Finite Group Databasebeta

← #193291

Subgroups — #193291

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34365C211yesmaximal, characteristic
2#21987C212yesmaximal
3C2×#1956C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5#8929C212yesmaximal, complemented
6#8928C212yesmaximal, complemented
7#8927C214yesmaximal, complemented
8#7898C212yesmaximal
9C22×C4⋊Q8C2211yescharacteristic
10C2×#432C2212yes—
11C22×#103—22no—
12C22×C4×C8C2211yescharacteristic
13#2381—22no—
14#2369C2212yes—
15#2369C2211yescharacteristic
16#2369C2211yescharacteristic
17#2344—22no—
18#2344—22no—
19#2331—22no—
20#2331—22no—
21#2310—22no—
22C2×#384C2211yescharacteristic
23#2283C2212yes—
24#1956—22no—
25#1956C2212yes—
26#1956C2212yes—
27#1948C2214yes—
28#1947C2214yes—
29#1853—22no—
30#1852—22no—
31#1847—22no—
32#1836—22no—
33#1558C418yescomplemented
34C2×#333C2212yes—
35C2×#333C2212yes—
36#1279C2214yes—
37#1279C2214yes—
38C23×Q8D412yes—
39C23×C8—21no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C414yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8Q814yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#117—22no—
48C2×#117—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C42C2311yescharacteristic
53#432—24no—
54#432C2×C418yes—
55C2×#107—24no—
56C2×#106—24no—
57C2×C4⋊C8C2314yes—
58C2×#103—24no—
59C2×#103—24no—
60C2×#103—48no—
61C2×C22⋊C8—22no—
62C2×#97—22no—
63C2×C4×C8—24no—
64C2×C4×C8—21no—
65C2×C4×C8C2312yes—
66#391—22no—
67#391—24no—
68#391—22no—
69#391—24no—
70#391—22no—
71#391—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#384—22no—
81#384—21no—
82#384C2312yes—
83#383—22no—
84#382D412yes—
85#382D412yes—
86#381—24no—
87#381—22no—
88#380D412yes—
89#380D412yes—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2312yes—
95C4×C4⋊C4C2312yes—
96C4×C22⋊C4—22no—
97#376—22no—
98#340—24no—
99#340—24no—
100#333C2×C418yescomplemented
101#333C2×C418yescomplemented
102C22×Q8—22no—
103C22×Q8—24no—
104C22×Q8—44no—
105C22×Q8—44no—
106C22×Q8C4⋊C418yes—
107C22×Q8C2×D412yes—
108C22×Q8C2×D412yes—
109C23×C4—22no—
110C23×C4—41no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C22×C8—24no—
114C22×C8—22no—
115C22×C8—24no—
116C22×C8—42no—
117C22×C8—42no—
118C4⋊Q8—28no—
119C4⋊Q8—28no—
120C4⋊Q8—24no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8C4⋊C418yes—
124C4⋊Q8C22×C412yes—
125C4⋊Q8C22×C412yes—
126C4⋊Q8C2×Q812yes—
127C4⋊Q8C2×Q812yes—
128#117—84no—
129#117—84no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—84no—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—24no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42C22×C412yes—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D412yes—
158C2×C42C2×Q812yes—
159C2×C42#4114yes—
160C2×C42C2411yescharacteristic
161#107—84no—
162#106—84no—
163C4⋊C8—24no—
164C4⋊C8C22×C418yes—
165#103—48no—
166#103—48no—
167C22⋊C8—28no—
168#97—28no—
169C4×C8—24no—
170C4×C8—42no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183C24#12011yescharacteristic
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—28no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—416no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—88no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8C2×C4⋊C414yes—
196C2×Q8C2×C4⋊C414yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—28no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—84no—
212C22×C4—82no—
213C22×C4C2×C4⋊C412yes—
214C22×C4C4×D414yes—
215C22×C4#12012yes—
216C22×C4#12011yescharacteristic
217C22×C4C22⋊Q814yes—
218C22×C4#12312yes—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D412yes—
221C2×C8—24no—
222C2×C8—28no—
223C2×C8—48no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—44no—
227C2×C8—44no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—216no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—84no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C42—44no—
248C42—44no—
249C42—84no—
250C42—82no—
251C42—82no—
252C42C2×C4⋊C412yes—
253C42C4×D414yes—
254C42C22⋊Q814yes—
255C42#12312yes—
256C42C23×C411yescharacteristic
257C42C22×Q811yescharacteristic
258C42C2×#4112yes—
259C23—22no—
260C23—22no—
261C23#38512yes—
262C23#45312yes—
263C23#45512yes—
264C23C2×#12011yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—82no—
268Q8—82no—
269Q8—88no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—416no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—164no—
302C2×C4#38514yes—
303C2×C4#38512yes—
304C2×C4C22×C4⋊C411yescharacteristic
305C2×C4C2×C4×D412yes—
306C2×C4C2×C22⋊Q812yes—
307C2×C4C2×#12311yescharacteristic
308C2×C4#57714yes—
309C2×C4#57814yes—
310C8—44no—
311C8—48no—
312C22—24no—
313C22—24no—
314C22—22no—
315C22—22no—
316C22#185614yes—
317C22C2×#38511yescharacteristic
318C22#301012yes—
319C22#301112yes—
320C22C2×#45311yescharacteristic
321C22C2×#45511yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—24no—
326C4—44no—
327C4—44no—
328C4—82no—
329C4—82no—
330C4—82no—
331C4—82no—
332C4—162no—
333C4#294714yes—
334C4#308214yes—
335C2—24no—
336C2#1868411yescharacteristic
337C2#1868511yescharacteristic
338C2#1869012yes—
339C2C2×#185611yescharacteristic
340C2#2087712yes—
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8929C2semidirect
6. #8928C2semidirect
7. #8927C2semidirect
33. #1558C4semidirect
100. #333C2×C4semidirect
101. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343652
#219872
C2×#19562
C2×#15582
#89292
#89282
#89272
#78982