Finite Group Databasebeta

← #120977

Subgroups — #120977

Order 512.

423 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21363C211yesmaximal, characteristic
2#8110C211yesmaximal, characteristic
3#8098C211yesmaximal, characteristic
4#8086C211yesmaximal, characteristic
5#8070C211yesmaximal, characteristic
6#7968C211yesmaximal, characteristic
7#7950C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18#1776—22no—
19#1305—21no—
20#1305—21no—
21#1303—22no—
22#1302—22no—
23#1300—22no—
24#1287C414yescomplemented
25#1285—22no—
26C22×C4⋊C4—21no—
27C22×C42—21no—
28C22×C42C2311yescharacteristic
29#432—21no—
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432C2×C412yes—
38#432C2×C412yes—
39#432D412yes—
40#432Q812yes—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431C2×C412yes—
49C2×#107—22no—
50C2×#106—22no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423Q812yes—
55C2×C4⋊C8—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60C2×C22⋊C8—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16—21no—
65C2×#97D412yes—
66#383—21no—
67#382—21no—
68#381—21no—
69#380D411yescharacteristic
70#380D411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4D412yes—
76C4×C22⋊C4—21no—
77#376—21no—
78C2×#95—21no—
79#340—22no—
80#340—22no—
81C23×C4—41no—
82C23×C4C2×D411yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—22no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C22×C8—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—41no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—22no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42—82no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×Q811yescharacteristic
106C2×C42C2×Q811yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C2×C42#4111yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113#107—44no—
114#106—44no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135#97—22no—
136#97—22no—
137#97—22no—
138#97—41no—
139#97—41no—
140#97C4⋊C414yes—
141C4×C8—22no—
142C4×C8—41no—
143C4×C8—41no—
144C4×C8—41no—
145#95—21no—
146#95—21no—
147#95—22no—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152C24#12311yescharacteristic
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—81no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4#11711yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×Q811yescharacteristic
166C22×C4#12011yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4#12411yescharacteristic
174C22×C4#12411yescharacteristic
175C22×C4#12511yescharacteristic
176C22×C4#12511yescharacteristic
177C22×C4#12511yescharacteristic
178C22×C4#12611yescharacteristic
179C22×C4#12611yescharacteristic
180C22×C4#12611yescharacteristic
181M16—81no—
182M16—81no—
183M16—82no—
184M16—82no—
185M16—82no—
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C4⋊C4—24no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—82no—
201C42—82no—
202C42—82no—
203C42C2×C4⋊C411yescharacteristic
204C42#11711yescharacteristic
205C42C4×D411yescharacteristic
206C42C4×D411yescharacteristic
207C42C22⋊Q811yescharacteristic
208C42C22⋊Q811yescharacteristic
209C42C22⋊Q811yescharacteristic
210C42#12311yescharacteristic
211C42#12311yescharacteristic
212C42#12311yescharacteristic
213C42#12411yescharacteristic
214C42#12411yescharacteristic
215C42#12511yescharacteristic
216C42#12511yescharacteristic
217C42#12611yescharacteristic
218C42C4⋊Q811yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23#38711yescharacteristic
224C23#39411yescharacteristic
225C23#40011yescharacteristic
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
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—44no—
247C2×C4—82no—
248C2×C4—88no—
249C2×C4—82no—
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#38811yescharacteristic
260C2×C4#38811yescharacteristic
261C2×C4#38911yescharacteristic
262C2×C4#39311yescharacteristic
263C2×C4#39711yescharacteristic
264C2×C4#39811yescharacteristic
265C2×C4#39811yescharacteristic
266C2×C4#40011yescharacteristic
267C2×C4#40011yescharacteristic
268C8—44no—
269C8—82no—
270C8—82no—
271C8—82no—
272C8—82no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22#144211yescharacteristic
284C22#204811yescharacteristic
285C22#204911yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—84no—
298C4—161no—
299C4—161no—
300C4#203112yes—
301C2—21no—
302C2—21no—
303C2—41no—
304C2—41no—
305C2#1126511yescharacteristic
306C2#1127411yescharacteristic
307C2#1128211yescharacteristic
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
24. #1287C4semidirect

Maximal subgroups

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

subgroupindex
#213632
#81102
#80982
#80862
#80702
#79682
#79502