Finite Group Databasebeta

← #223236

Subgroups — #223236

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#22749C212yesmaximal
3#9905C212yesmaximal, complemented
4#9135C212yesmaximal, complemented
5#9128C212yesmaximal, complemented
6#8724C212yesmaximal, complemented
7#8594C211yesmaximal, characteristic, complemented
8#8232C211yesmaximal, characteristic, complemented
9#8043C212yesmaximal, complemented
10C2×#554C412yes—
11C2×#431C2212yes—
12C2×#423C2212yes—
13#2868C2211yescharacteristic
14#2857C2212yes—
15#2567C412yes—
16#2552—22no—
17#2502C414yes—
18#2440—22no—
19#2440—22no—
20D4×C4⋊C4—22no—
21C4×C4⋊Q8C2211yescharacteristic
22C2×#384C2212yes—
23C42×D4C2211yescharacteristic
24#1924—22no—
25#1916—22no—
26#1870—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31#1753—22no—
32C2×#333C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C22×M16—22no—
43#554—48no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—22no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8C2×C414yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×D4—22no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2×C414yes—
57C22×C42C2312yes—
58#431—24no—
59#431—42no—
60#423—24no—
61#423—42no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#398—24no—
73#398—24no—
74#395—22no—
75#395—22no—
76#391C2×C412yes—
77#391C2×C414yes—
78#389—24no—
79#389—22no—
80#389—22no—
81#389—22no—
82#385—24no—
83#385—24no—
84#384—22no—
85#384—22no—
86#384—22no—
87#384C2×C414yes—
88#384C2×C412yes—
89#384C2311yescharacteristic
90#384C2312yes—
91#380—22no—
92#380—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×C4—24no—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C2×M16—44no—
116C2×M16—44no—
117C2×M16—82no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—44no—
126C4⋊Q8—82no—
127C4⋊Q8—82no—
128#125—48no—
129C22⋊Q8—416no—
130C22⋊Q8—48no—
131C4×Q8—44no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—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⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C414yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C22⋊C412yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2411yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—44no—
181C4⋊C8#4114yes—
182C22⋊C8—44no—
183C22⋊C8—44no—
184C22⋊C8—82no—
185#97—82no—
186C4×C8—44no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192#95C22⋊C411yescharacteristic
193#95C22⋊C411yescharacteristic
194C24C2×C22⋊C412yes—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—84no—
201C2×D4—24no—
202C2×D4—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—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—48no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4#13714yes—
230C22×C4#13714yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233M16—88no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—48no—
259C4⋊C4—44no—
260C4⋊C4—48no—
261C4⋊C4—84no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C4⋊C4#10014yes—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—48no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—44no—
282C42C2×#4112yes—
283C42H(5,𝔽2)11yescharacteristic
284C422−1+411yescharacteristic
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23#41914yes—
292C23C22×C22⋊C411yescharacteristic
293Q8—82no—
294Q8—82no—
295Q8—164no—
296D4—28no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—168no—
331C2×C4C2×#10012yes—
332C2×C4C2×#10014yes—
333C2×C4#41914yes—
334C2×C4C4×#4111yescharacteristic
335C2×C4#52011yescharacteristic
336C2×C4#54612yes—
337C2×C4#54712yes—
338C2×C4C2×#13712yes—
339C2×C4C2×#13712yes—
340C2×C4#57612yes—
341C2×C4#57712yes—
342C8—44no—
343C8—82no—
344C8—82no—
345C8—82no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22#233911yescharacteristic
354C22C22×#10011yescharacteristic
355C22#288911yescharacteristic
356C22C2×#41912yes—
357C22#289712yes—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—24no—
362C4—24no—
363C4—42no—
364C4—41no—
365C4—42no—
366C4—82no—
367C4—84no—
368C4—82no—
369C4#331412yes—
370C4#331612yes—
371C4#334412yes—
372C4#334612yes—
373C2—44no—
374C2#1774611yescharacteristic
375C2#1776112yes—
376C2#1778211yescharacteristic
377C2#2247911yescharacteristic
378C2#2249712yes—
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. #9905C2semidirect
4. #9135C2semidirect
5. #9128C2semidirect
6. #8724C2semidirect
7. #8594C2semidirect
8. #8232C2semidirect
9. #8043C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#227492
#99052
#91352
#91282
#87242
#85942
#82322
#80432