Finite Group Databasebeta

← #315735

Subgroups — #315735

Order 512.

730 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic
2#27961C211yesmaximal, characteristic
3#27960C211yesmaximal, characteristic
4#27960C211yesmaximal, characteristic
5#27948C211yesmaximal, characteristic
6C2×#1834C211yesmaximal, characteristic
7#10682C212yesmaximal, complemented
8#10681C212yesmaximal, complemented
9#9835C212yesmaximal, complemented
10#8821C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#467—22no—
14C22×C4×C8C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2100—22no—
20#2099—22no—
21#2099—22no—
22#2084—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1969—22no—
28#1956C2211yescharacteristic
29#1956C2211yescharacteristic
30#1853—22no—
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1834—22no—
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1834C2212yes—
39#1824—22no—
40#1823—22no—
41#1823—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×#322C2212yes—
46#1273C2212yes—
47C22×Q16—22no—
48C23×C8D412yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C42C2311yescharacteristic
60#494—24no—
61#467—48no—
62C2×#107—22no—
63C2×#106—24no—
64C2×#103—22no—
65C2×#103—22no—
66C2×C22⋊C8D412yes—
67C2×C22⋊C8D412yes—
68C2×C4×C8—24no—
69C2×C4×C8D412yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#391—21no—
75#391—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#384C2312yes—
89#384C2312yes—
90#380—21no—
91#380—21no—
92#380—21no—
93#380—21no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336D412yes—
104#336D412yes—
105#322—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×Q16—44no—
114C2×Q16—44no—
115C2×Q16—44no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—42no—
119C22×C8C2×D412yes—
120C22×C8C2×D414yes—
121C22×C8C2×D414yes—
122C22×C8C2×D412yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
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—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—82no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2411yescharacteristic
167#107—44no—
168#106—84no—
169#103—44no—
170#103—44no—
171C22⋊C8—24no—
172C22⋊C8—24no—
173C4×C8—24no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8#4114yes—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184C24#12711yescharacteristic
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4#12711yescharacteristic
213C22×C4#12711yescharacteristic
214C22×C4#12711yescharacteristic
215C22×C4#13514yes—
216C22×C4AGL(1,ℤ/8)14yes—
217C22×C4#13714yes—
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221Q16—88no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—48no—
234C2×C8#12314yes—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42—22no—
265C42C2×#4112yes—
266C422−1+412yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#49512yes—
272C23#49612yes—
273C23#49712yes—
274C23C2×#12711yescharacteristic
275Q8—82no—
276Q8—82no—
277Q8—82no—
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—164no—
308C2×C4—164no—
309C2×C4#49512yes—
310C2×C4#49612yes—
311C2×C4#49712yes—
312C2×C4C2×#12311yescharacteristic
313C2×C4#53611yescharacteristic
314C2×C4#54712yes—
315C2×C4#54712yes—
316C2×C4C2×#13512yes—
317C2×C4C2×AGL(1,ℤ/8)12yes—
318C2×C4C2×#13712yes—
319C2×C4#57512yes—
320C2×C4#57712yes—
321C2×C4#57812yes—
322C8—24no—
323C8—44no—
324C8—44no—
325C8—42no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22#172014yes—
333C22#250111yescharacteristic
334C22C2×#49511yescharacteristic
335C22C2×#49611yescharacteristic
336C22C2×#49711yescharacteristic
337C22#315311yescharacteristic
338C22#315411yescharacteristic
339C22#315611yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—161no—
344C4—161no—
345C4—162no—
346C4—162no—
347C4—161no—
348C4—161no—
349C4#309412yes—
350C4#323412yes—
351C4#333614yes—
352C4#334414yes—
353C2—24no—
354C2C2×#172011yescharacteristic
355C2#1963711yescharacteristic
356C2#1968111yescharacteristic
357C2#1970011yescharacteristic
358C2#2791811yescharacteristic
359C2#2793111yescharacteristic
360C2#2793311yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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. #10682C2semidirect
8. #10681C2semidirect
9. #9835C2semidirect
10. #8821C2semidirect

Maximal subgroups

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

subgroupindex
#331722
#279612
#279602
#279482
C2×#18342
#106822
#106812
#98352
#88212
#81452