Finite Group Databasebeta

← #229315

Subgroups — #229315

Order 512.

629 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#23161C211yesmaximal, characteristic
2#23096C212yesmaximal
3#22725C211yesmaximal, characteristic
4C2×#1674C211yesmaximal, characteristic
5#9836C211yesmaximal, characteristic
6#9617C211yesmaximal, characteristic
7#9561C212yesmaximal
8#8986C212yesmaximal
9#8922C212yesmaximal
10#8285C211yesmaximal, characteristic
11#8203C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#462C2212yes—
14C2×C8×Q8C2211yescharacteristic
15#2365C2212yes—
16#2355C2212yes—
17#1990C2212yes—
18#1988C2212yes—
19#1969—22no—
20#1956C2212yes—
21#1949—22no—
22#1947C2212yes—
23#1947C2212yes—
24#1943—21no—
25#1928—22no—
26#1921—22no—
27#1878—22no—
28#1846—21no—
29#1844C2211yescharacteristic
30#1776—21no—
31C4×#106—21no—
32C4×#106C2211yescharacteristic
33#1770C2211yescharacteristic
34C8×C4⋊C4C2212yes—
35C4×#103—22no—
36C4×#103C2211yescharacteristic
37#1674C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×Q8⋊C8C2211yescharacteristic
41C2×#322C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1277C2211yescharacteristic
46C22×Q16—22no—
47C2×C4⋊Q8C2312yes—
48C2×C4×Q8—22no—
49C2×C4×Q8C2311yescharacteristic
50#500—22no—
51#500C2×C414yes—
52#462—24no—
53#462C2×C418yes—
54C8×Q8—22no—
55C8×Q8C2×C414yes—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#106—21no—
61C2×#106—21no—
62C2×#106D411yescharacteristic
63C2×#106D411yescharacteristic
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103D412yes—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#391—22no—
80#389—22no—
81#387—42no—
82#387—42no—
83#384C2312yes—
84#382—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2312yes—
89C4×C4⋊C4C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—42no—
95#340D412yes—
96#340D412yes—
97#336—22no—
98#336—21no—
99#335C2×C414yes—
100#332C2×C418yes—
101Q8⋊C8—22no—
102#322—22no—
103C22×Q8—22no—
104C22×Q8C2×D411yescharacteristic
105C2×Q16—28no—
106C2×Q16—82no—
107C2×Q16—82no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8C2×D412yes—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8C22×C414yes—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—44no—
122C4×Q8—82no—
123C4×Q8C22×C412yes—
124C4×Q8#4112yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D412yes—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42C2411yescharacteristic
142#107—24no—
143#107—24no—
144#107—82no—
145#107—82no—
146#106—22no—
147#106—42no—
148#106—81no—
149#106Q1614yescomplemented
150C4⋊C8—24no—
151C4⋊C8—21no—
152C4⋊C8—22no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8C22×C412yes—
156C4⋊C8C22×C414yes—
157#103—28no—
158#103—42no—
159#103—44no—
160#103—82no—
161#97—42no—
162C4×C8—24no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—82no—
168C4×C8C22×C412yes—
169C4×C8#4112yes—
170#95—22no—
171#95—21no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177C2×Q8—21no—
178C2×Q8—24no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8C4×D412yes—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4H(5,𝔽2)11yescharacteristic
197Q16—88no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8C4×D412yes—
215C2×C8C4×D412yes—
216C2×C8C2×Q1612yes—
217C2×C8#13512yes—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4C4×D412yes—
237C4⋊C4C2×Q1614yes—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—84no—
245C42—162no—
246C42C23×C411yescharacteristic
247C42C2×#4111yescharacteristic
248C42C2×#4111yescharacteristic
249C422−1+411yescharacteristic
250C23D4211yescharacteristic
251Q8—22no—
252Q8—22no—
253Q8—42no—
254Q8—44no—
255Q8—44no—
256Q8—82no—
257Q8—82no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4C2×C4×D411yescharacteristic
279C2×C4C2×C4×D411yescharacteristic
280C2×C4#52011yescharacteristic
281C2×C4#54711yescharacteristic
282C2×C4#54811yescharacteristic
283C2×C4#55311yescharacteristic
284C2×C4C22×Q1611yescharacteristic
285C2×C4C2×#13511yescharacteristic
286C2×C4#57512yes—
287C2×C4#57612yes—
288C2×C4#57711yescharacteristic
289C2×C4#57811yescharacteristic
290C8—22no—
291C8—42no—
292C8—42no—
293C8—44no—
294C8—81no—
295C8—81no—
296C8C4×Q1614yes—
297C22#236111yescharacteristic
298C22#326812yes—
299C22#328411yescharacteristic
300C22D4×Q1611yescharacteristic
301C22#328912yes—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—44no—
306C4—42no—
307C4—82no—
308C4—82no—
309C4—84no—
310C4—162no—
311C4C2×C4×Q1611yescharacteristic
312C4#294311yescharacteristic
313C4#294711yescharacteristic
314C4#294811yescharacteristic
315C4#334712yes—
316C4#336311yescharacteristic
317C4#336411yescharacteristic
318C2#1981512yes—
319C2#1987912yes—
320C2#2299811yescharacteristic
321C2#2303111yescharacteristic
322C2#2304711yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #1674C4semidirect
149. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#231612
#230962
#227252
C2×#16742
#98362
#96172
#95612
#89862
#89222
#82852
#82032