Finite Group Databasebeta

← #120812

Subgroups — #120812

Order 512.

428 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17810C211yesmaximal, characteristic
2#8098C211yesmaximal, characteristic
3#8096C211yesmaximal, characteristic
4#8021C211yesmaximal, characteristic
5#7964C211yesmaximal, characteristic
6#7908C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1573—22no—
16#1572—22no—
17#1565—22no—
18#1303—22no—
19#1303C412yescomplemented
20#1303C412yescomplemented
21#1294—22no—
22#1287—22no—
23#1280—22no—
24#1274—22no—
25C22×C4⋊C4—21no—
26C22×C42—21no—
27C22×C42C2311yescharacteristic
28#432—21no—
29#432—21no—
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432D412yes—
35#432Q812yes—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431C2×C412yes—
45#431D412yes—
46#431D412yes—
47#431Q812yes—
48#423—21no—
49#423C2×C412yes—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53#407—21no—
54#407—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C4×M16—21no—
59C4×M16—21no—
60C2×#97—21no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#383—21no—
64#382—21no—
65#381—21no—
66#380—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4C2×C411yescharacteristic
72C4×C4⋊C4C2×C411yescharacteristic
73C4×C22⋊C4—21no—
74#376—21no—
75C2×#95D412yes—
76#335—24no—
77#334—24no—
78C23×C4—22no—
79C23×C4C2×D411yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4#4111yescharacteristic
82C2×M16—22no—
83C2×M16—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42—82no—
99C2×C42C22×C411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×Q811yescharacteristic
104C2×C42C2×Q811yescharacteristic
105C2×C42#4111yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8C4⋊C414yes—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131#97—22no—
132#97—22no—
133#97—41no—
134#97—41no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—41no—
140C4×C8—41no—
141#95—21no—
142#95—21no—
143#95—22no—
144#95—22no—
145#95—41no—
146#95—41no—
147#95—41no—
148C24#12311yescharacteristic
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—81no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4#11711yescharacteristic
162C22×C4C4×D411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170C22×C4#12311yescharacteristic
171C22×C4#12411yescharacteristic
172C22×C4#12411yescharacteristic
173C22×C4#12511yescharacteristic
174C22×C4#12511yescharacteristic
175C22×C4#12611yescharacteristic
176C22×C4#12611yescharacteristic
177C22×C4#12611yescharacteristic
178C22×C4C4⋊Q811yescharacteristic
179M16—42no—
180M16—42no—
181M16—81no—
182M16—81no—
183C2×C8—24no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C42—42no—
198C42—42no—
199C42—82no—
200C42—82no—
201C42C2×C4⋊C411yescharacteristic
202C42#11711yescharacteristic
203C42C4×D411yescharacteristic
204C42C4×Q811yescharacteristic
205C42#12011yescharacteristic
206C42C22⋊Q811yescharacteristic
207C42C22⋊Q811yescharacteristic
208C42C22⋊Q811yescharacteristic
209C42#12311yescharacteristic
210C42#12311yescharacteristic
211C42#12411yescharacteristic
212C42#12411yescharacteristic
213C42#12511yescharacteristic
214C42#12511yescharacteristic
215C42#12511yescharacteristic
216C42#12611yescharacteristic
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23#38811yescharacteristic
223C23#39711yescharacteristic
224C23#39811yescharacteristic
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
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—44no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4#38211yescharacteristic
257C2×C4#38511yescharacteristic
258C2×C4#38711yescharacteristic
259C2×C4#38711yescharacteristic
260C2×C4#38811yescharacteristic
261C2×C4#38911yescharacteristic
262C2×C4#39311yescharacteristic
263C2×C4#39411yescharacteristic
264C2×C4#39811yescharacteristic
265C2×C4#40011yescharacteristic
266C2×C4#40011yescharacteristic
267C2×C4#40011yescharacteristic
268C2×C4#44312yes—
269C2×C4#44412yes—
270C8—44no—
271C8—82no—
272C8—82no—
273C8—82no—
274C8—82no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#144211yescharacteristic
287C22#193711yescharacteristic
288C22#193811yescharacteristic
289C22#208111yescharacteristic
290C22#208211yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—84no—
302C4—161no—
303C4—161no—
304C4#186612yes—
305C4#192212yes—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—41no—
311C2#1127711yescharacteristic
312C2#1128311yescharacteristic
313C2#1129111yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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. #1303C4semidirect
20. #1303C4semidirect

Maximal subgroups

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

subgroupindex
#178102
#80982
#80962
#80212
#79642
#79082
#78582