Finite Group Databasebeta

← #508860

Subgroups — #508860

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#33173C211yesmaximal, characteristic
3#13539C212yesmaximal, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic, complemented
6#13408C212yesmaximal, complemented
7#13364C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13060C212yesmaximal, complemented
10#13059C212yesmaximal
11C2×#1374C211yesmaximal, characteristic, complemented
12C23×C16C2212yes—
13C2×#500C2211yescharacteristic
14C2×#495C2211yescharacteristic
15C2×#467C2211yescharacteristic
16C2×#466C2211yescharacteristic
17C2×#465—21no—
18C22×C4×C8C2211yescharacteristic
19#2857—21no—
20#2850—21no—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#357C2212yes—
24C2×#357C2212yes—
25#1990—21no—
26#1970C2211yescharacteristic
27#1969C2211yescharacteristic
28#1968—21no—
29#1958—21no—
30#1942—21no—
31#1942C2212yes—
32#1846C2211yescharacteristic
33#1845C2211yescharacteristic
34#1808—21no—
35#1808C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2211yescharacteristic
42#1374C2212yes—
43C22×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—21no—
49C2×#124—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—41no—
52C2×#121—21no—
53C22×C42D411yescharacteristic
54C22×C16—24no—
55C22×C16—42no—
56C22×C16—42no—
57C22×C16C2312yes—
58C22×C16C2312yes—
59#500—42no—
60#495—42no—
61#467—24no—
62#466—24no—
63#465—44no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107C2312yes—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#106—21no—
71C2×#103—21no—
72C2×#103—21no—
73C2×#102—21no—
74C2×#102—21no—
75C2×C4×C8—42no—
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391—21no—
81#391—41no—
82#390—21no—
83#389—22no—
84#389—41no—
85#388—41no—
86#388—41no—
87#386—41no—
88#386—41no—
89#384—21no—
90#382—21no—
91#380—22no—
92#380—41no—
93#358—42no—
94#358—42no—
95#357—42no—
96#357—42no—
97#340—22no—
98#340—22no—
99C22×Q8—21no—
100C22×Q8—41no—
101C22×D4—21no—
102C22×D4—41no—
103C23×C4D811yescharacteristic
104C23×C4D811yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×Q16—22no—
107C2×Q16—21no—
108C2×Q16—22no—
109C2×Q16—21no—
110C2×SD16—42no—
111C2×SD16—42no—
112C2×SD16—42no—
113C2×D8—22no—
114C2×D8—21no—
115C2×D8—22no—
116C2×D8—21no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8SD1614yes—
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8#4114yes—
127C22×C8C2411yescharacteristic
128C4⋊Q8—42no—
129C4⋊Q8—42no—
130#124—42no—
131#124—42no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—84no—
135#121—42no—
136#121—42no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42D811yescharacteristic
156C2×C42D811yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—48no—
166#107—22no—
167#107—22no—
168#107—22no—
169#107—22no—
170#107—22no—
171#107—22no—
172#107—42no—
173#107—42no—
174#106—42no—
175#103—42no—
176#103—42no—
177#102—42no—
178#102—42no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—41no—
182C4×C8—41no—
183#95—42no—
184#95—42no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189C24—41no—
190C24C2×D811yescharacteristic
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—82no—
207C2×D4—82no—
208C2×D4—82no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4#12412yes—
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×SD1612yes—
231C22×C4C22×D411yescharacteristic
232Q16—42no—
233Q16—42no—
234SD16—84no—
235D8—42no—
236D8—42no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—44no—
246C2×C8#12412yes—
247C2×C8C2×SD1612yes—
248C2×C8C2×SD1612yes—
249C2×C8C2×#4112yes—
250C2×C8H(5,𝔽2)11yescharacteristic
251C2×C8H(5,𝔽2)12yes—
252C2×C82−1+411yescharacteristic
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—84no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—41no—
278C42—41no—
279C16—42no—
280C16—42no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23#48612yes—
293C23#50912yes—
294C23#51012yes—
295C23C22×D811yescharacteristic
296Q8—81no—
297Q8—81no—
298Q8—81no—
299Q8—81no—
300Q8—164no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305D4—164no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4#48612yes—
340C2×C4#50812yes—
341C2×C4#50812yes—
342C2×C4C2×#12411yescharacteristic
343C2×C4#53411yescharacteristic
344C2×C4#53711yescharacteristic
345C2×C4C22×SD1611yescharacteristic
346C2×C4#57511yescharacteristic
347C2×C4#57511yescharacteristic
348C8—22no—
349C8—22no—
350C8—22no—
351C8#53812yes—
352C8#53912yes—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—161no—
365C22—161no—
366C22#224314yes—
367C22C2×#48611yescharacteristic
368C22#318611yescharacteristic
369C22#324511yescharacteristic
370C22C2×#50811yescharacteristic
371C22C2×#50811yescharacteristic
372C22C2×#50911yescharacteristic
373C22C2×#51011yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—162no—
380C4—161no—
381C4—161no—
382C4—162no—
383C4—161no—
384C4#263911yescharacteristic
385C4#313511yescharacteristic
386C4#318711yescharacteristic
387C4#324611yescharacteristic
388C2—24no—
389C2—321no—
390C2#3018511yescharacteristic
391C2#3369711yescharacteristic
392C2C2×#224311yescharacteristic
393C2#3375411yescharacteristic
394C2#3375611yescharacteristic
395C2#3381311yescharacteristic
396C2#3381411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33209C2semidirect
3. #13539C2semidirect
4. #13481C2semidirect
5. #13480C2semidirect
6. #13408C2semidirect
7. #13364C2semidirect
8. #13363C2semidirect
9. #13060C2semidirect
11. C2×#1374C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#331732
#135392
#134812
#134802
#134082
#133642
#133632
#130602
#130592
C2×#13742