Finite Group Databasebeta

← #530395

Subgroups — #530395

Order 512.

3058 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2337C212yesmaximal
2C2×C4×#124C211yesmaximal, characteristic
3#14688C211yesmaximal, characteristic, complemented
4#14488C214yesmaximal, complemented
5#14466C212yesmaximal, complemented
6#14438C212yesmaximal
7#14296C212yesmaximal, complemented
8#14118C214yesmaximal, complemented
9#13926C212yesmaximal, complemented
10#13880C212yesmaximal, complemented
11#13876C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13#13872C212yesmaximal
14C22⋊C4×C4⋊C4C212yesmaximal, complemented
15C4×#384C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20#2337—216no—
21C4×#124C4116yescomplemented
22C2×#391C2214yes—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#388C2212yes—
28C2×#388C2212yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#384C2212yes—
33C2×#384C2212yes—
34C2×#383C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2214yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C43C2211yescharacteristic
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2314yes—
81C2×#124C2×C418yes—
82C2×#124D418yes—
83C2×#124Q818yes—
84C2×C4×Q8—44no—
85C2×C4×Q8C2×C418yescomplemented
86C2×C4×D4—44no—
87C2×C4×D4C2×C418yes—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#391—216no—
123#391C2×C418yescomplemented
124#390C2×C418yes—
125#388—216no—
126#388—216no—
127#388C2×C4116yescomplemented
128#388C2×C4116yescomplemented
129#386—216no—
130#385—216no—
131#385—48no—
132#384—28no—
133#384—44no—
134#383—28no—
135#382C2×C418yescomplemented
136#380—28no—
137#380—44no—
138#380—44no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4—42no—
142C4×C4⋊C4—42no—
143C4×C4⋊C4—44no—
144C4×C22⋊C4—216no—
145C4×C22⋊C4C2×C4116yescomplemented
146C4×C22⋊C4C2×C4116yescomplemented
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C43C2×C418yescomplemented
172C25C2412yes—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8C22×C414yes—
176C22×Q8C2×D414yes—
177C22×Q8C2×Q814yes—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4C22×C414yes—
181C22×D4C2×D414yes—
182C22×D4C2×Q814yes—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C22×C418yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211#124C4⋊C4164yescomplemented
212C4×Q8—416no—
213C4×D4—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—432no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4#4118yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4C2×D418yes—
270C2×C22⋊C4C2×D418yes—
271C2×C22⋊C4C2×Q818yes—
272C2×C22⋊C4C2×Q818yes—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C22×C414yes—
293C2×C42C22×C418yes—
294C2×C42C22×C414yes—
295C2×C42C22×C414yes—
296C2×C42C22×C414yes—
297C2×C42C22×C414yes—
298C2×C42C22×C414yes—
299C2×C42C22×C414yes—
300C2×C42C2×D414yes—
301C2×C42C2×Q814yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C22×C418yes—
325#95C22×C418yes—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24C23×C414yes—
339C24C22×D414yes—
340C24C22×Q814yes—
341C24C2511yescharacteristic
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8C2×C4⋊C4116yescomplemented
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4C2×C4⋊C4116yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4C23×C414yes—
428C22×C4C23×C412yes—
429C22×C4C23×C414yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C412yes—
433C22×C4C23×C414yes—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q814yes—
445C22×C4C2×#4118yes—
446C22×C4C2×#4114yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C4⋊C4—216no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4C2×C4⋊C4132yescomplemented
471C22⋊C4C2×C4⋊C4132yescomplemented
472C42—28no—
473C42—28no—
474C42—216no—
475C42—28no—
476C42—28no—
477C42—44no—
478C42—44no—
479C42—88no—
480C42—84no—
481C42—84no—
482C42C2×C4⋊C4116yescomplemented
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23C22×C4⋊C418yes—
518C23C24×C411yescharacteristic
519C23C23×D411yescharacteristic
520C23C23×Q811yescharacteristic
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)12yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23#58512yes—
528C23#58512yes—
529Q8—416no—
530D4—416no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—432no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—832no—
609C2×C4C22×C4⋊C414yes—
610C2×C4C22×C4⋊C414yes—
611C2×C4C22×C4⋊C414yes—
612C2×C4C22×C4⋊C418yes—
613C2×C4C4×#4114yes—
614C2×C4#51814yes—
615C2×C4#52014yes—
616C2×C4#52014yes—
617C2×C4#53818yes—
618C2×C4#54218yes—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22C23×C4⋊C411yescharacteristic
635C22C2×C4×#4111yescharacteristic
636C22C2×#51811yescharacteristic
637C22C2×#52011yescharacteristic
638C22C2×#52011yescharacteristic
639C22#342811yescharacteristic
640C22#342811yescharacteristic
641C22C2×#53812yes—
642C22C2×#54212yes—
643C22D4×#4112yes—
644C22Q8×#4112yes—
645C22#349612yes—
646C22#349811yescharacteristic
647C22#349911yescharacteristic
648C22#350112yes—
649C22#350311yescharacteristic
650C22#350311yescharacteristic
651C22#351312yes—
652C22#351612yes—
653C22#354612yes—
654C22#354712yes—
655C22#356212yes—
656C22#356512yes—
657C4—24no—
658C4—24no—
659C4—28no—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—48no—
667C4—84no—
668C4—84no—
669C4—84no—
670C4#233718yescomplemented
671C2—48no—
672C2C2×#233711yescharacteristic
673C2C4⋊C4×#4111yescharacteristic
674C2#3418111yescharacteristic
675C2#3418311yescharacteristic
676C2#3424111yescharacteristic
677C2#3436711yescharacteristic
678C2#3438511yescharacteristic
679C2#3939612yes—
680C2#3939812yes—
681C2#3965212yes—
682C2#3965412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14688C2semidirect
4. #14488C2semidirect
5. #14466C2semidirect
7. #14296C2semidirect
8. #14118C2semidirect
9. #13926C2semidirect
10. #13880C2semidirect
11. #13876C2semidirect
12. #13876C2semidirect
14. C22⋊C4×C4⋊C4C2semidirect
15. C4×#384C2semidirect
21. C4×#124C4semidirect
85. C2×C4×Q8C2×C4semidirect
123. #391C2×C4semidirect
127. #388C2×C4semidirect
128. #388C2×C4semidirect
135. #382C2×C4semidirect
145. C4×C22⋊C4C2×C4semidirect
146. C4×C22⋊C4C2×C4semidirect
171. C43C2×C4semidirect
211. #124C4⋊C4semidirect
346. C2×Q8C2×C4⋊C4semidirect
470. C22⋊C4C2×C4⋊C4semidirect
471. C22⋊C4C2×C4⋊C4semidirect
482. C42C2×C4⋊C4semidirect
670. C4#2337semidirect

Maximal subgroups

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

subgroupindex
C2×#23372
C2×C4×#1242
#146882
#144882
#144662
#144382
#142962
#141182
#139262
#138802
#138762
#138722
C22⋊C4×C4⋊C42
C4×#3842