Finite Group Databasebeta

← #510915

Subgroups — #510915

Order 512.

529 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27932C211yesmaximal, characteristic
2#27789C211yesmaximal, characteristic, complemented
3#13529C211yesmaximal, characteristic, complemented
4#13483C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic
7#13365C211yesmaximal, characteristic, complemented
8#13305C211yesmaximal, characteristic
9#13281C211yesmaximal, characteristic, complemented
10#13237C211yesmaximal, characteristic, complemented
11#13178C211yesmaximal, characteristic, complemented
12#13131C211yesmaximal, characteristic, complemented
13#13120C211yesmaximal, characteristic, complemented
14#12993C211yesmaximal, characteristic, complemented
15#12920C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#481—21no—
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2504—21no—
23#2489—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#2092—21no—
38#2092—21no—
39#2035—21no—
40#2026—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1927—21no—
46#1846C2211yescharacteristic
47#1846C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×#121—21no—
65C22×C4⋊C4—21no—
66C2×SD32—22no—
67C2×SD32—41no—
68C2×SD32—41no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#499—22no—
74#499—41no—
75#481—44no—
76#467—22no—
77#467—41no—
78#466—41no—
79#466—41no—
80C2×#107—21no—
81C2×#107—41no—
82C2×#107—41no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#400—41no—
102#397—41no—
103#397—41no—
104#397—41no—
105#391—21no—
106#391—21no—
107#390—21no—
108#390—21no—
109#388—41no—
110#386—41no—
111#386—41no—
112#382—21no—
113#382—21no—
114#382—41no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—41no—
119#367—41no—
120#366—41no—
121C4⋊C16—22no—
122#358—41no—
123#358—41no—
124#358—41no—
125#357—22no—
126#357—41no—
127#357—41no—
128C22⋊C16—22no—
129C22⋊C16—41no—
130C4×C16—41no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C22×D4—21no—
137C22×D4—41no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2411yescharacteristic
162#125—42no—
163#125—81no—
164C22⋊Q8—42no—
165C22⋊Q8—81no—
166#121—42no—
167#121—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4D811yescharacteristic
188C2×C22⋊C4D811yescharacteristic
189C2×C42—41no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192SD32—44no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16#4112yes—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—42no—
217#107#4112yes—
218#106—42no—
219#106—42no—
220#106—42no—
221#103—42no—
222#103—42no—
223#103—81no—
224#102—42no—
225#102—42no—
226#102—42no—
227C22⋊C8—22no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C24C2×D811yescharacteristic
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—82no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—82no—
275C22×C4—161no—
276C22×C4—161no—
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C22×D411yescharacteristic
283Q16—41no—
284Q16—41no—
285Q16—41no—
286Q16—41no—
287D8—41no—
288D8—41no—
289D8—41no—
290D8—41no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8AGL(1,ℤ/8)12yes—
304C2×C8C2×#4111yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8H(5,𝔽2)11yescharacteristic
307C2×C8H(5,𝔽2)11yescharacteristic
308C2×C8H(5,𝔽2)11yescharacteristic
309C2×C82−1+411yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C22⋊C4—22no—
327C22⋊C4—41no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—161no—
332C22⋊C4—161no—
333C22⋊C4—161no—
334C22⋊C4—161no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—82no—
340C16—22no—
341C16—41no—
342C16—41no—
343C16—41no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#50812yes—
358C23C22×D811yescharacteristic
359Q8—81no—
360Q8—81no—
361Q8—81no—
362Q8—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—164no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—164no—
409C2×C4#53511yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57811yescharacteristic
415C8—22no—
416C8—41no—
417C8—41no—
418C8#53811yescharacteristic
419C8#54211yescharacteristic
420C8#55911yescharacteristic
421C8#56211yescharacteristic
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22#320511yescharacteristic
437C22#329411yescharacteristic
438C22#333411yescharacteristic
439C22C2×#50811yescharacteristic
440C22#341411yescharacteristic
441C22#341611yescharacteristic
442C22#341611yescharacteristic
443C4—41no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#271011yescharacteristic
456C4#322511yescharacteristic
457C4#330811yescharacteristic
458C4#335411yescharacteristic
459C2—42no—
460C2—321no—
461C2#3204611yescharacteristic
462C2#3377711yescharacteristic
463C2#3378811yescharacteristic
464C2#3386811yescharacteristic
465C2#3387411yescharacteristic
466C2#3390411yescharacteristic
467C2#3391811yescharacteristic
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
2. #27789C2semidirect
3. #13529C2semidirect
4. #13483C2semidirect
5. #13449C2semidirect
7. #13365C2semidirect
9. #13281C2semidirect
10. #13237C2semidirect
11. #13178C2semidirect
12. #13131C2semidirect
13. #13120C2semidirect
14. #12993C2semidirect
15. #12920C2semidirect

Maximal subgroups

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

subgroupindex
#279322
#277892
#135292
#134832
#134492
#133802
#133652
#133052
#132812
#132372
#131782
#131312
#131202
#129932
#129202