Finite Group Databasebeta

← #225503

Subgroups — #225503

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#444C211yesmaximal, characteristic
2#17899C211yesmaximal, characteristic
3#17898C211yesmaximal, characteristic
4#17799C211yesmaximal, characteristic, complemented
5#17798C211yesmaximal, characteristic, complemented
6#9608C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#8842C211yesmaximal, characteristic, complemented
9#8842C211yesmaximal, characteristic, complemented
10#8841C211yesmaximal, characteristic, complemented
11#8841C211yesmaximal, characteristic, complemented
12#8808C211yesmaximal, characteristic, complemented
13#8605C211yesmaximal, characteristic, complemented
14#8604C211yesmaximal, characteristic, complemented
15#8268C211yesmaximal, characteristic, complemented
16#2989C2211yescharacteristic
17#2989C2211yescharacteristic
18#2989C2211yescharacteristic
19#2989C2211yescharacteristic
20C2×#444C411yescharacteristic
21C2×#444C411yescharacteristic
22C4×#135C2211yescharacteristic
23C4×#131C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1842C2211yescharacteristic
26#1831C2211yescharacteristic
27C4×#108C2211yescharacteristic
28#1773C411yescharacteristic
29#1773C411yescharacteristic
30#1765C411yescharacteristic
31#1765C411yescharacteristic
32#1765C411yescharacteristic
33#1765C411yescharacteristic
34C4×#104C2211yescharacteristic
35C4×#104C2211yescharacteristic
36C4×#97C2211yescharacteristic
37#1594—21no—
38#1593—21no—
39#1592—21no—
40#1592—21no—
41#1591—21no—
42#1590—21no—
43#1590—21no—
44#1589—21no—
45#1569—21no—
46#1569—21no—
47#1569—21no—
48#1569—21no—
49#1568—21no—
50#1568—21no—
51#1567—21no—
52#1566—21no—
53#1565C2211yescharacteristic
54#1565C2211yescharacteristic
55#1565C2211yescharacteristic
56#1565C2211yescharacteristic
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#335C2211yescharacteristic
61C2×#334C2211yescharacteristic
62C2×#334C2211yescharacteristic
63#1477C2211yescharacteristic
64#1477C2211yescharacteristic
65#1477C2211yescharacteristic
66#1477C2211yescharacteristic
67#1450C2211yescharacteristic
68#1450C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C2×#131C2×C411yescharacteristic
77C2×#131C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C4×#41C2311yescharacteristic
81C4×#41C2311yescharacteristic
82#446—21no—
83#446—21no—
84#446—21no—
85#446—21no—
86#444—24no—
87#444—24no—
88C4×Q16D411yescharacteristic
89C4×Q16Q811yescharacteristic
90C4×SD16D411yescharacteristic
91C4×SD16D411yescharacteristic
92C4×SD16Q811yescharacteristic
93C4×SD16Q811yescharacteristic
94C4×D8D411yescharacteristic
95C4×D8Q811yescharacteristic
96#436—21no—
97#436—21no—
98#436—21no—
99#436—21no—
100#435—41no—
101#435—41no—
102#435—41no—
103#435—41no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111#431C2311yescharacteristic
112C2×#108C2×C411yescharacteristic
113C2×#108C2×C411yescharacteristic
114#427C2×C411yescharacteristic
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119C2×C4⋊C8—41no—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×#104C2×C411yescharacteristic
125C2×#104C2×C411yescharacteristic
126C2×#104C2×C411yescharacteristic
127C2×#104C2×C411yescharacteristic
128#416C2×C411yescharacteristic
129#416C2×C411yescharacteristic
130#405—21no—
131#405—21no—
132#405—21no—
133#405—21no—
134#405C2×C411yescharacteristic
135#405C2×C411yescharacteristic
136#405C2×C411yescharacteristic
137#405C2×C411yescharacteristic
138C4×M16—21no—
139C4×M16—21no—
140C4×M16C2311yescharacteristic
141C4×M16C2311yescharacteristic
142C2×#97C2×C411yescharacteristic
143C2×#97C2×C411yescharacteristic
144C2×#97C2×C411yescharacteristic
145C2×#97C2×C411yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C43C2311yescharacteristic
151#341C2×C411yescharacteristic
152#341C2×C411yescharacteristic
153#339C2×C411yescharacteristic
154#339C2×C411yescharacteristic
155#339C2×C411yescharacteristic
156#339C2×C411yescharacteristic
157#336—41no—
158#335—21no—
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#335—21no—
165#335—21no—
166#335—21no—
167#335D411yescharacteristic
168#335D411yescharacteristic
169#334—21no—
170#334—21no—
171#334—21no—
172#334—21no—
173#334—21no—
174#334—21no—
175#334—21no—
176#334—21no—
177#334—21no—
178#334D411yescharacteristic
179#334D411yescharacteristic
180Q8⋊C8—21no—
181Q8⋊C8—21no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82—21no—
192C2×#41C22×C411yescharacteristic
193C2×#41C22×C411yescharacteristic
194#135—22no—
195#135—22no—
196C2×Q16C4⋊C411yescharacteristic
197C2×SD16C4⋊C411yescharacteristic
198C2×SD16C4⋊C411yescharacteristic
199C2×D8C4⋊C411yescharacteristic
200#131—22no—
201#131—22no—
202#131—22no—
203#131—22no—
204#131—22no—
205#131—22no—
206#131—41no—
207#131—41no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—41no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16C22×C411yescharacteristic
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C4×Q8C2×D411yescharacteristic
233C4×Q8C2×D411yescharacteristic
234C4×Q8C2×Q811yescharacteristic
235C4×Q8C2×Q811yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4C2×D411yescharacteristic
241C4×D4C2×D411yescharacteristic
242C4×D4C2×Q811yescharacteristic
243C4×D4C2×Q811yescharacteristic
244#117—21no—
245#117—21no—
246#117C22×C411yescharacteristic
247#117C22×C411yescharacteristic
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2411yescharacteristic
255#108—22no—
256#108M1614yescomplemented
257#107C4⋊C411yescharacteristic
258#107C4⋊C411yescharacteristic
259#106C4⋊C411yescharacteristic
260#106C4⋊C411yescharacteristic
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—82no—
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278C4⋊C8C2×D411yescharacteristic
279C4⋊C8C2×D411yescharacteristic
280#104—22no—
281#104—22no—
282#104—22no—
283#104—22no—
284#103C4⋊C411yescharacteristic
285#103C4⋊C411yescharacteristic
286#103C4⋊C411yescharacteristic
287#103C4⋊C411yescharacteristic
288#102C4⋊C411yescharacteristic
289#102C4⋊C411yescharacteristic
290#102C4⋊C411yescharacteristic
291#102C4⋊C411yescharacteristic
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97—21no—
304#97—22no—
305#97—22no—
306#97—21no—
307#97—21no—
308#97—41no—
309#97—41no—
310#97—41no—
311#97—41no—
312#97—41no—
313#97M1614yescomplemented
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×Q811yescharacteristic
329C4×C8C2×Q811yescharacteristic
330C4×C8#4111yescharacteristic
331C4×C8#4111yescharacteristic
332#41—21no—
333#41—21no—
334#41—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338#41—21no—
339#41—21no—
340C2×Q8C2×C4⋊C411yescharacteristic
341C2×Q8C2×C4⋊C411yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4C2×C4⋊C411yescharacteristic
345C2×D4C2×C4⋊C411yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4C23×C411yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357Q16—22no—
358SD16—22no—
359SD16—22no—
360D8—22no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—21no—
365M16—21no—
366M16—21no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—42no—
371M16—161no—
372M16—161no—
373M16—161no—
374M16—161no—
375M16C2×M1612yes—
376M16C2×M1612yes—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—81no—
400C2×C8—81no—
401C2×C8—81no—
402C2×C8—81no—
403C2×C8C2×C4⋊C411yescharacteristic
404C2×C8C2×C4⋊C411yescharacteristic
405C2×C8C2×C4⋊C411yescharacteristic
406C2×C8C2×C4⋊C411yescharacteristic
407C2×C8C4×D411yescharacteristic
408C2×C8C4×D411yescharacteristic
409C2×C8C4×D411yescharacteristic
410C2×C8C4×D411yescharacteristic
411C2×C8C2×M1612yes—
412C2×C8C2×M1612yes—
413C2×C8C2×M1612yes—
414C2×C8#13112yes—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4C2×C4⋊C411yescharacteristic
420C4⋊C4C2×C4⋊C411yescharacteristic
421C4⋊C4C2×C4⋊C411yescharacteristic
422C4⋊C4C2×C4⋊C411yescharacteristic
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—41no—
439C42—41no—
440C42—81no—
441C42C2×M1612yes—
442C42C22×D411yescharacteristic
443C42C22×D411yescharacteristic
444C42C22×Q811yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—21no—
448C23#52011yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—21no—
452Q8—21no—
453D4—21no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4C22×C4⋊C411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4D4211yescharacteristic
489C2×C4#54711yescharacteristic
490C2×C4D4×Q811yescharacteristic
491C2×C4#55411yescharacteristic
492C2×C4C22×M1611yescharacteristic
493C2×C4C22×M1611yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4#56811yescharacteristic
496C8—21no—
497C8—21no—
498C8—21no—
499C8—21no—
500C8—22no—
501C8—22no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—41no—
508C8—81no—
509C8—81no—
510C8—81no—
511C8—81no—
512C8—81no—
513C8—81no—
514C8#42312yes—
515C8#42312yes—
516C8#43612yes—
517C8#43612yes—
518C22—21no—
519C22—21no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22D4×C4⋊C411yescharacteristic
527C22#298911yescharacteristic
528C22#299111yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—81no—
540C4C2×#42311yescharacteristic
541C4#290611yescharacteristic
542C4C2×#43611yescharacteristic
543C4D4×M1611yescharacteristic
544C4#328811yescharacteristic
545C4#328911yescharacteristic
546C2—21no—
547C2—21no—
548C2—81no—
549C2—81no—
550C2#2265011yescharacteristic
551C2#2269311yescharacteristic
552C2#2269511yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC2×#135C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
4. #17799C2semidirect
5. #17798C2semidirect
6. #9608C2semidirect
7. #9607C2semidirect
8. #8842C2semidirect
9. #8842C2semidirect
10. #8841C2semidirect
11. #8841C2semidirect
12. #8808C2semidirect
13. #8605C2semidirect
14. #8604C2semidirect
15. #8268C2semidirect
256. #108M16semidirect
313. #97M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4442
#178992
#178982
#177992
#177982
#96082
#96072
#88422
#88412
#88082
#86052
#86042
#82682