Finite Group Databasebeta

← #355677

Subgroups — #355677

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C211yesmaximal, characteristic
2C2×#1924C211yesmaximal, characteristic
3C2×#1771C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic
5#13844C211yesmaximal, characteristic
6#10921C211yesmaximal, characteristic
7#10921C211yesmaximal, characteristic
8#10876C211yesmaximal, characteristic, complemented
9#10876C211yesmaximal, characteristic, complemented
10#10874C211yesmaximal, characteristic, complemented
11#10874C211yesmaximal, characteristic, complemented
12#10835C211yesmaximal, characteristic, complemented
13#10798C211yesmaximal, characteristic, complemented
14#10713C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C22×#117C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1924C418yescomplemented
34#1924C418yescomplemented
35#1771—24no—
36#1750—24no—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1444C412yes—
54#1442C412yes—
55#1439C412yes—
56#1439C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—41no—
71C2×#117—41no—
72C2×#117—41no—
73C2×#117—41no—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8C2×C414yes—
92C2×C4⋊C8C2×C414yes—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—41no—
98C2×C22⋊C8—41no—
99C2×C22⋊C8—41no—
100C2×C22⋊C8—41no—
101C2×C22⋊C8C2×C414yes—
102C2×C22⋊C8C2×C414yes—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—41no—
108C2×#97—41no—
109C2×#97—41no—
110C2×#97—41no—
111C2×#97C2×C414yes—
112C2×#97C2×C414yes—
113C2×#97D414yes—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—41no—
118C2×C4×C8—41no—
119C2×C4×C8—41no—
120C2×C4×C8—41no—
121C2×C4×C8—41no—
122C2×C4×C8—41no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—41no—
126C4×C4⋊C4—41no—
127C4×C22⋊C4—41no—
128C4×C22⋊C4—41no—
129C4×C22⋊C4C2×C414yes—
130C4×C22⋊C4C2×C414yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138C2×#95C2×C411yescharacteristic
139C2×#95C2×C411yescharacteristic
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—41no—
147#336—41no—
148#336—41no—
149#336—41no—
150#336—41no—
151#336—41no—
152#336C2×C414yes—
153#336C2×C414yes—
154#336C2×C414yes—
155#336C2×C414yes—
156C25C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8C22×C412yes—
226C22×C8C22×C412yes—
227C22×C8C22×C412yes—
228C22×C8C22×C412yes—
229C22×C8C22×C412yes—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C2×D412yes—
234C22×C8C2×D412yes—
235C22×C8#4112yes—
236C22×C8#4112yes—
237#117—44no—
238#117—44no—
239#117—44no—
240#117—44no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C4214yes—
259C2×C22⋊C4C4214yes—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C22×C412yes—
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C22×C412yes—
294C2×C42C2×D412yes—
295C2×C42#4112yes—
296C4⋊C8—28no—
297C4⋊C8—44no—
298C22⋊C8—28no—
299C22⋊C8—28no—
300C22⋊C8—44no—
301C22⋊C8—44no—
302#97—28no—
303#97—44no—
304#97—44no—
305C4×C8—44no—
306C4×C8—44no—
307C4×C8—44no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95C4214yes—
316#95C4214yes—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24C2×C4212yes—
328C24C23×C411yescharacteristic
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4C2×C4212yes—
405C22×C4C2×C4212yes—
406C22×C4C2×C4212yes—
407C22×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C4×D414yes—
410C22×C4#13114yes—
411C22×C4#13114yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
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—41no—
479C2×C8—41no—
480C2×C8C4×D414yes—
481C2×C8C4×D414yes—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C22×C4211yescharacteristic
537C23C2×C4×D411yescharacteristic
538C23C4×#4111yescharacteristic
539C23#51811yescharacteristic
540C23#51811yescharacteristic
541C23#51911yescharacteristic
542C23#52011yescharacteristic
543C23C2×#13112yes—
544C23C2×#13112yes—
545C23#56812yes—
546C23#56812yes—
547C2×C4—24no—
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—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4C2×C4×D411yescharacteristic
611C2×C4C2×C4×D411yescharacteristic
612C2×C4C4×#4111yescharacteristic
613C2×C4C4×#4111yescharacteristic
614C2×C4#51811yescharacteristic
615C2×C4#51811yescharacteristic
616C2×C4#52011yescharacteristic
617C2×C4#52011yescharacteristic
618C2×C4#52011yescharacteristic
619C2×C4#52011yescharacteristic
620C2×C4#52011yescharacteristic
621C2×C4#52011yescharacteristic
622C2×C4#53411yescharacteristic
623C2×C4#53511yescharacteristic
624C2×C4#53611yescharacteristic
625C2×C4#53711yescharacteristic
626C2×C4#53811yescharacteristic
627C2×C4#53811yescharacteristic
628C2×C4#53811yescharacteristic
629C2×C4#53911yescharacteristic
630C2×C4#53911yescharacteristic
631C2×C4#53911yescharacteristic
632C2×C4#54011yescharacteristic
633C2×C4#54111yescharacteristic
634C2×C4#54211yescharacteristic
635C2×C4#54211yescharacteristic
636C2×C4#54211yescharacteristic
637C2×C4#54211yescharacteristic
638C8—24no—
639C8—24no—
640C8—42no—
641C8—42no—
642C8—42no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22C42×D411yescharacteristic
658C22#227411yescharacteristic
659C22#227511yescharacteristic
660C22#232111yescharacteristic
661C22#232311yescharacteristic
662C22#233211yescharacteristic
663C22#233511yescharacteristic
664C22#287311yescharacteristic
665C22#287311yescharacteristic
666C22C4×#13111yescharacteristic
667C22C4×#13111yescharacteristic
668C22#293312yes—
669C22#293512yes—
670C22#296812yes—
671C22#296912yes—
672C22#297612yes—
673C22#297712yes—
674C22#298012yes—
675C22#298112yes—
676C22#298612yes—
677C22#298712yes—
678C22#299212yes—
679C22#299312yes—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—82no—
689C4#232211yescharacteristic
690C4#232411yescharacteristic
691C4#232511yescharacteristic
692C4#232611yescharacteristic
693C4#233411yescharacteristic
694C4#233411yescharacteristic
695C4#233511yescharacteristic
696C4#233711yescharacteristic
697C2—82no—
698C2#1378911yescharacteristic
699C2#2049011yescharacteristic
700C2#2049211yescharacteristic
701C2#2053911yescharacteristic
702C2#2054011yescharacteristic
703C2#2056111yescharacteristic
704C2#2056611yescharacteristic
705C2#2203712yes—
706C2#2210912yes—
707C2#2232412yes—
708C2#2236812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1771C2semidirect
8. #10876C2semidirect
9. #10876C2semidirect
10. #10874C2semidirect
11. #10874C2semidirect
12. #10835C2semidirect
13. #10798C2semidirect
15. #10696C2semidirect
33. #1924C4semidirect
34. #1924C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#17712
C2×#17502
#138442
#109212
#108762
#108742
#108352
#107982
#107132
#106962