Finite Group Databasebeta

← #6526554

Subgroups — #6526554

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#34978C211yesmaximal, characteristic
3#29357C212yesmaximal, complemented
4#29347C212yesmaximal, complemented
5#29224C212yesmaximal, complemented
6#29222C212yesmaximal, complemented
7#27823C212yesmaximal, complemented
8#27816C212yesmaximal, complemented
9#27737C212yesmaximal, complemented
10#27730C212yesmaximal, complemented
11#27723C214yesmaximal
12#27705C214yesmaximal
13#21894C211yesmaximal, characteristic
14#21834C211yesmaximal, characteristic
15#21389C211yesmaximal, characteristic
16#21313C211yesmaximal, characteristic
17#20550C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2862C2212yes—
23#2862C2212yes—
24#2642—22no—
25#2612—22no—
26#2571C2211yescharacteristic
27#2568—21no—
28#2568—21no—
29#2568C2212yes—
30#2568C2212yes—
31#2552C2211yescharacteristic
32#2552C2211yescharacteristic
33#2551C2211yescharacteristic
34#2503—22no—
35#2503C2214yes—
36#2441—21no—
37#2441C2212yes—
38#2439—21no—
39#2439C2212yes—
40#2396—22no—
41#2310—21no—
42#2310—21no—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2212yes—
45C4×#125—21no—
46C4×#125C2212yes—
47#2285C2211yescharacteristic
48C4×#117C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2214yes—
53#2101C2214yes—
54#2074C2214yes—
55#2072C2214yes—
56#2026C2214yes—
57#2024C2214yes—
58#1993C2214yes—
59#1991C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1988C2211yescharacteristic
63#1988C2212yes—
64#1988C2211yescharacteristic
65#1988C2212yes—
66#1988C2212yes—
67#1987C2212yes—
68#1987C2212yes—
69#1987C2212yes—
70#1987C2212yes—
71#1981C2212yes—
72#1981C2211yescharacteristic
73#1981C2211yescharacteristic
74#1935C2214yes—
75#1935C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1894C2214yes—
79#1873C2212yes—
80#1873C2212yes—
81#1836C2214yes—
82#1835C2214yes—
83#1824C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1770C2212yes—
87C8×C4⋊C4C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92#1753C2214yes—
93#1736C2212yes—
94#1736C2212yes—
95C23×Q8—21no—
96C22×M16C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125—21no—
110C2×#125—21no—
111C2×#125C2312yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117D411yescharacteristic
120C2×#117D411yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126#431—24no—
127#423—24no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#400—24no—
143#400—24no—
144#400—24no—
145#400—24no—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#396—22no—
152#395—22no—
153#393—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—21no—
187#382—22no—
188#382—22no—
189#382—21no—
190#382C2312yes—
191#382C2314yes—
192#382C2314yes—
193#381—24no—
194#381—24no—
195#380—21no—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214#376—21no—
215#376D412yes—
216C43—21no—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2412yes—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×M16—24no—
247C2×M16—42no—
248C2×M16—42no—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#125—24no—
266#125—42no—
267#125—44no—
268#125—44no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—84no—
276#117—22no—
277#117—22no—
278#117—42no—
279#117—42no—
280#117—41no—
281#117—41no—
282#117—44no—
283#117—82no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42#4114yes—
341C2×C42#4114yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C4⋊C8—24no—
347C4⋊C8—24no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350#103—24no—
351#103—24no—
352#103—42no—
353#103—42no—
354#103—44no—
355#103—44no—
356C22⋊C8—24no—
357#97—44no—
358C4×C8—24no—
359C4×C8—44no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—82no—
403C2×Q8—816no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4#12414yes—
425C22×C4#12414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2511yescharacteristic
438M16—84no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C42−1+412yes—
488C4⋊C42−1+412yes—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—44no—
509C42—42no—
510C42—42no—
511C42#12414yes—
512C42#12414yes—
513C42C2×#4112yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C2×#12412yes—
524C23C23×D411yescharacteristic
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—88no—
530Q8—88no—
531Q8—84no—
532Q8—88no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—88no—
585C2×C4C2×#12412yes—
586C2×C4C2×#12412yes—
587C2×C4C2×#12412yes—
588C2×C4C2×#12412yes—
589C2×C4C2×#12412yes—
590C2×C4#57512yes—
591C2×C4#57512yes—
592C2×C4#57712yes—
593C2×C4#57812yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×2−1+411yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C2×C4#58511yescharacteristic
599C2×C4#58511yescharacteristic
600C8—42no—
601C8—42no—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22C22×#12411yescharacteristic
611C22#348911yescharacteristic
612C22#348911yescharacteristic
613C22C2×#57511yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57711yescharacteristic
616C22C2×#57811yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—84no—
633C4#313714yes—
634C4#314014yes—
635C4#350711yescharacteristic
636C4#350711yescharacteristic
637C4#350911yescharacteristic
638C4#351011yescharacteristic
639C2—42no—
640C2#3562811yescharacteristic
641C2C2×#313711yescharacteristic
642C2C2×#314011yescharacteristic
643C2#6213311yescharacteristic
644C2#6213511yescharacteristic
645C2#6214411yescharacteristic
646C2#6214611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29357C2semidirect
4. #29347C2semidirect
5. #29224C2semidirect
6. #29222C2semidirect
7. #27823C2semidirect
8. #27816C2semidirect
9. #27737C2semidirect
10. #27730C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#349782
#293572
#293472
#292242
#292222
#278232
#278162
#277372
#277302
#277232
#277052
#218942
#218342
#213892
#213132
#205502