Finite Group Databasebeta

← #208312

Subgroups — #208312

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 359 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#9705C212yesmaximal, complemented
6#9132C212yesmaximal
7#9131C212yesmaximal
8#8236C212yesmaximal, complemented
9#8233C212yesmaximal, complemented
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2647—21no—
16#2629—21no—
17#2507C414yes—
18#2497C414yes—
19#2446—22no—
20#2429—22no—
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×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C23×C8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C22⋊Q8—22no—
50C2×C4×Q8—22no—
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#391—22no—
73#391—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—24no—
78#389C2×C414yes—
79#389C2×C414yes—
80#389C2×C414yes—
81#388—24no—
82#387—24no—
83#387—24no—
84#384—21no—
85#384—21no—
86#384C2×C414yes—
87#384C2312yes—
88#382—21no—
89#380—22no—
90#380C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C414yes—
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#332—24no—
109#332—24no—
110#332—24no—
111#332—24no—
112C22×Q8—22no—
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—82no—
127C4⋊Q8—82no—
128#125—42no—
129#125—42no—
130C22⋊Q8—82no—
131C22⋊Q8—82no—
132C4×Q8—82no—
133C4×Q8—82no—
134#117—24no—
135#117—41no—
136#117—41no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C22⋊C4—24no—
150C2×C22⋊C4C22⋊C412yes—
151C2×C22⋊C4C22⋊C412yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—24no—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C22×C412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2411yescharacteristic
167C4⋊C8—24no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8#4114yes—
175C4⋊C8#4114yes—
176C22⋊C8—24no—
177C22⋊C8—44no—
178#97—42no—
179#97—82no—
180#97—82no—
181C4×C8—24no—
182C4×C8—42no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95C22⋊C412yes—
189#95C22⋊C412yes—
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192C24C2×C22⋊C411yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4#13714yes—
218C22×C4C23×C411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—84no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—48no—
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—84no—
249C4⋊C4—84no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—84no—
253C42—22no—
254C42—21no—
255C42—21no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—42no—
268C42—42no—
269C42—84no—
270C42C2×#4112yes—
271C42C2×#4112yes—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—21no—
276C23#41912yes—
277C23C22×C22⋊C411yescharacteristic
278Q8—82no—
279Q8—82no—
280Q8—82no—
281Q8—82no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—82no—
313C2×C4#41912yes—
314C2×C4C4×#4112yes—
315C2×C4#52511yescharacteristic
316C2×C4#52511yescharacteristic
317C2×C4#53212yes—
318C2×C4C2×#13712yes—
319C2×C4#57511yescharacteristic
320C2×C4#57511yescharacteristic
321C2×C4#57612yes—
322C2×C4#57712yes—
323C8—44no—
324C8—44no—
325C8—82no—
326C8—82no—
327C22—24no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22#229711yescharacteristic
333C22#288912yes—
334C22C2×#41911yescharacteristic
335C22#289611yescharacteristic
336C22#289712yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—41no—
344C4—41no—
345C4—84no—
346C4—84no—
347C4#311912yes—
348C4#312211yescharacteristic
349C4#312611yescharacteristic
350C4#318112yes—
351C4#318112yes—
352C2—42no—
353C2#1776112yes—
354C2#1776912yes—
355C2#2139911yescharacteristic
356C2#2140711yescharacteristic
357C2#2142911yescharacteristic
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. #9705C2semidirect
8. #8236C2semidirect
9. #8233C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#207862
#205472
#99072
#97052
#91322
#91312
#82362
#82332