Finite Group Databasebeta

← #229306

Subgroups — #229306

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23187C211yesmaximal, characteristic
2#22964C212yesmaximal
3Q8×#106C211yesmaximal, characteristic
4C2×#1670C211yesmaximal, characteristic
5#9839C211yesmaximal, characteristic
6#9618C211yesmaximal, characteristic
7#9565C212yesmaximal
8#8983C212yesmaximal
9#8925C212yesmaximal
10#8285C211yesmaximal, characteristic
11#8203C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#475C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C2×#441—21no—
16#2365C2212yes—
17Q8×C4⋊C4C2212yes—
18#1990C2212yes—
19#1990C2212yes—
20#1981C2212yes—
21#1971—21no—
22#1948—22no—
23#1948C2212yes—
24#1948C2212yes—
25#1941—22no—
26#1920—22no—
27#1878—22no—
28#1865—22no—
29#1844—21no—
30#1844C2211yescharacteristic
31C4×#106—21no—
32C4×#106C2211yescharacteristic
33#1770C2211yescharacteristic
34C8×C4⋊C4C2212yes—
35C4×#103—22no—
36C4×#103C2211yescharacteristic
37#1670C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×Q8⋊C8C2211yescharacteristic
41C2×#322C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1277C2211yescharacteristic
46C22×Q16—21no—
47C22×Q16—21no—
48C2×C4⋊Q8C2312yes—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2311yescharacteristic
52#494—22no—
53#494C2×C414yes—
54#475—24no—
55#475C2×C418yes—
56C8×Q8—22no—
57C8×Q8C2×C414yes—
58#441—44no—
59C2×#106—22no—
60C2×#106—21no—
61C2×#106—21no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106D411yescharacteristic
65C2×#106D411yescharacteristic
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103D412yes—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391—22no—
81#389—22no—
82#389—22no—
83#389—42no—
84#387—42no—
85#384—22no—
86#384C2312yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2312yes—
90C4×C4⋊C4C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—42no—
96#340D412yes—
97#340D412yes—
98#336—22no—
99#336—21no—
100#335C2×C414yes—
101#332C2×C418yes—
102Q8⋊C8—22no—
103#322—22no—
104C22×Q8—22no—
105C22×Q8C2×D411yescharacteristic
106C2×Q16—24no—
107C2×Q16—24no—
108C2×Q16—42no—
109C2×Q16—42no—
110C2×Q16—81no—
111C2×Q16—81no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8C2×D412yes—
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8C22×C414yes—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—44no—
126C4×Q8—44no—
127C4×Q8—82no—
128C4×Q8—82no—
129C4×Q8C22×C412yes—
130C4×Q8#4112yes—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D412yes—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—22no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42C2411yescharacteristic
150#106—22no—
151#106—28no—
152#106—42no—
153#106—82no—
154#106—82no—
155#106—81no—
156#106SD1614yes—
157C4⋊C8—24no—
158C4⋊C8—21no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8C22×C412yes—
163C4⋊C8C22×C414yes—
164#103—28no—
165#103—24no—
166#103—44no—
167#103—82no—
168#97—42no—
169C4×C8—24no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8—82no—
175C4×C8C22×C412yes—
176C4×C8#4112yes—
177#95—21no—
178#95—22no—
179#95—42no—
180#95—42no—
181#95—42no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—24no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8C4×D412yes—
191C2×Q8#13714yes—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4H(5,𝔽2)11yescharacteristic
202Q16—48no—
203Q16—84no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—22no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8C4×D412yes—
221C2×C8C4×D412yes—
222C2×C8C2×SD1612yes—
223C2×C8#13512yes—
224C4⋊C4—22no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—84no—
241C4⋊C4—84no—
242C4⋊C4—82no—
243C4⋊C4—162no—
244C4⋊C4C4×D412yes—
245C4⋊C4C2×SD1614yes—
246C42—22no—
247C42—22no—
248C42—41no—
249C42—42no—
250C42—42no—
251C42—41no—
252C42—84no—
253C42—84no—
254C42C23×C411yescharacteristic
255C42C2×#4111yescharacteristic
256C42C2×#4111yescharacteristic
257C422−1+411yescharacteristic
258C23D4211yescharacteristic
259Q8—24no—
260Q8—22no—
261Q8—44no—
262Q8—44no—
263Q8—82no—
264Q8—82no—
265Q8#44114yescomplemented
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4C2×C4×D411yescharacteristic
287C2×C4C2×C4×D411yescharacteristic
288C2×C4#52011yescharacteristic
289C2×C4#54711yescharacteristic
290C2×C4#54811yescharacteristic
291C2×C4#55311yescharacteristic
292C2×C4C22×SD1611yescharacteristic
293C2×C4C2×#13511yescharacteristic
294C2×C4C2×#13712yes—
295C2×C4#57611yescharacteristic
296C2×C4#57712yes—
297C2×C4#57711yescharacteristic
298C8—22no—
299C8—42no—
300C8—42no—
301C8—44no—
302C8—81no—
303C8—81no—
304C8C4×SD1614yes—
305C22#236111yescharacteristic
306C22#326612yes—
307C22D4×SD1611yescharacteristic
308C22#328711yescharacteristic
309C22#328912yes—
310C4—21no—
311C4—21no—
312C4—24no—
313C4—42no—
314C4—44no—
315C4—82no—
316C4—82no—
317C4—84no—
318C4—162no—
319C4C2×C4×SD1611yescharacteristic
320C4C2×#44111yescharacteristic
321C4#294611yescharacteristic
322C4#295211yescharacteristic
323C4#334412yes—
324C4#336211yescharacteristic
325C4#336511yescharacteristic
326C2#1980512yes—
327C2#1986912yes—
328C2#2299711yescharacteristic
329C2#2302811yescharacteristic
330C2#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. #1670C4semidirect
265. Q8#441semidirect

Maximal subgroups

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

subgroupindex
#231872
#229642
Q8×#1062
C2×#16702
#98392
#96182
#95652
#89832
#89252
#82852
#82032