Finite Group Databasebeta

← #223255

Subgroups — #223255

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2#22750C212yesmaximal
3#9903C212yesmaximal, complemented
4#9136C212yesmaximal, complemented
5#9127C212yesmaximal, complemented
6#8729C212yesmaximal, complemented
7#8594C211yesmaximal, characteristic, complemented
8#8232C211yesmaximal, characteristic, complemented
9#8044C212yesmaximal, complemented
10C2×#556C412yes—
11C2×#432C2212yes—
12C22×C4⋊C8C2212yes—
13#2868C2211yescharacteristic
14#2857C2212yes—
15#2568C412yes—
16#2547—22no—
17#2499C414yes—
18#2496—22no—
19#2496—22no—
20#2365—22no—
21C4×C4⋊Q8C2211yescharacteristic
22#2285C2212yes—
23C42×D4C2211yescharacteristic
24#1924—22no—
25#1917—22no—
26#1869—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31#1752—22no—
32C2×#333C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C23×C8—22no—
43#556—48no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#126—24no—
48C2×#124—22no—
49C2×C22⋊Q8C2×C414yes—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×#117C2×C414yes—
55C22×C42C2312yes—
56#432—24no—
57#432—42no—
58C2×C4⋊C8—24no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—42no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—22no—
69C2×#97—22no—
70#400—22no—
71#400—22no—
72#397—24no—
73#397—24no—
74#391—24no—
75#391C2×C412yes—
76#391C2×C414yes—
77#389—22no—
78#389—22no—
79#389—22no—
80#387—24no—
81#387—24no—
82#384—22no—
83#384C2×C412yes—
84#384C2311yescharacteristic
85#384C2312yes—
86#382C2×C414yes—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4D412yes—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98C4×C22⋊C4D412yes—
99C43C2311yescharacteristic
100#336—22no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105#333—24no—
106#333—41no—
107#333—41no—
108C22×Q8—22no—
109C22×Q8C22×C412yes—
110C22×D4C22⋊C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—44no—
119C22×C8—44no—
120C22×C8—82no—
121C4⋊Q8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127#126—416no—
128#124—48no—
129C22⋊Q8—48no—
130C4×Q8—44no—
131C4×Q8—48no—
132C4×D4—24no—
133C4×D4—42no—
134C4×D4—42no—
135#117—48no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4C22⋊C412yes—
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C22⋊C412yes—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—22no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C412yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2411yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—44no—
175C4⋊C8#4114yes—
176C22⋊C8—44no—
177C22⋊C8—44no—
178C22⋊C8—82no—
179#97—44no—
180#97—82no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95C22⋊C412yes—
187#95C22⋊C412yes—
188#95C22⋊C411yescharacteristic
189#95C22⋊C411yescharacteristic
190C24C2×C22⋊C412yes—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—48no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×D4—42no—
198C2×D4#10014yes—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D412yes—
226C2×C8—24no—
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—42no—
237C2×C8—42no—
238C2×C8—88no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—48no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—24no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—41no—
272C42—48no—
273C42—44no—
274C42C2×#4112yes—
275C42H(5,𝔽2)11yescharacteristic
276C422−1+411yescharacteristic
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23C2×#10014yes—
284C23C22×C22⋊C411yescharacteristic
285Q8—82no—
286Q8—82no—
287Q8—88no—
288D4—44no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
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—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4C2×#10012yes—
325C2×C4C4×#4111yescharacteristic
326C2×C4#52011yescharacteristic
327C2×C4#54612yes—
328C2×C4#54712yes—
329C2×C4#57512yes—
330C2×C4#57512yes—
331C2×C4#57712yes—
332C2×C4#57812yes—
333C8—44no—
334C8—82no—
335C8—82no—
336C8—82no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22#233911yescharacteristic
345C22C22×#10011yescharacteristic
346C22#288911yescharacteristic
347C22#289612yes—
348C22#289812yes—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—42no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—84no—
358C4—82no—
359C4—82no—
360C4#331112yes—
361C4#331312yes—
362C4#334912yes—
363C4#335012yes—
364C2—44no—
365C2#1775411yescharacteristic
366C2#1776712yes—
367C2#1777911yescharacteristic
368C2#2247911yescharacteristic
369C2#2249412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9903C2semidirect
4. #9136C2semidirect
5. #9127C2semidirect
6. #8729C2semidirect
7. #8594C2semidirect
8. #8232C2semidirect
9. #8044C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#227502
#99032
#91362
#91272
#87292
#85942
#82322
#80442