Finite Group Databasebeta

← #325152

Subgroups — #325152

Order 512.

548 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#33256C211yesmaximal, characteristic
2#29212C211yesmaximal, characteristic
3#28693C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10658C212yesmaximal
6#10657C212yesmaximal
7#9904C211yesmaximal, characteristic
8#9902C211yesmaximal, characteristic
9#9612C212yesmaximal
10#9611C212yesmaximal
11#8273C211yesmaximal, characteristic
12C2×#501—21no—
13C2×C8×Q8—21no—
14#2859C2211yescharacteristic
15#2859C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2085—22no—
18#2059—22no—
19#1993—21no—
20#1992—21no—
21#1984—21no—
22#1948C2212yes—
23#1948C2212yes—
24#1947C2212yes—
25#1947C2212yes—
26#1941—21no—
27#1941—21no—
28#1941—21no—
29#1941—21no—
30#1939—21no—
31#1939C2212yes—
32#1847—21no—
33#1844C2211yescharacteristic
34#1844C2211yescharacteristic
35#1842C2211yescharacteristic
36#1834C2211yescharacteristic
37#1834C2211yescharacteristic
38C8×C4⋊C4—21no—
39C8×C4⋊C4C2212yes—
40#1767C2211yescharacteristic
41#1767C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×#334C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×#125—21no—
51C2×#125—21no—
52C2×C4×Q8—21no—
53#501—44no—
54C8×Q8—44no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#106—21no—
58C2×#106—21no—
59C2×#106—21no—
60C2×#106—21no—
61C2×#106—22no—
62C2×#106—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8—22no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389D412yes—
89#384D411yescharacteristic
90#384C2311yescharacteristic
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2311yescharacteristic
94#382—21no—
95#382—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4C2312yes—
98C43C2311yescharacteristic
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#336—22no—
108#336—22no—
109#336—21no—
110#336—21no—
111#334Q818yescomplemented
112C22×Q8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C4⋊Q8—22no—
121C4⋊Q8SD1614yes—
122#125—42no—
123#125—42no—
124#125—82no—
125C4×Q8—42no—
126C4×Q8—42no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C42—22no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—44no—
158#107—82no—
159#106—42no—
160#106—42no—
161#106—42no—
162#106—42no—
163#106—44no—
164#106—82no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—24no—
168C4⋊C8—42no—
169C4⋊C8—42no—
170C4⋊C8—44no—
171C4⋊C8C2×Q814yes—
172C4⋊C8C2×Q814yes—
173#97—42no—
174#97—42no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—41no—
179C4×C8—41no—
180C4×C8—44no—
181C4×C8C2×Q814yes—
182#95—22no—
183#95—22no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C2×Q8—24no—
189C2×Q8—42no—
190C2×Q8—42no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C2×#4111yescharacteristic
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8C4⋊Q812yes—
222C2×C8C4⋊Q812yes—
223C2×C8AGL(1,ℤ/8)12yes—
224C2×C8#13712yes—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4C2×SD1614yes—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—41no—
261C42—41no—
262C42—42no—
263C42—42no—
264C42C2×SD1612yes—
265C42#13512yes—
266C42C22×Q812yes—
267C422−1+412yes—
268C23C2×#12111yescharacteristic
269Q8—48no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4#46512yes—
301C2×C4#46512yes—
302C2×C4C2×C4⋊Q811yescharacteristic
303C2×C4#53611yescharacteristic
304C2×C4D4×Q812yes—
305C2×C4#55712yes—
306C2×C4C22×SD1611yescharacteristic
307C2×C4C2×#13511yescharacteristic
308C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
309C2×C4C2×#13711yescharacteristic
310C2×C4#57511yescharacteristic
311C2×C4#57511yescharacteristic
312C2×C4#57712yes—
313C8—22no—
314C8—44no—
315C8—44no—
316C8—41no—
317C8—41no—
318C8—82no—
319C8#50114yescomplemented
320C22#254611yescharacteristic
321C22#303611yescharacteristic
322C22#303711yescharacteristic
323C22C2×#46511yescharacteristic
324C22#305711yescharacteristic
325C22#306711yescharacteristic
326C22#306711yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—44no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4—161no—
335C4—161no—
336C4—161no—
337C4—161no—
338C4C2×#50111yescharacteristic
339C4#316611yescharacteristic
340C4#323511yescharacteristic
341C4#323711yescharacteristic
342C4Q8×SD1612yes—
343C4#338112yes—
344C2#1925411yescharacteristic
345C2#1926011yescharacteristic
346C2#1932811yescharacteristic
347C2#1934011yescharacteristic
348C2#2867711yescharacteristic
349C2#2869311yescharacteristic
350C2#2870111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
111. #334Q8semidirect
319. C8#501semidirect

Maximal subgroups

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

subgroupindex
#332562
#292122
#286932
C8×C4⋊Q82
#106582
#106572
#99042
#99022
#96122
#96112
#82732