Finite Group Databasebeta

← #223264

Subgroups — #223264

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36182C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#22751C211yesmaximal, characteristic, complemented
4#9900C211yesmaximal, characteristic, complemented
5#9899C211yesmaximal, characteristic, complemented
6#9142C211yesmaximal, characteristic, complemented
7#9142C211yesmaximal, characteristic, complemented
8#9142C211yesmaximal, characteristic, complemented
9#9141C211yesmaximal, characteristic, complemented
10#8727C211yesmaximal, characteristic, complemented
11#8724C211yesmaximal, characteristic, complemented
12#8588C211yesmaximal, characteristic, complemented
13C4×#331C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic, complemented
15#8040C211yesmaximal, characteristic, complemented
16C2×#553—21no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2859C2211yescharacteristic
22#2856C2211yescharacteristic
23#2856C2211yescharacteristic
24#2566—21no—
25#2512—21no—
26#2512C412yes—
27#2504—21no—
28#2500—21no—
29#2422—21no—
30#2422C412yes—
31#2416—21no—
32#2416C412yes—
33#2361—21no—
34#2361C412yes—
35C4×#127C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1924—21no—
41#1924—21no—
42#1923—21no—
43#1917—21no—
44#1870—21no—
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770—21no—
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55C2×#331C2211yescharacteristic
56C2×#331C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72#553—44no—
73#553—44no—
74C2×#127C2311yescharacteristic
75C2×#126—22no—
76C2×#125—21no—
77C2×#123—22no—
78C2×#121—22no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121—21no—
83C2×#121C2×C412yes—
84C2×#121C2×C412yes—
85C2×C4×D4—22no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C412yes—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×#117—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2×C412yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#432—22no—
99#432—22no—
100#432—22no—
101#432—41no—
102#432—41no—
103#432—41no—
104#423—22no—
105#423—41no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129#397—22no—
130#397—22no—
131#396—22no—
132#396—22no—
133#394—22no—
134#394—22no—
135#390—21no—
136#390—21no—
137#390—21no—
138#390—21no—
139#390—21no—
140#390—21no—
141#390C2×C412yes—
142#390C2×C412yes—
143#390C2×C412yes—
144#389—21no—
145#389—21no—
146#389C2×C412yes—
147#389C2×C412yes—
148#388—22no—
149#388—22no—
150#386—22no—
151#386—22no—
152#385—22no—
153#385—22no—
154#384—21no—
155#384—21no—
156#384C2×C412yes—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#382—21no—
161#382—21no—
162#380—21no—
163#380C2×C412yes—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C22⋊C4C2×C411yescharacteristic
170C4×C22⋊C4C2×C411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175C43C2311yescharacteristic
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#331—22no—
185#331—22no—
186#331—41no—
187#331—41no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4C22⋊C411yescharacteristic
193C22×D4C22×C412yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C2×M16—42no—
203C2×M16—42no—
204C2×M16—81no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213#127—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#126—48no—
218#125—42no—
219#125—42no—
220#123—48no—
221#121—44no—
222#121—42no—
223#121—42no—
224#121—42no—
225#121—42no—
226#121—42no—
227#121—42no—
228#121—48no—
229#121—81no—
230#121—81no—
231#121—82no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—42no—
237C4×D4—48no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—81no—
241C4×D4—81no—
242C4×D4—82no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—81no—
259C2×C4⋊C4—81no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321C4⋊C8—81no—
322C4⋊C8#4112yes—
323C4⋊C8#4112yes—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328C22⋊C8—81no—
329C22⋊C8—81no—
330C22⋊C8—81no—
331#97—42no—
332#97—42no—
333#97—42no—
334C4×C8—42no—
335C4×C8—81no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346C24—22no—
347C24C2×C22⋊C411yescharacteristic
348C24C2×C22⋊C411yescharacteristic
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—82no—
373C2×D4—82no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
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—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428M16—84no—
429C2×C8—22no—
430C2×C8—22no—
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—41no—
451C2×C8—41no—
452C2×C8—41no—
453C4⋊C4—22no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—41no—
512C42—44no—
513C42AGL(1,ℤ/8)12yes—
514C42AGL(1,ℤ/8)12yes—
515C42C2×#4111yescharacteristic
516C42C2×#4111yescharacteristic
517C42H(5,𝔽2)11yescharacteristic
518C422−1+411yescharacteristic
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23#41312yes—
540C23C22×C22⋊C411yescharacteristic
541D4—44no—
542D4—82no—
543D4—82no—
544D4—88no—
545D4—81no—
546D4—81no—
547D4—81no—
548D4—81no—
549D4—81no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—164no—
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—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
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—21no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
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×C4—42no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—164no—
614C2×C4—164no—
615C2×C4#41812yes—
616C2×C4#41812yes—
617C2×C4C4×#4111yescharacteristic
618C2×C4#52011yescharacteristic
619C2×C4#54611yescharacteristic
620C2×C4#54611yescharacteristic
621C2×C4#54711yescharacteristic
622C2×C4#54711yescharacteristic
623C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
624C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
625C2×C4#57511yescharacteristic
626C2×C4#57511yescharacteristic
627C2×C4#57611yescharacteristic
628C2×C4#57711yescharacteristic
629C2×C4#57711yescharacteristic
630C2×C4#57811yescharacteristic
631C8—42no—
632C8—42no—
633C8—81no—
634C8—81no—
635C8—81no—
636C8—81no—
637C8—81no—
638C8—81no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#233911yescharacteristic
661C22C2×#41311yescharacteristic
662C22#288811yescharacteristic
663C22C2×#41811yescharacteristic
664C22#289611yescharacteristic
665C22#289711yescharacteristic
666C22#289811yescharacteristic
667C4—22no—
668C4—22no—
669C4—21no—
670C4—21no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#332211yescharacteristic
683C4#332611yescharacteristic
684C4#333011yescharacteristic
685C4#333211yescharacteristic
686C4#335111yescharacteristic
687C4#335311yescharacteristic
688C4#335411yescharacteristic
689C4#335511yescharacteristic
690C2—42no—
691C2—81no—
692C2—162no—
693C2#1773111yescharacteristic
694C2#1775811yescharacteristic
695C2#1776811yescharacteristic
696C2#1777511yescharacteristic
697C2#2248211yescharacteristic
698C2#2250011yescharacteristic
699C2#2250111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36182C2
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
2. #22799C2semidirect
3. #22751C2semidirect
4. #9900C2semidirect
5. #9899C2semidirect
6. #9142C2semidirect
7. #9142C2semidirect
8. #9142C2semidirect
9. #9141C2semidirect
10. #8727C2semidirect
11. #8724C2semidirect
12. #8588C2semidirect
13. C4×#331C2semidirect
14. #8042C2semidirect
15. #8040C2semidirect

Maximal subgroups

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

subgroupindex
#361822
#227992
#227512
#99002
#98992
#91422
#91412
#87272
#87242
#85882
C4×#3312
#80422
#80402