Finite Group Databasebeta

← #223231

Subgroups — #223231

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#22797C212yesmaximal
3#9906C212yesmaximal, complemented
4#9136C212yesmaximal, complemented
5#9128C212yesmaximal, complemented
6#8727C212yesmaximal, complemented
7#8587C211yesmaximal, characteristic, complemented
8#8232C211yesmaximal, characteristic, complemented
9#8046C212yesmaximal, complemented
10C2×#432C2212yes—
11C2×#431C2212yes—
12#2859C2211yescharacteristic
13#2857C2212yes—
14#2536C412yes—
15#2508—22no—
16#2495—22no—
17#2489C414yes—
18#2485—22no—
19#2356C412yes—
20#2346—22no—
21C4×C4⋊Q8C2211yescharacteristic
22#2285C2212yes—
23C42×D4C2211yescharacteristic
24#1955—22no—
25#1924—22no—
26#1923—22no—
27#1842C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1771—22no—
31#1767C2212yes—
32C2×#333C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C2×C4⋊Q8C2311yescharacteristic
43C2×#125C2×C412yes—
44C2×C22⋊Q8C2×C414yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×D4C2×C411yescharacteristic
47C2×C4×D4D412yes—
48C2×#117—22no—
49C22×C42C2312yes—
50#432—24no—
51#432—42no—
52#431—24no—
53#431—42no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65C2×#97—22no—
66C2×C4×C8—22no—
67#400—24no—
68#398—22no—
69#397—24no—
70#394—22no—
71#391C2×C412yes—
72#391C2×C414yes—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#388—24no—
79#388—24no—
80#387—24no—
81#384C2311yescharacteristic
82#384C2312yes—
83#383—22no—
84#382—22no—
85#382—24no—
86#382C2×C412yes—
87#380—22no—
88#380—22no—
89#380—22no—
90#380C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C414yes—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4D412yes—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98C4×C22⋊C4D412yes—
99#376—22no—
100C43C2311yescharacteristic
101#336—22no—
102#336—22no—
103#336—22no—
104#333—24no—
105#333—41no—
106#333—41no—
107C22×Q8C22×C412yes—
108C22×D4C22⋊C411yescharacteristic
109C23×C4C22×C412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119#125—82no—
120C22⋊Q8—84no—
121C4×Q8—44no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125#117—44no—
126#117—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C414yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C22⋊C412yes—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—24no—
148C2×C42—22no—
149C2×C42C22×C411yescharacteristic
150C2×C42C22×C412yes—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2411yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—44no—
165C4⋊C8—82no—
166C4⋊C8#4114yes—
167C22⋊C8—44no—
168C22⋊C8—44no—
169C22⋊C8—82no—
170C22⋊C8—82no—
171#97—44no—
172#97—44no—
173C4×C8—44no—
174#95—24no—
175#95—22no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—22no—
181#95C22⋊C411yescharacteristic
182#95C22⋊C412yes—
183#95C22⋊C412yes—
184#95C22⋊C411yescharacteristic
185C24C2×C22⋊C412yes—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×D4—41no—
191C2×D4—42no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D412yes—
219C2×C8—24no—
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—42no—
229C2×C8—42no—
230C2×C8—42no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C22⋊C4—24no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—48no—
250C22⋊C4—84no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—41no—
263C42—48no—
264C42—44no—
265C42C2×#4112yes—
266C42H(5,𝔽2)11yescharacteristic
267C422−1+411yescharacteristic
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23C22×C22⋊C411yescharacteristic
276Q8—82no—
277Q8—82no—
278D4—44no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
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—22no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#41312yes—
315C2×C4C4×#4111yescharacteristic
316C2×C4#52011yescharacteristic
317C2×C4#54612yes—
318C2×C4#54712yes—
319C2×C4#57512yes—
320C2×C4#57512yes—
321C2×C4#57712yes—
322C2×C4#57812yes—
323C8—44no—
324C8—82no—
325C8—82no—
326C8—82no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22#233911yescharacteristic
335C22C2×#41311yescharacteristic
336C22#288811yescharacteristic
337C22#289612yes—
338C22#289812yes—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—82no—
348C4—84no—
349C4—82no—
350C4#331512yes—
351C4#331712yes—
352C4#334812yes—
353C4#334912yes—
354C2—82no—
355C2#1775711yescharacteristic
356C2#1776812yes—
357C2#1778011yescharacteristic
358C2#2248311yescharacteristic
359C2#2249612yes—
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. #9906C2semidirect
4. #9136C2semidirect
5. #9128C2semidirect
6. #8727C2semidirect
7. #8587C2semidirect
8. #8232C2semidirect
9. #8046C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#227972
#99062
#91362
#91282
#87272
#85872
#82322
#80462