Finite Group Databasebeta

← #506165

Subgroups — #506165

Order 512.

501 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27837C211yesmaximal, characteristic, complemented
2#27791C211yesmaximal, characteristic
3#13528C211yesmaximal, characteristic, complemented
4#13494C211yesmaximal, characteristic, complemented
5#13451C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic
7#13283C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13112C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic
13#12834C211yesmaximal, characteristic, complemented
14#12674C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19#2494—21no—
20#2491—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×#346C2211yescharacteristic
31C2×#346C2211yescharacteristic
32#2101—21no—
33#2077—21no—
34#2033—21no—
35#2023—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1925—21no—
41#1919—21no—
42#1846C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#124—41no—
63C2×#121—21no—
64C22×C16—22no—
65C22×C16—22no—
66C22×C16—41no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500—41no—
73#466—22no—
74#466—41no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×C22⋊C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#401—41no—
91#397—41no—
92#394—41no—
93#393—41no—
94#390—21no—
95#390—21no—
96#389—21no—
97#389—21no—
98#388—41no—
99#388—41no—
100#387—41no—
101#382—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108#367—22no—
109#366—22no—
110#366—22no—
111#366—41no—
112#357—22no—
113#357—22no—
114#357—22no—
115#357—41no—
116C22⋊C16—22no—
117#346—22no—
118#346—41no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—41no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4C2×D411yescharacteristic
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150#124—84no—
151#121—42no—
152#121—81no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4D811yescharacteristic
172C2×C22⋊C4D811yescharacteristic
173C2×C42—41no—
174C2×C42—41no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—44no—
187C2×C16#4112yes—
188C2×C16#4112yes—
189C2×C16#4112yes—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—42no—
204#107#4112yes—
205#106—42no—
206#102—42no—
207#102—42no—
208#102—81no—
209C22⋊C8—22no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95D811yescharacteristic
224#95D811yescharacteristic
225C24—41no—
226C24C2×D811yescharacteristic
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—161no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—161no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C22×D411yescharacteristic
258D8—41no—
259D8—41no—
260D8—41no—
261D8—41no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8#13512yes—
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8H(5,𝔽2)11yescharacteristic
280C2×C82−1+411yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—161no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—161no—
307C22⋊C4—161no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—82no—
313C42—161no—
314C16—22no—
315C16—22no—
316C16—22no—
317C16—41no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23#50912yes—
332C23C22×D811yescharacteristic
333Q8—164no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—164no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#53211yescharacteristic
380C2×C4#54611yescharacteristic
381C2×C4#54711yescharacteristic
382C2×C4C2×#13511yescharacteristic
383C2×C4#57711yescharacteristic
384C2×C4#57811yescharacteristic
385C8—22no—
386C8—41no—
387C8—41no—
388C8#52911yescharacteristic
389C8#54211yescharacteristic
390C8#55111yescharacteristic
391C8#55611yescharacteristic
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22#316911yescharacteristic
407C22#329411yescharacteristic
408C22#333411yescharacteristic
409C22C2×#50911yescharacteristic
410C22#341411yescharacteristic
411C22#341511yescharacteristic
412C22#341711yescharacteristic
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#252111yescharacteristic
426C4#317811yescharacteristic
427C4#330011yescharacteristic
428C4#334011yescharacteristic
429C2—42no—
430C2—321no—
431C2#2806511yescharacteristic
432C2#3373911yescharacteristic
433C2#3374611yescharacteristic
434C2#3388011yescharacteristic
435C2#3388511yescharacteristic
436C2#3391011yescharacteristic
437C2#3391411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27837C2semidirect
3. #13528C2semidirect
4. #13494C2semidirect
5. #13451C2semidirect
7. #13283C2semidirect
8. #13149C2semidirect
9. #13149C2semidirect
10. #13130C2semidirect
11. #13112C2semidirect
13. #12834C2semidirect
14. #12674C2semidirect
15. #12611C2semidirect

Maximal subgroups

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

subgroupindex
#278372
#277912
#135282
#134942
#134512
#133802
#132832
#131492
#131302
#131122
#130592
#128342
#126742
#126112