Finite Group Databasebeta

← #357332

Subgroups — #357332

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C4×#388C211yesmaximal, characteristic
4#10921C211yesmaximal, characteristic
5#10921C211yesmaximal, characteristic
6#10876C211yesmaximal, characteristic, complemented
7#10876C211yesmaximal, characteristic, complemented
8#10876C211yesmaximal, characteristic, complemented
9#10874C211yesmaximal, characteristic, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10832C211yesmaximal, characteristic, complemented
12#10797C211yesmaximal, characteristic, complemented
13#10795C211yesmaximal, characteristic, complemented
14#10714C211yesmaximal, characteristic
15#10714C211yesmaximal, characteristic
16C24×C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#388C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1923C418yescomplemented
33#1850C418yescomplemented
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
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
52#1437C412yes—
53#1436C412yes—
54#1434C412yes—
55C4×#95C2211yescharacteristic
56C4×#95C2211yescharacteristic
57C24×C4C2311yescharacteristic
58C23×C8—41no—
59C23×C8—41no—
60C23×C8—41no—
61C23×C8—41no—
62C23×C8—41no—
63C23×C8—41no—
64C23×C8C2×C414yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8C2×C414yes—
94C2×C4⋊C8C2×C414yes—
95C2×C22⋊C8—41no—
96C2×C22⋊C8—41no—
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×C4×C8—41no—
105C2×C4×C8—41no—
106C2×C4×C8—41no—
107C2×C4×C8—41no—
108C2×C4×C8—41no—
109C2×C4×C8—41no—
110C2×C4×C8—41no—
111C2×C4×C8—41no—
112C2×C4×C8C2×C414yes—
113#388C8116yescomplemented
114C4×C4⋊C4—41no—
115C4×C4⋊C4—41no—
116C4×C22⋊C4—41no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119C4×C22⋊C4—41no—
120C4×C22⋊C4C2×C414yes—
121C4×C22⋊C4C2×C414yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C43—41no—
132C43—41no—
133#336—41no—
134#336—41no—
135#336—41no—
136#336—41no—
137#336—41no—
138#336—41no—
139#336—41no—
140#336—41no—
141#336—41no—
142#336—41no—
143#336—41no—
144#336—41no—
145#336—41no—
146#336—41no—
147#336C2×C414yes—
148#336C2×C414yes—
149#336C2×C414yes—
150#336C2×C414yes—
151C25C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
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—44no—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8—44no—
230C22×C8—44no—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C2×C818yescomplemented
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C4214yes—
248C2×C22⋊C4C4214yes—
249C2×C22⋊C4C2×C818yes—
250C2×C22⋊C4C2×C818yes—
251C2×C22⋊C4C2×C818yes—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
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—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C2×C818yescomplemented
290C2×C42C22×C412yes—
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C22×C412yes—
294C4⋊C8—44no—
295C4⋊C8—44no—
296C4⋊C8—44no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299C22⋊C8—44no—
300C4×C8—44no—
301C4×C8—44no—
302C4×C8—44no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95C4214yes—
309#95C4214yes—
310#95C2×C818yescomplemented
311#95C2×C818yescomplemented
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C2×C4212yes—
322C24C22×C814yes—
323C24C23×C411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
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—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
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×C4C2×C4212yes—
397C22×C4C2×C4212yes—
398C22×C4C2×C4212yes—
399C22×C4C2×C4212yes—
400C22×C4C2×C4212yes—
401C22×C4C22×C814yes—
402C22×C4C22×C814yes—
403C22×C4C22×C814yes—
404C22×C4C22×C814yes—
405C22×C4C22×C814yes—
406C22×C4C22×C814yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419C22×C42−1+411yescharacteristic
420C22×C42−1+411yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
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—416no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
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—41no—
477C2×C8—41no—
478C4⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4C4×C8116yescomplemented
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—48no—
492C42—48no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×C4×C814yes—
532C23C22×C4211yescharacteristic
533C23#51811yescharacteristic
534C23#51811yescharacteristic
535C23#52011yescharacteristic
536C23#52011yescharacteristic
537C23#52011yescharacteristic
538C23#52011yescharacteristic
539C23C23×C812yes—
540C23#56812yes—
541C23#56812yes—
542C23#56812yes—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
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—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
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×C4C2×C4×C814yes—
605C2×C4C2×C4×C814yes—
606C2×C4#51811yescharacteristic
607C2×C4#51811yescharacteristic
608C2×C4#51811yescharacteristic
609C2×C4#51811yescharacteristic
610C2×C4#51811yescharacteristic
611C2×C4#51911yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4#52011yescharacteristic
614C2×C4#52011yescharacteristic
615C2×C4#52011yescharacteristic
616C2×C4#52011yescharacteristic
617C2×C4#52011yescharacteristic
618C2×C4#56011yescharacteristic
619C2×C4#56011yescharacteristic
620C2×C4#56011yescharacteristic
621C2×C4#56011yescharacteristic
622C2×C4H(3,𝔽4)11yescharacteristic
623C2×C4#56211yescharacteristic
624C2×C4#56211yescharacteristic
625C2×C4#56211yescharacteristic
626C2×C4#56211yescharacteristic
627C2×C4#56211yescharacteristic
628C2×C4#56211yescharacteristic
629C2×C4#56311yescharacteristic
630C2×C4#56311yescharacteristic
631C2×C4#56311yescharacteristic
632C2×C4#56311yescharacteristic
633C2×C4#56311yescharacteristic
634C8—42no—
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22#227411yescharacteristic
654C22#227511yescharacteristic
655C22#227511yescharacteristic
656C22#237611yescharacteristic
657C22#237711yescharacteristic
658C22#237911yescharacteristic
659C22#238311yescharacteristic
660C22C22×C4×C811yescharacteristic
661C22#287311yescharacteristic
662C22#287511yescharacteristic
663C22#287511yescharacteristic
664C22#297212yes—
665C22#297212yes—
666C22#297812yes—
667C22#298012yes—
668C22#298812yes—
669C22#298812yes—
670C22#298812yes—
671C22#298812yes—
672C22#299512yes—
673C22#299512yes—
674C22#299512yes—
675C22#299512yes—
676C4—24no—
677C4—24no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4#237511yescharacteristic
686C4#237611yescharacteristic
687C4#237611yescharacteristic
688C4#237711yescharacteristic
689C4#237811yescharacteristic
690C4#237811yescharacteristic
691C4#237911yescharacteristic
692C4#238311yescharacteristic
693C2—44no—
694C2#1383111yescharacteristic
695C2#2054311yescharacteristic
696C2#2055411yescharacteristic
697C2#2055511yescharacteristic
698C2#2055511yescharacteristic
699C2#2056311yescharacteristic
700C2#2056411yescharacteristic
701C2#2331412yes—
702C2#2332612yes—
703C2#2334612yes—
704C2#2337512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC22C22×C4×C8
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10876C2semidirect
7. #10876C2semidirect
8. #10876C2semidirect
9. #10874C2semidirect
10. #10834C2semidirect
11. #10832C2semidirect
12. #10797C2semidirect
13. #10795C2semidirect
32. #1923C4semidirect
33. #1850C4semidirect
113. #388C8semidirect
241. C2×C4⋊C4C2×C8semidirect
289. C2×C42C2×C8semidirect
310. #95C2×C8semidirect
311. #95C2×C8semidirect
482. C22⋊C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#18502
C4×#3882
#109212
#108762
#108742
#108342
#108322
#107972
#107952
#107142