Finite Group Databasebeta

← #315036

Subgroups — #315036

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28827C211yesmaximal, characteristic
2#28737C211yesmaximal, characteristic
3#27816C212yesmaximal
4#10422C212yesmaximal
5#10357C212yesmaximal
6#9825C211yesmaximal, characteristic
7#9733C211yesmaximal, characteristic
8#9566C212yesmaximal
9#8645C211yesmaximal, characteristic
10C8×#107C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2568C2212yes—
15#2552C2212yes—
16#2101—22no—
17#2083—22no—
18#2056—22no—
19#2026—22no—
20#1991—22no—
21#1990C2212yes—
22#1990C2212yes—
23#1988C2212yes—
24#1981C2212yes—
25#1969—21no—
26#1969—21no—
27#1947C2212yes—
28#1942—21no—
29#1933—22no—
30#1845—21no—
31#1845C2211yescharacteristic
32#1803C2212yes—
33C4×#107C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C22×Q16—21no—
46C22×Q16—21no—
47C2×C4⋊Q8C2312yes—
48C2×#125—22no—
49C2×C4×Q8C2311yescharacteristic
50#494—22no—
51#494—22no—
52C8×Q8—22no—
53C8×Q8—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#107D411yescharacteristic
58C2×#107D411yescharacteristic
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103D411yescharacteristic
67C2×#103D411yescharacteristic
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—22no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#400—42no—
75#398—42no—
76#391—22no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#384—22no—
82#384C2312yes—
83#382—22no—
84C4×C4⋊C4C2312yes—
85C4×C4⋊C4C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—42no—
93#340D412yes—
94#340D411yescharacteristic
95#340D411yescharacteristic
96#336—21no—
97#336—22no—
98#335—22no—
99#332—24no—
100Q8⋊C8—22no—
101C82—22no—
102C22×Q8—22no—
103C22×Q8C2×D411yescharacteristic
104C2×Q16—42no—
105C2×Q16—42no—
106C2×Q16—42no—
107C2×Q16—42no—
108C2×Q16—81no—
109C2×Q16—81no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122#125—44no—
123#125—82no—
124C4×Q8—21no—
125C4×Q8—21no—
126C4×Q8—21no—
127C4×Q8—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—21no—
145C2×C42C2411yescharacteristic
146#107—22no—
147#107—22no—
148#107—42no—
149#107—82no—
150#107—81no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—42no—
157C4⋊C8#4112yes—
158#103—22no—
159#103—82no—
160#103—82no—
161#103—82no—
162#103Q1614yescomplemented
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—42no—
168C4×C8—42no—
169C4×C8—82no—
170C4×C8#4112yes—
171C4×C8#4112yes—
172#95—21no—
173#95—22no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8C2×Q1612yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4H(5,𝔽2)11yescharacteristic
202Q16—84no—
203Q16—84no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—22no—
209C2×C8—21no—
210C2×C8—22no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8#12412yes—
224C2×C8#12412yes—
225C2×C8C2×Q1612yes—
226C2×C8#13512yes—
227C2×C8#13512yes—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4—82no—
249C4⋊C4—162no—
250C4⋊C4C2×Q1612yes—
251C4⋊C4#13512yes—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—42no—
257C42—41no—
258C42C2×#4111yescharacteristic
259C42C2×#4111yescharacteristic
260C42C2×#4111yescharacteristic
261C422−1+411yescharacteristic
262C23D4211yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—41no—
266Q8—41no—
267Q8—82no—
268Q8—82no—
269Q8—82no—
270Q8—82no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4C2×#12411yescharacteristic
298C2×C4#53211yescharacteristic
299C2×C4#54211yescharacteristic
300C2×C4#54711yescharacteristic
301C2×C4#54811yescharacteristic
302C2×C4#55511yescharacteristic
303C2×C4C22×Q1611yescharacteristic
304C2×C4C2×#13511yescharacteristic
305C2×C4C2×#13511yescharacteristic
306C2×C4C2×#13511yescharacteristic
307C2×C4#57611yescharacteristic
308C2×C4#57611yescharacteristic
309C2×C4#57711yescharacteristic
310C2×C4#57811yescharacteristic
311C8—22no—
312C8—22no—
313C8—42no—
314C8—41no—
315C8—42no—
316C8—81no—
317C8—82no—
318C8#48814yes—
319C22#249111yescharacteristic
320C22#328011yescharacteristic
321C22#328411yescharacteristic
322C22D4×Q1611yescharacteristic
323C22#328711yescharacteristic
324C22#329011yescharacteristic
325C22#329311yescharacteristic
326C4—21no—
327C4—21no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—82no—
332C4—82no—
333C4—81no—
334C4—162no—
335C4—162no—
336C4C2×#48811yescharacteristic
337C4#313611yescharacteristic
338C4#317011yescharacteristic
339C4#317511yescharacteristic
340C4#333811yescharacteristic
341C4#334211yescharacteristic
342C4#336111yescharacteristic
343C4#336511yescharacteristic
344C2#1999011yescharacteristic
345C2#1999511yescharacteristic
346C2#2000711yescharacteristic
347C2#2001011yescharacteristic
348C2#2777011yescharacteristic
349C2#2779211yescharacteristic
350C2#2780811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
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
162. #103Q16semidirect

Maximal subgroups

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

subgroupindex
#288272
#287372
#278162
#104222
#103572
#98252
#97332
#95662
#86452
C8×#1072
C8×#1032