Finite Group Databasebeta

← #324993

Subgroups — #324993

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33230C211yesmaximal, characteristic
2#29346C211yesmaximal, characteristic
3#28689C211yesmaximal, characteristic, complemented
4#21713C211yesmaximal, characteristic, complemented
5#10592C211yesmaximal, characteristic, complemented
6#10591C211yesmaximal, characteristic
7#10588C211yesmaximal, characteristic, complemented
8#10587C211yesmaximal, characteristic
9#9896C211yesmaximal, characteristic, complemented
10#9698C211yesmaximal, characteristic
11#9571C211yesmaximal, characteristic
12#9569C211yesmaximal, characteristic, complemented
13#9563C211yesmaximal, characteristic
14#9561C211yesmaximal, characteristic, complemented
15#8275C211yesmaximal, characteristic, complemented
16C2×#500—21no—
17C2×#489—21no—
18C2×#489—21no—
19C2×#446—21no—
20#2868C2211yescharacteristic
21#2858C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23#2083—21no—
24#2083—21no—
25#2071—21no—
26#2023—21no—
27#1990—21no—
28#1988C2211yescharacteristic
29#1988C2211yescharacteristic
30#1985—21no—
31#1982C2211yescharacteristic
32#1982C2211yescharacteristic
33#1959—21no—
34#1959—21no—
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1956C2211yescharacteristic
38#1956C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1947C2211yescharacteristic
43#1940—21no—
44#1939C2211yescharacteristic
45#1939C2211yescharacteristic
46#1934—21no—
47#1934—21no—
48#1842C2211yescharacteristic
49#1771—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×#124—21no—
74C2×#124—21no—
75C2×C4×Q8—21no—
76#500—44no—
77#489—44no—
78#489—44no—
79#446—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#97—21no—
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#389—21no—
122#389D411yescharacteristic
123#389D411yescharacteristic
124#384—21no—
125#384—21no—
126#384—21no—
127#384D411yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C43C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#334—22no—
150#332—22no—
151#332—22no—
152#331—22no—
153#331—22no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—41no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—41no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—82no—
181#127—21no—
182#127—21no—
183#127—21no—
184#127—21no—
185#124—42no—
186#124—42no—
187#124—42no—
188#124—42no—
189C4×Q8—42no—
190C4×Q8—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224#107—42no—
225#107—42no—
226#107—42no—
227#107—42no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8#4112yes—
243C4⋊C8#4112yes—
244#103—42no—
245#103—42no—
246#103—42no—
247#103—42no—
248#102—42no—
249#102—42no—
250#102—42no—
251#102—42no—
252#102—42no—
253#102—42no—
254#102—81no—
255#102—81no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—42no—
260#97D814yes—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8—42no—
267C4×C8—81no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—81no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8C2×D812yes—
344C2×C8C2×D812yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4AGL(1,ℤ/8)12yes—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42C2×D812yes—
390C42#13512yes—
391C42AGL(1,ℤ/8)12yes—
392C42#13712yes—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C422−1+411yescharacteristic
396C422−1+411yescharacteristic
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23C2×#12111yescharacteristic
405Q8—84no—
406Q8—81no—
407Q8—81no—
408Q8—81no—
409Q8—81no—
410Q8—164no—
411D4—81no—
412D4—81no—
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—161no—
420D4—161no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#45912yes—
453C2×C4#45912yes—
454C2×C4#46412yes—
455C2×C4#46412yes—
456C2×C4#53211yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#54811yescharacteristic
459C2×C4#54811yescharacteristic
460C2×C4#55611yescharacteristic
461C2×C4#55611yescharacteristic
462C2×C4C22×D811yescharacteristic
463C2×C4C2×#13511yescharacteristic
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57711yescharacteristic
469C2×C4#57811yescharacteristic
470C8—22no—
471C8—22no—
472C8—42no—
473C8—42no—
474C8—41no—
475C8—41no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—81no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22—161no—
486C22—161no—
487C22#254511yescharacteristic
488C22C2×#45911yescharacteristic
489C22C2×#46411yescharacteristic
490C22#305511yescharacteristic
491C22#306311yescharacteristic
492C22#306811yescharacteristic
493C22#306911yescharacteristic
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—42no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4#316911yescharacteristic
508C4#318211yescharacteristic
509C4#322611yescharacteristic
510C4#322811yescharacteristic
511C4#336711yescharacteristic
512C4#337111yescharacteristic
513C4#337311yescharacteristic
514C4#337711yescharacteristic
515C2—321no—
516C2#1916211yescharacteristic
517C2#1917711yescharacteristic
518C2#1923111yescharacteristic
519C2#1924411yescharacteristic
520C2#2864911yescharacteristic
521C2#2865511yescharacteristic
522C2#2867211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28689C2semidirect
4. #21713C2semidirect
5. #10592C2semidirect
7. #10588C2semidirect
9. #9896C2semidirect
12. #9569C2semidirect
14. #9561C2semidirect
15. #8275C2semidirect

Maximal subgroups

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

subgroupindex
#332302
#293462
#286892
#217132
#105922
#105912
#105882
#105872
#98962
#96982
#95712
#95692
#95632
#95612
#82752