Finite Group Databasebeta

← #241389

Subgroups — #241389

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5#10581C214yesmaximal, complemented
6#10577C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#107—21no—
13C22×C4⋊C8C2211yescharacteristic
14#2851—22no—
15#2646—22no—
16#2571—21no—
17#2571C2212yes—
18#2570—22no—
19#2570—22no—
20#2552—21no—
21#2552C2212yes—
22#2317—21no—
23#2317—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1948—22no—
29#1947—22no—
30#1947C2214yes—
31#1947C2214yes—
32#1853—22no—
33#1852—22no—
34#1845—21no—
35#1844C2212yes—
36#1834—21no—
37#1834C2212yes—
38C2×#340—21no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×C8—21no—
44C2×#125—22no—
45C2×#117—21no—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#431—22no—
55#431—22no—
56#431Q814yes—
57#431Q814yes—
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—44no—
61C2×#106—24no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8Q814yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8C2312yes—
72#400—24no—
73#400—24no—
74#398—24no—
75#398—24no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#387—24no—
83#387—24no—
84#385—24no—
85#385—24no—
86#384—22no—
87#384—22no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384D411yescharacteristic
92#384D411yescharacteristic
93#384C2312yes—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#382—22no—
97#382—22no—
98#382—22no—
99#382—21no—
100#382—21no—
101#380—21no—
102#380—21no—
103#380D412yes—
104#380D412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2312yes—
113C2×#95—21no—
114#340—24no—
115#340—24no—
116#340—24no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—24no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—42no—
128C22×C8—42no—
129#125—84no—
130#117—42no—
131#117—42no—
132#117—42no—
133#117—42no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—82no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q812yes—
179C2×C42C2×Q812yes—
180C2×C42C2×Q812yes—
181C2×C42#4112yes—
182C2×C42C2411yescharacteristic
183#107—48no—
184#107—84no—
185#106—84no—
186C4⋊C8—24no—
187C4⋊C8—24no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8C2×Q814yes—
191C4⋊C8C2×Q814yes—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C4×C8—24no—
195C4×C8—22no—
196C4×C8C2×Q814yes—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207C24#12011yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4C22⋊Q812yes—
232C22×C4C22⋊Q812yes—
233C22×C4C22⋊Q812yes—
234C22×C4C4⋊Q812yes—
235C22×C4C4⋊Q812yes—
236C22×C4C4⋊Q812yes—
237C22×C4#13512yes—
238C22×C4#13512yes—
239C22×C4AGL(1,ℤ/8)12yes—
240C22×C4#13712yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—48no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4—82no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C42—42no—
288C42—42no—
289C42—44no—
290C42—44no—
291C42—82no—
292C42C22⋊Q812yes—
293C42C22⋊Q812yes—
294C42C22⋊Q812yes—
295C42C4⋊Q812yes—
296C42C4⋊Q812yes—
297C42C4⋊Q812yes—
298C42C22×Q811yescharacteristic
299C42C22×Q811yescharacteristic
300C42C22×Q811yescharacteristic
301C42C2×#4111yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23#39512yes—
307C23#44912yes—
308C23#45212yes—
309C23C2×#12011yescharacteristic
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—164no—
344C2×C4#39512yes—
345C2×C4#39512yes—
346C2×C4#39512yes—
347C2×C4#44912yes—
348C2×C4#45212yes—
349C2×C4#47814yes—
350C2×C4#47914yes—
351C2×C4#49914yes—
352C2×C4#50114yes—
353C2×C4C2×C22⋊Q811yescharacteristic
354C2×C4C2×C22⋊Q811yescharacteristic
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4C2×C4⋊Q811yescharacteristic
357C2×C4C2×C4⋊Q811yescharacteristic
358C2×C4C2×C4⋊Q811yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4C2×#13511yescharacteristic
361C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
362C2×C4C2×#13711yescharacteristic
363C2×C4#57612yes—
364C2×C4#57812yes—
365C8—44no—
366C8—44no—
367C8—44no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22#205914yes—
375C22C2×#39511yescharacteristic
376C22C2×#44911yescharacteristic
377C22C2×#45211yescharacteristic
378C22#300811yescharacteristic
379C22#301111yescharacteristic
380C22#302111yescharacteristic
381C22#302211yescharacteristic
382C4—22no—
383C4—21no—
384C4—21no—
385C4—22no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4—162no—
393C4#205914yes—
394C4C2×#47811yescharacteristic
395C4C2×#47911yescharacteristic
396C4#308312yes—
397C4C2×#49911yescharacteristic
398C4C2×#50111yescharacteristic
399C4#316412yes—
400C2—24no—
401C2#1889711yescharacteristic
402C2#1890811yescharacteristic
403C2#1891811yescharacteristic
404C2#1892711yescharacteristic
405C2C2×#205911yescharacteristic
406C2#2355511yescharacteristic
407C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10581C2semidirect
6. #10577C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#219822
#219812
C2×#19472
#105812
#105772
#80392
#78932
#78922