Finite Group Databasebeta

← #222089

Subgroups — #222089

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#22751C211yesmaximal, characteristic
3#22648C211yesmaximal, characteristic
4C2×#1559C211yesmaximal, characteristic, complemented
5C2×#1550C211yesmaximal, characteristic, complemented
6#9705C211yesmaximal, characteristic, complemented
7#9692C211yesmaximal, characteristic, complemented
8#9139C211yesmaximal, characteristic, complemented
9#9138C211yesmaximal, characteristic, complemented
10#9137C211yesmaximal, characteristic, complemented
11#9131C211yesmaximal, characteristic, complemented
12#8590C211yesmaximal, characteristic, complemented
13#8343C211yesmaximal, characteristic, complemented
14#8343C211yesmaximal, characteristic, complemented
15#8235C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#435—21no—
19C2×#432C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2868C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2552—21no—
27#2542—21no—
28#2538C412yes—
29#2441—21no—
30#2441—21no—
31#2429—21no—
32#2429—21no—
33#2423C412yes—
34#2423C412yes—
35#2365—21no—
36#2354—21no—
37Q8×C4⋊C4C412yes—
38C4×C4⋊Q8C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1940—21no—
41#1923—21no—
42#1917—21no—
43#1870—21no—
44#1851—21no—
45#1850—21no—
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55#1559—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—
75C23×C8—21no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533—41no—
81#533—41no—
82#533—41no—
83#533—41no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#125—21no—
90C2×#125—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2×C412yes—
96C2×C22⋊Q8C2×C412yes—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8C2×C412yes—
100C2×C4×Q8C2×C412yes—
101C2×C4×Q8C2×C412yes—
102C2×C4×D4C2×C411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×C4×D4D411yescharacteristic
105C2×#117—21no—
106C2×#117—21no—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#435—44no—
110#432—22no—
111#432—41no—
112#423—22no—
113#423—22no—
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—41no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×#97—21no—
137C2×#97—21no—
138C2×C4×C8—21no—
139#399—22no—
140#399—22no—
141#398—22no—
142#398—22no—
143#396—22no—
144#396—22no—
145#391—21no—
146#391—21no—
147#391C2×C412yes—
148#390—22no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389C2×C412yes—
157#389C2×C412yes—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#384—22no—
163#384—21no—
164#384—21no—
165#384C2×C412yes—
166#384C2×C412yes—
167#384C2311yescharacteristic
168#380—21no—
169#380—21no—
170#380C2×C412yes—
171#380C2×C412yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C22⋊C4C2×C411yescharacteristic
179C4×C22⋊C4C2×C411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C43C2311yescharacteristic
185#336—21no—
186#336—21no—
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#333—22no—
194#333—22no—
195#333—22no—
196#333D414yescomplemented
197#332—22no—
198#332—22no—
199#332—22no—
200#332—22no—
201#332—41no—
202#332—41no—
203#332—41no—
204#332—41no—
205C22×Q8—21no—
206C22×Q8—21no—
207C22×Q8C22×C412yes—
208C22×D4C22⋊C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C2×M16—42no—
216C2×M16—42no—
217C2×M16—42no—
218C2×M16—42no—
219C2×M16—81no—
220C2×M16—81no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8—81no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—42no—
243C4⋊Q8—42no—
244C4⋊Q8—81no—
245C4⋊Q8—81no—
246C4⋊Q8C2×D412yes—
247C4⋊Q8#4112yes—
248#125—42no—
249#125—42no—
250#125—42no—
251#125—42no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—42no—
255C22⋊Q8—42no—
256C22⋊Q8—42no—
257C22⋊Q8—42no—
258C22⋊Q8—42no—
259C22⋊Q8—42no—
260C22⋊Q8—81no—
261C22⋊Q8—81no—
262C22⋊Q8—81no—
263C22⋊Q8—81no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×Q8—42no—
272C4×Q8—44no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—42no—
278#117—42no—
279#117—42no—
280#117—42no—
281#117—42no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
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—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C4⋊C4C22⋊C411yescharacteristic
301C2×C4⋊C4C22⋊C411yescharacteristic
302C2×C4⋊C4C22×C412yes—
303C2×C4⋊C4C22×C412yes—
304C2×C4⋊C4C22×C412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C22⋊C4C22⋊C411yescharacteristic
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
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—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—42no—
354C4⋊C8—42no—
355C4⋊C8—42no—
356C4⋊C8—42no—
357C4⋊C8—81no—
358C4⋊C8C2×D412yes—
359C4⋊C8C2×D412yes—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—81no—
365#97—42no—
366#97—42no—
367C4×C8—81no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C22⋊C411yescharacteristic
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380C24C2×C22⋊C411yescharacteristic
381C24C2×C22⋊C411yescharacteristic
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
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×Q8C4×D414yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—41no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
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—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4C2×C22⋊C411yescharacteristic
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×C4#13712yes—
446C22×C4#13712yes—
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450M16—84no—
451M16—84no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
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—84no—
477C4⋊C4—22no—
478C4⋊C4—22no—
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—41no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
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—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—82no—
517C22⋊C4—82no—
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—21no—
535C42—21no—
536C42—22no—
537C42—41no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42AGL(1,ℤ/8)12yes—
542C42#13712yes—
543C42C22×D411yescharacteristic
544C42C2×#4111yescharacteristic
545C42H(5,𝔽2)11yescharacteristic
546C42H(5,𝔽2)11yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23#41912yes—
561C23C22×C22⋊C411yescharacteristic
562Q8—44no—
563Q8—44no—
564Q8—81no—
565Q8—81no—
566Q8—81no—
567Q8—81no—
568Q8—81no—
569Q8—81no—
570Q8—81no—
571Q8—81no—
572D4—44no—
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—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—24no—
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—21no—
605C2×C4—21no—
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—81no—
632C2×C4—81no—
633C2×C4#41312yes—
634C2×C4#41712yes—
635C2×C4#41712yes—
636C2×C4#41912yes—
637C2×C4C2×C4×D411yescharacteristic
638C2×C4#51811yescharacteristic
639C2×C4D4211yescharacteristic
640C2×C4D4211yescharacteristic
641C2×C4#54611yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
644C2×C4C2×#13711yescharacteristic
645C2×C4C2×#13711yescharacteristic
646C2×C4C2×#13711yescharacteristic
647C2×C4#57611yescharacteristic
648C2×C4#57711yescharacteristic
649C2×C4#57711yescharacteristic
650C2×C4#57811yescharacteristic
651C8—42no—
652C8—42no—
653C8—81no—
654C8—81no—
655C8—81no—
656C8—81no—
657C8—81no—
658C8—81no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22D4×C22⋊C411yescharacteristic
674C22C2×#41311yescharacteristic
675C22#288911yescharacteristic
676C22C2×#41911yescharacteristic
677C22C2×#41711yescharacteristic
678C22#289711yescharacteristic
679C22#289811yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—21no—
684C4—21no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—44no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#326611yescharacteristic
697C4#327011yescharacteristic
698C4#327411yescharacteristic
699C4#327811yescharacteristic
700C4#331211yescharacteristic
701C4#332011yescharacteristic
702C4#333211yescharacteristic
703C4#333311yescharacteristic
704C2—42no—
705C2—81no—
706C2#1774911yescharacteristic
707C2#1776111yescharacteristic
708C2#1776311yescharacteristic
709C2#1777511yescharacteristic
710C2#2241111yescharacteristic
711C2#2242511yescharacteristic
712C2#2243211yescharacteristic
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×#1559C2semidirect
5. C2×#1550C2semidirect
6. #9705C2semidirect
7. #9692C2semidirect
8. #9139C2semidirect
9. #9138C2semidirect
10. #9137C2semidirect
11. #9131C2semidirect
12. #8590C2semidirect
13. #8343C2semidirect
14. #8343C2semidirect
15. #8235C2semidirect
196. #333D4semidirect

Maximal subgroups

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

subgroupindex
#361342
#227512
#226482
C2×#15592
C2×#15502
#97052
#96922
#91392
#91382
#91372
#91312
#85902
#83432
#82352