Finite Group Databasebeta

← #315128

Subgroups — #315128

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28830C211yesmaximal, characteristic, complemented
2#28711C211yesmaximal, characteristic
3#27728C212yesmaximal
4#10413C212yesmaximal
5#10352C212yesmaximal, complemented
6#9836C211yesmaximal, characteristic, complemented
7#9730C211yesmaximal, characteristic, complemented
8#9568C212yesmaximal, complemented
9#8650C211yesmaximal, characteristic
10#8285C211yesmaximal, characteristic, complemented
11#8210C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#436C2211yescharacteristic
14#2571C2212yes—
15#2563C2212yes—
16#2101—22no—
17#2085—22no—
18#2075—22no—
19#2073—22no—
20#1990C2212yes—
21#1987C2212yes—
22#1987C2212yes—
23#1956C2212yes—
24#1947C2212yes—
25#1941—21no—
26#1934—22no—
27#1846—21no—
28#1844C2211yescharacteristic
29#1835—22no—
30#1809—21no—
31#1809—21no—
32#1802C2212yes—
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#102C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#333C2212yes—
39C2×D4⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C22×M16—21no—
46C2×C4⋊Q8C2312yes—
47C2×C4×D4C2311yescharacteristic
48#500—22no—
49#500—22no—
50#436—22no—
51#436—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—22no—
55C2×#106—21no—
56C2×#106—21no—
57C2×#106D411yescharacteristic
58C2×#106D411yescharacteristic
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—42no—
63C2×#102—22no—
64C2×#102D412yes—
65C2×C22⋊C8—21no—
66C2×C4×C8—22no—
67C2×C4×C8—21no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#400—42no—
72#393—42no—
73#390—22no—
74#389—22no—
75#389—22no—
76#384C2312yes—
77#382—22no—
78#382—22no—
79#382—22no—
80#380—22no—
81#380—22no—
82C4×C4⋊C4C2312yes—
83C4×C4⋊C4C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340D412yes—
93#340D412yes—
94#336—22no—
95#336—21no—
96#335—22no—
97#333—24no—
98D4⋊C8—22no—
99#322—22no—
100C22×Q8—22no—
101C22×D4C2×D411yescharacteristic
102C23×C4—21no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16—42no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8C2×D412yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D411yescharacteristic
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42C2411yescharacteristic
137#107—42no—
138#107—42no—
139#107—82no—
140#106—22no—
141#106—22no—
142#106—42no—
143#106—81no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8#4112yes—
150#103—84no—
151#102—42no—
152#102—44no—
153C22⋊C8—42no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—44no—
159C4×C8—42no—
160C4×C8#4112yes—
161C4×C8#4112yes—
162#95—22no—
163#95—21no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169C24—21no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—82no—
173C2×Q8—82no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—22no—
177C2×D4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4H(5,𝔽2)11yescharacteristic
193M16—84no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8#12412yes—
211C2×C8#12412yes—
212C2×C8#13512yes—
213C2×C8#13512yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—162no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—162no—
235C22⋊C4—162no—
236C42—41no—
237C42—42no—
238C42—42no—
239C42—41no—
240C42C2×#4111yescharacteristic
241C42C2×#4111yescharacteristic
242C42C2×#4111yescharacteristic
243C422−1+411yescharacteristic
244C23—41no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23D4211yescharacteristic
250Q8—84no—
251Q8—162no—
252D4—42no—
253D4—41no—
254D4—41no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4C2×#12411yescharacteristic
279C2×C4#53211yescharacteristic
280C2×C4#54211yescharacteristic
281C2×C4#54711yescharacteristic
282C2×C4#54811yescharacteristic
283C2×C4#55511yescharacteristic
284C2×C4C2×#13511yescharacteristic
285C2×C4C2×#13511yescharacteristic
286C2×C4#57512yes—
287C2×C4#57711yescharacteristic
288C2×C4#57812yes—
289C2×C4#57811yescharacteristic
290C8—22no—
291C8—42no—
292C8—42no—
293C8—81no—
294C8—81no—
295C8—82no—
296C8#48814yes—
297C22—81no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22—82no—
302C22#249111yescharacteristic
303C22#326912yes—
304C22#328311yescharacteristic
305C22#328411yescharacteristic
306C22#328812yes—
307C4—21no—
308C4—21no—
309C4—81no—
310C4—82no—
311C4—82no—
312C4—82no—
313C4—162no—
314C4—162no—
315C4C2×#48811yescharacteristic
316C4#313711yescharacteristic
317C4#317711yescharacteristic
318C4#318311yescharacteristic
319C4#335212yes—
320C4#336011yescharacteristic
321C4#336411yescharacteristic
322C2—161no—
323C2#1986412yes—
324C2#1988012yes—
325C2#2776811yescharacteristic
326C2#2780611yescharacteristic
327C2#2781111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×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
1. #28830C2semidirect
5. #10352C2semidirect
6. #9836C2semidirect
7. #9730C2semidirect
8. #9568C2semidirect
10. #8285C2semidirect

Maximal subgroups

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

subgroupindex
#288302
#287112
#277282
#104132
#103522
#98362
#97302
#95682
#86502
#82852
#82102