Finite Group Databasebeta

← #120357

Subgroups — #120357

Order 512.

427 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21965C211yesmaximal, characteristic
2#17809C211yesmaximal, characteristic
3#8113C211yesmaximal, characteristic
4#8096C211yesmaximal, characteristic
5#7980C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7945C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2283C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—22no—
16#1949—22no—
17#1809—22no—
18#1574—22no—
19#1571C412yescomplemented
20#1570C412yescomplemented
21#1565—22no—
22#1306—22no—
23#1303—22no—
24#1289—22no—
25#1286—22no—
26#1284—22no—
27C22×M16—21no—
28C22×C42C2311yescharacteristic
29#432—21no—
30#432—21no—
31#432—21no—
32#432Q812yes—
33#431—21no—
34#431—21no—
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431C2×C412yes—
40#431Q812yes—
41C2×#107—22no—
42C2×#106—22no—
43#423—21no—
44#423—21no—
45#423—21no—
46#423—21no—
47#423C2×C412yes—
48#423D412yes—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2×C411yescharacteristic
51C2×C4⋊C8C2×C411yescharacteristic
52C2×C4⋊C8D412yes—
53#407—21no—
54#407D412yes—
55C2×C22⋊C8—21no—
56C4×M16—21no—
57C4×M16—21no—
58C2×C4×C8—21no—
59#383—21no—
60#383—21no—
61#383Q812yes—
62#382—21no—
63#382—21no—
64#382—21no—
65#381—21no—
66#381—21no—
67#381—21no—
68#380—21no—
69#380—21no—
70#380D412yes—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4D412yes—
74C4×C22⋊C4—21no—
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77#376—21no—
78#376—21no—
79#376—21no—
80#340—24no—
81#335—24no—
82#334—24no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4#4111yescharacteristic
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—81no—
91C2×M16—81no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C42—22no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22×C411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×Q811yescharacteristic
109C2×C42C2×Q811yescharacteristic
110C2×C42C2×Q811yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116#107—82no—
117#106—82no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—41no—
129C4⋊C8C22⋊C414yes—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8C4⋊C414yes—
136#97—22no—
137#97—41no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—41no—
141C4×C8—41no—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154C24#12111yescharacteristic
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—81no—
162C22×C4C2×C4⋊C411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4C4×Q811yescharacteristic
165C22×C4#12011yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4C22⋊Q811yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4#12311yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12411yescharacteristic
177C22×C4#12511yescharacteristic
178C22×C4#12611yescharacteristic
179C22×C4C4⋊Q811yescharacteristic
180M16—44no—
181M16—82no—
182M16—82no—
183M16—82no—
184C2×C8—24no—
185C2×C8—24no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—81no—
192C2×C8—81no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C42—82no—
203C42—82no—
204C42—82no—
205C42C2×C22⋊C411yescharacteristic
206C42#11711yescharacteristic
207C42C4×D411yescharacteristic
208C42C4×Q811yescharacteristic
209C42#12111yescharacteristic
210C42C22⋊Q811yescharacteristic
211C42C22⋊Q811yescharacteristic
212C42C22⋊Q811yescharacteristic
213C42C22⋊Q811yescharacteristic
214C42#12311yescharacteristic
215C42#12511yescharacteristic
216C42#12511yescharacteristic
217C42#12511yescharacteristic
218C42#12611yescharacteristic
219C42C4⋊Q811yescharacteristic
220C42C4⋊Q811yescharacteristic
221C23—22no—
222C23—41no—
223C23—41no—
224C23#38911yescharacteristic
225C23#39411yescharacteristic
226C23#39611yescharacteristic
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#38011yescharacteristic
258C2×C4#38611yescharacteristic
259C2×C4#38711yescharacteristic
260C2×C4#38711yescharacteristic
261C2×C4#38911yescharacteristic
262C2×C4#39111yescharacteristic
263C2×C4#39511yescharacteristic
264C2×C4#39611yescharacteristic
265C2×C4#39811yescharacteristic
266C2×C4#39811yescharacteristic
267C2×C4#39811yescharacteristic
268C2×C4#40011yescharacteristic
269C8—44no—
270C8—44no—
271C8—82no—
272C8—81no—
273C8—81no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—82no—
283C22#144111yescharacteristic
284C22#205312yes—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—162no—
297C4—162no—
298C4—162no—
299C4#190411yescharacteristic
300C4#190511yescharacteristic
301C2—22no—
302C2—41no—
303C2—41no—
304C2#1121311yescharacteristic
305C2#1123611yescharacteristic
306C2#1125111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1571C4semidirect
20. #1570C4semidirect

Maximal subgroups

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

subgroupindex
#219652
#178092
#81132
#80962
#79802
#79622
#79452