Finite Group Databasebeta

← #208309

Subgroups — #208309

Order 512.

778 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#35284C211yesmaximal, characteristic
2#20786C211yesmaximal, characteristic
3#20547C211yesmaximal, characteristic
4#9907C212yesmaximal, complemented
5#9702C212yesmaximal, complemented
6#9128C212yesmaximal
7#9127C212yesmaximal
8#8235C212yesmaximal, complemented
9#8232C212yesmaximal, complemented
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2647—21no—
16#2629—21no—
17#2507—22no—
18#2497—22no—
19#2446C414yes—
20#2429C414yes—
21#2330—21no—
22#2314—21no—
23C4×C4⋊Q8C2212yes—
24C4×#117C2211yescharacteristic
25#1840—21no—
26#1831C2212yes—
27#1818—21no—
28#1797C2212yes—
29#1770—22no—
30#1767C2211yescharacteristic
31#1767C2212yes—
32#1767C2212yes—
33C4×C4⋊C8C2211yescharacteristic
34#1752—22no—
35#1750—21no—
36#1746—21no—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#333C2212yes—
40C2×#333C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C23×C8—21no—
46C2×C4⋊Q8C2×C414yes—
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C22⋊Q8C2×C414yes—
50C2×C4×Q8C2×C414yes—
51C2×#117D411yescharacteristic
52C2×#117D411yescharacteristic
53C22×C42C2311yescharacteristic
54#431—24no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—24no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×#97—21no—
64C2×#97—22no—
65C2×#97—22no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68#400—24no—
69#398—24no—
70#398—24no—
71#396—24no—
72#391C2×C414yes—
73#391C2×C414yes—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—24no—
80#389C2×C414yes—
81#388—24no—
82#387—24no—
83#387—24no—
84#384—21no—
85#384—21no—
86#384—22no—
87#384C2312yes—
88#382—21no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4D412yes—
97#376D411yescharacteristic
98#376D411yescharacteristic
99C43C2311yescharacteristic
100C43C2311yescharacteristic
101#336—22no—
102#336—22no—
103#336—21no—
104#336—21no—
105#336—22no—
106#336—22no—
107#336—22no—
108#333—24no—
109#333—24no—
110#333—24no—
111#333—24no—
112C22×Q8C22×C414yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—24no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—82no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—84no—
127#125—42no—
128#125—42no—
129C22⋊Q8—84no—
130C4×Q8—84no—
131#117—24no—
132#117—41no—
133#117—41no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22×C414yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C22⋊C412yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C412yes—
154C2×C42C22×C412yes—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2411yescharacteristic
164C4⋊C8—24no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8#4114yes—
172C4⋊C8#4114yes—
173C22⋊C8—24no—
174C22⋊C8—44no—
175#97—42no—
176#97—82no—
177#97—82no—
178C4×C8—24no—
179C4×C8—42no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95C22⋊C412yes—
186#95C22⋊C412yes—
187#95C22⋊C412yes—
188#95C22⋊C412yes—
189C24C2×C22⋊C411yescharacteristic
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—44no—
193C2×Q8—44no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4#13714yes—
213C22×C4C23×C411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—84no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—48no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C22⋊C4—24no—
246C22⋊C4—42no—
247C22⋊C4—84no—
248C42—22no—
249C42—21no—
250C42—21no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42—42no—
264C42—84no—
265C42C2×#4112yes—
266C42C2×#4112yes—
267C23—22no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23#41912yes—
272C23C22×C22⋊C411yescharacteristic
273Q8—84no—
274Q8—84no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—82no—
306C2×C4#41912yes—
307C2×C4C4×#4112yes—
308C2×C4#52511yescharacteristic
309C2×C4#52511yescharacteristic
310C2×C4#53212yes—
311C2×C4C2×#13712yes—
312C2×C4#57511yescharacteristic
313C2×C4#57511yescharacteristic
314C2×C4#57712yes—
315C2×C4#57812yes—
316C8—44no—
317C8—44no—
318C8—82no—
319C8—82no—
320C22—24no—
321C22—22no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22#229711yescharacteristic
326C22#288812yes—
327C22C2×#41911yescharacteristic
328C22#289611yescharacteristic
329C22#289812yes—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—84no—
339C4—84no—
340C4#312012yes—
341C4#312311yescharacteristic
342C4#312711yescharacteristic
343C4#318112yes—
344C4#318112yes—
345C2—42no—
346C2#1776012yes—
347C2#1776812yes—
348C2#2139811yescharacteristic
349C2#2140711yescharacteristic
350C2#2142811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9702C2semidirect
8. #8235C2semidirect
9. #8232C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#207862
#205472
#99072
#97022
#91282
#91272
#82352
#82322