Finite Group Databasebeta

← #121165

Subgroups — #121165

Order 512.

388 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8089C211yesmaximal, characteristic
2#8018C211yesmaximal, characteristic
3#8018C211yesmaximal, characteristic
4#7981C211yesmaximal, characteristic
5#7952C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7916C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#383C2211yescharacteristic
15#1302—22no—
16#1294—21no—
17#1294—21no—
18#1294—21no—
19#1294—21no—
20#1289—22no—
21#1285—22no—
22#1285C414yescomplemented
23#1280—22no—
24C2×#117—21no—
25C2×#117—21no—
26C22×C42C2311yescharacteristic
27#432—21no—
28#432—21no—
29#432—21no—
30#432—21no—
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432C2×C412yes—
37#432C2×C412yes—
38#432D412yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8D412yes—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55#384—21no—
56#384—21no—
57#383—21no—
58#383—21no—
59#383—21no—
60#383—21no—
61#382—21no—
62#382—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4C2×C412yes—
70C4×C4⋊C4Q812yes—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4#4111yescharacteristic
80C22×C8—22no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84#117—81no—
85#117—81no—
86#117—81no—
87#117—81no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42C22×C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×Q811yescharacteristic
101C2×C42#4111yescharacteristic
102C2×C42#4111yescharacteristic
103C2×C42#4111yescharacteristic
104C2×C42#4111yescharacteristic
105C2×C42#4111yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108C2×C42#4111yescharacteristic
109C2×C42#4111yescharacteristic
110C4⋊C8—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126#97—22no—
127#97—22no—
128#97—22no—
129#97—41no—
130#97—41no—
131#97—41no—
132C4×C8—41no—
133C4×C8—41no—
134#95—22no—
135#95—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146C24#12111yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4#11711yescharacteristic
166C22×C4C4×D411yescharacteristic
167C22×C4C4×D411yescharacteristic
168C22×C4#12111yescharacteristic
169C22×C4#12111yescharacteristic
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4C22⋊Q811yescharacteristic
173C22×C4#12311yescharacteristic
174C22×C4#12311yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12311yescharacteristic
177C22×C4#12411yescharacteristic
178C22×C4#12411yescharacteristic
179C22×C4#12611yescharacteristic
180C22×C4#12611yescharacteristic
181C22×C4#12611yescharacteristic
182C22×C4#12611yescharacteristic
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C42—82no—
202C42—82no—
203C42#11711yescharacteristic
204C42#11711yescharacteristic
205C42C4×D411yescharacteristic
206C42C4×Q811yescharacteristic
207C42#12311yescharacteristic
208C42#12311yescharacteristic
209C42#12411yescharacteristic
210C42#12511yescharacteristic
211C42#12511yescharacteristic
212C42#12511yescharacteristic
213C42#12611yescharacteristic
214C42#12611yescharacteristic
215C42#12611yescharacteristic
216C42#12611yescharacteristic
217C42#12611yescharacteristic
218C42#12611yescharacteristic
219C23—41no—
220C23—41no—
221C23#38811yescharacteristic
222C23#39611yescharacteristic
223C23#39711yescharacteristic
224C23#47111yescharacteristic
225C23#47111yescharacteristic
226C23#47211yescharacteristic
227C23#47311yescharacteristic
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
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—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—161no—
265C2×C4—161no—
266C2×C4—161no—
267C2×C4—161no—
268C2×C4#38311yescharacteristic
269C2×C4#38711yescharacteristic
270C2×C4#38711yescharacteristic
271C2×C4#38711yescharacteristic
272C2×C4#38811yescharacteristic
273C2×C4#38811yescharacteristic
274C2×C4#39711yescharacteristic
275C2×C4#39711yescharacteristic
276C2×C4#40011yescharacteristic
277C2×C4#40011yescharacteristic
278C2×C4#40011yescharacteristic
279C2×C4#40111yescharacteristic
280C8—82no—
281C8—82no—
282C8—82no—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22#144311yescharacteristic
295C22#193011yescharacteristic
296C22#193111yescharacteristic
297C22#206311yescharacteristic
298C22#206411yescharacteristic
299C22#207811yescharacteristic
300C22#207911yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—162no—
314C4—162no—
315C4—161no—
316C4—161no—
317C2—21no—
318C2—21no—
319C2—21no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2#1130311yescharacteristic
324C2#1130711yescharacteristic
325C2#1130911yescharacteristic
subgroupG/H
center, socleC22#1443
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
22. #1285C4semidirect

Maximal subgroups

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

subgroupindex
#80892
#80182
#79812
#79522
#79162