Finite Group Databasebeta

← #505095

Subgroups — #505095

Order 512.

491 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#25035C211yesmaximal, characteristic
2#22612C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13494C211yesmaximal, characteristic
5#13475C211yesmaximal, characteristic, complemented
6#13382C211yesmaximal, characteristic
7#13177C211yesmaximal, characteristic, complemented
8#13168C211yesmaximal, characteristic, complemented
9#13133C211yesmaximal, characteristic, complemented
10#13121C211yesmaximal, characteristic
11#13109C211yesmaximal, characteristic, complemented
12#12993C211yesmaximal, characteristic
13#12938C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic
15#12638C211yesmaximal, characteristic, complemented
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C22×#107C2211yescharacteristic
19#2423—21no—
20#2345—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2072—21no—
34#2036—21no—
35#1969C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1921—21no—
40#1921—21no—
41#1863—21no—
42#1845C2211yescharacteristic
43#1824—21no—
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×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C22×C4⋊C4—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#467—22no—
69#467—41no—
70C4×Q16—22no—
71C4×Q16—41no—
72C2×#107—21no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86C2×C22⋊C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#399—41no—
90#398—41no—
91#391—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#388—41no—
96#387—41no—
97#387—41no—
98#387—41no—
99#384—21no—
100#380—21no—
101#380—21no—
102#380—41no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C22⋊C4—41no—
106C4×C22⋊C4D411yescharacteristic
107#367—41no—
108#367—41no—
109#366—22no—
110#366—22no—
111C4⋊C16—22no—
112#358—22no—
113#358—22no—
114#358—22no—
115#358—22no—
116C22⋊C16—22no—
117C22⋊C16—41no—
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—21no—
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16#4112yes—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C22⋊Q8—42no—
148C22⋊Q8—81no—
149C4×Q8—42no—
150C4×Q8—81no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
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—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊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×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
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—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16#4112yes—
190#107—24no—
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—81no—
204#107#4112yes—
205#107#4112yes—
206#103—42no—
207#103—42no—
208#103—42no—
209C22⋊C8—41no—
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
225C24C2×D811yescharacteristic
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—161no—
248C22×C4—161no—
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C22×D411yescharacteristic
255Q16—22no—
256Q16—22no—
257Q16—41no—
258Q16—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8C2×#4111yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C82−1+411yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—41no—
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—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—161no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C4⋊C4—161no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—161no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—82no—
310C42—161no—
311C42—161no—
312C16—22no—
313C16—41no—
314C16—41no—
315C16—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23#51012yes—
323C23C22×D811yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
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—164no—
371C2×C4#51012yes—
372C2×C4#52511yescharacteristic
373C2×C4#54611yescharacteristic
374C2×C4#54711yescharacteristic
375C2×C4#57511yescharacteristic
376C2×C4#57711yescharacteristic
377C2×C4#57811yescharacteristic
378C8—41no—
379C8—41no—
380C8—41no—
381C8#52911yescharacteristic
382C8#54211yescharacteristic
383C8#55211yescharacteristic
384C8#55311yescharacteristic
385C22—22no—
386C22—22no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#310011yescharacteristic
393C22#329411yescharacteristic
394C22#333411yescharacteristic
395C22C2×#51011yescharacteristic
396C22C2×#51011yescharacteristic
397C22#341511yescharacteristic
398C22#341611yescharacteristic
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—161no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4#247011yescharacteristic
413C4#312111yescharacteristic
414C4#331511yescharacteristic
415C4#335111yescharacteristic
416C2—42no—
417C2#2715811yescharacteristic
418C2#3367311yescharacteristic
419C2#3367811yescharacteristic
420C2#3388711yescharacteristic
421C2#3390011yescharacteristic
422C2#3391111yescharacteristic
423C2#3391311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
3. #13537C2semidirect
5. #13475C2semidirect
7. #13177C2semidirect
8. #13168C2semidirect
9. #13133C2semidirect
11. #13109C2semidirect
13. #12938C2semidirect
15. #12638C2semidirect

Maximal subgroups

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

subgroupindex
#250352
#226122
#135372
#134942
#134752
#133822
#131772
#131682
#131332
#131212
#131092
#129932
#129382
#129192
#126382