Finite Group Databasebeta

← #109728

Subgroups — #109728

Order 512.

480 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#20539C211yesmaximal, characteristic
3C2×#1308C211yesmaximal, characteristic
4#8046C211yesmaximal, characteristic
5#8030C211yesmaximal, characteristic
6C2×#1287C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1949—22no—
16#1853C412yes—
17#1852C412yes—
18#1847—21no—
19#1836—21no—
20#1771—22no—
21#1771—22no—
22#1750—22no—
23#1308C414yescomplemented
24#1296—21no—
25#1296—21no—
26#1295—22no—
27#1287C414yescomplemented
28#1280—22no—
29C2×#117—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432C2×C412yes—
43#432C2×C412yes—
44#432C2×C412yes—
45#432C2×C412yes—
46#432C2×C412yes—
47#432Q812yes—
48#432Q812yes—
49#431—21no—
50#431—21no—
51#431—21no—
52#431C2×C412yes—
53C2×#107—22no—
54C2×#106—22no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8C2×C412yes—
64C2×C22⋊C8D412yes—
65C2×#97—21no—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97—22no—
69C2×#97D412yes—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
72#384D411yescharacteristic
73#384D411yescharacteristic
74#382—21no—
75#382—21no—
76#382—21no—
77#382—21no—
78#382—21no—
79#380—21no—
80#380C2×C412yes—
81C4×C4⋊C4—21no—
82C4×C4⋊C4D412yes—
83C2×#95—21no—
84C2×#95—21no—
85#340—22no—
86#340—22no—
87#336—24no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—41no—
97C22×C8—41no—
98#117—81no—
99#117—81no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4C22⋊C412yes—
112C2×C22⋊C4—22no—
113C2×C42—22no—
114C2×C42C22×C411yescharacteristic
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×Q811yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#107—82no—
127#106—82no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8C4214yes—
147C22⋊C8C4⋊C414yes—
148#97—22no—
149#97—22no—
150#97—22no—
151#97—22no—
152#97—22no—
153#97—22no—
154#97—41no—
155#97—82no—
156#97C4⋊C414yes—
157C4×C8—22no—
158C4×C8—41no—
159C4×C8—82no—
160#95—22no—
161#95—22no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166C24C4×D411yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×C4211yescharacteristic
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C2×C4⋊C411yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4#11711yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4C4×Q811yescharacteristic
191C22×C4C4×Q811yescharacteristic
192C22×C4C4×Q811yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12511yescharacteristic
200C22×C4#12611yescharacteristic
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C22⋊C4—82no—
227C42—42no—
228C42—42no—
229C42C2×C4⋊C411yescharacteristic
230C42#11711yescharacteristic
231C42C4×D411yescharacteristic
232C42C4×D411yescharacteristic
233C42C4×D411yescharacteristic
234C42C4×D411yescharacteristic
235C42C4×D411yescharacteristic
236C42C4×Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42#12311yescharacteristic
241C42#12411yescharacteristic
242C42#12511yescharacteristic
243C42#12611yescharacteristic
244C42C4⋊Q811yescharacteristic
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23C4×C4⋊C411yescharacteristic
252C23#38611yescharacteristic
253C23#38711yescharacteristic
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—41no—
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—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—162no—
291C2×C4—162no—
292C2×C4C4×C22⋊C411yescharacteristic
293C2×C4C4×C4⋊C411yescharacteristic
294C2×C4#38511yescharacteristic
295C2×C4#38511yescharacteristic
296C2×C4#38711yescharacteristic
297C2×C4#38711yescharacteristic
298C2×C4#38711yescharacteristic
299C2×C4#38811yescharacteristic
300C2×C4#38811yescharacteristic
301C2×C4#38911yescharacteristic
302C2×C4#38911yescharacteristic
303C2×C4#39111yescharacteristic
304C8—44no—
305C8—44no—
306C8—44no—
307C8—82no—
308C8—82no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22#143611yescharacteristic
320C22#188111yescharacteristic
321C22#188111yescharacteristic
322C22#190411yescharacteristic
323C22#190511yescharacteristic
324C22#191411yescharacteristic
325C22#191511yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—82no—
340C4—161no—
341C4—161no—
342C2—41no—
343C2—41no—
344C2#933311yescharacteristic
345C2#933411yescharacteristic
346C2#933511yescharacteristic
347C2#933611yescharacteristic
348C2#1090011yescharacteristic
349C2#1090311yescharacteristic
350C2#1091611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
23. #1308C4semidirect
27. #1287C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#205392
C2×#13082
#80462
#80302
C2×#12872
#79142