Finite Group Databasebeta

← #189327

Subgroups — #189327

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#21983C211yesmaximal, characteristic, complemented
3#17803C211yesmaximal, characteristic, complemented
4#17588C211yesmaximal, characteristic
5#17571C211yesmaximal, characteristic
6#8681C211yesmaximal, characteristic, complemented
7#8680C211yesmaximal, characteristic, complemented
8#8679C211yesmaximal, characteristic, complemented
9#8678C211yesmaximal, characteristic, complemented
10#8677C211yesmaximal, characteristic, complemented
11#8676C211yesmaximal, characteristic, complemented
12#8675C211yesmaximal, characteristic, complemented
13#8674C211yesmaximal, characteristic, complemented
14#7962C211yesmaximal, characteristic, complemented
15C2×#1286C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2994—21no—
18#2994—21no—
19#2993C412yescomplemented
20#2993C412yescomplemented
21#2991—21no—
22#2990—21no—
23#2989C412yescomplemented
24#2988C412yescomplemented
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2933—21no—
28#2933—21no—
29#2931C2211yescharacteristic
30#2931C2211yescharacteristic
31#2925—21no—
32#2920C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35#2906—21no—
36C2×#423C2211yescharacteristic
37#2283C2211yescharacteristic
38C2×#382C2211yescharacteristic
39#1949—21no—
40#1949—21no—
41#1853—21no—
42#1852—21no—
43#1846C2211yescharacteristic
44#1846C2211yescharacteristic
45#1835C2211yescharacteristic
46#1835C2211yescharacteristic
47#1572—21no—
48#1572—21no—
49#1571—21no—
50#1570—21no—
51#1564C2211yescharacteristic
52#1564C2211yescharacteristic
53#1564C2211yescharacteristic
54#1564C2211yescharacteristic
55#1518C2211yescharacteristic
56#1518C2211yescharacteristic
57#1518C2211yescharacteristic
58#1518C2211yescharacteristic
59#1517C2211yescharacteristic
60#1517C2211yescharacteristic
61#1517C2211yescharacteristic
62#1517C2211yescharacteristic
63#1516—22no—
64#1512—22no—
65#1504—22no—
66#1500—22no—
67#1286C2211yescharacteristic
68#1286C2211yescharacteristic
69#1286C2211yescharacteristic
70#1286C2211yescharacteristic
71#1286C2211yescharacteristic
72#1286C2211yescharacteristic
73#1286C2211yescharacteristic
74#1286C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77C22×M16—21no—
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#529C2×C412yes—
89#529D412yes—
90#525—21no—
91#525—21no—
92#525D412yes—
93C4×#41—21no—
94C4×#41—21no—
95C4×#41C2×C412yes—
96C22×C4⋊C4—21no—
97C22×C42C2311yescharacteristic
98#446—21no—
99#446—21no—
100#446C2×C412yescomplemented
101#446C2×C412yescomplemented
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8C2×C412yescomplemented
105C8×Q8C2×C412yescomplemented
106#436—22no—
107#436—22no—
108#436—21no—
109#436—21no—
110#436C2×C412yescomplemented
111#436C2×C412yescomplemented
112#435—22no—
113#435—22no—
114#435—22no—
115#435—22no—
116C8×D4—22no—
117C8×D4—22no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4C2×C412yescomplemented
121C8×D4C2×C412yescomplemented
122#433—22no—
123#433—22no—
124#432—22no—
125#432—21no—
126#432—21no—
127#431—22no—
128#431—21no—
129#431—21no—
130#431—21no—
131#431—21no—
132#431C2×C411yescharacteristic
133#431C2×C411yescharacteristic
134#431C2×C411yescharacteristic
135#431C2×C411yescharacteristic
136#431D411yescharacteristic
137#431D411yescharacteristic
138#431D411yescharacteristic
139#431D411yescharacteristic
140#431C2311yescharacteristic
141#431C2311yescharacteristic
142#431C2311yescharacteristic
143#431C2311yescharacteristic
144C2×#107—21no—
145C2×#107—21no—
146C2×#106—21no—
147C2×#106—21no—
148#424—21no—
149#424—21no—
150#424—22no—
151#424—22no—
152#423—21no—
153#423—21no—
154#423—22no—
155#423—21no—
156#423C2×C412yes—
157#423C2311yescharacteristic
158#423C2311yescharacteristic
159C2×C4⋊C8—21no—
160C2×C4⋊C8—22no—
161C2×C4⋊C8—21no—
162C2×C4⋊C8C2×C412yes—
163C2×C4⋊C8D411yescharacteristic
164C2×C4⋊C8D411yescharacteristic
165#408—21no—
166#408—21no—
167#407—21no—
168C2×C22⋊C8—21no—
169C2×C22⋊C8D411yescharacteristic
170C2×C22⋊C8D411yescharacteristic
171#405—21no—
172#405—21no—
173C4×M16—21no—
174C2×C4×C8—21no—
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177#383—21no—
178#382—21no—
179#382—21no—
180#382—21no—
181#382C2311yescharacteristic
182#382C2311yescharacteristic
183#382C2311yescharacteristic
184#382C2311yescharacteristic
185#381—21no—
186#380—21no—
187C4×C4⋊C4—21no—
188C4×C22⋊C4—21no—
189#376—21no—
190C2×#95—21no—
191C2×#95—21no—
192#340—21no—
193#340—21no—
194#340—21no—
195#340—21no—
196#335—21no—
197#335—21no—
198#335—21no—
199#335—21no—
200#334—21no—
201#334—21no—
202#334—21no—
203#334—21no—
204#328—22no—
205#328—22no—
206#328—22no—
207#328—22no—
208#327—22no—
209#327—22no—
210#327—22no—
211#327—22no—
212C2×#41—21no—
213C2×#41—21no—
214C2×#41C4⋊C412yes—
215C23×C4—41no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4#4111yescharacteristic
219#131—81no—
220#131—81no—
221#131—81no—
222#131—81no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—41no—
229C2×M16—81no—
230C2×M16—81no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—41no—
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C4⋊Q8C2×Q811yescharacteristic
240C4⋊Q8C2×Q811yescharacteristic
241C4⋊Q8C2×Q811yescharacteristic
242#127C2×Q811yescharacteristic
243#126—22no—
244#126—22no—
245#125—21no—
246#125—21no—
247#125C4⋊C412yes—
248#124—21no—
249#124—21no—
250#124C4⋊C412yes—
251#123—22no—
252#123—22no—
253#123—22no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8C4⋊C412yes—
259C22⋊Q8C2×D412yes—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264#121C4⋊C412yes—
265#121C2×D412yes—
266#120—22no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8C4⋊C412yes—
272C4×Q8C22×C412yes—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C4⋊C412yes—
284C4×D4C22×C412yes—
285#117—21no—
286#117—21no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117C4⋊C412yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—81no—
297C2×C4⋊C4—81no—
298C2×C22⋊C4—41no—
299C2×C42—22no—
300C2×C42—41no—
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×Q811yescharacteristic
308C2×C42C2×Q811yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42#4111yescharacteristic
312C2×C42C2411yescharacteristic
313#107—81no—
314#107—81no—
315#106—81no—
316#106—81no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—41no—
330C4⋊C8—41no—
331C4⋊C8C22⋊C411yescharacteristic
332C4⋊C8C22⋊C411yescharacteristic
333C4⋊C8C22⋊C411yescharacteristic
334C4⋊C8C22⋊C411yescharacteristic
335C4⋊C8C22⋊C411yescharacteristic
336C4⋊C8C22⋊C411yescharacteristic
337C4⋊C8C22⋊C411yescharacteristic
338C4⋊C8C22⋊C411yescharacteristic
339C4⋊C8C22×C412yes—
340C4⋊C8C22×C412yes—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—21no—
349C22⋊C8—41no—
350C22⋊C8—41no—
351C22⋊C8C22⋊C411yescharacteristic
352C22⋊C8C22⋊C411yescharacteristic
353C22⋊C8C22⋊C411yescharacteristic
354C22⋊C8C22⋊C411yescharacteristic
355C22⋊C8C22⋊C411yescharacteristic
356C22⋊C8C22⋊C411yescharacteristic
357C22⋊C8C22⋊C411yescharacteristic
358C22⋊C8C22⋊C411yescharacteristic
359#97—22no—
360#97—22no—
361#97—41no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—41no—
369C4×C8C22×C411yescharacteristic
370C4×C8C22×C411yescharacteristic
371C4×C8C22×C411yescharacteristic
372C4×C8C22×C411yescharacteristic
373C4×C8C2×D411yescharacteristic
374C4×C8C2×D411yescharacteristic
375C4×C8C2×D411yescharacteristic
376C4×C8C2×D411yescharacteristic
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—41no—
386#95—41no—
387#95—41no—
388#95—41no—
389C24#12111yescharacteristic
390#41—44no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—82no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C2×C4⋊C412yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4C2×C4⋊C412yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—81no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C4⋊C411yescharacteristic
424C22×C4C2×C4⋊C411yescharacteristic
425C22×C4#11711yescharacteristic
426C22×C4#11711yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4C22⋊Q811yescharacteristic
433C22×C4C22⋊Q811yescharacteristic
434C22×C4C22⋊Q811yescharacteristic
435C22×C4C22⋊Q811yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4111yescharacteristic
439M16—41no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—82no—
444M16—82no—
445M16—82no—
446M16—82no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—82no—
462C2×C8—82no—
463C2×C8—82no—
464C2×C8—82no—
465C2×C8C2×C22⋊C411yescharacteristic
466C2×C8C2×C22⋊C411yescharacteristic
467C2×C8C2×C22⋊C411yescharacteristic
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—82no—
492C4⋊C4C2×C4⋊C412yes—
493C4⋊C4C2×C4⋊C412yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—82no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—82no—
508C42C2×C22⋊C411yescharacteristic
509C42C2×C22⋊C411yescharacteristic
510C42C2×C4⋊C411yescharacteristic
511C42C2×C4⋊C411yescharacteristic
512C42#11711yescharacteristic
513C42#11711yescharacteristic
514C42#12111yescharacteristic
515C42#12111yescharacteristic
516C42C22⋊Q811yescharacteristic
517C42C22⋊Q811yescharacteristic
518C42C22⋊Q811yescharacteristic
519C42C22⋊Q811yescharacteristic
520C42C23×C411yescharacteristic
521C42C22×D411yescharacteristic
522C42C22×Q811yescharacteristic
523C42C2×#4111yescharacteristic
524C23—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23#38011yescharacteristic
530C23#38011yescharacteristic
531C23C2×#12111yescharacteristic
532Q8—42no—
533Q8—42no—
534Q8—41no—
535Q8—41no—
536D4—42no—
537D4—42no—
538D4—44no—
539D4—42no—
540D4—42no—
541D4—41no—
542D4—41no—
543D4—82no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—164no—
586C2×C4#38011yescharacteristic
587C2×C4#38011yescharacteristic
588C2×C4#38011yescharacteristic
589C2×C4#38011yescharacteristic
590C2×C4#38011yescharacteristic
591C2×C4#38011yescharacteristic
592C2×C4C22×C22⋊C411yescharacteristic
593C2×C4C22×C4⋊C411yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4#57611yescharacteristic
599C2×C4#57711yescharacteristic
600C2×C4#57711yescharacteristic
601C2×C4#57811yescharacteristic
602C8—21no—
603C8—21no—
604C8—21no—
605C8—21no—
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—42no—
611C8—42no—
612C8—81no—
613C8—81no—
614C8—81no—
615C8—81no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22#181212yes—
629C22C2×#38011yescharacteristic
630C22#304511yescharacteristic
631C22#304611yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—161no—
646C4—161no—
647C4—161no—
648C4—161no—
649C4#289711yescharacteristic
650C4#289811yescharacteristic
651C4#308311yescharacteristic
652C4#308411yescharacteristic
653C2—21no—
654C2—21no—
655C2—22no—
656C2—41no—
657C2—82no—
658C2#2065211yescharacteristic
659C2C2×#181211yescharacteristic
660C2#2067511yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ 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. #21983C2semidirect
3. #17803C2semidirect
6. #8681C2semidirect
7. #8680C2semidirect
8. #8679C2semidirect
9. #8678C2semidirect
10. #8677C2semidirect
11. #8676C2semidirect
12. #8675C2semidirect
13. #8674C2semidirect
14. #7962C2semidirect
15. C2×#1286C2semidirect
19. #2993C4semidirect
20. #2993C4semidirect
23. #2989C4semidirect
24. #2988C4semidirect
100. #446C2×C4semidirect
101. #446C2×C4semidirect
104. C8×Q8C2×C4semidirect
105. C8×Q8C2×C4semidirect
110. #436C2×C4semidirect
111. #436C2×C4semidirect
120. C8×D4C2×C4semidirect
121. C8×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602682
#219832
#178032
#175882
#175712
#86812
#86802
#86792
#86782
#86772
#86762
#86752
#86742
#79622
C2×#12862