Finite Group Databasebeta

← #188970

Subgroups — #188970

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60265C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic, complemented
3#21962C211yesmaximal, characteristic, complemented
4#17585C211yesmaximal, characteristic
5#17558C211yesmaximal, characteristic
6#8629C212yesmaximal, complemented
7#8628C212yesmaximal, complemented
8#8627C212yesmaximal, complemented
9#8626C212yesmaximal, complemented
10#8110C211yesmaximal, characteristic, complemented
11#8105C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2994—22no—
14#2992—22no—
15#2990—21no—
16#2990—21no—
17#2989—21no—
18#2988—21no—
19#2962C412yescomplemented
20#2935C412yescomplemented
21#2929C2212yes—
22#2929C2212yes—
23#2925C411yescharacteristic, complemented
24#2925C411yescharacteristic, complemented
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28#2906C411yescharacteristic, complemented
29#2906C411yescharacteristic, complemented
30C22×C4⋊C8C2211yescharacteristic
31#2285C2211yescharacteristic
32#2283C2211yescharacteristic
33#1950—21no—
34#1950—21no—
35#1950—22no—
36#1949—22no—
37#1948—21no—
38#1947—21no—
39#1810C2212yes—
40#1810C2212yes—
41#1808C2212yes—
42#1807C2212yes—
43#1518—21no—
44#1517—21no—
45#1515—21no—
46#1514—21no—
47#1513C2211yescharacteristic
48#1513C2211yescharacteristic
49#1513C2211yescharacteristic
50#1513C2211yescharacteristic
51#1513C2212yes—
52#1513C2212yes—
53#1509—21no—
54#1509—21no—
55#1500—21no—
56#1500—21no—
57#1305C2212yes—
58#1305C2212yes—
59#1305C2212yes—
60#1305C2212yes—
61C2×#131C2×C412yes—
62C2×#131C2×C412yes—
63C23×C8C2312yes—
64#533C2×C411yescharacteristic
65#533D411yescharacteristic
66#533Q811yescharacteristic
67#532C2×C411yescharacteristic
68#532D411yescharacteristic
69#532Q811yescharacteristic
70#529—21no—
71#529—21no—
72#529—21no—
73#529—21no—
74#529D411yescharacteristic
75#529D411yescharacteristic
76#529Q811yescharacteristic
77#529Q811yescharacteristic
78#525—21no—
79#525—21no—
80#525C2311yescharacteristic
81#525C2311yescharacteristic
82C4×#41—21no—
83C4×#41—21no—
84C4×#41C2×C411yescharacteristic
85C4×#41C2×C411yescharacteristic
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88#446—22no—
89#446—22no—
90#446—22no—
91C8×Q8—22no—
92#436—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4—22no—
104#433—22no—
105#433—22no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122C2×#108—22no—
123C2×#108—22no—
124C2×#107—22no—
125C2×#106—22no—
126#424—22no—
127#424—22no—
128#424C2×C412yes—
129#424C2×C412yes—
130#423—21no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423C2×C411yescharacteristic
135#423C2×C411yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×C411yescharacteristic
139C2×C4⋊C8C2×C411yescharacteristic
140C2×C4⋊C8C2×C412yes—
141C2×C4⋊C8D411yescharacteristic
142C2×C4⋊C8D411yescharacteristic
143C2×C4⋊C8D412yes—
144C2×C4⋊C8D411yescharacteristic
145C2×C4⋊C8D411yescharacteristic
146C2×C4⋊C8D412yes—
147#408C2×C412yes—
148#408C2×C412yes—
149#407C2311yescharacteristic
150#407C2312yes—
151#407C2311yescharacteristic
152#407C2312yes—
153#405C2×C412yes—
154#405C2×C412yes—
155C4×M16—21no—
156C4×M16—21no—
157C4×M16C2×C412yes—
158#384—21no—
159#383—21no—
160#382—21no—
161#382—21no—
162#381—21no—
163#380—21no—
164#380C2311yescharacteristic
165#380C2312yes—
166#380C2311yescharacteristic
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C22⋊C4—21no—
171#376—21no—
172#341—22no—
173#341—22no—
174#340—22no—
175#340—22no—
176#328—21no—
177#328—21no—
178#328—21no—
179#328—21no—
180#328—22no—
181#328—22no—
182#327—21no—
183#327—21no—
184#327—21no—
185#327—21no—
186#327—22no—
187#327—22no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41C22×C411yescharacteristic
191C2×#41C22×C411yescharacteristic
192C23×C4#4111yescharacteristic
193C23×C4#4111yescharacteristic
194C23×C4C2411yescharacteristic
195#131—82no—
196#131—82no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—21no—
200C2×M16—22no—
201C2×M16—21no—
202C2×M16C22×C412yes—
203C2×M16C22×C412yes—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—82no—
207C22×C8C22×C412yes—
208C22×C8C22×C412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C4⋊Q8C4⋊C411yescharacteristic
214C4⋊Q8C4⋊C411yescharacteristic
215C4⋊Q8C4⋊C411yescharacteristic
216#127C4⋊C411yescharacteristic
217#126—21no—
218#126—21no—
219#126—21no—
220#126—21no—
221#125—21no—
222#125—21no—
223#125C4⋊C411yescharacteristic
224#125C4⋊C411yescharacteristic
225#124—21no—
226#124—21no—
227#124C4⋊C411yescharacteristic
228#124C4⋊C411yescharacteristic
229#123—21no—
230#123—21no—
231#123—21no—
232#123—21no—
233#123—21no—
234#123—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8C2×D411yescharacteristic
240C22⋊Q8C2×D411yescharacteristic
241C22⋊Q8C2×Q811yescharacteristic
242C22⋊Q8C2×Q811yescharacteristic
243#121—21no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121C2×D411yescharacteristic
248#121C2×D411yescharacteristic
249#121C2×Q811yescharacteristic
250#121C2×Q811yescharacteristic
251#120—21no—
252#120—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8C4⋊C411yescharacteristic
258C4×Q8C4⋊C411yescharacteristic
259C4×Q8C4⋊C411yescharacteristic
260C4×Q8C4⋊C411yescharacteristic
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4C4⋊C411yescharacteristic
278C4×D4C4⋊C411yescharacteristic
279C4×D4C4⋊C411yescharacteristic
280C4×D4C4⋊C411yescharacteristic
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—82no—
290#117C22×C411yescharacteristic
291#117C22×C411yescharacteristic
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—41no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×Q811yescharacteristic
313C2×C42C2×Q811yescharacteristic
314C2×C42#4111yescharacteristic
315C2×C42#4111yescharacteristic
316#108—82no—
317#108—82no—
318#107—82no—
319#106—82no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—41no—
329C4⋊C8—41no—
330C4⋊C8—41no—
331C4⋊C8—41no—
332C4⋊C8C22⋊C411yescharacteristic
333C4⋊C8C22⋊C411yescharacteristic
334C4⋊C8C22⋊C411yescharacteristic
335C4⋊C8C22⋊C411yescharacteristic
336C4⋊C8C22⋊C412yes—
337C4⋊C8C22⋊C412yes—
338C4⋊C8C22⋊C412yes—
339C4⋊C8C22⋊C412yes—
340C4⋊C8C22⋊C412yes—
341C4⋊C8C22⋊C412yes—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—22no—
351C22⋊C8—21no—
352C22⋊C8—22no—
353C22⋊C8—21no—
354#97—22no—
355#97—22no—
356#97—22no—
357#97—41no—
358#97—41no—
359C4×C8—22no—
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—41no—
363C4×C8—41no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—41no—
368#95—41no—
369#95—41no—
370#95—41no—
371C24C2×#4111yescharacteristic
372#41—44no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—81no—
378#41—81no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8C2×C4⋊C411yescharacteristic
382C2×Q8C2×C4⋊C411yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4C2×C4⋊C411yescharacteristic
392C2×D4C2×C4⋊C411yescharacteristic
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4#11711yescharacteristic
408C22×C4#11711yescharacteristic
409C22×C4#11711yescharacteristic
410C22×C4#11711yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4C22⋊Q811yescharacteristic
416C22×C4C22⋊Q811yescharacteristic
417C22×C4C22⋊Q811yescharacteristic
418C22×C4C22⋊Q811yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q811yescharacteristic
424C22×C4C2×#4111yescharacteristic
425M16—42no—
426M16—42no—
427M16—41no—
428M16—42no—
429M16—41no—
430M16—41no—
431M16—42no—
432M16—41no—
433M16—82no—
434M16—82no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—82no—
447C2×C8—82no—
448C2×C8C2×C22⋊C412yes—
449C2×C8C2×C22⋊C412yes—
450C2×C8C2×C22⋊C412yes—
451C2×C8C2×C22⋊C412yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4C2×C4⋊C411yescharacteristic
477C4⋊C4C2×C4⋊C411yescharacteristic
478C4⋊C4C2×C4⋊C411yescharacteristic
479C4⋊C4C2×C4⋊C411yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—82no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—82no—
508C42—82no—
509C42C2×C22⋊C411yescharacteristic
510C42C2×C22⋊C411yescharacteristic
511C42C2×C22⋊C411yescharacteristic
512C42C2×C22⋊C411yescharacteristic
513C42C2×C4⋊C411yescharacteristic
514C42C2×C4⋊C411yescharacteristic
515C42C2×C4⋊C411yescharacteristic
516C42C2×C4⋊C411yescharacteristic
517C42#12111yescharacteristic
518C42#12111yescharacteristic
519C42#12111yescharacteristic
520C42#12111yescharacteristic
521C42C22⋊Q811yescharacteristic
522C42C22⋊Q811yescharacteristic
523C42C22⋊Q811yescharacteristic
524C42C22⋊Q811yescharacteristic
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—41no—
530C23—41no—
531C23C2×#11711yescharacteristic
532C23C2×#12111yescharacteristic
533C23C2×C22⋊Q811yescharacteristic
534Q8—41no—
535Q8—41no—
536Q8—42no—
537Q8—81no—
538D4—41no—
539D4—41no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—81no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4#38011yescharacteristic
597C2×C4#38011yescharacteristic
598C2×C4#38011yescharacteristic
599C2×C4#38011yescharacteristic
600C2×C4#38011yescharacteristic
601C2×C4#38011yescharacteristic
602C2×C4#38011yescharacteristic
603C2×C4#38011yescharacteristic
604C2×C4C22×C22⋊C411yescharacteristic
605C2×C4C22×C4⋊C411yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
608C8—22no—
609C8—22no—
610C8—22no—
611C8—22no—
612C8—42no—
613C8—42no—
614C8—81no—
615C8—81no—
616C8—82no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—42no—
626C22—42no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22C2×#38011yescharacteristic
634C22#306811yescharacteristic
635C22#306911yescharacteristic
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—161no—
653C4—162no—
654C4—161no—
655C4#181412yes—
656C4#288311yescharacteristic
657C4#288411yescharacteristic
658C2—21no—
659C2—21no—
660C2—41no—
661C2—41no—
662C2—81no—
663C2—81no—
664C2#2063411yescharacteristic
665C2C2×#181411yescharacteristic
666C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21969C2semidirect
3. #21962C2semidirect
6. #8629C2semidirect
7. #8628C2semidirect
8. #8627C2semidirect
9. #8626C2semidirect
10. #8110C2semidirect
11. #8105C2semidirect
19. #2962C4semidirect
20. #2935C4semidirect
23. #2925C4semidirect
24. #2925C4semidirect
28. #2906C4semidirect
29. #2906C4semidirect

Maximal subgroups

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

subgroupindex
#602652
#219692
#219622
#175852
#175582
#86292
#86282
#86272
#86262
#81102
#81052