Finite Group Databasebeta

← #188969

Subgroups — #188969

Order 512.

991 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60233C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21941C211yesmaximal, characteristic, complemented
4#17592C211yesmaximal, characteristic
5#17559C211yesmaximal, characteristic
6#8629C212yesmaximal, complemented
7#8628C212yesmaximal, complemented
8#8627C212yesmaximal, complemented
9#8626C212yesmaximal, complemented
10#8108C211yesmaximal, characteristic, complemented
11C2×#1305C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13#2977—21no—
14#2977—21no—
15#2976—22no—
16#2974—22no—
17#2973—21no—
18#2972—21no—
19#2933C412yescomplemented
20#2933C412yescomplemented
21#2929C2212yes—
22#2929C2212yes—
23#2924C412yescomplemented
24#2880C412yescomplemented
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2280C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1855—21no—
32#1855—21no—
33#1855—22no—
34#1854—22no—
35#1853—21no—
36#1852—21no—
37#1810C2212yescomplemented
38#1810C2212yescomplemented
39#1808C2212yescomplemented
40#1807C2212yescomplemented
41#1520—21no—
42#1519—21no—
43#1513C2212yes—
44#1513C2212yes—
45#1513C2212yes—
46#1513C2212yes—
47#1511—21no—
48#1510—21no—
49#1504—22no—
50#1501—22no—
51#1305C2212yescomplemented
52#1305C2212yescomplemented
53#1305C2211yescharacteristic, complemented
54#1305C2211yescharacteristic, complemented
55#1305C2211yescharacteristic, complemented
56#1305C2211yescharacteristic, complemented
57C22×#41C2×C411yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#131C2×C412yescomplemented
60C2×#131C2×C412yescomplemented
61C22×M16—21no—
62C22×M16—21no—
63C22×M16C2×C412yes—
64C23×C8C2312yes—
65#525—21no—
66#525—21no—
67#525—21no—
68#525—21no—
69#525C2×C412yes—
70#525D412yes—
71#525Q812yes—
72#525C2312yes—
73C2×#123—21no—
74C2×C22⋊Q8D411yescharacteristic
75C2×C22⋊Q8Q811yescharacteristic
76C2×#121D411yescharacteristic
77C2×#121Q811yescharacteristic
78C2×#120—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×#117—21no—
82C2×#117C2×C411yescharacteristic
83C22×C4⋊C4—21no—
84C22×C22⋊C4—21no—
85#436—22no—
86#436—22no—
87#435—22no—
88#435—22no—
89#435—22no—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95C8×D4—22no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101#433—22no—
102#433—22no—
103#432—22no—
104#431—22no—
105C2×#108—22no—
106C2×#108—22no—
107C2×#107—22no—
108C2×#106—22no—
109#424C2×C412yescomplemented
110#424C2×C412yescomplemented
111#408—22no—
112#408—22no—
113#408C2×C412yescomplemented
114#408C2×C412yescomplemented
115#407—21no—
116#407—21no—
117#407—22no—
118#407—21no—
119#407—21no—
120#407—21no—
121#407—21no—
122#407—21no—
123#407—21no—
124#407C2×C412yes—
125#407C2311yescharacteristic
126#407C2312yes—
127#407C2311yescharacteristic
128#407C2311yescharacteristic
129#407C2311yescharacteristic
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8C2×C412yes—
138C2×C22⋊C8C2×C411yescharacteristic
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×C22⋊C8D412yes—
143C2×C22⋊C8D411yescharacteristic
144C2×C22⋊C8D411yescharacteristic
145C2×C22⋊C8D412yes—
146#405C2×C412yescomplemented
147#405C2×C412yescomplemented
148#381—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380C2311yescharacteristic
154#380C2312yes—
155#380C2311yescharacteristic
156#379—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C2×#95—21no—
160#341—22no—
161#341—22no—
162#340—22no—
163#340—22no—
164#328—22no—
165#328—22no—
166#328—22no—
167#328—22no—
168#327—22no—
169#327—22no—
170#327—22no—
171#327—22no—
172C25#4111yescharacteristic
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—82no—
176C2×#41C4⋊C412yes—
177C2×#41C22×C412yes—
178C22×Q8C4⋊C411yescharacteristic
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C4⋊C411yescharacteristic
183C23×C4—21no—
184C23×C4—41no—
185C23×C4—82no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×Q811yescharacteristic
195C23×C4C2×Q811yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4C2411yescharacteristic
200#131—82no—
201#131—82no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—41no—
209C2×M16—41no—
210C2×M16—82no—
211C2×M16—82no—
212C2×M16—82no—
213C2×M16C22×C412yes—
214C2×M16C22×C412yes—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—41no—
219C22×C8—41no—
220C22×C8—82no—
221C22×C8C22×C412yes—
222C22×C8C22×C412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227#123—22no—
228#123—22no—
229#123—22no—
230#123—22no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8C4⋊C412yes—
236C22⋊Q8C4⋊C412yes—
237C22⋊Q8C2×D412yes—
238C22⋊Q8C2×Q812yes—
239#121—21no—
240#121—21no—
241#121—21no—
242#121—21no—
243#121C4⋊C412yes—
244#121C4⋊C412yes—
245#121C2×D412yes—
246#121C2×Q812yes—
247#120—22no—
248#120—22no—
249#120—22no—
250#120—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259#117—21no—
260#117—21no—
261#117—82no—
262#117C4⋊C412yes—
263#117C22×C412yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4C4⋊C411yescharacteristic
272C2×C4⋊C4C4⋊C411yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—82no—
285C2×C42—21no—
286C2×C42—41no—
287#108—82no—
288#108—82no—
289#107—82no—
290#106—82no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—22no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8C22⋊C412yes—
316C22⋊C8C22⋊C412yes—
317C22⋊C8C22⋊C412yes—
318C22⋊C8C22⋊C412yes—
319C22⋊C8C22⋊C411yescharacteristic
320C22⋊C8C22⋊C411yescharacteristic
321C22⋊C8C22⋊C411yescharacteristic
322C22⋊C8C22⋊C411yescharacteristic
323C22⋊C8C22⋊C412yes—
324C22⋊C8C22⋊C412yes—
325#97—22no—
326#97—22no—
327C4×C8—22no—
328C4×C8—22no—
329#95—22no—
330#95—21no—
331#95—21no—
332#95—41no—
333#95—41no—
334C24—21no—
335C24—41no—
336C24—42no—
337C24—41no—
338C24—81no—
339C24#11711yescharacteristic
340C24#11711yescharacteristic
341C24#12111yescharacteristic
342C24#12111yescharacteristic
343C24C22⋊Q811yescharacteristic
344C24C22⋊Q811yescharacteristic
345C24C2×#4111yescharacteristic
346#41—42no—
347#41—42no—
348#41—82no—
349#41—82no—
350#41—82no—
351C2×Q8—21no—
352C2×Q8—81no—
353C2×Q8C2×C4⋊C412yes—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—44no—
362C2×D4—81no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—82no—
366C2×D4C2×C4⋊C412yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C4⋊C411yescharacteristic
398C22×C4C2×C4⋊C411yescharacteristic
399C22×C4C2×C4⋊C411yescharacteristic
400C22×C4C2×C4⋊C411yescharacteristic
401C22×C4#11711yescharacteristic
402C22×C4#11711yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4C22⋊Q811yescharacteristic
410C22×C4C22⋊Q811yescharacteristic
411C22×C4C22⋊Q811yescharacteristic
412C22×C4C22⋊Q811yescharacteristic
413C22×C4C22⋊Q811yescharacteristic
414C22×C4C22⋊Q811yescharacteristic
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421M16—42no—
422M16—42no—
423M16—42no—
424M16—42no—
425M16—42no—
426M16—42no—
427M16—82no—
428M16—82no—
429M16—82no—
430M16—82no—
431M16—82no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—82no—
444C2×C8—82no—
445C2×C8—82no—
446C2×C8—82no—
447C2×C8—82no—
448C2×C8C2×C22⋊C412yes—
449C2×C8C2×C22⋊C412yes—
450C2×C8C2×C22⋊C412yes—
451C2×C8C2×C22⋊C412yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—82no—
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—
474C22⋊C4—22no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—82no—
481C42—22no—
482C42—22no—
483C42—82no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—81no—
511C23—81no—
512C23—84no—
513C23—84no—
514C23#38011yescharacteristic
515C23#38011yescharacteristic
516C23#38011yescharacteristic
517C23#38011yescharacteristic
518C23C2×#11711yescharacteristic
519C23C2×#12111yescharacteristic
520C23C2×C22⋊Q811yescharacteristic
521Q8—42no—
522Q8—81no—
523D4—42no—
524D4—44no—
525D4—44no—
526D4—44no—
527D4—42no—
528D4—42no—
529D4—81no—
530D4—82no—
531D4—82no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—164no—
579C2×C4#38011yescharacteristic
580C2×C4#38011yescharacteristic
581C2×C4#38011yescharacteristic
582C2×C4#38011yescharacteristic
583C2×C4C22×C22⋊C411yescharacteristic
584C2×C4C22×C4⋊C411yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C8—22no—
588C8—22no—
589C8—22no—
590C8—22no—
591C8—42no—
592C8—42no—
593C8—82no—
594C8—81no—
595C8—81no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—162no—
629C22#181412yes—
630C22C2×#38011yescharacteristic
631C22#306811yescharacteristic
632C22#306911yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—161no—
644C4—161no—
645C4—162no—
646C4#288312yes—
647C2—21no—
648C2—21no—
649C2—22no—
650C2—41no—
651C2—41no—
652C2—41no—
653C2—41no—
654C2—41no—
655C2—82no—
656C2#2063311yescharacteristic
657C2C2×#181411yescharacteristic
658C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21941C2semidirect
6. #8629C2semidirect
7. #8628C2semidirect
8. #8627C2semidirect
9. #8626C2semidirect
10. #8108C2semidirect
11. C2×#1305C2semidirect
19. #2933C4semidirect
20. #2933C4semidirect
23. #2924C4semidirect
24. #2880C4semidirect
37. #1810C22semidirect
38. #1810C22semidirect
39. #1808C22semidirect
40. #1807C22semidirect
51. #1305C22semidirect
52. #1305C22semidirect
53. #1305C22semidirect
54. #1305C22semidirect
55. #1305C22semidirect
56. #1305C22semidirect
59. C2×#131C2×C4semidirect
60. C2×#131C2×C4semidirect
109. #424C2×C4semidirect
110. #424C2×C4semidirect
113. #408C2×C4semidirect
114. #408C2×C4semidirect
146. #405C2×C4semidirect
147. #405C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602332
#219482
#219412
#175922
#175592
#86292
#86282
#86272
#86262
#81082
C2×#13052