Finite Group Databasebeta

← #225748

Subgroups — #225748

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#108C211yesmaximal, characteristic, complemented
2#22717C211yesmaximal, characteristic
3#18040C211yesmaximal, characteristic
4M16×SD16C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9625C211yesmaximal, characteristic, complemented
7#9624C211yesmaximal, characteristic, complemented
8#8909C211yesmaximal, characteristic, complemented
9#8908C211yesmaximal, characteristic
10#8907C211yesmaximal, characteristic, complemented
11#8906C211yesmaximal, characteristic
12#8828C211yesmaximal, characteristic, complemented
13#8661C211yesmaximal, characteristic, complemented
14#8658C211yesmaximal, characteristic
15#8297C211yesmaximal, characteristic, complemented
16Q8×M16C2211yescharacteristic
17Q8×M16C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20D4×M16C2211yescharacteristic
21D4×M16C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#1950—21no—
24#1950—21no—
25#1950C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1855—21no—
30#1855—21no—
31#1848—21no—
32#1848C2211yescharacteristic
33#1837C2211yescharacteristic
34#1810—21no—
35C4×#108—21no—
36C4×#108C2211yescharacteristic
37C2×#341—21no—
38#1592—21no—
39#1590—21no—
40#1588—21no—
41#1585—21no—
42#1583—21no—
43#1582—21no—
44SD16⋊C8—21no—
45SD16⋊C8C411yescharacteristic
46SD16⋊C8C411yescharacteristic
47SD16⋊C8C411yescharacteristic
48SD16⋊C8C411yescharacteristic
49SD16⋊C8C411yescharacteristic
50SD16⋊C8C411yescharacteristic
51SD16⋊C8C411yescharacteristic
52SD16⋊C8C411yescharacteristic
53C8×SD16—21no—
54#1567C2211yescharacteristic
55#1567C2211yescharacteristic
56#1566C2211yescharacteristic
57#1566C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#335C2211yescharacteristic
60#1487C2211yescharacteristic
61#1487C2211yescharacteristic
62#1486C2211yescharacteristic
63#1486C2211yescharacteristic
64C2×Q8⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1301C2211yescharacteristic
70#1301C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C22×SD16Q811yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78C23×C8—21no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#446—21no—
82#446—21no—
83#446—41no—
84#446—41no—
85C8×Q8—21no—
86C8×Q8—21no—
87C8×Q8—21no—
88C8×Q8—21no—
89C8×Q8C2×C411yescharacteristic
90C8×Q8C2×C411yescharacteristic
91C8×Q8C2×C411yescharacteristic
92C8×Q8C2×C411yescharacteristic
93C4×SD16C2×C411yescharacteristic
94C4×SD16C2×C411yescharacteristic
95C4×SD16C2×C411yescharacteristic
96C4×SD16C2×C411yescharacteristic
97C4×SD16D411yescharacteristic
98C4×SD16D411yescharacteristic
99C4×SD16D411yescharacteristic
100C4×SD16D411yescharacteristic
101#436—21no—
102#436—21no—
103#435—41no—
104#435—41no—
105C8×D4—21no—
106C8×D4—21no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4C2×C411yescharacteristic
110C8×D4C2×C411yescharacteristic
111C8×D4C2×C411yescharacteristic
112C8×D4C2×C411yescharacteristic
113#431C2311yescharacteristic
114#431C2311yescharacteristic
115C2×#108—21no—
116C2×#108—21no—
117C2×#108—21no—
118C2×#108—21no—
119C2×#108—21no—
120C2×#108—21no—
121C2×#108D411yescharacteristic
122C2×#108D411yescharacteristic
123C2×#106Q811yescharacteristic
124#423—21no—
125#423—21no—
126#423C2311yescharacteristic
127#423C2311yescharacteristic
128#423C2311yescharacteristic
129#423C2311yescharacteristic
130C2×C4⋊C8—21no—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×#103Q811yescharacteristic
134C2×#102Q811yescharacteristic
135#407—21no—
136#407—21no—
137C2×C22⋊C8—21no—
138C4×M16—21no—
139C4×M16—21no—
140C4×M16C2311yescharacteristic
141C4×M16C2311yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#341—21no—
147#341—21no—
148#341—21no—
149#341—21no—
150#341—21no—
151#341—21no—
152#341D411yescharacteristic
153#341D411yescharacteristic
154#335—21no—
155#335—21no—
156#335—21no—
157#335—41no—
158#335C2×C411yescharacteristic
159#335C2×C411yescharacteristic
160#335C2×C411yescharacteristic
161#335C2×C411yescharacteristic
162#334—21no—
163#334—21no—
164#334—21no—
165#334—41no—
166Q8⋊C8—21no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8C2×C411yescharacteristic
171Q8⋊C8C2×C411yescharacteristic
172Q8⋊C8C2×C411yescharacteristic
173Q8⋊C8C2×C411yescharacteristic
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8C2×C411yescharacteristic
179D4⋊C8C2×C411yescharacteristic
180D4⋊C8C2×C411yescharacteristic
181D4⋊C8C2×C411yescharacteristic
182#322—21no—
183#322—21no—
184#322—21no—
185#322C2×C411yescharacteristic
186#322C2×C411yescharacteristic
187#322C2×C411yescharacteristic
188#322C2×C411yescharacteristic
189C82—21no—
190C22×Q8C2×Q811yescharacteristic
191C22×D4C2×Q811yescharacteristic
192C23×C4—21no—
193C2×SD16—24no—
194C2×SD16C4⋊C411yescharacteristic
195C2×SD16C4⋊C411yescharacteristic
196C2×SD16C4⋊C411yescharacteristic
197C2×SD16C4⋊C411yescharacteristic
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—41no—
203C2×M16—41no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—81no—
207C2×M16—81no—
208C2×M16C2×D411yescharacteristic
209C2×M16C2×D411yescharacteristic
210C2×M16C2×D411yescharacteristic
211C2×M16C2×D411yescharacteristic
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—41no—
220C22×C8—41no—
221C22×C8—81no—
222C22×C8—81no—
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×Q811yescharacteristic
226C22×C8C2×Q811yescharacteristic
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22×C411yescharacteristic
230C4×Q8C22×C411yescharacteristic
231C4×Q8C2×D411yescharacteristic
232C4×Q8C2×D411yescharacteristic
233C4×D4—21no—
234C4×D4—21no—
235C4×D4C22×C411yescharacteristic
236C4×D4C22×C411yescharacteristic
237C4×D4C2×D411yescharacteristic
238C4×D4C2×D411yescharacteristic
239C2×C4⋊C4—21no—
240C2×C4⋊C4C2×Q811yescharacteristic
241C2×C4⋊C4C2×Q811yescharacteristic
242C2×C22⋊C4—21no—
243C2×C42—21no—
244C2×C42C2411yescharacteristic
245#108—22no—
246#108—41no—
247#108—42no—
248#108—81no—
249#108—81no—
250#108—81no—
251#108—81no—
252#108—81no—
253#106C4⋊C411yescharacteristic
254#106C4⋊C411yescharacteristic
255#106C4⋊C411yescharacteristic
256#106C4⋊C411yescharacteristic
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281#103C4⋊C411yescharacteristic
282#103C4⋊C411yescharacteristic
283#103C4⋊C411yescharacteristic
284#103C4⋊C411yescharacteristic
285#102C4⋊C411yescharacteristic
286#102C4⋊C411yescharacteristic
287#102C4⋊C411yescharacteristic
288#102C4⋊C411yescharacteristic
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299#97—21no—
300#97—21no—
301#97—41no—
302#97—41no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—41no—
313C4×C8—41no—
314C4×C8C22×C411yescharacteristic
315C4×C8C22×C411yescharacteristic
316C4×C8C22×C411yescharacteristic
317C4×C8C22×C411yescharacteristic
318C4×C8C22×C411yescharacteristic
319C4×C8C22×C411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8#4111yescharacteristic
323C4×C8#4111yescharacteristic
324C24—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8C2×C4⋊C411yescharacteristic
329C2×Q8C2×C4⋊C411yescharacteristic
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4C2×C4⋊C411yescharacteristic
334C2×D4C2×C4⋊C411yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q811yescharacteristic
348C22×C42−1+411yescharacteristic
349SD16—22no—
350SD16—22no—
351SD16—22no—
352SD16#10814yescomplemented
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—21no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—82no—
369M16—82no—
370M16#13712yes—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—81no—
398C2×C8—81no—
399C2×C8—81no—
400C2×C8—81no—
401C2×C8—82no—
402C2×C8—81no—
403C2×C8—82no—
404C2×C8C2×C4⋊C411yescharacteristic
405C2×C8C2×C4⋊C411yescharacteristic
406C2×C8C2×C4⋊C411yescharacteristic
407C2×C8C2×C4⋊C411yescharacteristic
408C2×C8C4×D411yescharacteristic
409C2×C8C4×D411yescharacteristic
410C2×C8C4×D411yescharacteristic
411C2×C8C4×D411yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4C2×C4⋊C411yescharacteristic
419C4⋊C4C2×C4⋊C411yescharacteristic
420C4⋊C4C2×C4⋊C411yescharacteristic
421C4⋊C4C2×C4⋊C411yescharacteristic
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42C23×C411yescharacteristic
431C42C22×D411yescharacteristic
432C42C2×#4111yescharacteristic
433C42H(5,𝔽2)11yescharacteristic
434C23—21no—
435C23—21no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—81no—
440C23D4×Q811yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—21no—
444Q8—42no—
445Q8C2×#10812yes—
446D4—21no—
447D4—21no—
448D4—21no—
449D4—42no—
450D4C2×#10812yes—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—81no—
472C2×C4C22×C4⋊C411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4#52011yescharacteristic
475C2×C4D4211yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4#55411yescharacteristic
478C2×C4C2×#13711yescharacteristic
479C2×C4#57511yescharacteristic
480C2×C4#57611yescharacteristic
481C2×C4#57711yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—22no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8C2×#10812yes—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—82no—
515C22—82no—
516C22D4×C4⋊C411yescharacteristic
517C22#338511yescharacteristic
518C22#338811yescharacteristic
519C4—21no—
520C4—22no—
521C4—21no—
522C4—21no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4C22×#10811yescharacteristic
529C4#291611yescharacteristic
530C4C4×#13711yescharacteristic
531C4#294611yescharacteristic
532C4#326611yescharacteristic
533C4#326811yescharacteristic
534C2—21no—
535C2—21no—
536C2—42no—
537C2—81no—
538C2D4×#10811yescharacteristic
539C2#2267811yescharacteristic
540C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC22×D4C2×Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×#108C2semidirect
6. #9625C2semidirect
7. #9624C2semidirect
8. #8909C2semidirect
10. #8907C2semidirect
12. #8828C2semidirect
13. #8661C2semidirect
15. #8297C2semidirect
352. SD16#108semidirect

Maximal subgroups

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

subgroupindex
Q8×#1082
#227172
#180402
M16×SD162
C2×SD16⋊C82
#96252
#96242
#89092
#89082
#89072
#89062
#88282
#86612
#86582
#82972