Finite Group Databasebeta

← #222094

Subgroups — #222094

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36151C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3#22648C211yesmaximal, characteristic
4C2×#1561C211yesmaximal, characteristic, complemented
5C2×#1550C211yesmaximal, characteristic, complemented
6#9701C211yesmaximal, characteristic, complemented
7#9697C211yesmaximal, characteristic, complemented
8#9139C211yesmaximal, characteristic, complemented
9#9135C211yesmaximal, characteristic, complemented
10#9135C211yesmaximal, characteristic, complemented
11#9129C211yesmaximal, characteristic, complemented
12#8586C211yesmaximal, characteristic, complemented
13#8345C211yesmaximal, characteristic, complemented
14#8343C211yesmaximal, characteristic, complemented
15#8236C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#436—21no—
19C2×#435—21no—
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2862C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2536—21no—
28#2489—21no—
29#2489C412yes—
30#2453—21no—
31#2453C412yes—
32#2421—21no—
33#2421—21no—
34#2415—21no—
35#2415C412yes—
36#2362—21no—
37#2362C412yes—
38#2351—21no—
39C4×C4⋊Q8C2211yescharacteristic
40C42×D4C2211yescharacteristic
41#1940—21no—
42#1939—21no—
43#1851—21no—
44#1851—21no—
45#1851—21no—
46#1850—21no—
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55#1561—24no—
56#1550—24no—
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#332C2211yescharacteristic
67C2×#332C2211yescharacteristic
68C2×#332C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×M16—21no—
74C22×M16—21no—
75C22×M16—21no—
76C23×C8—21no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533—22no—
81#533—41no—
82#533—41no—
83#533—41no—
84#533—41no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—21no—
89C2×#125C2×C412yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2×C412yes—
95C2×C22⋊Q8C2×C412yes—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8C2×C412yes—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×#117—21no—
105C2×#117C2×C412yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#436—44no—
109#435—44no—
110#423—22no—
111#423—22no—
112#423—22no—
113#423—41no—
114#423—41no—
115#423—41no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—41no—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×#97—21no—
139C2×C4×C8—21no—
140#400—22no—
141#400—22no—
142#398—22no—
143#397—22no—
144#397—22no—
145#394—22no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391C2×C412yes—
151#391C2×C412yes—
152#391C2×C412yes—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389C2×C412yes—
160#389C2×C412yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#384C2311yescharacteristic
168#382—21no—
169#382—21no—
170#382—22no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380C2×C412yes—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2×C412yes—
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C43C2311yescharacteristic
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#333—22no—
195#333—22no—
196#333—41no—
197#333—41no—
198#332—22no—
199#332—22no—
200#332—22no—
201#332—22no—
202#332—22no—
203#332—41no—
204#332—41no—
205#332D414yescomplemented
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8C22×C412yes—
209C22×D4C22⋊C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C2×M16—42no—
217C2×M16—42no—
218C2×M16—42no—
219C2×M16—42no—
220C2×M16—42no—
221C2×M16—42no—
222C2×M16—81no—
223C2×M16—81no—
224C2×M16—81no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—81no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8C2×D412yes—
247C4⋊Q8#4112yes—
248#125—24no—
249#125—42no—
250#125—42no—
251#125—42no—
252C22⋊Q8—24no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255C22⋊Q8—42no—
256C22⋊Q8—42no—
257C22⋊Q8—42no—
258C22⋊Q8—42no—
259C22⋊Q8—42no—
260C22⋊Q8—42no—
261C22⋊Q8—42no—
262C22⋊Q8—44no—
263C4×Q8—24no—
264C4×Q8—42no—
265C4×Q8—42no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×Q8—42no—
272C4×Q8—42no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—42no—
278#117—24no—
279#117—42no—
280#117—42no—
281#117—42no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C22⋊C411yescharacteristic
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22⋊C411yescharacteristic
298C2×C4⋊C4C22×C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4C22⋊C411yescharacteristic
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2411yescharacteristic
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—42no—
350C4⋊C8—42no—
351C4⋊C8—42no—
352C4⋊C8—42no—
353C4⋊C8—42no—
354C4⋊C8—42no—
355C4⋊C8—81no—
356C4⋊C8—81no—
357C4⋊C8C2×D412yes—
358C4⋊C8C2×D412yes—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363#97—42no—
364C4×C8—42no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381C24C2×C22⋊C411yescharacteristic
382C24C2×C22⋊C411yescharacteristic
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—41no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4AGL(1,ℤ/8)12yes—
447C22×C4#13712yes—
448C22×C4C23×C411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451M16—84no—
452M16—84no—
453M16—84no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—84no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4C4×D414yes—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—41no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—41no—
535C42—41no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42AGL(1,ℤ/8)12yes—
540C42#13712yes—
541C42C22×D411yescharacteristic
542C42C2×#4111yescharacteristic
543C42H(5,𝔽2)11yescharacteristic
544C42H(5,𝔽2)11yescharacteristic
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23#41712yes—
559C23C22×C22⋊C411yescharacteristic
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—82no—
565Q8—82no—
566Q8—81no—
567Q8—81no—
568Q8—81no—
569Q8—81no—
570D4—44no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4#41312yes—
632C2×C4#41712yes—
633C2×C4#41712yes—
634C2×C4#41712yes—
635C2×C4C2×C4×D411yescharacteristic
636C2×C4#51811yescharacteristic
637C2×C4D4211yescharacteristic
638C2×C4D4211yescharacteristic
639C2×C4#54611yescharacteristic
640C2×C4#54611yescharacteristic
641C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
642C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
643C2×C4C2×#13711yescharacteristic
644C2×C4C2×#13711yescharacteristic
645C2×C4#57711yescharacteristic
646C2×C4#57711yescharacteristic
647C2×C4#57811yescharacteristic
648C2×C4#57811yescharacteristic
649C8—42no—
650C8—42no—
651C8—81no—
652C8—81no—
653C8—81no—
654C8—81no—
655C8—81no—
656C8—81no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22D4×C22⋊C411yescharacteristic
672C22C2×#41311yescharacteristic
673C22#288911yescharacteristic
674C22C2×#41711yescharacteristic
675C22C2×#41711yescharacteristic
676C22#289811yescharacteristic
677C22#289811yescharacteristic
678C4—22no—
679C4—22no—
680C4—21no—
681C4—21no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#326511yescharacteristic
695C4#327011yescharacteristic
696C4#327211yescharacteristic
697C4#327611yescharacteristic
698C4#330811yescharacteristic
699C4#331611yescharacteristic
700C4#333011yescharacteristic
701C4#333111yescharacteristic
702C2—42no—
703C2—81no—
704C2#1775211yescharacteristic
705C2#1776311yescharacteristic
706C2#1776311yescharacteristic
707C2#1777811yescharacteristic
708C2#2241011yescharacteristic
709C2#2242111yescharacteristic
710C2#2243111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1561C2semidirect
5. C2×#1550C2semidirect
6. #9701C2semidirect
7. #9697C2semidirect
8. #9139C2semidirect
9. #9135C2semidirect
10. #9135C2semidirect
11. #9129C2semidirect
12. #8586C2semidirect
13. #8345C2semidirect
14. #8343C2semidirect
15. #8236C2semidirect
205. #332D4semidirect

Maximal subgroups

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

subgroupindex
#361512
#226502
#226482
C2×#15612
C2×#15502
#97012
#96972
#91392
#91352
#91292
#85862
#83452
#83432
#82362