Finite Group Databasebeta

← #121170

Subgroups — #121170

Order 512.

388 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8089C211yesmaximal, characteristic
2#8030C211yesmaximal, characteristic
3#8026C211yesmaximal, characteristic
4#8017C211yesmaximal, characteristic
5#7984C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1302—22no—
16#1295—22no—
17#1294—22no—
18#1294—21no—
19#1294—21no—
20#1289—21no—
21#1289—21no—
22#1285C414yescomplemented
23#1282—22no—
24C2×#117—21no—
25C22×C4⋊C4—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#432—21no—
37#432C2×C412yes—
38#432C2×C412yes—
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×#97D412yes—
54C2×C4×C8—21no—
55#384—21no—
56#383—21no—
57#382—21no—
58#382—21no—
59#382—21no—
60#382—21no—
61#382—21no—
62#382Q812yes—
63#381—21no—
64#380—21no—
65#380—21no—
66#380D411yescharacteristic
67#380D411yescharacteristic
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4C2×C412yes—
71C4×C22⋊C4—21no—
72#376—21no—
73C2×#95—21no—
74C2×#95—21no—
75C23×C4—41no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4#4111yescharacteristic
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8—41no—
83#117—81no—
84#117—81no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—41no—
97C2×C42—22no—
98C2×C42—41no—
99C2×C42C22×C411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×Q811yescharacteristic
102C2×C42#4111yescharacteristic
103C2×C42#4111yescharacteristic
104C2×C42#4111yescharacteristic
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—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127#97—22no—
128#97—22no—
129#97—22no—
130#97—22no—
131#97—41no—
132#97—41no—
133C4×C8—41no—
134C4×C8—41no—
135#95—22no—
136#95—22no—
137#95—22no—
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—81no—
156C22×C4—81no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4#11711yescharacteristic
162C22×C4C4×D411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4C22⋊Q811yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170C22×C4#12311yescharacteristic
171C22×C4#12311yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4#12411yescharacteristic
174C22×C4#12411yescharacteristic
175C22×C4#12611yescharacteristic
176C22×C4#12611yescharacteristic
177C22×C4#12611yescharacteristic
178C22×C4#12611yescharacteristic
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—82no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—82no—
201C42—82no—
202C42#11711yescharacteristic
203C42#11711yescharacteristic
204C42C4×D411yescharacteristic
205C42C4×Q811yescharacteristic
206C42#12311yescharacteristic
207C42#12311yescharacteristic
208C42#12411yescharacteristic
209C42#12511yescharacteristic
210C42#12511yescharacteristic
211C42#12511yescharacteristic
212C42#12611yescharacteristic
213C42#12611yescharacteristic
214C42#12611yescharacteristic
215C42#12611yescharacteristic
216C42#12611yescharacteristic
217C42#12611yescharacteristic
218C23—41no—
219C23—41no—
220C23#38811yescharacteristic
221C23#39611yescharacteristic
222C23#39711yescharacteristic
223C23#47111yescharacteristic
224C23#47111yescharacteristic
225C23#47211yescharacteristic
226C23#47311yescharacteristic
227C2×C4—21no—
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—41no—
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—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—162no—
264C2×C4—162no—
265C2×C4#38311yescharacteristic
266C2×C4#38711yescharacteristic
267C2×C4#38711yescharacteristic
268C2×C4#38711yescharacteristic
269C2×C4#38811yescharacteristic
270C2×C4#38811yescharacteristic
271C2×C4#39711yescharacteristic
272C2×C4#39711yescharacteristic
273C2×C4#40011yescharacteristic
274C2×C4#40011yescharacteristic
275C2×C4#40011yescharacteristic
276C2×C4#40111yescharacteristic
277C8—82no—
278C8—82no—
279C8—82no—
280C8—82no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#144311yescharacteristic
292C22#193011yescharacteristic
293C22#193111yescharacteristic
294C22#206311yescharacteristic
295C22#206411yescharacteristic
296C22#207811yescharacteristic
297C22#207911yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—162no—
311C4—161no—
312C4—161no—
313C4—162no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—21no—
318C2—21no—
319C2—21no—
320C2#1130311yescharacteristic
321C2#1130711yescharacteristic
322C2#1131411yescharacteristic
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
#80302
#80262
#80172
#79842
#79522
#79282