Finite Group Databasebeta

← #230099

Subgroups — #230099

Order 512.

566 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23171C211yesmaximal, characteristic, complemented
2#22717C211yesmaximal, characteristic
3#19105C211yesmaximal, characteristic
4#18000C211yesmaximal, characteristic
5#17995C211yesmaximal, characteristic
6#9859C211yesmaximal, characteristic, complemented
7#9632C211yesmaximal, characteristic, complemented
8#9591C211yesmaximal, characteristic, complemented
9#9589C211yesmaximal, characteristic, complemented
10#8989C211yesmaximal, characteristic
11#8989C211yesmaximal, characteristic
12#8933C211yesmaximal, characteristic, complemented
13#8931C211yesmaximal, characteristic, complemented
14#8297C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic
16#3160C2211yescharacteristic
17#3101C2211yescharacteristic
18#3101C2211yescharacteristic
19#2962C2211yescharacteristic
20#2962C2211yescharacteristic
21D4×M16C2211yescharacteristic
22D4×M16C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24#1986C2211yescharacteristic
25#1986C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1946—21no—
29#1922—21no—
30#1922—21no—
31#1855—21no—
32#1855—21no—
33#1837C2211yescharacteristic
34#1810—21no—
35C4×#108C2211yescharacteristic
36#1773—21no—
37#1770C2211yescharacteristic
38C4×#102C2211yescharacteristic
39C2×#341—21no—
40#1675C411yescharacteristic, complemented
41#1675C411yescharacteristic, complemented
42#1673C411yescharacteristic, complemented
43#1673C411yescharacteristic, complemented
44#1671C411yescharacteristic, complemented
45#1671C411yescharacteristic, complemented
46#1671C411yescharacteristic, complemented
47#1671C411yescharacteristic, complemented
48#1592—21no—
49#1590—21no—
50#1588—21no—
51#1585—21no—
52#1583—21no—
53#1582—21no—
54SD16⋊C8—21no—
55C8×SD16—21no—
56#1571C2211yescharacteristic
57#1570C2211yescharacteristic
58#1564C2211yescharacteristic
59#1542C2211yescharacteristic
60#1542C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1493C2211yescharacteristic
64#1493C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75C22×M16—21no—
76C22×M16—21no—
77C23×C8—21no—
78#533C2311yescharacteristic
79#533C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81#500C2×C411yescharacteristic
82#500C2×C411yescharacteristic
83#499—21no—
84#499—21no—
85#498C2×C411yescharacteristic
86#498C2×C411yescharacteristic
87#483—21no—
88#483—21no—
89#476C2×C411yescharacteristic
90#476C2×C411yescharacteristic
91#476C2×C411yescharacteristic
92#476C2×C411yescharacteristic
93#461C2×C411yescharacteristic
94#461C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#461C2×C411yescharacteristic
97#450—21no—
98#450—21no—
99#446—21no—
100#446—21no—
101#446—41no—
102#446—41no—
103C8×Q8—21no—
104C8×Q8—21no—
105C4×SD16—21no—
106C4×SD16—21no—
107C4×SD16—21no—
108C4×SD16—21no—
109#436—21no—
110#436—21no—
111#435—41no—
112#435—41no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4C2×C411yescharacteristic
118C8×D4C2×C411yescharacteristic
119C8×D4C2×C411yescharacteristic
120C8×D4C2×C411yescharacteristic
121#431C2311yescharacteristic
122#431C2311yescharacteristic
123C2×#108—21no—
124C2×#108—21no—
125C2×#108D411yescharacteristic
126C2×#108D411yescharacteristic
127#427—21no—
128#427—21no—
129#424—21no—
130#424—21no—
131#423C2311yescharacteristic
132#423C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×#102—21no—
136#407—21no—
137#407—21no—
138C2×C22⋊C8—21no—
139#405—21no—
140#405—21no—
141C4×M16C2311yescharacteristic
142C4×M16C2311yescharacteristic
143C2×#97—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148#341—21no—
149#341—21no—
150#341—21no—
151#341—21no—
152#341D411yescharacteristic
153#341D411yescharacteristic
154#340—21no—
155#339—21no—
156#339—21no—
157#339—21no—
158#339—21no—
159#336—21no—
160#335—21no—
161#335—21no—
162#335C2×C411yescharacteristic
163#335C2×C411yescharacteristic
164#334—21no—
165#334—21no—
166#334C2×C411yescharacteristic
167#334C2×C411yescharacteristic
168#333C2×C411yescharacteristic
169#333C2×C411yescharacteristic
170#333C2×C411yescharacteristic
171#333C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—41no—
181Q8⋊C8—41no—
182D4⋊C8D411yescharacteristic
183D4⋊C8D411yescharacteristic
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191C82—21no—
192C22×D4#4111yescharacteristic
193C23×C4—21no—
194C2×SD16—21no—
195C2×SD16—21no—
196C2×SD16—21no—
197C2×SD16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—41no—
201C2×M16—41no—
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—81no—
205C2×M16—81no—
206C2×M16C2×D411yescharacteristic
207C2×M16C2×D411yescharacteristic
208C2×M16C2×D411yescharacteristic
209C2×M16C2×D411yescharacteristic
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—41no—
219C22×C8—41no—
220C22×C8—81no—
221C22×C8—81no—
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227C4⋊Q8C22×C411yescharacteristic
228#125—21no—
229#125—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4C22×C411yescharacteristic
241C4×D4C22×C411yescharacteristic
242C4×D4C2×D411yescharacteristic
243C4×D4C2×D411yescharacteristic
244#117—21no—
245#117—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4#4111yescharacteristic
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42C2411yescharacteristic
251#108—22no—
252#108—41no—
253#108—81no—
254#108—81no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#106—21no—
260#106—21no—
261#106—21no—
262#106—21no—
263#106—21no—
264#106—21no—
265#106—21no—
266#106—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
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C2×D411yescharacteristic
284C4⋊C8C2×D411yescharacteristic
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293#102—42no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304#97—21no—
305#97—21no—
306#97—41no—
307#97—41no—
308#97—42no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—41no—
317C4×C8—41no—
318C4×C8C22×C411yescharacteristic
319C4×C8C22×C411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8#4111yescharacteristic
323C4×C8#4111yescharacteristic
324#95—21no—
325C24—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4C4×D411yescharacteristic
334C2×D4C4×D411yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C42−1+411yescharacteristic
348SD16—42no—
349SD16—42no—
350SD16—81no—
351SD16—81no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—21no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—82no—
364M16—82no—
365M16#13712yes—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
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—42no—
385C2×C8—41no—
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—81no—
395C2×C8—81no—
396C2×C8—81no—
397C2×C8—81no—
398C2×C8—82no—
399C2×C8—82no—
400C2×C8C4×D411yescharacteristic
401C2×C8C4×D411yescharacteristic
402C2×C8C4×D411yescharacteristic
403C2×C8C4×D411yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4C4×D411yescharacteristic
420C4⋊C4C4×D411yescharacteristic
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42C23×C411yescharacteristic
432C42C22×D411yescharacteristic
433C42C2×#4111yescharacteristic
434C42H(5,𝔽2)11yescharacteristic
435C23—21no—
436C23—21no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—81no—
441C23#54811yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—81no—
447Q8—81no—
448D4—21no—
449D4—21no—
450D4—21no—
451D4—42no—
452D4#44412yes—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—41no—
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—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#52011yescharacteristic
478C2×C4D4211yescharacteristic
479C2×C4#54711yescharacteristic
480C2×C4#55311yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57711yescharacteristic
485C8—22no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—22no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
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—
516C22#236111yescharacteristic
517C22#337211yescharacteristic
518C22#337611yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—41no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—161no—
529C4C4×#13711yescharacteristic
530C4#294911yescharacteristic
531C4C2×#44411yescharacteristic
532C4#295711yescharacteristic
533C4#326611yescharacteristic
534C4#326711yescharacteristic
535C2—21no—
536C2—21no—
537C2—42no—
538C2—81no—
539C2#2301211yescharacteristic
540C2#2301911yescharacteristic
541C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC22×D4#41
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. #23171C2semidirect
6. #9859C2semidirect
7. #9632C2semidirect
8. #9591C2semidirect
9. #9589C2semidirect
12. #8933C2semidirect
13. #8931C2semidirect
14. #8297C2semidirect
40. #1675C4semidirect
41. #1675C4semidirect
42. #1673C4semidirect
43. #1673C4semidirect
44. #1671C4semidirect
45. #1671C4semidirect
46. #1671C4semidirect
47. #1671C4semidirect

Maximal subgroups

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

subgroupindex
#231712
#227172
#191052
#180002
#179952
#98592
#96322
#95912
#95892
#89892
#89332
#89312
#82972
#82042