Finite Group Databasebeta

← #211963

Subgroups — #211963

Order 512.

537 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#17979C211yesmaximal, characteristic, complemented
3#17944C211yesmaximal, characteristic, complemented
4#17940C211yesmaximal, characteristic
5#17329C211yesmaximal, characteristic, complemented
6#9723C211yesmaximal, characteristic, complemented
7#9634C211yesmaximal, characteristic, complemented
8#9588C211yesmaximal, characteristic, complemented
9#9587C211yesmaximal, characteristic, complemented
10#9214C211yesmaximal, characteristic, complemented
11#9213C211yesmaximal, characteristic
12#8671C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8245C211yesmaximal, characteristic, complemented
15#8244C211yesmaximal, 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#1588—21no—
39#1588—21no—
40#1587—21no—
41#1585—21no—
42#1585—21no—
43#1582—21no—
44#1581—21no—
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1567C2211yescharacteristic
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#1452C2211yescharacteristic
63#1451—21no—
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
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#335—21no—
152#335—21no—
153#335C2×C411yescharacteristic
154#335C2×C411yescharacteristic
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#322—21no—
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
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—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—81no—
357M16—81no—
358M16—82no—
359M16—81no—
360M16—81no—
361C2×C8—21no—
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—22no—
372C2×C8—41no—
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—21no—
471C8—41no—
472C8—41no—
473C8—42no—
474C8—42no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—81no—
484C8—81no—
485C8—81no—
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
509C4#306811yescharacteristic
510C4#306911yescharacteristic
511C2—41no—
512C2—81no—
513C2—161no—
514C2#2151911yescharacteristic
515C2#2154511yescharacteristic
516C2#2155711yescharacteristic
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. #17979C2semidirect
3. #17944C2semidirect
5. #17329C2semidirect
6. #9723C2semidirect
7. #9634C2semidirect
8. #9588C2semidirect
9. #9587C2semidirect
10. #9214C2semidirect
12. #8671C2semidirect
13. #8296C2semidirect
14. #8245C2semidirect
15. #8244C2semidirect

Maximal subgroups

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

subgroupindex
#191272
#179792
#179442
#179402
#173292
#97232
#96342
#95882
#95872
#92142
#92132
#86712
#82962
#82452
#82442