Finite Group Databasebeta

← #189012

Subgroups — #189012

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic, complemented
3#17816C211yesmaximal, characteristic, complemented
4#17678C211yesmaximal, characteristic
5#17662C211yesmaximal, characteristic
6#8635C211yesmaximal, characteristic, complemented
7#8635C211yesmaximal, characteristic, complemented
8#8634C211yesmaximal, characteristic, complemented
9#8634C211yesmaximal, characteristic, complemented
10#8633C211yesmaximal, characteristic, complemented
11#8632C211yesmaximal, characteristic, complemented
12#8631C211yesmaximal, characteristic, complemented
13#8630C211yesmaximal, characteristic, complemented
14#8121C211yesmaximal, characteristic, complemented
15C2×#1307C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2994—21no—
18#2994—21no—
19#2993—21no—
20#2993—21no—
21#2991—21no—
22#2990—21no—
23#2989—21no—
24#2988—21no—
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2933C412yes—
28#2933C412yes—
29#2931C2211yescharacteristic
30#2931C2211yescharacteristic
31#2925C412yes—
32#2920C2211yescharacteristic
33C2×#432C2211yescharacteristic
34C2×#432C2211yescharacteristic
35C2×#432C2211yescharacteristic
36C2×#431C2211yescharacteristic
37#2906C412yes—
38C2×#423C2211yescharacteristic
39#1573—21no—
40#1573—21no—
41#1573—21no—
42#1573—21no—
43#1572—21no—
44#1572—21no—
45#1571—21no—
46#1570—21no—
47#1568C2211yescharacteristic
48#1568C2211yescharacteristic
49#1568C2211yescharacteristic
50#1568C2211yescharacteristic
51#1568C2211yescharacteristic
52#1568C2211yescharacteristic
53#1568C2211yescharacteristic
54#1568C2211yescharacteristic
55#1537C2211yescharacteristic
56#1537C2211yescharacteristic
57#1537C2211yescharacteristic
58#1537C2211yescharacteristic
59#1536C2211yescharacteristic
60#1536C2211yescharacteristic
61#1536C2211yescharacteristic
62#1536C2211yescharacteristic
63#1535—22no—
64#1532—22no—
65#1532—22no—
66#1524—22no—
67#1307C2211yescharacteristic
68#1307C2211yescharacteristic
69#1307C2211yescharacteristic
70#1307C2211yescharacteristic
71#1307C2211yescharacteristic
72#1307C2211yescharacteristic
73#1307C2211yescharacteristic
74#1307C2211yescharacteristic
75C2×#131C2×C412yes—
76C2×#131C2×C412yes—
77C22×M16C2×C412yes—
78#533Q811yescharacteristic
79#533Q811yescharacteristic
80#533C2311yescharacteristic
81#532Q811yescharacteristic
82#532Q811yescharacteristic
83#532C2311yescharacteristic
84#529—21no—
85#529—21no—
86#529—21no—
87#529—21no—
88#529D412yes—
89#529D412yes—
90#525—21no—
91#525—21no—
92#525C2×C412yes—
93C4×#41—21no—
94C4×#41—21no—
95C4×#41C2×C412yes—
96C22×C42C2311yescharacteristic
97#446—21no—
98#446—21no—
99#446—21no—
100#446—21no—
101C8×Q8—21no—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105#436—22no—
106#436—22no—
107#436—21no—
108#436—21no—
109#436—21no—
110#436—21no—
111#435—22no—
112#435—22no—
113#435—22no—
114#435—22no—
115C8×D4—22no—
116C8×D4—22no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—21no—
121#433—22no—
122#433—22no—
123#432—22no—
124#432—21no—
125#432—21no—
126#432—21no—
127#432—21no—
128#432—21no—
129#432—21no—
130#432—21no—
131#432—21no—
132#432—21no—
133#432—21no—
134#432C2311yescharacteristic
135#432C2311yescharacteristic
136#432C2311yescharacteristic
137#432C2311yescharacteristic
138#432C2311yescharacteristic
139#432C2311yescharacteristic
140#432C2311yescharacteristic
141#432C2311yescharacteristic
142#431—22no—
143#431—21no—
144#431—21no—
145#431D411yescharacteristic
146#431D411yescharacteristic
147#431D411yescharacteristic
148#431D411yescharacteristic
149#431D411yescharacteristic
150#431D411yescharacteristic
151#431D411yescharacteristic
152#431D411yescharacteristic
153#424—22no—
154#424—22no—
155#424C2×C412yes—
156#424C2×C412yes—
157#423—21no—
158#423—21no—
159#423—22no—
160#423—21no—
161#423C2×C412yes—
162#423C2311yescharacteristic
163#423C2311yescharacteristic
164C2×C4⋊C8—21no—
165C2×C4⋊C8—21no—
166C2×C4⋊C8—22no—
167C2×C4⋊C8—21no—
168C2×C4⋊C8—21no—
169C2×C4⋊C8—21no—
170C2×C4⋊C8C2×C411yescharacteristic
171C2×C4⋊C8C2×C411yescharacteristic
172#408C2×C412yes—
173#408C2×C412yes—
174#407—21no—
175C2×C22⋊C8—21no—
176C2×C22⋊C8—21no—
177C2×C22⋊C8—21no—
178C2×C22⋊C8C2×C411yescharacteristic
179C2×C22⋊C8C2×C411yescharacteristic
180#405C2×C412yes—
181#405C2×C412yes—
182C4×M16—21no—
183C2×#97—21no—
184C2×#97—21no—
185C2×#97—21no—
186C2×C4×C8C2311yescharacteristic
187C2×C4×C8C2311yescharacteristic
188#335—21no—
189#335—21no—
190#335—21no—
191#335—21no—
192#335—21no—
193#335—21no—
194#335—21no—
195#335—21no—
196#334—21no—
197#334—21no—
198#334—21no—
199#334—21no—
200#334—21no—
201#334—21no—
202#334—21no—
203#334—21no—
204#330—22no—
205#330—22no—
206#330—22no—
207#330—22no—
208#329—22no—
209#329—22no—
210#329—22no—
211#329—22no—
212C2×#41—21no—
213C2×#41—21no—
214C2×#41C22×C412yes—
215C23×C4C22×C411yescharacteristic
216C23×C4#4111yescharacteristic
217C23×C4#4111yescharacteristic
218#131—82no—
219#131—82no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—41no—
224C2×M16—82no—
225C2×M16C22×C412yes—
226C2×M16C22×C412yes—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—41no—
231C22×C8—41no—
232C22×C8—41no—
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C4⋊Q8C2×Q811yescharacteristic
238C4⋊Q8C2×Q811yescharacteristic
239C4⋊Q8C2×Q811yescharacteristic
240#127C2×Q811yescharacteristic
241#126—22no—
242#126—22no—
243#125—21no—
244#125—21no—
245#125C2×D412yes—
246#124—21no—
247#124—21no—
248#124C2×D412yes—
249#123—22no—
250#123—22no—
251#123—22no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8C4⋊C412yes—
257C22⋊Q8C4⋊C412yes—
258#121—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121C4⋊C412yes—
263#121C4⋊C412yes—
264#120—22no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4⋊C412yes—
270C4×Q8C4⋊C412yes—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C4⋊C412yes—
282C4×D4C4⋊C412yes—
283#117—21no—
284#117—21no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117C22×C412yes—
289C2×C42—22no—
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×Q811yescharacteristic
298C2×C42C2×Q811yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42C2411yescharacteristic
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323C4⋊C8—41no—
324C4⋊C8C22⋊C411yescharacteristic
325C4⋊C8C22⋊C411yescharacteristic
326C4⋊C8C22⋊C411yescharacteristic
327C4⋊C8C22⋊C411yescharacteristic
328C4⋊C8C22⋊C411yescharacteristic
329C4⋊C8C22⋊C411yescharacteristic
330C4⋊C8C22⋊C411yescharacteristic
331C4⋊C8C22⋊C411yescharacteristic
332C22⋊C8—22no—
333C22⋊C8—22no—
334C22⋊C8—22no—
335C22⋊C8—22no—
336C22⋊C8—21no—
337C22⋊C8—21no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8—41no—
345C22⋊C8—41no—
346C22⋊C8—41no—
347C22⋊C8—41no—
348C22⋊C8C22⋊C411yescharacteristic
349C22⋊C8C22⋊C411yescharacteristic
350C22⋊C8C22⋊C411yescharacteristic
351C22⋊C8C22⋊C411yescharacteristic
352C22⋊C8C22⋊C411yescharacteristic
353C22⋊C8C22⋊C411yescharacteristic
354C22⋊C8C22⋊C411yescharacteristic
355C22⋊C8C22⋊C411yescharacteristic
356#97—22no—
357#97—22no—
358#97—21no—
359#97—21no—
360#97—21no—
361#97—21no—
362#97—21no—
363#97—21no—
364#97—21no—
365#97—21no—
366#97—41no—
367#97—41no—
368#97—41no—
369C4×C8—22no—
370C4×C8—22no—
371C4×C8—41no—
372C4×C8C2×D411yescharacteristic
373C4×C8C2×D411yescharacteristic
374C4×C8C2×D411yescharacteristic
375C4×C8C2×D411yescharacteristic
376C4×C8C2×D411yescharacteristic
377C4×C8C2×D411yescharacteristic
378C4×C8C2×D411yescharacteristic
379C4×C8C2×D411yescharacteristic
380C24#11711yescharacteristic
381#41—44no—
382#41—42no—
383#41—42no—
384#41—82no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8C2×C4⋊C412yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4C2×C4⋊C412yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—81no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C4⋊C411yescharacteristic
404C22×C4C2×C4⋊C411yescharacteristic
405C22×C4#11711yescharacteristic
406C22×C4#11711yescharacteristic
407C22×C4#11711yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4C22⋊Q811yescharacteristic
413C22×C4C22⋊Q811yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419M16—42no—
420M16—42no—
421M16—82no—
422M16—82no—
423M16—82no—
424M16—82no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—82no—
440C2×C8—82no—
441C2×C8—82no—
442C2×C8—82no—
443C2×C8—82no—
444C2×C8—82no—
445C2×C8C2×C22⋊C411yescharacteristic
446C2×C8C2×C22⋊C411yescharacteristic
447C2×C8C2×C22⋊C411yescharacteristic
448C2×C8C2×C22⋊C411yescharacteristic
449C2×C8C2×C22⋊C411yescharacteristic
450C2×C8C2×C22⋊C411yescharacteristic
451C2×C8C2×C22⋊C411yescharacteristic
452C2×C8C2×C22⋊C411yescharacteristic
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4C2×C4⋊C412yes—
464C4⋊C4C2×C4⋊C412yes—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42C2×C22⋊C411yescharacteristic
478C42C2×C22⋊C411yescharacteristic
479C42C2×C4⋊C411yescharacteristic
480C42C2×C4⋊C411yescharacteristic
481C42#12111yescharacteristic
482C42#12111yescharacteristic
483C42#12111yescharacteristic
484C42#12111yescharacteristic
485C42C22⋊Q811yescharacteristic
486C42C22⋊Q811yescharacteristic
487C42C22⋊Q811yescharacteristic
488C42C22⋊Q811yescharacteristic
489C42C22×D411yescharacteristic
490C42C22×D411yescharacteristic
491C42C22×D411yescharacteristic
492C42C22×Q811yescharacteristic
493C23—22no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23#38011yescharacteristic
498C23#38011yescharacteristic
499C23C2×#11711yescharacteristic
500Q8—42no—
501Q8—42no—
502Q8—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—44no—
507D4—44no—
508D4—82no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4#38011yescharacteristic
538C2×C4#38011yescharacteristic
539C2×C4#38011yescharacteristic
540C2×C4#38011yescharacteristic
541C2×C4#38011yescharacteristic
542C2×C4#38011yescharacteristic
543C2×C4C22×C22⋊C411yescharacteristic
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—21no—
557C8—42no—
558C8—42no—
559C8—81no—
560C8—81no—
561C8—81no—
562C8—81no—
563C8—81no—
564C8—81no—
565C8—81no—
566C8—81no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22#181412yes—
580C22C2×#38011yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4#288811yescharacteristic
596C4#288911yescharacteristic
597C2—22no—
598C2—41no—
599C2—41no—
600C2—82no—
601C2#2064011yescharacteristic
602C2C2×#181411yescharacteristic
603C2#2067511yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17818C2semidirect
3. #17816C2semidirect
6. #8635C2semidirect
7. #8635C2semidirect
8. #8634C2semidirect
9. #8634C2semidirect
10. #8633C2semidirect
11. #8632C2semidirect
12. #8631C2semidirect
13. #8630C2semidirect
14. #8121C2semidirect
15. C2×#1307C2semidirect

Maximal subgroups

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

subgroupindex
#602682
#178182
#178162
#176782
#176622
#86352
#86342
#86332
#86322
#86312
#86302
#81212
C2×#13072