Finite Group Databasebeta

← #192080

Subgroups — #192080

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34059C211yesmaximal, characteristic
2#21363C212yesmaximal
3C2×#1467C211yesmaximal, characteristic
4#13942C211yesmaximal, characteristic
5#8857C212yesmaximal
6#8856C212yesmaximal
7#8855C214yesmaximal
8#8017C212yesmaximal
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2320—24no—
12#2316C2211yescharacteristic
13#2316C2211yescharacteristic
14#2313—21no—
15#2313C2212yes—
16#2310—21no—
17#2295—24no—
18C2×#391—21no—
19#2283—21no—
20#2283C2212yes—
21C2×#382C2211yescharacteristic
22C4×#117—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1949—24no—
25#1853C2214yes—
26#1852C2214yes—
27#1776—24no—
28#1467C418yescomplemented
29C2×#324C2212yes—
30C2×#324C2212yes—
31#1441C2212yes—
32#1441C2212yes—
33C4⋊#95—21no—
34#1434—21no—
35#1434—21no—
36#1294C2214yes—
37#1294C2214yes—
38C23×Q8D411yescharacteristic
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C414yes—
43C2×C4×Q8Q814yes—
44C2×#117—21no—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C42—21no—
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42C2311yescharacteristic
55#432—24no—
56#432C2×C418yes—
57C2×#107—24no—
58C2×#106—24no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#391—24no—
63#391—24no—
64#391—44no—
65#389—24no—
66#389—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#383—22no—
72#383—22no—
73#382—22no—
74#382—24no—
75#382D414yes—
76#381—22no—
77#381—22no—
78#380—21no—
79#380—21no—
80#380C2312yes—
81#380C2312yes—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90#376—22no—
91#376—22no—
92#376—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95C2312yes—
100C43—22no—
101#340—24no—
102#340—24no—
103#324C2×C418yes—
104#324C2×C418yes—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8C4⋊C414yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C4×Q8—28no—
123C4×Q8—28no—
124C4×Q8—28no—
125C4×Q8—28no—
126#117—42no—
127#117—42no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C4⋊C414yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C2×D414yes—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×Q812yes—
151C2×C4⋊C4C2×Q812yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42C4⋊C414yes—
168C2×C42C4⋊C414yes—
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×Q812yes—
173C2×C42#4114yes—
174#107—48no—
175#106—48no—
176C4⋊C8—28no—
177C22⋊C8—24no—
178C22⋊C8C22×C418yes—
179#97—28no—
180#95—24no—
181#95—22no—
182#95—22no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95—22no—
190#95—24no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—41no—
195#95—41no—
196#95—42no—
197#95—44no—
198C24C22×D411yescharacteristic
199C2×Q8—28no—
200C2×Q8—28no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C2×C4⋊C412yes—
245C22×C4C2×C4⋊C412yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C412yes—
248C22×C4C2×C4⋊C412yes—
249C22×C4C2×C4⋊C412yes—
250C22×C4#12011yescharacteristic
251C22×C4#12012yes—
252C22×C4#12011yescharacteristic
253C22×C4#12312yes—
254C22×C4#12312yes—
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4#13712yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×Q811yescharacteristic
260C22×C4C2×#4112yes—
261C2×C8—28no—
262C2×C8—44no—
263C2×C8—44no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—48no—
287C42—44no—
288C42—44no—
289C42—84no—
290C42—84no—
291C42C4×D414yes—
292C42C4×D414yes—
293C42C22⋊Q814yes—
294C42C22⋊Q814yes—
295C23—22no—
296C23—22no—
297C23C22×C4⋊C411yescharacteristic
298C23C2×#12011yescharacteristic
299C23C2×#12311yescharacteristic
300C23C2×AGL(1,ℤ/8)11yescharacteristic
301C23C2×#13711yescharacteristic
302C23#57511yescharacteristic
303C23#57511yescharacteristic
304Q8—416no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#38512yes—
361C2×C4#38512yes—
362C2×C4#38514yes—
363C2×C4#42814yes—
364C2×C4C2×C4×D412yes—
365C2×C4C2×C22⋊Q812yes—
366C8—48no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22C2×#38511yescharacteristic
376C22C2×#42811yescharacteristic
377C22#291111yescharacteristic
378C22#300611yescharacteristic
379C22#300711yescharacteristic
380C22#302012yes—
381C22#302512yes—
382C22#309411yescharacteristic
383C22#309511yescharacteristic
384C4—24no—
385C4—24no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—84no—
394C4—84no—
395C4—84no—
396C4—162no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2#1816012yes—
402C2#1816112yes—
403C2#2084611yescharacteristic
404C2#2085711yescharacteristic
405C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#340592
#213632
C2×#14672
#139422
#88572
#88562
#88552
#80172