Finite Group Databasebeta

← #316206

Subgroups — #316206

Order 512.

415 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29880C211yesmaximal, characteristic
2#21995C211yesmaximal, characteristic
3#10636C211yesmaximal, characteristic, complemented
4#10636C211yesmaximal, characteristic, complemented
5#10633C211yesmaximal, characteristic, complemented
6#10633C211yesmaximal, characteristic, complemented
7#9778C211yesmaximal, characteristic, complemented
8#9771C211yesmaximal, characteristic, complemented
9#9194C211yesmaximal, characteristic
10#9193C211yesmaximal, characteristic
11#9192C211yesmaximal, characteristic
12#9191C211yesmaximal, characteristic
13#8794C211yesmaximal, characteristic, complemented
14#8791C211yesmaximal, characteristic, complemented
15#8158C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17#2921C2211yescharacteristic
18C2×#433C2211yescharacteristic
19#2905C2211yescharacteristic
20#2628C2211yescharacteristic
21#2628C2211yescharacteristic
22#2087—21no—
23#2086—21no—
24#2076—21no—
25#2075—21no—
26#1993C2211yescharacteristic
27#1993C2211yescharacteristic
28#1950—21no—
29#1950—21no—
30#1855—21no—
31#1855—21no—
32#1849C2211yescharacteristic
33#1848C2211yescharacteristic
34#1838C2211yescharacteristic
35#1837C2211yescharacteristic
36#1835C2211yescharacteristic
37#1549C2211yescharacteristic
38#1549C2211yescharacteristic
39#1549C2211yescharacteristic
40#1549C2211yescharacteristic
41#1548C2211yescharacteristic
42#1548C2211yescharacteristic
43#1548C2211yescharacteristic
44#1548C2211yescharacteristic
45#1458C2211yescharacteristic
46#1458C2211yescharacteristic
47#1452C2211yescharacteristic
48#1450C2211yescharacteristic
49#1286C2211yescharacteristic
50#1286C2211yescharacteristic
51#1286C2211yescharacteristic
52#1286C2211yescharacteristic
53#1286C2211yescharacteristic
54#1286C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1281C2211yescharacteristic
58#1281C2211yescharacteristic
59C22×M16—21no—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×#117C2311yescharacteristic
63#500—21no—
64#500—21no—
65#500—21no—
66#500—21no—
67#433—21no—
68#433—21no—
69#433—21no—
70#433—21no—
71#433—41no—
72#433—41no—
73#431C2311yescharacteristic
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77C2×#108—21no—
78C2×#108—21no—
79C2×#108—21no—
80C2×#108—21no—
81C2×#107—21no—
82C2×#107—21no—
83#424—41no—
84#424—41no—
85#423—21no—
86#423C2311yescharacteristic
87#423C2311yescharacteristic
88#423C2311yescharacteristic
89#423C2311yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92#407—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C4×M16—21no—
96C2×#97C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#400—21no—
99#400—21no—
100#399—21no—
101#399—21no—
102#395—21no—
103#395—21no—
104#393—21no—
105#393—21no—
106#389—21no—
107#389—21no—
108#382C2311yescharacteristic
109#382C2311yescharacteristic
110#380—21no—
111#380—21no—
112#341—21no—
113#341—21no—
114#341—21no—
115#341—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#333—41no—
123#333—41no—
124#333—41no—
125#333—41no—
126#332—41no—
127#332—41no—
128#332—41no—
129#332—41no—
130#324—21no—
131#324—21no—
132#323—21no—
133#323—21no—
134#322—21no—
135#322—21no—
136#322—21no—
137C82—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C23×C4C2×D411yescharacteristic
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—41no—
146C2×M16—81no—
147C2×M16—81no—
148C22×C8—41no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161#117—21no—
162#117—21no—
163#117—21no—
164#117—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4#4111yescharacteristic
174C2×C4⋊C4#4111yescharacteristic
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4#4111yescharacteristic
178C2×C22⋊C4#4111yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2411yescharacteristic
181#108—81no—
182#108—81no—
183#108—81no—
184#108—81no—
185#107—41no—
186#107—41no—
187#107—41no—
188#107—41no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—42no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—42no—
215#97—21no—
216#97—21no—
217#97—41no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—41no—
223C4×C8#4111yescharacteristic
224C4×C8#4111yescharacteristic
225C4×C8#4111yescharacteristic
226C4×C8#4111yescharacteristic
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235C24#12411yescharacteristic
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—161no—
241C2×Q8—161no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4#12411yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—41no—
259M16—41no—
260M16—41no—
261M16—41no—
262M16—41no—
263M16—41no—
264M16—41no—
265M16—41no—
266M16—82no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—82no—
286C2×C8#12311yescharacteristic
287C2×C8#12311yescharacteristic
288C2×C8#12311yescharacteristic
289C2×C8#12311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—161no—
307C4⋊C4—161no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C42—21no—
315C42—21no—
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C42H(5,𝔽2)11yescharacteristic
319C42H(5,𝔽2)11yescharacteristic
320C23—21no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23C2×#12411yescharacteristic
326Q8—161no—
327Q8—161no—
328Q8—161no—
329Q8—161no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4C2×#12311yescharacteristic
356C2×C4#53511yescharacteristic
357C2×C4#55311yescharacteristic
358C2×C4#55311yescharacteristic
359C2×C4#55311yescharacteristic
360C2×C4#55311yescharacteristic
361C2×C4#57711yescharacteristic
362C2×C4#57711yescharacteristic
363C2×C4#57811yescharacteristic
364C2×C4#57811yescharacteristic
365C8—21no—
366C8—21no—
367C8—21no—
368C8—21no—
369C8—41no—
370C8—41no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22—81no—
386C22#250411yescharacteristic
387C22#313711yescharacteristic
388C22#313711yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—161no—
397C4—161no—
398C4—161no—
399C4—161no—
400C4#309711yescharacteristic
401C4#309711yescharacteristic
402C4#322211yescharacteristic
403C4#322311yescharacteristic
404C2—21no—
405C2—21no—
406C2—81no—
407C2#2798511yescharacteristic
408C2#2798811yescharacteristic
409C2#2798911yescharacteristic
subgroupG/H
center, socleC22#2504
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10636C2semidirect
4. #10636C2semidirect
5. #10633C2semidirect
6. #10633C2semidirect
7. #9778C2semidirect
8. #9771C2semidirect
13. #8794C2semidirect
14. #8791C2semidirect

Maximal subgroups

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

subgroupindex
#298802
#219952
#106362
#106332
#97782
#97712
#91942
#91932
#91922
#91912
#87942
#87912
#81582