Finite Group Databasebeta

← #120365

Subgroups — #120365

Order 512.

439 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21963C211yesmaximal, characteristic
2#17817C211yesmaximal, characteristic
3#8109C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8004C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7857C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—22no—
16#1948—21no—
17#1947—21no—
18#1809—22no—
19#1574—22no—
20#1573—22no—
21#1567C412yescomplemented
22#1566C412yescomplemented
23#1305—22no—
24#1303—21no—
25#1303—21no—
26#1292—22no—
27#1286—22no—
28#1274—22no—
29C22×M16—21no—
30C22×M16—21no—
31C2×#117—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431D412yes—
43#431Q812yes—
44#431Q812yes—
45C2×#107—22no—
46C2×#106—22no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423C2×C411yescharacteristic
55#423C2×C411yescharacteristic
56#423C2×C412yes—
57#423D412yes—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8D412yes—
62C2×C4⋊C8Q812yes—
63#407—21no—
64#407—21no—
65C2×C22⋊C8—21no—
66C4×M16—21no—
67C4×M16—21no—
68C2×C4×C8C2×C412yes—
69#384—21no—
70#383—21no—
71#382—21no—
72#382—21no—
73#381—21no—
74#380—21no—
75#380—21no—
76#380D412yes—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D411yescharacteristic
80C4×C4⋊C4D411yescharacteristic
81C4×C22⋊C4—21no—
82#376—21no—
83#340—22no—
84#340—22no—
85#335—24no—
86#334—24no—
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—82no—
98C2×M16—81no—
99C2×M16—81no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103#117—81no—
104#117—81no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—41no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42C2×Q811yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127#107—82no—
128#106—82no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—22no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8C22⋊C414yes—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#97—41no—
152#97—41no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—41no—
156C4×C8C4⋊C414yes—
157#95—22no—
158#95—21no—
159#95—21no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—41no—
164C24#12311yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4#11711yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4C4×Q811yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12511yescharacteristic
191C22×C4#12611yescharacteristic
192C22×C4C4⋊Q811yescharacteristic
193M16—82no—
194M16—82no—
195M16—81no—
196M16—81no—
197M16—82no—
198M16—81no—
199M16—81no—
200M16—81no—
201M16—81no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—82no—
211C2×C8—82no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—82no—
228C42C2×C22⋊C411yescharacteristic
229C42C2×C4⋊C411yescharacteristic
230C42C4×D411yescharacteristic
231C42C4×Q811yescharacteristic
232C42#12111yescharacteristic
233C42#12111yescharacteristic
234C42C22⋊Q811yescharacteristic
235C42C22⋊Q811yescharacteristic
236C42C22⋊Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42#12311yescharacteristic
240C42#12511yescharacteristic
241C42#12611yescharacteristic
242C42C4⋊Q811yescharacteristic
243C42C4⋊Q811yescharacteristic
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23#38711yescharacteristic
249C23#39411yescharacteristic
250C23#39811yescharacteristic
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4#38011yescharacteristic
286C2×C4#38611yescharacteristic
287C2×C4#38711yescharacteristic
288C2×C4#38911yescharacteristic
289C2×C4#38911yescharacteristic
290C2×C4#39111yescharacteristic
291C2×C4#39511yescharacteristic
292C2×C4#39611yescharacteristic
293C2×C4#39611yescharacteristic
294C2×C4#39811yescharacteristic
295C2×C4#39811yescharacteristic
296C2×C4#40011yescharacteristic
297C2×C4#47111yescharacteristic
298C2×C4#47111yescharacteristic
299C2×C4#47211yescharacteristic
300C2×C4#47311yescharacteristic
301C8—44no—
302C8—44no—
303C8—82no—
304C8—82no—
305C8—81no—
306C8—81no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—81no—
316C22—81no—
317C22#144111yescharacteristic
318C22#205111yescharacteristic
319C22#205211yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—161no—
330C4—162no—
331C4—161no—
332C4#190411yescharacteristic
333C4#190511yescharacteristic
334C4#195111yescharacteristic
335C4#195211yescharacteristic
336C4#206311yescharacteristic
337C4#206411yescharacteristic
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#1121311yescharacteristic
343C2#1124511yescharacteristic
344C2#1124911yescharacteristic
subgroupG/H
center, socleC22#1441
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
21. #1567C4semidirect
22. #1566C4semidirect

Maximal subgroups

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

subgroupindex
#219632
#178172
#81092
#80932
#80042
#79572
#78572