Finite Group Databasebeta

← #507368

Subgroups — #507368

Order 512.

692 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#33209C211yesmaximal, characteristic, complemented
2#21224C211yesmaximal, characteristic
3#13481C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13364C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic, complemented
7#13148C212yesmaximal, complemented
8#13147C212yesmaximal, complemented
9#12868C212yesmaximal, complemented
10#12733C212yesmaximal
11C4×C22⋊C16C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#416—21no—
16C22×C4×C8C2211yescharacteristic
17#2857—21no—
18#2297—21no—
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×C22⋊C16C2212yes—
24C2×C4×C16C2212yes—
25#1990—21no—
26#1970C2211yescharacteristic
27#1970C2211yescharacteristic
28#1969C2211yescharacteristic
29#1969C2211yescharacteristic
30#1968—21no—
31#1968—21no—
32#1942—21no—
33#1942C2212yes—
34#1845C2211yescharacteristic
35#1808—21no—
36#1808C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×#41—21no—
44C2×#135—22no—
45C2×#135—22no—
46C2×#135—24no—
47C2×#135—42no—
48C22×Q16C2311yescharacteristic
49C22×SD16—21no—
50C22×D8C2311yescharacteristic
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C22⋊Q8—41no—
55C2×#117—21no—
56C22×C42D411yescharacteristic
57C22×C16C2312yes—
58C22×C16C2312yes—
59#500—42no—
60#427—24no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107C2312yes—
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#106—21no—
68#416—44no—
69C2×#103—21no—
70C2×#103—21no—
71C2×#102—21no—
72C2×#102—21no—
73C2×C4×C8—42no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#391—21no—
80#391—41no—
81#390—21no—
82#390—21no—
83#389—22no—
84#389—41no—
85#386—41no—
86#386—41no—
87#384—21no—
88#380—22no—
89#380—41no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—41no—
92#358—24no—
93#358—42no—
94#357—24no—
95#357—42no—
96C22⋊C16—24no—
97C4×C16—42no—
98#340—22no—
99#340—22no—
100C2×#41—24no—
101C2×#41—42no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8—21no—
107C22×Q8—41no—
108C22×D4—21no—
109C22×D4—41no—
110C23×C4—41no—
111C23×C4D811yescharacteristic
112C23×C4D811yescharacteristic
113C23×C4C2×D411yescharacteristic
114#135—48no—
115#135—48no—
116C2×Q16—22no—
117C2×Q16—21no—
118C2×Q16—22no—
119C2×Q16#4112yes—
120C2×SD16—24no—
121C2×SD16—42no—
122C2×SD16—42no—
123C2×D8—22no—
124C2×D8—21no—
125C2×D8—22no—
126C2×D8#4112yes—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8SD1614yes—
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8#4114yes—
137C22×C8C2411yescharacteristic
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C22⋊Q8—84no—
141#117—42no—
142#117—42no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42D811yescharacteristic
161C2×C42D811yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16—22no—
170#107—22no—
171#107—22no—
172#107—22no—
173#107—22no—
174#107—22no—
175#107—22no—
176#107—42no—
177#107—42no—
178#106—42no—
179#103—42no—
180#103—42no—
181#102—42no—
182#102—42no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—42no—
192#95—42no—
193C24—41no—
194C24C2×D811yescharacteristic
195#41—48no—
196#41—48no—
197#41—84no—
198#41—84no—
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—82no—
216C2×D4—82no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4#12412yes—
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×SD1612yes—
242C22×C4C22×D411yescharacteristic
243Q16—24no—
244Q16—42no—
245SD16—48no—
246D8—24no—
247D8—42no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—44no—
257C2×C8#12412yes—
258C2×C8C2×SD1612yes—
259C2×C8C2×SD1612yes—
260C2×C8C2×#4111yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C8C2×#4112yes—
263C2×C8H(5,𝔽2)12yes—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C22⋊C4—84no—
277C22⋊C4—82no—
278C22⋊C4—82no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—41no—
287C42—41no—
288C42—82no—
289C16—42no—
290C16—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23#48612yes—
305C23C22×D811yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—81no—
309Q8—81no—
310Q8—164no—
311D4—42no—
312D4—42no—
313D4—88no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—88no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4#48612yes—
349C2×C4#50812yes—
350C2×C4#50812yes—
351C2×C4#50912yes—
352C2×C4#51012yes—
353C2×C4#52511yescharacteristic
354C2×C4C2×#12411yescharacteristic
355C2×C4#53411yescharacteristic
356C2×C4C22×SD1611yescharacteristic
357C2×C4#57511yescharacteristic
358C2×C4#57511yescharacteristic
359C8—22no—
360C8—22no—
361C8—22no—
362C8#55112yes—
363C8#55112yes—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22#310011yescharacteristic
379C22C2×#48611yescharacteristic
380C22#318611yescharacteristic
381C22C2×#50811yescharacteristic
382C22C2×#50811yescharacteristic
383C22C2×#50911yescharacteristic
384C22C2×#51011yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—82no—
391C4—162no—
392C4—161no—
393C4—161no—
394C4—162no—
395C4#256111yescharacteristic
396C4#310111yescharacteristic
397C4#314011yescharacteristic
398C4#318711yescharacteristic
399C2—42no—
400C2—162no—
401C2#2910311yescharacteristic
402C2#3366611yescharacteristic
403C2#3366711yescharacteristic
404C2#3369711yescharacteristic
405C2#3370711yescharacteristic
406C2#3375411yescharacteristic
407C2#3375611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
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. #13481C2semidirect
4. #13480C2semidirect
5. #13364C2semidirect
6. #13363C2semidirect
7. #13148C2semidirect
8. #13147C2semidirect
9. #12868C2semidirect
11. C4×C22⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#212242
#134812
#134802
#133642
#133632
#131482
#131472
#128682
#127332
C4×C22⋊C162