Finite Group Databasebeta

← #211975

Subgroups — #211975

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19159C211yesmaximal, characteristic
2#17957C211yesmaximal, characteristic, complemented
3#17945C211yesmaximal, characteristic
4#17944C211yesmaximal, characteristic, complemented
5#17335C211yesmaximal, characteristic, complemented
6#9728C211yesmaximal, characteristic, complemented
7#9635C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#9212C211yesmaximal, characteristic, complemented
11#9211C211yesmaximal, characteristic
12#8672C211yesmaximal, characteristic, complemented
13C8×#108C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8245C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3107C2211yescharacteristic
18#3106C2211yescharacteristic
19#2965C2211yescharacteristic
20#2965C2211yescharacteristic
21#2960C2211yescharacteristic
22#2931C2211yescharacteristic
23C4×#131C2211yescharacteristic
24C4×#108C2211yescharacteristic
25#1704C411yescharacteristic
26#1703C411yescharacteristic
27#1702C411yescharacteristic
28#1701C411yescharacteristic
29#1692C411yescharacteristic
30#1691C411yescharacteristic
31#1686C411yescharacteristic
32#1685C411yescharacteristic
33#1591—21no—
34#1590—21no—
35#1590—21no—
36#1590—21no—
37#1590—21no—
38#1589—21no—
39#1588—21no—
40#1587—21no—
41#1586—21no—
42#1585—21no—
43C8×SD16—21no—
44C8×SD16—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#1455—21no—
60#1453—21no—
61#1451C2211yescharacteristic
62#1451C2211yescharacteristic
63#1450C2211yescharacteristic
64C8×M16—21no—
65C8×M16—21no—
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×C82C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77#533C2311yescharacteristic
78#532C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#498C2×C411yescharacteristic
81#492C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#485—21no—
85#482—21no—
86#477C2×C411yescharacteristic
87#476C2×C411yescharacteristic
88#475C2×C411yescharacteristic
89#474C2×C411yescharacteristic
90#465—21no—
91#465—21no—
92#462C2×C411yescharacteristic
93#461C2×C411yescharacteristic
94#460C2×C411yescharacteristic
95#459C2×C411yescharacteristic
96#452—21no—
97#449—21no—
98#446—21no—
99#446—21no—
100#446C2×C411yescharacteristic
101#446C2×C411yescharacteristic
102#446D411yescharacteristic
103#446D411yescharacteristic
104C8×Q8—21no—
105C8×Q8—21no—
106C4×Q16—21no—
107C4×SD16—21no—
108C4×SD16—21no—
109C4×SD16—21no—
110C4×SD16—21no—
111C4×D8—21no—
112#436—21no—
113#436—21no—
114#436C2×C411yescharacteristic
115#436C2×C411yescharacteristic
116#436D411yescharacteristic
117#436D411yescharacteristic
118#435—21no—
119#435—21no—
120#435—21no—
121#435—21no—
122C8×D4—21no—
123C8×D4—21no—
124#431—21no—
125#431—21no—
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130C2×#108—21no—
131#424—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134#408—21no—
135#405—21no—
136#405—22no—
137#405—21no—
138C4×M16—21no—
139C4×M16—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—
188C82—21no—
189C82—21no—
190C82—21no—
191C82D411yescharacteristic
192C82D411yescharacteristic
193C82D411yescharacteristic
194C82D411yescharacteristic
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—42no—
204#131—42no—
205#131—82no—
206C2×M16—22no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—41no—
217C22×C8—21no—
218C22×C8—22no—
219C22×C8—41no—
220C22×C8#4111yescharacteristic
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C22×C8#4111yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227#127C22×C411yescharacteristic
228#125—21no—
229#124—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8C22×C411yescharacteristic
237C4×Q8C2×D411yescharacteristic
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C22×C411yescharacteristic
243C4×D4C2×D411yescharacteristic
244#117—21no—
245#117—21no—
246#117#4111yescharacteristic
247C2×C42—21no—
248C2×C42C2411yescharacteristic
249#108—24no—
250#107—21no—
251#107—21no—
252#106—21no—
253#106—21no—
254#106—21no—
255#106—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—41no—
265C4⋊C8—41no—
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C22×C411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274#103—21no—
275#103—21no—
276#103—21no—
277#103—21no—
278#103—21no—
279#103—21no—
280#102—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
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—22no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97—41no—
304#97#4112yes—
305C4×C8—22no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—41no—
315C4×C8C22×C411yescharacteristic
316C4×C8C22×C411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324C4×C8C2×D411yescharacteristic
325#41—41no—
326#41—41no—
327#41—81no—
328#41—81no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C4×D411yescharacteristic
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—41no—
338C2×D4C4×D411yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346Q16—81no—
347SD16—42no—
348SD16—42no—
349SD16—81no—
350SD16—81no—
351D8—81no—
352M16—22no—
353M16—22no—
354M16—42no—
355M16—42no—
356M16—44no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—42no—
362M16—42no—
363M16—82no—
364M16—81no—
365M16—81no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
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—22no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—44no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—82no—
389C2×C8C4×D411yescharacteristic
390C2×C8C4×D411yescharacteristic
391C2×C8C4×D411yescharacteristic
392C2×C8C4×D411yescharacteristic
393C2×C8#12111yescharacteristic
394C2×C8#12111yescharacteristic
395C2×C8#12111yescharacteristic
396C2×C8#12111yescharacteristic
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C4×D411yescharacteristic
408C4⋊C4C4×D411yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—41no—
419C42C23×C411yescharacteristic
420C42C22×D411yescharacteristic
421C42C22×D411yescharacteristic
422C42C2×#4111yescharacteristic
423C23—21no—
424C23—41no—
425C23#52911yescharacteristic
426Q8—21no—
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432Q8—81no—
433D4—21no—
434D4—21no—
435D4—42no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—81no—
441D4—82no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C4×#4111yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4#52511yescharacteristic
466C2×C4#53211yescharacteristic
467C8—22no—
468C8—22no—
469C8—21no—
470C8—21no—
471C8—21no—
472C8—21no—
473C8—21no—
474C8—21no—
475C8—42no—
476C8—42no—
477C8—42no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—81no—
487C8—81no—
488C8#44312yes—
489C8#47112yes—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C4×#12111yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4C2×#44311yescharacteristic
510C4#295511yescharacteristic
511C4#295611yescharacteristic
512C4#295611yescharacteristic
513C4C2×#47111yescharacteristic
514C4#306711yescharacteristic
515C2—41no—
516C2—81no—
517C2—161no—
518C2#2151711yescharacteristic
519C2#2154911yescharacteristic
520C2C4×#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. #17957C2semidirect
4. #17944C2semidirect
5. #17335C2semidirect
6. #9728C2semidirect
7. #9635C2semidirect
8. #9586C2semidirect
9. #9585C2semidirect
10. #9212C2semidirect
12. #8672C2semidirect
13. C8×#108C2semidirect
14. #8246C2semidirect
15. #8245C2semidirect

Maximal subgroups

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

subgroupindex
#191592
#179572
#179452
#179442
#173352
#97282
#96352
#95862
#95852
#92122
#92112
#86722
C8×#1082
#82462
#82452