Finite Group Databasebeta

← #242546

Subgroups — #242546

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21961C211yesmaximal, characteristic
2#21954C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#21864C211yesmaximal, characteristic
5C2×#1571C211yesmaximal, characteristic
6C2×#1552C211yesmaximal, characteristic
7#10594C214yesmaximal, complemented
8#10592C214yesmaximal, complemented
9#8042C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2285C2211yescharacteristic
16#2285C2211yescharacteristic
17#1990C2212yes—
18#1988C2212yes—
19#1981C2212yes—
20#1956C2212yes—
21#1947—22no—
22#1947C2214yes—
23#1940—22no—
24#1939C2214yes—
25#1894—22no—
26#1873—22no—
27#1808—22no—
28#1805—22no—
29#1803—22no—
30#1798—22no—
31#1571—24no—
32C2×#335C2214yes—
33#1552—24no—
34C2×#332C2212yes—
35C2×#332C2212yes—
36#1296C2214yes—
37C22×M16—21no—
38C23×C8—21no—
39#533—22no—
40#533—22no—
41#533—22no—
42#533Q814yes—
43C2×C4⋊Q8C2311yescharacteristic
44C2×C4⋊Q8C2311yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#432—22no—
55#432—22no—
56#431—22no—
57#431—22no—
58C2×#107—24no—
59#423—22no—
60#423—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8D414yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×#103—24no—
71C2×#103—24no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D412yes—
74C2×#97—21no—
75C2×C4×C8C2312yes—
76#384D412yes—
77#384C2312yes—
78#382—21no—
79#382—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380D412yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2312yes—
88#340—24no—
89#340—24no—
90#340—24no—
91#336—24no—
92#335—28no—
93#332—24no—
94#332—24no—
95C22×Q8—21no—
96C22×Q8—21no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—42no—
101C2×M16—82no—
102C22×C8—22no—
103C22×C8—24no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—82no—
107C22×C8C2×D414yes—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8C2×Q812yes—
115C4⋊Q8C2×Q812yes—
116#125—24no—
117#125—42no—
118C22⋊Q8—24no—
119C22⋊Q8—24no—
120C22⋊Q8—42no—
121C22⋊Q8—42no—
122C4×Q8—24no—
123C4×Q8—24no—
124C4×Q8—42no—
125C4×Q8—42no—
126#117—24no—
127#117—42no—
128#117—82no—
129#117—82no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C2×D414yes—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C42—22no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D412yes—
151C2×C42C2×Q812yes—
152C2×C42#4112yes—
153C2×C42#4112yes—
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156#107—84no—
157C4⋊C8—24no—
158C4⋊C8—24no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—42no—
165C4⋊C8—84no—
166C4⋊C8C2×D414yes—
167#103—84no—
168#103—84no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171#97—42no—
172C4×C8—22no—
173C4×C8—22no—
174#95—22no—
175#95—22no—
176#95—41no—
177#95—41no—
178C24#12111yescharacteristic
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4C22⋊Q812yes—
202C22×C4C22⋊Q812yes—
203C22×C4C22⋊Q812yes—
204C22×C4#12612yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208M16—84no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—44no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—84no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C22⋊C4—44no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C42—44no—
242C42—44no—
243C42—82no—
244C42#12112yes—
245C42#12112yes—
246C42C22⋊Q812yes—
247C42C22⋊Q812yes—
248C42C22⋊Q812yes—
249C42#12612yes—
250C42C22×D411yescharacteristic
251C42C22×Q811yescharacteristic
252C42C2×#4111yescharacteristic
253C42C2×#4111yescharacteristic
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23#39612yes—
259C23C2×#12111yescharacteristic
260Q8—44no—
261Q8—44no—
262Q8—81no—
263Q8—81no—
264Q8—82no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#39612yes—
289C2×C4#39612yes—
290C2×C4#39612yes—
291C2×C4C2×#12111yescharacteristic
292C2×C4C2×#12111yescharacteristic
293C2×C4C2×C22⋊Q811yescharacteristic
294C2×C4C2×C22⋊Q811yescharacteristic
295C2×C4C2×C22⋊Q811yescharacteristic
296C2×C4C2×#12611yescharacteristic
297C2×C4#57612yes—
298C2×C4#57612yes—
299C2×C4#57812yes—
300C2×C4#57812yes—
301C8—44no—
302C8—44no—
303C8—82no—
304C8—82no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22C2×#39611yescharacteristic
311C22#304111yescharacteristic
312C22#304411yescharacteristic
313C22#306112yes—
314C22#306912yes—
315C4—21no—
316C4—21no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—84no—
321C4—82no—
322C4—82no—
323C4—162no—
324C4#304111yescharacteristic
325C4#304411yescharacteristic
326C4#306112yes—
327C4#308312yes—
328C4#308312yes—
329C2—42no—
330C2#1951112yes—
331C2#1953412yes—
332C2#2360311yescharacteristic
333C2#2361111yescharacteristic
334C2#2361111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10594C2semidirect
8. #10592C2semidirect

Maximal subgroups

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

subgroupindex
#219612
#219542
#218942
#218642
C2×#15712
C2×#15522
#105942
#105922
#80422