Finite Group Databasebeta

← #121942

Subgroups — #121942

Order 512.

429 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21360C211yesmaximal, characteristic
2#8052C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7945C211yesmaximal, characteristic
5#7945C211yesmaximal, characteristic
6#7940C211yesmaximal, characteristic
7#7936C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2283C2211yescharacteristic
12#2283C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1949—22no—
16#1854—22no—
17C4×#107C412yescomplemented
18C4×#106C412yescomplemented
19#1297—22no—
20#1289—22no—
21#1284—22no—
22#1284—22no—
23#1284—22no—
24#1283—22no—
25C2×#117—21no—
26C22×C42C2311yescharacteristic
27#432—21no—
28#432—21no—
29#432—21no—
30#432—21no—
31#431—21no—
32#431—21no—
33#431—21no—
34#431—21no—
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39C2×#107—22no—
40C2×#106—22no—
41#423—21no—
42#423—21no—
43C2×C4⋊C8D412yes—
44#407D412yes—
45#407D412yes—
46C2×C22⋊C8D412yes—
47C4×M16—21no—
48C4×M16—21no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51#383—21no—
52#383—21no—
53#383Q812yes—
54#382—21no—
55#382—21no—
56#382—21no—
57#381—21no—
58#381—21no—
59#381—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4C2×C412yes—
67C4×C4⋊C4C2×C412yes—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73#376—21no—
74#376—21no—
75#376—21no—
76#376Q812yes—
77C43—21no—
78#340—22no—
79#340—22no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—22no—
84C2×M16—22no—
85C22×C8C22⋊C412yes—
86C22×C8C22⋊C412yes—
87#117—44no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42C22×C411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×Q811yescharacteristic
105C2×C42C2×Q811yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114#107—44no—
115#106—44no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128#97—41no—
129#97—41no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—41no—
135C4×C8—41no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24#12711yescharacteristic
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—81no—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4C22⋊Q811yescharacteristic
164C22×C4C22⋊Q811yescharacteristic
165C22×C4C22⋊Q811yescharacteristic
166C22×C4C22⋊Q811yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170C22×C4#12311yescharacteristic
171C22×C4#12311yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4#12311yescharacteristic
174C22×C4#12311yescharacteristic
175C22×C4#12411yescharacteristic
176M16—44no—
177M16—82no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C4⋊C4—24no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C22⋊C4—44no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C42—44no—
195C42—82no—
196C42—82no—
197C42—82no—
198C42—82no—
199C42—82no—
200C42#11711yescharacteristic
201C42#11711yescharacteristic
202C42C4×Q811yescharacteristic
203C42C4×Q811yescharacteristic
204C42#12311yescharacteristic
205C42#12311yescharacteristic
206C42#12411yescharacteristic
207C42#12511yescharacteristic
208C42#12511yescharacteristic
209C42#12511yescharacteristic
210C42#12511yescharacteristic
211C42#12611yescharacteristic
212C42#12611yescharacteristic
213C42#12611yescharacteristic
214C42#12611yescharacteristic
215C42C4⋊Q811yescharacteristic
216C23—22no—
217C23—41no—
218C23—41no—
219C23#39011yescharacteristic
220C23#39911yescharacteristic
221C23#39911yescharacteristic
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4#38111yescharacteristic
253C2×C4#38711yescharacteristic
254C2×C4#38711yescharacteristic
255C2×C4#38711yescharacteristic
256C2×C4#38711yescharacteristic
257C2×C4#39111yescharacteristic
258C2×C4#39711yescharacteristic
259C2×C4#39711yescharacteristic
260C2×C4#39811yescharacteristic
261C2×C4#39811yescharacteristic
262C2×C4#40011yescharacteristic
263C2×C4#40011yescharacteristic
264C8—42no—
265C8—42no—
266C8—82no—
267C8—82no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—82no—
277C22#144511yescharacteristic
278C22#197612yes—
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—84no—
291C4—162no—
292C4—162no—
293C4—162no—
294C4#182712yes—
295C2—22no—
296C2—41no—
297C2—41no—
298C2#1134811yescharacteristic
299C2#1136111yescharacteristic
300C2#1136411yescharacteristic
subgroupG/H
center, socleC22#1445
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
17. C4×#107C4semidirect
18. C4×#106C4semidirect

Maximal subgroups

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

subgroupindex
#213602
#80522
#79802
#79452
#79402
#79362