Finite Group Databasebeta

← #212002

Subgroups — #212002

Order 512.

553 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#17979C211yesmaximal, characteristic, complemented
3#17945C211yesmaximal, characteristic
4#17941C211yesmaximal, characteristic, complemented
5#17336C211yesmaximal, characteristic, complemented
6#9728C211yesmaximal, characteristic, complemented
7#9636C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#9212C211yesmaximal, characteristic, complemented
11#9211C211yesmaximal, characteristic
12#8673C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8243C211yesmaximal, 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#1593—21no—
34#1592—21no—
35#1590—21no—
36#1589—21no—
37#1588—21no—
38#1588—21no—
39#1586—21no—
40#1585—21no—
41#1585—21no—
42#1585—21no—
43#1584—21no—
44#1583—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#1455—21no—
60#1453—21no—
61#1451—21no—
62#1451—21no—
63#1451C2211yescharacteristic
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
66#1450C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
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#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—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—42no—
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—81no—
348SD16—81no—
349SD16—81no—
350SD16—81no—
351D8—81no—
352M16—22no—
353M16—44no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—81no—
360M16—81no—
361M16—81no—
362M16—81no—
363M16—82no—
364M16—81no—
365M16—81no—
366C2×C8—22no—
367C2×C8—22no—
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—22no—
379C2×C8—41no—
380C2×C8—41no—
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—21no—
476C8—21no—
477C8—42no—
478C8—42no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—81no—
488C8—81no—
489C8—81no—
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—
509C4#295611yescharacteristic
510C4#295611yescharacteristic
511C4#295611yescharacteristic
512C4#295711yescharacteristic
513C4#306811yescharacteristic
514C4#306911yescharacteristic
515C2—41no—
516C2—81no—
517C2—161no—
518C2#2151911yescharacteristic
519C2#2154911yescharacteristic
520C2#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
4. #17941C2semidirect
5. #17336C2semidirect
6. #9728C2semidirect
7. #9636C2semidirect
8. #9586C2semidirect
9. #9585C2semidirect
10. #9212C2semidirect
12. #8673C2semidirect
13. #8296C2semidirect
14. #8246C2semidirect
15. #8243C2semidirect

Maximal subgroups

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

subgroupindex
#191592
#179792
#179452
#179412
#173362
#97282
#96362
#95862
#95852
#92122
#92112
#86732
#82962
#82462
#82432