Finite Group Databasebeta

← #192099

Subgroups — #192099

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21359C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4C2×#1461C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#8855C218yesmaximal
7#8026C211yesmaximal, characteristic
8#8026C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2356—22no—
13#2356—22no—
14#2355—22no—
15#2355—22no—
16#2313C2214yes—
17#2310—21no—
18C2×#391—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283—22no—
22C2×#382C2211yescharacteristic
23#2274—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1948—22no—
26#1947—22no—
27#1853C2214yes—
28#1852C2214yes—
29C4×#107—22no—
30C4×#106—22no—
31#1461C418yescomplemented
32C2×#324C2214yes—
33#1441C2214yes—
34C4⋊#95—21no—
35#1434—22no—
36#1294C2214yes—
37#1294C2214yes—
38C23×Q8D411yescharacteristic
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8C2×C414yes—
46C2×C4×Q8Q814yes—
47C2×#117—21no—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#431—22no—
57#431—22no—
58#431C2×C414yes—
59#431C2×C414yes—
60C2×#107—24no—
61C2×#106—24no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68#391—24no—
69#391—24no—
70#391—44no—
71#387—28no—
72#387—28no—
73#384D412yes—
74#384D412yes—
75#383—24no—
76#382—24no—
77#382—21no—
78#382—21no—
79#382—22no—
80#382—22no—
81#381—24no—
82#380—21no—
83#380—21no—
84#380C2312yes—
85#380C2312yes—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4D412yes—
96C4×C4⋊C4D412yes—
97C4×C22⋊C4—24no—
98#376—24no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95C2312yes—
104#340—24no—
105#340—24no—
106#324C2×C4116yes—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8C4⋊C414yes—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C4×Q8—24no—
125C4×Q8—24no—
126C4×Q8—24no—
127C4×Q8—24no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×Q8—24no—
131C4×Q8—24no—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C4⋊C414yes—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×Q814yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—24no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—28no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42C4⋊C418yes—
171C2×C42C22×C412yes—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q812yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179#107—48no—
180#106—48no—
181C4⋊C8—24no—
182C4⋊C8—24no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8C22×C414yes—
186C22⋊C8C22×C414yes—
187C4×C8—24no—
188C4×C8—24no—
189#95—28no—
190#95—28no—
191#95—28no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—48no—
199#95—48no—
200#95—41no—
201#95—41no—
202#95—42no—
203C24C22×D411yescharacteristic
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—28no—
207C2×Q8—28no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C2×C4⋊C414yes—
240C22×C4C2×C4⋊C414yes—
241C22×C4C2×C4⋊C412yes—
242C22×C4C2×C4⋊C412yes—
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4#13512yes—
250C22×C4#13512yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4#13712yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—44no—
262C2×C8—44no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—84no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C42—28no—
285C42—28no—
286C42—44no—
287C42—44no—
288C42—88no—
289C42—82no—
290C42—82no—
291C42C4×D412yes—
292C42C4×D412yes—
293C42C4×D412yes—
294C42C4×D412yes—
295C42C22⋊Q812yes—
296C42C22⋊Q812yes—
297C42C22⋊Q812yes—
298C42C22⋊Q812yes—
299C23—22no—
300C23—22no—
301C23C22×C4⋊C411yescharacteristic
302C23C2×#12011yescharacteristic
303C23C2×#12311yescharacteristic
304C23C2×#13511yescharacteristic
305C23C2×#13511yescharacteristic
306C23C2×AGL(1,ℤ/8)11yescharacteristic
307C23C2×#13711yescharacteristic
308Q8—48no—
309Q8—48no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#38512yes—
350C2×C4#38512yes—
351C2×C4#38512yes—
352C2×C4#38512yes—
353C2×C4#42714yes—
354C2×C4#42814yes—
355C2×C4#44912yes—
356C2×C4#44912yes—
357C2×C4#45212yes—
358C2×C4#45212yes—
359C2×C4#48214yes—
360C2×C4#48514yes—
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×C4×D411yescharacteristic
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4C2×C22⋊Q811yescharacteristic
365C8—44no—
366C8—44no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22C2×#38511yescharacteristic
376C22C2×#42711yescharacteristic
377C22C2×#42811yescharacteristic
378C22C2×#44911yescharacteristic
379C22C2×#45211yescharacteristic
380C22#302012yes—
381C22#302512yes—
382C22C2×#48211yescharacteristic
383C22C2×#48511yescharacteristic
384C4—22no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—44no—
392C4—84no—
393C4—84no—
394C4—84no—
395C4—162no—
396C4#185414yes—
397C4#185414yes—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#1821412yes—
403C2#1821512yes—
404C2C2×#185411yescharacteristic
405C2#2085711yescharacteristic
406C2#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
31. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#213592
#213582
C2×#14612
#139422
#88552
#80262