Finite Group Databasebeta

← #230856

Subgroups — #230856

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23678C211yesmaximal, characteristic
2#23173C211yesmaximal, characteristic
3#22822C211yesmaximal, characteristic, complemented
4#9661C211yesmaximal, characteristic, complemented
5#9655C211yesmaximal, characteristic, complemented
6#9644C211yesmaximal, characteristic
7#9602C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic, complemented
10#8945C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8264C211yesmaximal, characteristic, complemented
13#8202C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8182C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C8×D4—21no—
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2295C2211yescharacteristic
23#2084C411yescharacteristic
24#2072C411yescharacteristic
25#2065C411yescharacteristic
26#2065C411yescharacteristic
27#2055C411yescharacteristic
28#2055C411yescharacteristic
29#2041C411yescharacteristic
30#2014C411yescharacteristic
31#1984C2211yescharacteristic
32#1984C2211yescharacteristic
33#1954—21no—
34#1954C2211yescharacteristic
35#1954C2211yescharacteristic
36#1917—21no—
37#1916—21no—
38#1869—21no—
39#1869—21no—
40#1842C2211yescharacteristic
41#1832—21no—
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1768—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C2×#127C2×C411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C4×SD16—41no—
78C4×SD16—41no—
79C4×SD16—41no—
80C4×SD16—41no—
81C8×D4—22no—
82C8×D4—22no—
83C8×D4—41no—
84C8×D4—41no—
85C8×D4—44no—
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8C2×C411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#390—21no—
124#390—21no—
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#387—21no—
130#387—21no—
131#384C2×C411yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134#376—21no—
135C43C2311yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#336D411yescharacteristic
152#336D411yescharacteristic
153#334—41no—
154#334—41no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—22no—
173C2×SD16—41no—
174C2×SD16—41no—
175C2×SD16—41no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—81no—
187C22×C8—81no—
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196#127—22no—
197#127—22no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—81no—
213C4×D4—81no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42C2411yescharacteristic
239#106—41no—
240#106—41no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—41no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253C4⋊C8SD1614yescomplemented
254C4⋊C8#4112yes—
255#103—41no—
256#103—41no—
257#102—22no—
258#102—22no—
259#102—22no—
260#102—41no—
261#102—41no—
262#102—41no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—81no—
268C22⋊C8—81no—
269#97—81no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—42no—
282C4×C8—81no—
283C4×C8#4112yes—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291C24—21no—
292C24—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4#13112yes—
316C2×D4#13112yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342SD16—44no—
343SD16—44no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
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—41no—
383C2×C8—84no—
384C2×C8—84no—
385C2×C8#13112yes—
386C2×C8C2×SD1612yes—
387C2×C8C2×SD1612yes—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—81no—
411C22⋊C4—81no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42C2×SD1612yes—
425C42#13512yes—
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23C2×C4×D411yescharacteristic
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449D4—22no—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—48no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—84no—
459C2×C4—21no—
460C2×C4—21no—
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—21no—
470C2×C4—21no—
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—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4C4×SD1612yes—
498C2×C4C4×SD1612yes—
499C2×C4C4×#4111yescharacteristic
500C2×C4#51811yescharacteristic
501C2×C4#54611yescharacteristic
502C2×C4#55011yescharacteristic
503C2×C4#55311yescharacteristic
504C2×C4#55511yescharacteristic
505C2×C4C2×#13111yescharacteristic
506C2×C4C2×#13111yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4#56811yescharacteristic
509C2×C4C22×SD1611yescharacteristic
510C2×C4C2×#13511yescharacteristic
511C2×C4#57611yescharacteristic
512C2×C4#57711yescharacteristic
513C8—22no—
514C8—22no—
515C8—22no—
516C8—42no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—42no—
521C8—41no—
522C8—41no—
523C8—81no—
524C8—81no—
525C8—81no—
526C8—81no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22#236311yescharacteristic
542C22#292911yescharacteristic
543C22#293511yescharacteristic
544C22C2×C4×SD1611yescharacteristic
545C22#294611yescharacteristic
546C22#295511yescharacteristic
547C22#295711yescharacteristic
548C4—21no—
549C4—21no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4#296611yescharacteristic
561C4#296811yescharacteristic
562C4#297611yescharacteristic
563C4#297711yescharacteristic
564C4#329511yescharacteristic
565C4#330411yescharacteristic
566C4#339211yescharacteristic
567C4#339611yescharacteristic
568C2—82no—
569C2—82no—
570C2#1790211yescharacteristic
571C2#1791811yescharacteristic
572C2#1806711yescharacteristic
573C2#1808011yescharacteristic
574C2#2308911yescharacteristic
575C2#2309511yescharacteristic
576C2#2311511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22822C2semidirect
4. #9661C2semidirect
5. #9655C2semidirect
7. #9602C2semidirect
8. #9546C2semidirect
9. #9542C2semidirect
10. #8945C2semidirect
11. #8945C2semidirect
12. #8264C2semidirect
13. #8202C2semidirect
14. C8×#102C2semidirect
15. #8182C2semidirect
253. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#236782
#231732
#228222
#96612
#96552
#96442
#96022
#95462
#95422
#89452
#82642
#82022
C8×#1022
#81822