Finite Group Databasebeta

← #211969

Subgroups — #211969

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19127C211yesmaximal, characteristic
2#17976C211yesmaximal, characteristic, complemented
3#17944C211yesmaximal, characteristic, complemented
4#17940C211yesmaximal, characteristic
5#17331C211yesmaximal, characteristic, complemented
6#9728C211yesmaximal, characteristic, complemented
7#9635C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#9214C211yesmaximal, characteristic, complemented
11#9213C211yesmaximal, characteristic
12#8670C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8241C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3107C2211yescharacteristic
18#3106C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2931C2211yescharacteristic
23#2922C2211yescharacteristic
24C4×#108C2211yescharacteristic
25#1684C411yescharacteristic
26#1683C411yescharacteristic
27#1682C411yescharacteristic
28#1681C411yescharacteristic
29#1672C411yescharacteristic
30#1671C411yescharacteristic
31#1666C411yescharacteristic
32#1665C411yescharacteristic
33#1594—21no—
34#1592—21no—
35#1591—21no—
36#1590—21no—
37#1588—21no—
38#1587—21no—
39#1583—21no—
40#1583—21no—
41#1582—21no—
42#1582—21no—
43#1582—21no—
44#1581—21no—
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1566C2211yescharacteristic
48#1564C2211yescharacteristic
49#1544C2211yescharacteristic
50#1543C2211yescharacteristic
51#1526C2211yescharacteristic
52#1525C2211yescharacteristic
53#1493C2211yescharacteristic
54#1492C2211yescharacteristic
55#1489C2211yescharacteristic
56#1489C2211yescharacteristic
57#1485C2211yescharacteristic
58#1484C2211yescharacteristic
59#1454—21no—
60#1453—21no—
61#1452—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450C2211yescharacteristic
65C8×M16—21no—
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76#533C2311yescharacteristic
77#532C2311yescharacteristic
78C4×#41C2311yescharacteristic
79#498C2×C411yescharacteristic
80#492C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#486C2×C411yescharacteristic
83#485—21no—
84#482—21no—
85#477C2×C411yescharacteristic
86#476C2×C411yescharacteristic
87#475C2×C411yescharacteristic
88#474C2×C411yescharacteristic
89#465—21no—
90#465—21no—
91#462C2×C411yescharacteristic
92#461C2×C411yescharacteristic
93#460C2×C411yescharacteristic
94#459C2×C411yescharacteristic
95#452—21no—
96#449—21no—
97#446—21no—
98#446—21no—
99#446D411yescharacteristic
100#446D411yescharacteristic
101C8×Q8—21no—
102C8×Q8—21no—
103C8×Q8C2×C411yescharacteristic
104C8×Q8C2×C411yescharacteristic
105C4×Q16—21no—
106C4×SD16—21no—
107C4×SD16—21no—
108C4×SD16—21no—
109C4×SD16—21no—
110C4×D8—21no—
111#436—21no—
112#436—21no—
113#436D411yescharacteristic
114#436D411yescharacteristic
115#435—21no—
116#435—21no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—21no—
121C8×D4C2×C411yescharacteristic
122C8×D4C2×C411yescharacteristic
123#433—22no—
124#431—21no—
125#431—21no—
126#431—21no—
127#431—21no—
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132C2×#108—21no—
133#424—21no—
134#424—21no—
135#423C2311yescharacteristic
136#423C2311yescharacteristic
137#408—21no—
138#408—21no—
139#405—21no—
140C4×M16—21no—
141C4×M16C2311yescharacteristic
142C4×M16C2311yescharacteristic
143C4×M16C2311yescharacteristic
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#341—21no—
149#335—21no—
150#335—21no—
151#335C2×C411yescharacteristic
152#335C2×C411yescharacteristic
153#334—21no—
154#334—21no—
155#334—21no—
156#334—21no—
157#334C2×C411yescharacteristic
158#334C2×C411yescharacteristic
159#333C2×C411yescharacteristic
160#333C2×C411yescharacteristic
161#332C2×C411yescharacteristic
162#332C2×C411yescharacteristic
163#332C2×C411yescharacteristic
164#332C2×C411yescharacteristic
165#331C2×C411yescharacteristic
166#331C2×C411yescharacteristic
167#330—22no—
168#329—22no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41—21no—
196C2×#41#4111yescharacteristic
197C2×Q16—21no—
198C2×SD16—21no—
199C2×SD16—21no—
200C2×SD16—21no—
201C2×SD16—21no—
202C2×D8—21no—
203#131—82no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—41no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—41no—
216C22×C8#4111yescharacteristic
217C22×C8#4111yescharacteristic
218C22×C8#4111yescharacteristic
219C22×C8#4111yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223#127C22×C411yescharacteristic
224#125—21no—
225#124—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#121—21no—
229#121—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8C22×C411yescharacteristic
233C4×Q8C2×D411yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C22×C411yescharacteristic
239C4×D4C2×D411yescharacteristic
240#117—21no—
241#117—21no—
242#117#4111yescharacteristic
243C2×C42—21no—
244C2×C42C2411yescharacteristic
245#108—42no—
246#107—21no—
247#107—21no—
248#106—21no—
249#106—21no—
250#106—21no—
251#106—21no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8C22×C411yescharacteristic
265C4⋊C8C22×C411yescharacteristic
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C2×D411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280#102—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296#97—21no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—41no—
301#97#4112yes—
302C4×C8—21no—
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×C8C22×C411yescharacteristic
314C4×C8C22×C411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323#41—41no—
324#41—41no—
325#41—81no—
326#41—81no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8C4×D411yescharacteristic
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4C4×D411yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344Q16—81no—
345SD16—81no—
346SD16—81no—
347SD16—81no—
348SD16—81no—
349D8—81no—
350M16—22no—
351M16—42no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—42no—
356M16—41no—
357M16—41no—
358M16—82no—
359M16—81no—
360M16—81no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—81no—
383C2×C8—81no—
384C2×C8—82no—
385C2×C8C4×D411yescharacteristic
386C2×C8C4×D411yescharacteristic
387C2×C8C4×D411yescharacteristic
388C2×C8C4×D411yescharacteristic
389C2×C8#12111yescharacteristic
390C2×C8#12111yescharacteristic
391C2×C8#12111yescharacteristic
392C2×C8#12111yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4C4×D411yescharacteristic
404C4⋊C4C4×D411yescharacteristic
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—41no—
415C42C23×C411yescharacteristic
416C42C22×D411yescharacteristic
417C42C22×D411yescharacteristic
418C42C2×#4111yescharacteristic
419C23—21no—
420C23—41no—
421C23#52911yescharacteristic
422Q8—21no—
423Q8—21no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—81no—
429D4—21no—
430D4—21no—
431D4—42no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—81no—
437D4—82no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C4×#4111yescharacteristic
460C2×C4C2×#12111yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4#53211yescharacteristic
463C8—21no—
464C8—21no—
465C8—21no—
466C8—21no—
467C8—21no—
468C8—21no—
469C8—21no—
470C8—41no—
471C8—41no—
472C8—42no—
473C8—42no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—81no—
483C8—81no—
484C8—81no—
485C8#47112yes—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22C4×#12111yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4#295611yescharacteristic
506C4#295711yescharacteristic
507C4#295711yescharacteristic
508C4#295711yescharacteristic
509C4C2×#47111yescharacteristic
510C4#306711yescharacteristic
511C2—41no—
512C2—81no—
513C2—161no—
514C2#2151911yescharacteristic
515C2#2154711yescharacteristic
516C2C4×#47111yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3106C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17976C2semidirect
3. #17944C2semidirect
5. #17331C2semidirect
6. #9728C2semidirect
7. #9635C2semidirect
8. #9586C2semidirect
9. #9585C2semidirect
10. #9214C2semidirect
12. #8670C2semidirect
13. #8297C2semidirect
14. #8246C2semidirect
15. #8241C2semidirect

Maximal subgroups

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

subgroupindex
#191272
#179762
#179442
#179402
#173312
#97282
#96352
#95862
#95852
#92142
#92132
#86702
#82972
#82462
#82412