D3—41no—
89C4×C3⋊D3—41no—
90#634—22no—
91#634—41no—
92#554—91noSylow
93C2×C3×Q8C2×D314yes—
94C22×C3×C4C2×D314yes—
95C2×#67—34no—
96C22×Dic3—34no—
97C22×Dic3—34no—
98C2×C4×D3—34no—
99C2×C4×D3—34no—
100C2×C4×D3—34no—
101C2×Dic6—34no—
102C2×Dic6—34no—
103C3×C4⋊C4C2×D314yes—
104C3×C4⋊C4C2×D314yes—
105C3×C4⋊C4C2×D314yes—
106C3×C22⋊C4C2×D314yes—
107C3×C22⋊C4C2×D314yes—
108#217—34no—
109(C2×D3)⋊C4—34no—
110(C2×D3)⋊C4—34no—
111(C2×D3)⋊C4—34no—
112C4⋊Dic3—34no—
113C4⋊Dic3—34no—
114Dic3⋊C4—34no—
115Dic3⋊C4—34no—
116Dic3⋊C4—34no—
117Dic3⋊C4—34no—
118Dic3⋊C4—34no—
119Dic3⋊C4—34no—
120Dic3⋊C4—34no—
121Dic3⋊C4—34no—
122Dic3⋊C4—34no—
123Dic3⋊C4—34no—
124Dic3⋊C4—34no—
125C4×Dic3—34no—
126C4×Dic3—34no—
127C4×Dic3—34no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32C2×Q812yes—
131C22×C32C2411yescharacteristic
132C2×C3⋊D3—21no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3C2×Q812yes—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C3⋊Dic3—21no—
142C3⋊Dic3—21no—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3⋊Dic3—21no—
146C3⋊Dic3C2×Q812yes—
147C3⋊Dic3#4112yes—
148C4⋊Q8—91no—
149#125—91no—
150#125—91no—
151C22⋊Q8—91no—
152C22⋊Q8—91no—
153C22⋊Q8—91no—
154C22⋊Q8—91no—
155C22⋊Q8—91no—
156C22⋊Q8C3⋊D311yescharacteristic, complemented
157C4×Q8—91no—
158C4×D4—91no—
159C4×D4—91no—
160C4×D4—91no—
161C2×C4⋊C4—91no—
162C2×C4⋊C4—91no—
163C23×C3C22×D314yes—
164C22×D3—34no—
165C3×Q8—44no—
166C2×C3×C4—44no—
167C2×C3×C4C22×D314yes—
168C2×C3×C4C22×D314yes—
169C2×C3×C4C22×D314yes—
170C2×C3×C4C22×D314yes—
171C2×C3×C4C22×D314yes—
172C2×C3×C4C22×D314yes—
173#67—316no—
174C2×Dic3—34no—
175C2×Dic3—34no—
176C2×Dic3—34no—
177C2×Dic3—34no—
178C2×Dic3—34no—
179C2×Dic3—34no—
180C2×Dic3—34no—
181C2×Dic3—68no—
182C2×Dic3—124no—
183C4×D3—68no—
184C4×D3—124no—
185C4×D3—124no—
186Dic6—68no—
187Dic6—124no—
188C2×C32—22no—
189C2×C32C22×Q811yescharacteristic
190C2×C32C2×#4111yescharacteristic
191C2×C32H(5,𝔽2)11yescharacteristic
192C3⋊D3—22no—
193C2×Q8—91no—
194C2×Q8—91no—
195C2×Q8C2×C3⋊D311yescharacteristic
196C2×D4—91no—
197C22×C4—91no—
198C22×C4—91no—
199C22×C4—91no—
200C22×C4—91no—
201C22×C4—91no—
202C22×C4C2×C3⋊D311yescharacteristic
203C4⋊C4—91no—
204C4⋊C4—91no—
205C4⋊C4—91no—
206C4⋊C4—91no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C4⋊C4—91no—
210C4⋊C4—91no—
211C4⋊C4—91no—
212C4⋊C4—91no—
213C4⋊C4—91no—
214C4⋊C4—91no—
215C4⋊C4—91no—
216C4⋊C4C2×C3⋊D311yescharacteristic
217C4⋊C4C2×C3⋊D311yescharacteristic
218C4⋊C4C2×C3⋊D311yescharacteristic
219C22⋊C4—91no—
220C22⋊C4—91no—
221C22⋊C4—91no—
222C22⋊C4—91no—
223C22⋊C4C2×C3⋊D311yescharacteristic
224C22⋊C4C2×C3⋊D311yescharacteristic
225C42—91no—
226C42—91no—
227C42—91no—
228C22×C3—24no—
229C22×C3—24no—
230C22×C3D3×Q818yes—
231C22×C3C23×D314yes—
232C2×D3—38no—
233C2×D3—64no—
234C2×D3—64no—
235C3×C4—24no—
236C3×C4—24no—
237C3×C4—24no—
238C3×C4—24no—
239C3×C4—24no—
240C3×C4—24no—
241Dic3—38no—
242Dic3—38no—
243Dic3—64no—
244Dic3—64no—
245Dic3—64no—
246Dic3—64no—
247Dic3—64no—
248C32#55411yescharacteristic, complemented, Sylow
249C23—91no—
250C23C22×C3⋊D311yescharacteristic
251Q8—41no—
252Q8—182no—
253Q8—361no—
254D4—94no—
255C2×C4—41no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4—361no—
266C2×C4—361no—
267C2×C4—361no—
268C2×C4C22×C3⋊D311yescharacteristic
269C2×C4C22×C3⋊D311yescharacteristic
270C2×C4C22×C3⋊D311yescharacteristic
271C2×C4C22×C3⋊D311yescharacteristic
272C2×C4C22×C3⋊D311yescharacteristic
273C2×C4C22×C3⋊D311yescharacteristic
274C2×C3—28no—
275C2×C3#100414yes—
276C2×C3C2×D3×Q814yes—
277C2×C3D3×#4114yes—
278D3—68no—
279C22—21no—
280C22—21no—
281C22—92no—
282C22—181no—
283C22—181no—
284C22Q8×C3⋊D312yes—
285C22C23×C3⋊D311yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—92no—
293C4—92no—
294C4—181no—
295C4—181no—
296C4—181no—
297C4—181no—
298C4—181no—
299C3#563614yescomplemented
300C2—22no—
301C2—182no—
302C2#6436111yescharacteristic
303C2C2×Q8×C3⋊D311yescharacteristic
304C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radicalC2×#638C22
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. #64155C2semidirect
3. #64153C2semidirect
5. C2×#3750C2semidirect
6. #64105C2semidirect
7. #64105C2semidirect
8. C4⋊C4×C3⋊D3C2semidirect
9. #64098C2semidirect
10. #64098C2semidirect
11. #64096C2semidirect
14. #64087C2semidirect
15. #64087C2semidirect
52. C3×C22⋊Q8D3semidirect
77. #638Q8semidirect
156. C22⋊Q8C3⋊D3semidirect
248. C32#554semidirect
299. C3#5636semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641552
#641532
C4×#6382
C2×#37502
#641052
C4⋊C4×C3⋊D32
#640982
#640962
#640902
#640872
#56363