Finite Group Databasebeta

← #206864

Subgroups — #206864

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34978C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic
3#20462C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9876C211yesmaximal, characteristic, complemented
6#9875C211yesmaximal, characteristic, complemented
7#9874C211yesmaximal, characteristic, complemented
8#9873C211yesmaximal, characteristic, complemented
9#9872C211yesmaximal, characteristic, complemented
10#9871C211yesmaximal, characteristic, complemented
11#9870C211yesmaximal, characteristic, complemented
12#7902C211yesmaximal, characteristic
13#7900C211yesmaximal, characteristic
14#7893C211yesmaximal, characteristic
15#7891C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2612—21no—
21#2612—21no—
22#2571C411yescharacteristic
23#2552C411yescharacteristic
24#2552C411yescharacteristic
25#2551C411yescharacteristic
26#2441—21no—
27#2441C411yescharacteristic
28#2441C411yescharacteristic
29#2439—21no—
30#2439C411yescharacteristic
31#2439C411yescharacteristic
32#2396—21no—
33#2396—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#125—21no—
38C4×#125C2211yescharacteristic
39C4×#125C2211yescharacteristic
40#2285C2211yescharacteristic
41C4×#117C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#1940—21no—
44#1940—21no—
45#1850—21no—
46#1850—21no—
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1768—21no—
52#1768—21no—
53#1750—21no—
54#1750—21no—
55#1744C2211yescharacteristic
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C23×C8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#125—21no—
80C2×#125C2×C411yescharacteristic
81C2×#125C2×C411yescharacteristic
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×#117—21no—
85C2×#117C2×C411yescharacteristic
86C2×#117C2×C411yescharacteristic
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C42C2311yescharacteristic
90#432—22no—
91#432—22no—
92#431—22no—
93#431—22no—
94#431Q814yescomplemented
95#431Q814yescomplemented
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#400—21no—
122#400—21no—
123#399—21no—
124#399—21no—
125#398—21no—
126#398—21no—
127#398—21no—
128#398—21no—
129#396—21no—
130#396—21no—
131#396—21no—
132#396—21no—
133#395—21no—
134#395—21no—
135#393—21no—
136#393—21no—
137#391—21no—
138#391—21no—
139#391—21no—
140#391—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—21no—
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#389C2×C411yescharacteristic
152#389C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#389C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#387—21no—
162#387—21no—
163#387—21no—
164#387—21no—
165#384—21no—
166#384C2×C411yescharacteristic
167#384C2×C411yescharacteristic
168#384C2×C411yescharacteristic
169#384C2×C411yescharacteristic
170#382—21no—
171#382—21no—
172#382C2×C411yescharacteristic
173#382C2×C411yescharacteristic
174#380—21no—
175#380C2×C411yescharacteristic
176#380C2×C411yescharacteristic
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2×C411yescharacteristic
185C4×C4⋊C4C2×C411yescharacteristic
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4C2×C411yescharacteristic
188C4×C4⋊C4D411yescharacteristic
189C4×C4⋊C4D411yescharacteristic
190C4×C4⋊C4C2311yescharacteristic
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2311yescharacteristic
193C4×C4⋊C4C2311yescharacteristic
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4C2×C411yescharacteristic
197C4×C22⋊C4C2×C411yescharacteristic
198C4×C22⋊C4D411yescharacteristic
199C4×C22⋊C4D411yescharacteristic
200#376—21no—
201#376D411yescharacteristic
202#376D411yescharacteristic
203C43—21no—
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214C22×Q8—21no—
215C22×Q8—21no—
216C23×C4C22×C411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—42no—
228C22×C8—81no—
229C22×C8—81no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—41no—
233C4⋊Q8—41no—
234C4⋊Q8—44no—
235#125—22no—
236#125—22no—
237#125—41no—
238#125—41no—
239#125—44no—
240C4×Q8—81no—
241C4×Q8—81no—
242C4×Q8—81no—
243C4×Q8—81no—
244#117—22no—
245#117—22no—
246#117—42no—
247#117—42no—
248#117—41no—
249#117—41no—
250#117—81no—
251#117—81no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×Q812yes—
318C2×C42C2×Q812yes—
319C2×C42#4112yes—
320C2×C42#4112yes—
321C2×C42C2411yescharacteristic
322C4⋊C8—24no—
323C4⋊C8—24no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—42no—
333C4⋊C8—81no—
334C4⋊C8—81no—
335C4⋊C8—81no—
336C4⋊C8—81no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C22⋊C8—42no—
340#97—42no—
341#97—42no—
342#97—42no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—42no—
348C4×C8—42no—
349C4×C8C2×Q812yes—
350C4×C8C2×Q812yes—
351C4×C8C2×Q812yes—
352C4×C8C2×Q812yes—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
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#95C22⋊C411yescharacteristic
374#95C22⋊C411yescharacteristic
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381C24C2×C22⋊C411yescharacteristic
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—81no—
387C2×Q8—81no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C4×Q812yes—
418C22×C4C4×Q812yes—
419C22×C4C22⋊Q812yes—
420C22×C4C22⋊Q812yes—
421C22×C4#12412yes—
422C22×C4C4⋊Q812yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—84no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4#13112yes—
492C4⋊C4#13112yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—22no—
518C42—22no—
519C42—41no—
520C42—41no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42C4×Q812yes—
534C42C4×Q812yes—
535C42C22⋊Q812yes—
536C42C22⋊Q812yes—
537C42#12412yes—
538C42C4⋊Q812yes—
539C42C22×Q811yescharacteristic
540C42C22×Q811yescharacteristic
541C42C2×#4111yescharacteristic
542C42C2×#4111yescharacteristic
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#39112yes—
550C23C22×C22⋊C411yescharacteristic
551Q8—84no—
552Q8—84no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
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#39112yes—
618C2×C4#39112yes—
619C2×C4#39112yes—
620C2×C4#42112yes—
621C2×C4#42112yes—
622C2×C4C2×C4×Q811yescharacteristic
623C2×C4C2×C4×Q811yescharacteristic
624C2×C4C2×C22⋊Q811yescharacteristic
625C2×C4C2×C22⋊Q811yescharacteristic
626C2×C4C2×#12411yescharacteristic
627C2×C4C2×C4⋊Q811yescharacteristic
628C2×C4C2×#13111yescharacteristic
629C2×C4C2×#13111yescharacteristic
630C2×C4#56811yescharacteristic
631C2×C4#56811yescharacteristic
632C2×C4#57611yescharacteristic
633C2×C4#57711yescharacteristic
634C2×C4#57711yescharacteristic
635C2×C4#57811yescharacteristic
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C8—81no—
641C8—81no—
642C8—81no—
643C8—81no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×#39111yescharacteristic
652C22#288011yescharacteristic
653C22#288011yescharacteristic
654C22#289711yescharacteristic
655C22#289811yescharacteristic
656C22C2×#42111yescharacteristic
657C22C2×#42111yescharacteristic
658C4—22no—
659C4—22no—
660C4—21no—
661C4—21no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4#296011yescharacteristic
680C4#296111yescharacteristic
681C4#296211yescharacteristic
682C4#296211yescharacteristic
683C4#313611yescharacteristic
684C4#313711yescharacteristic
685C4#316311yescharacteristic
686C4#316411yescharacteristic
687C2—42no—
688C2#1752111yescharacteristic
689C2#1752311yescharacteristic
690C2#1752411yescharacteristic
691C2#1752611yescharacteristic
692C2#2130411yescharacteristic
693C2#2132611yescharacteristic
694C2#2132911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊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
4. #9877C2semidirect
5. #9876C2semidirect
6. #9875C2semidirect
7. #9874C2semidirect
8. #9873C2semidirect
9. #9872C2semidirect
10. #9871C2semidirect
11. #9870C2semidirect
94. #431Q8semidirect
95. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#349782
#219762
#204622
#98772
#98762
#98752
#98742
#98732
#98722
#98712
#98702
#79022
#79002
#78932
#78912