Finite Group Databasebeta

← #226184

Subgroups — #226184

Order 512.

565 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
1#22733C211yesmaximal, characteristic, complemented
2#22717C211yesmaximal, characteristic
3#18040C211yesmaximal, characteristic
4M16×SD16C211yesmaximal, characteristic
5C2×#1585C211yesmaximal, 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#8665C211yesmaximal, characteristic, complemented
14#8658C211yesmaximal, characteristic
15#8296C211yesmaximal, characteristic, complemented
16Q8×M16C2211yescharacteristic
17Q8×M16C2211yescharacteristic
18C2×#446C2211yescharacteristic
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#1849—21no—
32#1848C2211yescharacteristic
33#1837C2211yescharacteristic
34#1810—21no—
35#1778—21no—
36C4×#108C2211yescharacteristic
37C2×#341—21no—
38#1592—21no—
39#1590—21no—
40#1588—21no—
41#1585—21no—
42#1585C411yescharacteristic
43#1585C411yescharacteristic
44#1585C411yescharacteristic
45#1585C411yescharacteristic
46#1585C411yescharacteristic
47#1585C411yescharacteristic
48#1585C411yescharacteristic
49#1585C411yescharacteristic
50#1583—21no—
51#1582—21no—
52SD16⋊C8—21no—
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—21no—
84#446—21no—
85#446—41no—
86#446—41no—
87#446C2×C411yescharacteristic
88#446C2×C411yescharacteristic
89#446C2×C411yescharacteristic
90#446C2×C411yescharacteristic
91C8×Q8—21no—
92C8×Q8—21no—
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×#97—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—42no—
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—21no—
302#97—21no—
303#97—21no—
304#97—21no—
305#97—41no—
306#97—41no—
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—41no—
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—41no—
352SD16—41no—
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—21no—
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—41no—
386C2×C8—41no—
387C2×C8—42no—
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—41no—
398C2×C8—81no—
399C2×C8—81no—
400C2×C8—81no—
401C2×C8—81no—
402C2×C8—82no—
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—21no—
445Q8—81no—
446D4—21no—
447D4—21no—
448D4—21no—
449D4—42no—
450D4#43012yes—
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—41no—
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—81no—
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—22no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—21no—
491C8—22no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—81no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C8—81no—
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—21no—
521C4—21no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4C2×#43011yescharacteristic
529C4#291611yescharacteristic
530C4C4×#13711yescharacteristic
531C4#294611yescharacteristic
532C4#326611yescharacteristic
533C4#326811yescharacteristic
534C2—21no—
535C2—21no—
536C2—42no—
537C2—81no—
538C2#2266811yescharacteristic
539C2#2267811yescharacteristic
540C2#2269411yescharacteristic
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. #22733C2semidirect
6. #9625C2semidirect
7. #9624C2semidirect
8. #8909C2semidirect
10. #8907C2semidirect
12. #8828C2semidirect
13. #8665C2semidirect
15. #8296C2semidirect

Maximal subgroups

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

subgroupindex
#227332
#227172
#180402
M16×SD162
C2×#15852
#96252
#96242
#89092
#89082
#89072
#89062
#88282
#86652
#86582
#82962