Finite Group Databasebeta

← #223248

Subgroups — #223248

Order 512.

865 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#22648C212yesmaximal
3#9906C212yesmaximal, complemented
4#9135C212yesmaximal, complemented
5#9127C212yesmaximal, complemented
6#8726C212yesmaximal, complemented
7#8587C211yesmaximal, characteristic, complemented
8#8232C211yesmaximal, characteristic, complemented
9#8045C212yesmaximal, complemented
10C2×#435—22no—
11C2×#423C2212yes—
12C22×C4⋊C8C2212yes—
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2536C412yes—
16#2508—22no—
17#2495—22no—
18#2489C414yes—
19#2485—22no—
20#2356C412yes—
21#2346—22no—
22C4×C4⋊Q8C2211yescharacteristic
23#2285C2212yes—
24C42×D4C2211yescharacteristic
25#1940—22no—
26#1851—22no—
27#1850—22no—
28#1842C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1767C2212yes—
32C2×#333C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C22×M16—22no—
43C23×C8—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#125C2×C412yes—
46C2×C22⋊Q8C2×C414yes—
47C2×C4×Q8C2×C412yes—
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C2×#117—22no—
51C22×C42C2312yes—
52#435—48no—
53#423—24no—
54#423—42no—
55C2×C4⋊C8—24no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
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×#97—22no—
68#400—24no—
69#398—22no—
70#397—24no—
71#394—22no—
72#391C2×C412yes—
73#391C2×C414yes—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#388—24no—
80#388—24no—
81#387—24no—
82#384C2311yescharacteristic
83#384C2312yes—
84#383—22no—
85#382—22no—
86#382—24no—
87#382C2×C412yes—
88#380—22no—
89#380—22no—
90#380—22no—
91#380C2×C414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4D412yes—
97C4×C22⋊C4C2×C412yes—
98C4×C22⋊C4D412yes—
99C4×C22⋊C4D412yes—
100#376—22no—
101C43C2311yescharacteristic
102#336—22no—
103#336—22no—
104#336—22no—
105#333—24no—
106#333—41no—
107#333—41no—
108C22×Q8C22×C412yes—
109C22×D4C22⋊C411yescharacteristic
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C2×M16—44no—
114C2×M16—44no—
115C2×M16—82no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—44no—
121C22×C8—44no—
122C22×C8—82no—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—82no—
127C22⋊Q8—84no—
128C4×Q8—44no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—42no—
132#117—44no—
133#117—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C414yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C22⋊C412yes—
149C2×C22⋊C4C22⋊C412yes—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—24no—
155C2×C42—22no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C412yes—
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2411yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—44no—
172C4⋊C8—44no—
173C4⋊C8—44no—
174C4⋊C8—82no—
175C4⋊C8#4114yes—
176C22⋊C8—44no—
177C22⋊C8—44no—
178#97—44no—
179#95—24no—
180#95—22no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—22no—
186#95C22⋊C411yescharacteristic
187#95C22⋊C412yes—
188#95C22⋊C412yes—
189#95C22⋊C411yescharacteristic
190C24C2×C22⋊C412yes—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×D4—24no—
196C2×D4—41no—
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—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C412yes—
220C22×C4AGL(1,ℤ/8)14yes—
221C22×C4#13714yes—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D412yes—
224M16—88no—
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—42no—
237C2×C8—88no—
238C4⋊C4—24no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—48no—
257C22⋊C4—84no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—41no—
270C42—44no—
271C42—48no—
272C42C2×#4112yes—
273C42H(5,𝔽2)11yescharacteristic
274C422−1+411yescharacteristic
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23#41714yes—
282C23C22×C22⋊C411yescharacteristic
283Q8—82no—
284Q8—82no—
285D4—44no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
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—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4#41312yes—
321C2×C4#41714yes—
322C2×C4C4×#4111yescharacteristic
323C2×C4#52011yescharacteristic
324C2×C4#54612yes—
325C2×C4#54712yes—
326C2×C4C2×AGL(1,ℤ/8)12yes—
327C2×C4C2×#13712yes—
328C2×C4#57712yes—
329C2×C4#57812yes—
330C8—44no—
331C8—82no—
332C8—82no—
333C8—82no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#233911yescharacteristic
342C22C2×#41311yescharacteristic
343C22#288811yescharacteristic
344C22C2×#41712yes—
345C22#289812yes—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—24no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—82no—
355C4—84no—
356C4—82no—
357C4#331012yes—
358C4#331212yes—
359C4#334312yes—
360C4#334412yes—
361C2—44no—
362C2#1775211yescharacteristic
363C2#1776412yes—
364C2#1778011yescharacteristic
365C2#2248311yescharacteristic
366C2#2249112yes—
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. #9135C2semidirect
5. #9127C2semidirect
6. #8726C2semidirect
7. #8587C2semidirect
8. #8232C2semidirect
9. #8045C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#226482
#99062
#91352
#91272
#87262
#85872
#82322
#80452