Finite Group Databasebeta

← #208525

Subgroups — #208525

Order 512.

869 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34974C211yesmaximal, characteristic
2#21924C211yesmaximal, characteristic, complemented
3#20366C211yesmaximal, characteristic, complemented
4#9874C211yesmaximal, characteristic, complemented
5#9873C211yesmaximal, characteristic, complemented
6#9645C211yesmaximal, characteristic, complemented
7#9640C211yesmaximal, characteristic, complemented
8#9078C211yesmaximal, characteristic
9#9077C211yesmaximal, characteristic, complemented
10#9072C211yesmaximal, characteristic
11#9071C211yesmaximal, characteristic, complemented
12#8183C211yesmaximal, characteristic, complemented
13#8179C211yesmaximal, characteristic, complemented
14#8175C211yesmaximal, characteristic, complemented
15#8171C211yesmaximal, characteristic, complemented
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2532—21no—
19#2529—21no—
20#2512C411yescharacteristic
21#2508C411yescharacteristic
22#2495C411yescharacteristic
23#2491C411yescharacteristic
24#2479—21no—
25#2474—21no—
26#2451C411yescharacteristic
27#2448C411yescharacteristic
28#2426C411yescharacteristic
29#2423C411yescharacteristic
30#2396—21no—
31#2395—21no—
32C4×C4⋊Q8C2211yescharacteristic
33C4×#126—21no—
34C4×#124—21no—
35C4×#124C2211yescharacteristic
36C4×C22⋊Q8C2211yescharacteristic
37C4×#121C2211yescharacteristic
38#2293C2211yescharacteristic
39C42×Q8C2211yescharacteristic
40C4×#117C2211yescharacteristic
41#1924—21no—
42#1923—21no—
43#1917—21no—
44#1916—21no—
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1767C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1746—21no—
52#1746—21no—
53#1744C2211yescharacteristic
54#1744C2211yescharacteristic
55#1744C2211yescharacteristic
56#1744C2211yescharacteristic
57#1729—21no—
58#1729—21no—
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#126—21no—
79C2×#124—21no—
80C2×#124C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×C4×Q8—21no—
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C2×#117D411yescharacteristic
91C2×#117D411yescharacteristic
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C4×M16—22no—
102C4×M16—22no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#400—21no—
114#400—21no—
115#399—21no—
116#399—21no—
117#398—21no—
118#398—21no—
119#398—21no—
120#397—21no—
121#397—21no—
122#397—21no—
123#396—21no—
124#396—21no—
125#395—21no—
126#394—21no—
127#394—21no—
128#393—21no—
129#391—21no—
130#391—21no—
131#391C2×C411yescharacteristic
132#391C2×C411yescharacteristic
133#390—21no—
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#389—21no—
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
152#388—21no—
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#386—21no—
162#385—21no—
163#384C2×C411yescharacteristic
164#383—21no—
165#382—21no—
166#382C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C4⋊C4C2311yescharacteristic
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4C2×C411yescharacteristic
188C4×C22⋊C4C2×C411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191#376D411yescharacteristic
192#376D411yescharacteristic
193C43—21no—
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336—21no—
202#336—21no—
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210Q8⋊C8—22no—
211Q8⋊C8—22no—
212Q8⋊C8—22no—
213Q8⋊C8—22no—
214D4⋊C8—22no—
215D4⋊C8—22no—
216D4⋊C8—22no—
217D4⋊C8—22no—
218C22×Q8—21no—
219C22×Q8C22×C411yescharacteristic
220C22×D4—21no—
221C22×D4C22×C411yescharacteristic
222C23×C4—21no—
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C2×M16—24no—
227C2×M16—42no—
228C2×M16—42no—
229C22×C8—24no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—42no—
239C22×C8—42no—
240C4⋊Q8—22no—
241C4⋊Q8—41no—
242#126—44no—
243#124—22no—
244#124—41no—
245#124—44no—
246C22⋊Q8—22no—
247C22⋊Q8—41no—
248#121—22no—
249#121—41no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—41no—
257C4×Q8—41no—
258C4×Q8—81no—
259C4×Q8—81no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—81no—
265C4×D4—81no—
266#117—42no—
267#117—42no—
268#117—81no—
269#117—81no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22⋊C411yescharacteristic
286C2×C4⋊C4C22⋊C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
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×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
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—81no—
348C4⋊C8—81no—
349C22⋊C8—81no—
350C22⋊C8—81no—
351#97—24no—
352#97—81no—
353C4×C8—24no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—81no—
359C4×C8#4112yes—
360C4×C8#4112yes—
361C4×C8#4112yes—
362C4×C8#4112yes—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389C24—21no—
390C24C2×C22⋊C411yescharacteristic
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8#13112yes—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—81no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
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—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452M16—48no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—48no—
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—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4#13112yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—42no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42C2×#4111yescharacteristic
565C42C2×#4111yescharacteristic
566C42C2×#4111yescharacteristic
567C42C2×#4111yescharacteristic
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23#42112yes—
582C23C22×C22⋊C411yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—84no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—84no—
593C2×C4—24no—
594C2×C4—24no—
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—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4#42112yes—
663C2×C4C4×#4111yescharacteristic
664C2×C4C4×#4111yescharacteristic
665C2×C4#52511yescharacteristic
666C2×C4#52511yescharacteristic
667C2×C4#53211yescharacteristic
668C2×C4#53211yescharacteristic
669C2×C4C2×#13111yescharacteristic
670C2×C4C2×#13111yescharacteristic
671C2×C4#56811yescharacteristic
672C2×C4#56811yescharacteristic
673C2×C4#57611yescharacteristic
674C2×C4#57711yescharacteristic
675C2×C4#57711yescharacteristic
676C2×C4#57811yescharacteristic
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C8—81no—
682C8—81no—
683C8—81no—
684C8—81no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22#229711yescharacteristic
700C22#288011yescharacteristic
701C22#288011yescharacteristic
702C22#289711yescharacteristic
703C22#289811yescharacteristic
704C22C2×#42111yescharacteristic
705C22C2×#42111yescharacteristic
706C4—21no—
707C4—21no—
708C4—22no—
709C4—41no—
710C4—41no—
711C4—42no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—42no—
720C4—41no—
721C4—81no—
722C4—81no—
723C4—81no—
724C4—81no—
725C4—81no—
726C4#296611yescharacteristic
727C4#296711yescharacteristic
728C4#296811yescharacteristic
729C4#296911yescharacteristic
730C4#317311yescharacteristic
731C4#317511yescharacteristic
732C4#317811yescharacteristic
733C4#317911yescharacteristic
734C2—42no—
735C2—161no—
736C2#1752111yescharacteristic
737C2#1752411yescharacteristic
738C2#1752511yescharacteristic
739C2#1752611yescharacteristic
740C2#2138111yescharacteristic
741C2#2141211yescharacteristic
742C2#2141711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21924C2semidirect
3. #20366C2semidirect
4. #9874C2semidirect
5. #9873C2semidirect
6. #9645C2semidirect
7. #9640C2semidirect
9. #9077C2semidirect
11. #9071C2semidirect
12. #8183C2semidirect
13. #8179C2semidirect
14. #8175C2semidirect
15. #8171C2semidirect

Maximal subgroups

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

subgroupindex
#349742
#219242
#203662
#98742
#98732
#96452
#96402
#90782
#90772
#90722
#90712
#81832
#81792
#81752
#81712