Finite Group Databasebeta

← #241327

Subgroups — #241327

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37623C211yesmaximal, characteristic
2#21962C212yesmaximal
3#20764C211yesmaximal, characteristic
4#10582C212yesmaximal, complemented
5#10581C212yesmaximal, complemented
6#10578C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8C2×#1296C211yesmaximal, characteristic
9#7895C212yesmaximal
10C22×C4⋊C8C2212yes—
11C2×C4×M16C2211yescharacteristic
12#2843—22no—
13#2660—22no—
14#2647—22no—
15#2627—22no—
16#2571C2211yescharacteristic
17#2552C2212yes—
18#2551C2211yescharacteristic
19#2442—22no—
20#2440—22no—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2283C2212yes—
24#1949—22no—
25#1949—22no—
26#1948C2212yes—
27#1948C2212yes—
28#1947C2212yes—
29#1947C2212yes—
30#1836—22no—
31#1834C2212yes—
32#1834C2212yes—
33#1808—22no—
34#1807—22no—
35#1736—22no—
36#1296C2212yes—
37#1296C2211yescharacteristic
38#1296C2211yescharacteristic
39#1279C2212yes—
40#1279C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43C22×M16—21no—
44C23×C8—22no—
45C2×C4⋊Q8—22no—
46C2×#125—22no—
47C2×C22⋊Q8—22no—
48C2×#117—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D412yes—
51C22×C42C2311yescharacteristic
52C2×#107—22no—
53C2×#107—22no—
54C2×#106—22no—
55C2×#106—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8Q818yescomplemented
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C4×M16—22no—
62C4×M16Q814yescomplemented
63C2×#97—21no—
64C2×C4×C8C2312yes—
65#400—24no—
66#400—22no—
67#400—22no—
68#398—24no—
69#398—24no—
70#398—24no—
71#398—24no—
72#395—24no—
73#395—22no—
74#395—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#385—24no—
80#384—22no—
81#384—22no—
82#384D412yes—
83#384C2312yes—
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#383—22no—
87#382—22no—
88#382—21no—
89#382D412yes—
90#382D412yes—
91#381—22no—
92#380—22no—
93#380—22no—
94#380D412yes—
95#380D412yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2312yes—
98C4×C4⋊C4C2312yes—
99C4×C22⋊C4—22no—
100#376—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107C22×Q8—22no—
108C23×C4—22no—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C2×M16—24no—
112C2×M16—42no—
113C2×M16—42no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—28no—
118C22×C8—82no—
119C22×C8—82no—
120C4⋊Q8—82no—
121C4⋊Q8—82no—
122#125—82no—
123#125—82no—
124C22⋊Q8—84no—
125#117—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4—82no—
141C2×C4⋊C4C2×D412yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×Q812yes—
156C2×C42C2×Q814yes—
157C2×C42#4112yes—
158C2×C42C2411yescharacteristic
159#107—82no—
160#107—82no—
161#106—82no—
162#106—82no—
163C4⋊C8—28no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8C2×Q814yes—
167C4⋊C8C2×Q814yes—
168#97—24no—
169C4×C8—22no—
170C4×C8C2×Q814yes—
171#95—22no—
172#95—22no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—42no—
184C24#12011yescharacteristic
185C2×Q8—82no—
186C2×Q8—84no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—84no—
202C22×C4#12012yes—
203C22×C4#12011yescharacteristic
204C22×C4C22⋊Q814yes—
205C22×C4C22⋊Q812yes—
206C22×C4C4⋊Q812yes—
207C22×C4C4⋊Q814yes—
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D412yes—
210M16—48no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—88no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C4⋊C4—82no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—82no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—82no—
255C42C22⋊Q812yes—
256C42C22⋊Q814yes—
257C42C4⋊Q812yes—
258C42C4⋊Q814yes—
259C42C22×Q811yescharacteristic
260C42C22×Q812yes—
261C42C2×#4111yescharacteristic
262C23—22no—
263C23—22no—
264C23#39512yes—
265C23#45412yes—
266C23#45512yes—
267C23C2×#12011yescharacteristic
268Q8—164no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—164no—
301C2×C4#39514yes—
302C2×C4#39512yes—
303C2×C4C2×C22⋊Q811yescharacteristic
304C2×C4C2×C22⋊Q812yes—
305C2×C4C2×C4⋊Q811yescharacteristic
306C2×C4C2×C4⋊Q812yes—
307C2×C4#57612yes—
308C2×C4#57712yes—
309C2×C4#57712yes—
310C2×C4#57812yes—
311C8—44no—
312C8—44no—
313C8—44no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—24no—
318C22#206114yes—
319C22C2×#39511yescharacteristic
320C22#301212yes—
321C22#301312yes—
322C22C2×#45511yescharacteristic
323C22C2×#45411yescharacteristic
324C4—24no—
325C4—21no—
326C4—21no—
327C4—22no—
328C4—82no—
329C4—82no—
330C4—82no—
331C4—82no—
332C4—82no—
333C4—82no—
334C4—162no—
335C4#308112yes—
336C4#308212yes—
337C4#316312yes—
338C4#316412yes—
339C2—24no—
340C2#1872011yescharacteristic
341C2#1872111yescharacteristic
342C2#1872612yes—
343C2#2355712yes—
344C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
4. #10582C2semidirect
5. #10581C2semidirect
6. #10578C2semidirect
7. #10575C2semidirect
58. C2×C4⋊C8Q8semidirect
62. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#376232
#219622
#207642
#105822
#105812
#105782
#105752
C2×#12962
#78952