Finite Group Databasebeta

← #121037

Subgroups — #121037

Order 512.

443 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20823C211yesmaximal, characteristic
2C2×#1302C211yesmaximal, characteristic
3#8034C211yesmaximal, characteristic
4#7984C211yesmaximal, characteristic
5#7981C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7926C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#97C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1847—21no—
16#1836—21no—
17#1808—21no—
18#1807—21no—
19#1776—22no—
20#1302—21no—
21#1302—21no—
22#1295C414yescomplemented
23#1289—22no—
24#1289—21no—
25#1289—21no—
26#1285—22no—
27#1282—22no—
28C23×C8—21no—
29C2×#117—21no—
30C2×#117—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44C2×#107—22no—
45C2×#106—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8Q812yes—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8D412yes—
52C2×#97—21no—
53C2×#97—21no—
54C2×#97—21no—
55C2×#97—21no—
56C2×#97C2×C412yes—
57C2×#97C2×C412yes—
58C2×#97D411yescharacteristic
59C2×#97D411yescharacteristic
60C2×#97Q812yes—
61C2×C4×C8—21no—
62#384—21no—
63#384—21no—
64#383—21no—
65#382—21no—
66#382—21no—
67#382C2×C412yes—
68#381—21no—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380D411yescharacteristic
74#380D411yescharacteristic
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D412yes—
80C4×C22⋊C4—21no—
81#376—21no—
82#340—22no—
83#340—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—44no—
95#117—81no—
96#117—81no—
97#117—81no—
98#117—81no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C42—22no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×Q811yescharacteristic
118C2×C42C2×Q811yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#107—44no—
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#97—22no—
141#97—22no—
142#97—22no—
143#97—22no—
144#97—21no—
145#97—21no—
146#97—41no—
147#97—41no—
148#97C4⋊C412yes—
149#97C4⋊C412yes—
150C4×C8—41no—
151C4×C8—41no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95—21no—
156#95—21no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161C24#12111yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4#11711yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12411yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12511yescharacteristic
187C22×C4#12511yescharacteristic
188C22×C4#12611yescharacteristic
189C22×C4C4⋊Q811yescharacteristic
190C2×C8—24no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C42—82no—
220C42—82no—
221C42—82no—
222C42C2×C4⋊C411yescharacteristic
223C42#11711yescharacteristic
224C42C4×D411yescharacteristic
225C42C4×Q811yescharacteristic
226C42C22⋊Q811yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42C22⋊Q811yescharacteristic
229C42#12311yescharacteristic
230C42#12411yescharacteristic
231C42#12411yescharacteristic
232C42#12511yescharacteristic
233C42#12511yescharacteristic
234C42#12511yescharacteristic
235C42#12611yescharacteristic
236C42#12611yescharacteristic
237C42#12611yescharacteristic
238C23—41no—
239C23—41no—
240C23#38811yescharacteristic
241C23#39411yescharacteristic
242C23#39811yescharacteristic
243C23#47111yescharacteristic
244C23#47111yescharacteristic
245C23#47211yescharacteristic
246C23#47311yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
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—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4#38211yescharacteristic
284C2×C4#38511yescharacteristic
285C2×C4#38711yescharacteristic
286C2×C4#38711yescharacteristic
287C2×C4#38811yescharacteristic
288C2×C4#38911yescharacteristic
289C2×C4#39311yescharacteristic
290C2×C4#39711yescharacteristic
291C2×C4#39811yescharacteristic
292C2×C4#40011yescharacteristic
293C2×C4#40011yescharacteristic
294C2×C4#40011yescharacteristic
295C8—24no—
296C8—42no—
297C8—82no—
298C8—82no—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22#144211yescharacteristic
311C22#193011yescharacteristic
312C22#193111yescharacteristic
313C22#205111yescharacteristic
314C22#205211yescharacteristic
315C22#208911yescharacteristic
316C22#209011yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—161no—
329C4—162no—
330C4—162no—
331C4—161no—
332C4#203112yes—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—21no—
338C2—21no—
339C2#1126211yescharacteristic
340C2#1127711yescharacteristic
341C2#1129011yescharacteristic
subgroupG/H
center, socleC22#1442
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. #1295C4semidirect

Maximal subgroups

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

subgroupindex
#208232
C2×#13022
#80342
#79842
#79812
#79522
#79262