Finite Group Databasebeta

← #208310

Subgroups — #208310

Order 512.

854 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#35319C211yesmaximal, characteristic
2#20786C211yesmaximal, characteristic
3#20547C211yesmaximal, characteristic
4#9907C212yesmaximal, complemented
5#9703C212yesmaximal, complemented
6#9130C212yesmaximal
7#9129C212yesmaximal
8#8237C212yesmaximal, complemented
9#8233C212yesmaximal, complemented
10C2×#541—21no—
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857C2212yes—
16#2642—21no—
17#2621—21no—
18#2506—22no—
19#2506C414yes—
20#2496—22no—
21#2496C414yes—
22#2327—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—
46#541—48no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2312yes—
50C2×#126—24no—
51C2×#125—24no—
52C2×#125—21no—
53C2×C22⋊Q8—22no—
54C2×C4×Q8—22no—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C42C2311yescharacteristic
58#431—24no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#400—24no—
73#400—24no—
74#398—24no—
75#397—24no—
76#391—22no—
77#391—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389C2×C414yes—
82#389C2×C414yes—
83#389C2×C414yes—
84#387—24no—
85#387—24no—
86#384—22no—
87#384C2×C414yes—
88#384C2312yes—
89#382—21no—
90#380—22no—
91#380C2×C414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C414yes—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98#376D411yescharacteristic
99#376D411yescharacteristic
100C43C2311yescharacteristic
101C43C2311yescharacteristic
102#336—22no—
103#336—22no—
104#336—21no—
105#336—21no—
106#336—22no—
107#336—22no—
108#336—22no—
109#332—24no—
110#332—24no—
111#332—24no—
112#332—24no—
113C22×Q8—22no—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—24no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—82no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129C4⋊Q8—44no—
130#126—416no—
131#125—44no—
132#125—416no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C4×Q8—44no—
136C4×Q8—44no—
137#117—24no—
138#117—41no—
139#117—41no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C22⋊C412yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C412yes—
160C2×C42C22×C412yes—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2411yescharacteristic
170C4⋊C8—24no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8#4114yes—
178C4⋊C8#4114yes—
179C22⋊C8—24no—
180C22⋊C8—44no—
181#97—42no—
182#97—82no—
183#97—82no—
184C4×C8—24no—
185C4×C8—42no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95C22⋊C412yes—
192#95C22⋊C412yes—
193#95C22⋊C412yes—
194#95C22⋊C412yes—
195C24C2×C22⋊C411yescharacteristic
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4AGL(1,ℤ/8)14yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—84no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—48no—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—48no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—48no—
256C42—22no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—42no—
271C42—42no—
272C42—48no—
273C42C2×#4112yes—
274C42C2×#4112yes—
275C23—22no—
276C23—21no—
277C23—22no—
278C23—21no—
279C23#41812yes—
280C23C22×C22⋊C411yescharacteristic
281Q8—82no—
282Q8—82no—
283Q8—82no—
284Q8—82no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
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—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—82no—
316C2×C4#41812yes—
317C2×C4C4×#4112yes—
318C2×C4#52511yescharacteristic
319C2×C4#52511yescharacteristic
320C2×C4#53212yes—
321C2×C4C2×AGL(1,ℤ/8)12yes—
322C2×C4#57511yescharacteristic
323C2×C4#57511yescharacteristic
324C2×C4#57712yes—
325C2×C4#57812yes—
326C8—44no—
327C8—44no—
328C8—82no—
329C8—82no—
330C22—24no—
331C22—22no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22#229711yescharacteristic
336C22#288912yes—
337C22C2×#41811yescharacteristic
338C22#289611yescharacteristic
339C22#289812yes—
340C4—22no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—84no—
349C4—84no—
350C4#312012yes—
351C4#312311yescharacteristic
352C4#312711yescharacteristic
353C4#318012yes—
354C4#318012yes—
355C2—42no—
356C2#1775912yes—
357C2#1776712yes—
358C2#2139811yescharacteristic
359C2#2140811yescharacteristic
360C2#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. #9703C2semidirect
8. #8237C2semidirect
9. #8233C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#207862
#205472
#99072
#97032
#91302
#91292
#82372
#82332