Finite Group Databasebeta

← #508840

Subgroups — #508840

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2C4×#466C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic, complemented
4#13493C212yesmaximal, complemented
5#13480C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13381C212yesmaximal
8#13363C211yesmaximal, characteristic, complemented
9#13110C212yesmaximal, complemented
10#12935C212yesmaximal, complemented
11#12844C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#466C2211yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2857—21no—
17C4×#121—21no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×C22⋊C16C2212yes—
23#1990—21no—
24#1970C2211yescharacteristic
25#1942—21no—
26#1942C2211yescharacteristic
27#1942C2212yes—
28#1934—21no—
29#1893—21no—
30#1845C2211yescharacteristic
31#1808—21no—
32#1808C2211yescharacteristic
33#1808C2212yes—
34C4×#107C2211yescharacteristic
35C4×#102—21no—
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2212yes—
42#1374C2211yescharacteristic
43C22×D8C2311yescharacteristic
44C23×C8D411yescharacteristic
45C23×C8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×C22⋊Q8—41no—
48C2×#121—21no—
49C2×C4×D4—21no—
50C2×C4×D4—41no—
51C22×C42D411yescharacteristic
52C22×C16C2312yes—
53C22×C16C2312yes—
54#500—42no—
55#466—22no—
56#466—22no—
57C4×D8—42no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107C2312yes—
61C2×#107C2312yes—
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#102—21no—
66C2×#102—21no—
67C2×#102—21no—
68C2×C4×C8—42no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—41no—
74#390—21no—
75#389—22no—
76#389—21no—
77#389—41no—
78#388—41no—
79#386—41no—
80#384—21no—
81#380—22no—
82#380—21no—
83#380—41no—
84C4×C4⋊C4—21no—
85C4×C22⋊C4—41no—
86#367—24no—
87#366—42no—
88#357—24no—
89#357—42no—
90C22⋊C16—42no—
91#340—22no—
92#340—21no—
93#340—22no—
94#340—21no—
95#340—21no—
96C22×Q8—41no—
97C22×D4—21no—
98C22×D4—41no—
99C23×C4—41no—
100C23×C4D811yescharacteristic
101C23×C4D811yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×D8—21no—
104C2×D8—22no—
105C2×D8—22no—
106C2×D8#4112yes—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2411yescharacteristic
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120C22⋊Q8—84no—
121#121—24no—
122#121—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—84no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42D811yescharacteristic
146C2×C42D811yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—22no—
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—22no—
155#107—21no—
156#107—22no—
157#107—22no—
158#107—22no—
159#107—22no—
160#107—22no—
161#107—22no—
162#107—42no—
163#107—42no—
164#107#4112yes—
165#102—24no—
166#102—42no—
167#102—42no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—41no—
171C4×C8—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175#95—42no—
176#95—42no—
177C24—41no—
178C24C2×D811yescharacteristic
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8—82no—
182C2×D4—22no—
183C2×D4—44no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—82no—
190C2×D4—82no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4C2×D811yescharacteristic
212C22×C4C2×D811yescharacteristic
213C22×C4C2×D811yescharacteristic
214C22×C4C2×D811yescharacteristic
215C22×C4C2×D811yescharacteristic
216C22×C4C22×D411yescharacteristic
217D8—24no—
218D8—42no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8#13712yes—
231C2×C8C2×#4111yescharacteristic
232C2×C8C2×#4111yescharacteristic
233C2×C8H(5,𝔽2)12yes—
234C2×C82−1+412yes—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C22⋊C4—44no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—84no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—41no—
261C42—41no—
262C42—82no—
263C16—42no—
264C16—42no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23#51012yes—
279C23C22×D811yescharacteristic
280Q8—164no—
281D4—42no—
282D4—42no—
283D4—88no—
284D4—81no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—164no—
320C2×C4#50812yes—
321C2×C4#50912yes—
322C2×C4#50912yes—
323C2×C4#52511yescharacteristic
324C2×C4#53411yescharacteristic
325C2×C4#53611yescharacteristic
326C2×C4C2×#13711yescharacteristic
327C2×C4#57511yescharacteristic
328C2×C4#57511yescharacteristic
329C8—22no—
330C8—41no—
331C8—41no—
332C8#53912yes—
333C8#54212yes—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22—161no—
345C22—161no—
346C22—161no—
347C22—161no—
348C22#310011yescharacteristic
349C22#318611yescharacteristic
350C22#322611yescharacteristic
351C22C2×#50811yescharacteristic
352C22C2×#50911yescharacteristic
353C22C2×#50911yescharacteristic
354C22C2×#51011yescharacteristic
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—82no—
361C4—161no—
362C4—162no—
363C4—161no—
364C4—162no—
365C4#263811yescharacteristic
366C4#311611yescharacteristic
367C4#319611yescharacteristic
368C4#324011yescharacteristic
369C2—42no—
370C2—162no—
371C2#3016511yescharacteristic
372C2#3366611yescharacteristic
373C2#3367011yescharacteristic
374C2#3375611yescharacteristic
375C2#3375811yescharacteristic
376C2#3379411yescharacteristic
377C2#3380511yescharacteristic
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. #13506C2semidirect
4. #13493C2semidirect
5. #13480C2semidirect
6. #13451C2semidirect
8. #13363C2semidirect
9. #13110C2semidirect
10. #12935C2semidirect
11. #12844C2semidirect

Maximal subgroups

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

subgroupindex
#332092
C4×#4662
#135062
#134932
#134802
#134512
#133812
#133632
#131102
#129352
#128442