Finite Group Databasebeta

← #507439

Subgroups — #507439

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2C4×#466C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic, complemented
5#13451C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic, complemented
7#13110C212yesmaximal, complemented
8#13061C212yesmaximal
9#12935C212yesmaximal, complemented
10#12836C212yesmaximal, complemented
11#12544C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#500C2211yescharacteristic
14C2×#466C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857—21no—
18C4×#121—21no—
19C2×#367C2212yes—
20C2×#366C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23#1990—21no—
24#1970C2211yescharacteristic
25#1942—21no—
26#1942C2211yescharacteristic
27#1942C2212yes—
28#1934—21no—
29#1893—21no—
30#1845C2211yescharacteristic
31#1808—21no—
32#1808C2212yes—
33#1808C2211yescharacteristic
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
52C2×M32—24no—
53C2×M32—24no—
54C2×M32—42no—
55C22×C16C2312yes—
56C22×C16C2312yes—
57#500—42no—
58#466—22no—
59#466Q814yescomplemented
60C4×D8—42no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107C2312yes—
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#102—21no—
69C2×#102—21no—
70C2×#102—21no—
71C2×C4×C8—42no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—41no—
77#390—21no—
78#389—22no—
79#389—21no—
80#389—41no—
81#388—41no—
82#386—41no—
83#384—21no—
84#380—22no—
85#380—21no—
86#380—41no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—41no—
89#367—24no—
90#366—42no—
91#357—24no—
92#357—42no—
93#340—22no—
94#340—21no—
95#340—22no—
96#340—21no—
97#340—21no—
98C22×Q8—41no—
99C22×D4—21no—
100C22×D4—41no—
101C23×C4—41no—
102C23×C4D811yescharacteristic
103C23×C4D811yescharacteristic
104C23×C4C2×D411yescharacteristic
105C2×D8—22no—
106C2×D8—21no—
107C2×D8—22no—
108C2×D8C2×Q812yes—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×Q812yes—
118C22×C8#4112yes—
119C22×C8C2411yescharacteristic
120C4⋊Q8—42no—
121C4⋊Q8—42no—
122C22⋊Q8—84no—
123#121—24no—
124#121—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—84no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42D811yescharacteristic
148C2×C42D811yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151M32—48no—
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—22no—
155C2×C16—22no—
156C2×C16—22no—
157C2×C16#4114yes—
158#107—21no—
159#107—22no—
160#107—22no—
161#107—22no—
162#107—22no—
163#107—22no—
164#107—22no—
165#107—42no—
166#107—42no—
167#107C2×Q812yes—
168#102—24no—
169#102—42no—
170#102—42no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—41no—
174C4×C8—41no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—42no—
179#95—42no—
180C24—41no—
181C24C2×D811yescharacteristic
182C2×Q8—82no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×D4—22no—
186C2×D4—44no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—82no—
193C2×D4—82no—
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—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4C22⋊Q812yes—
213C22×C4C2×D811yescharacteristic
214C22×C4C2×D811yescharacteristic
215C22×C4C2×D811yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4C2×D811yescharacteristic
218C22×C4#13712yes—
219C22×C4C22×D411yescharacteristic
220D8—24no—
221D8—42no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—44no—
232C2×C8C22⋊Q812yes—
233C2×C8#13512yes—
234C2×C8C22×Q811yescharacteristic
235C2×C8C2×#4111yescharacteristic
236C2×C8C2×#4112yes—
237C2×C8H(5,𝔽2)12yes—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C22⋊C4—44no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—84no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—41no—
264C42—41no—
265C42—82no—
266C16—24no—
267C16—42no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23#47412yes—
282C23C22×D811yescharacteristic
283Q8—164no—
284D4—42no—
285D4—42no—
286D4—88no—
287D4—81no—
288D4—81no—
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—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—164no—
322C2×C4#47412yes—
323C2×C4#50912yes—
324C2×C4#50912yes—
325C2×C4#51012yes—
326C2×C4C2×C22⋊Q811yescharacteristic
327C2×C4#53211yescharacteristic
328C2×C4#53411yescharacteristic
329C2×C4C2×#13511yescharacteristic
330C2×C4C2×#13711yescharacteristic
331C2×C4#57511yescharacteristic
332C8—22no—
333C8—22no—
334C8—41no—
335C8#55112yes—
336C8#55412yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—161no—
348C22—161no—
349C22—161no—
350C22—161no—
351C22C2×#47411yescharacteristic
352C22#316911yescharacteristic
353C22#318611yescharacteristic
354C22C2×#50911yescharacteristic
355C22C2×#50911yescharacteristic
356C22C2×#51011yescharacteristic
357C22#341411yescharacteristic
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—41no—
363C4—82no—
364C4—162no—
365C4—161no—
366C4—161no—
367C4—162no—
368C4#256411yescharacteristic
369C4#307611yescharacteristic
370C4#318111yescharacteristic
371C4#319011yescharacteristic
372C2—42no—
373C2—162no—
374C2#2915711yescharacteristic
375C2#3362711yescharacteristic
376C2#3362911yescharacteristic
377C2#3373711yescharacteristic
378C2#3375111yescharacteristic
379C2#3375811yescharacteristic
380C2#3376211yescharacteristic
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. #13538C2semidirect
4. #13527C2semidirect
5. #13451C2semidirect
6. #13363C2semidirect
7. #13110C2semidirect
9. #12935C2semidirect
10. #12836C2semidirect
11. #12544C2semidirect
59. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
C4×#4662
#135382
#135272
#134512
#133632
#131102
#130612
#129352
#128362
#125442