Finite Group Databasebeta

← #514144

Subgroups — #514144

Order 512.

3106 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C4⋊C4C211yesmaximal, characteristic
2#13934C212yesmaximal, complemented
3#13932C212yesmaximal, complemented
4#13930C212yesmaximal, complemented
5#13928C212yesmaximal, complemented
6#13923C212yesmaximal
7#13908C212yesmaximal
8#13883C212yesmaximal
9#13882C212yesmaximal
10#13848C212yesmaximal, complemented
11#13846C212yesmaximal, complemented
12C4×#380C211yesmaximal, characteristic
13C4×#380C211yesmaximal, characteristic
14C4×#380C211yesmaximal, characteristic
15C4×#387C212yesmaximal, complemented
16#13759C212yesmaximal, complemented
17#13753C211yesmaximal, characteristic
18#13753C211yesmaximal, characteristic
19C42×C22⋊C4C211yesmaximal, characteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#387C2214yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C4116yescomplemented
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×#376C2212yes—
48C2×#376C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C2×C43C2212yes—
52#1445C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2212yes—
75#1434C2212yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C4×#95C2212yes—
90C4×#95C2212yes—
91C4×#95C2212yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2311yescharacteristic
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4C2×C418yes—
102C22×C4⋊C4C2×C418yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—28no—
119C22×C42—44no—
120C22×C42—44no—
121C22×C42C2×C418yes—
122C22×C42C2×C418yes—
123C22×C42C2×C418yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#387—216no—
143#380—28no—
144#380—28no—
145#380—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—44no—
148C4×C4⋊C4—44no—
149C4×C4⋊C4C2×C418yes—
150C4×C4⋊C4C2×C418yes—
151C4×C4⋊C4C2×C418yes—
152C4×C4⋊C4C2×C418yes—
153C4×C4⋊C4C2×C418yes—
154C4×C4⋊C4C2×C418yes—
155C4×C4⋊C4C2×C418yes—
156C4×C4⋊C4C2×C418yes—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4Q818yes—
163#376—28no—
164#376—44no—
165C2×#95C2×C418yes—
166C2×#95C2×C418yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2312yes—
198C43—44no—
199C25C2411yescharacteristic
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C2×Q814yes—
227C23×C4#4114yes—
228C23×C4#4118yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×C4⋊C4—232no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—232no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C414yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C414yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4#4118yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4C2×Q818yes—
285C2×C22⋊C4#4118yes—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—232no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—416no—
309C2×C42—416no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42C22×C414yes—
318C2×C42C22×C414yes—
319C2×C42C22×C414yes—
320C2×C42C22×C414yes—
321C2×C42C22×C414yes—
322C2×C42C22×C414yes—
323C2×C42C22×C414yes—
324C2×C42C22×C414yes—
325C2×C42C22×C414yes—
326C2×C42C22×C414yes—
327C2×C42C22×C414yes—
328C2×C42C22×C414yes—
329C2×C42C2×Q818yes—
330C2×C42#4118yes—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95C22×C414yes—
354#95C22×C414yes—
355#95C22×C414yes—
356#95C22×C414yes—
357#95C22×C414yes—
358#95C22×C414yes—
359#95C22×C414yes—
360#95C22×C414yes—
361#95C2×Q818yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24C23×C412yes—
367C24C22×Q812yes—
368C24C2×#4114yes—
369C24C2×#4112yes—
370C24C2×#4114yes—
371C24C2511yescharacteristic
372C22×C4—232no—
373C22×C4—28no—
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—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4#117116yes—
457C22×C4C4×Q8116yes—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C412yes—
471C22×C4C23×C412yes—
472C22×C4C22×Q818yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4118yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—416no—
497C4⋊C4#117116yes—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C42—28no—
503C42—28no—
504C42—28no—
505C42—28no—
506C42—28no—
507C42—28no—
508C42—28no—
509C42—216no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—416no—
515C42—416no—
516C42—48no—
517C42—48no—
518C42—48no—
519C42#117116yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23C2×#11714yes—
533C23C2×C4×Q814yes—
534C23C4×#4114yes—
535C23#52918yes—
536C23#53318yes—
537C23C24×C411yescharacteristic
538C23C23×Q811yescharacteristic
539C23C22×#4112yes—
540C23C22×#4112yes—
541C23C22×#4112yes—
542C23C22×#4111yescharacteristic
543C23C22×#4112yes—
544C23C2×H(5,𝔽2)12yes—
545C23#58512yes—
546C2×C4—232no—
547C2×C4—22no—
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—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—232no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
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—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4C2×#11714yes—
627C2×C4C2×#11714yes—
628C2×C4C2×#11714yes—
629C2×C4C2×#11714yes—
630C2×C4C2×#11714yes—
631C2×C4C2×#11714yes—
632C2×C4C2×#11714yes—
633C2×C4C2×#11714yes—
634C2×C4C2×C4×Q814yes—
635C2×C4C2×C4×Q814yes—
636C2×C4#51814yes—
637C2×C4#55318yes—
638C2×C4#55418yes—
639C22—24no—
640C22—216no—
641C22—216no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22#229518yes—
648C22C22×#11711yescharacteristic
649C22C22×#11711yescharacteristic
650C22C22×#11711yescharacteristic
651C22C22×C4×Q811yescharacteristic
652C22C2×C4×#4111yescharacteristic
653C22C2×#51811yescharacteristic
654C22#342811yescharacteristic
655C22C2×#52912yes—
656C22C2×#53312yes—
657C22#344414yes—
658C22#345914yes—
659C22C2×#55312yes—
660C22C2×#55412yes—
661C22#347514yes—
662C22Q8×#4112yes—
663C22#348012yes—
664C22#348114yes—
665C4—24no—
666C4—24no—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—48no—
676C4—48no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—84no—
681C4—84no—
682C4#234518yes—
683C2—28no—
684C2#3401211yescharacteristic
685C2C2×#229511yescharacteristic
686C2#3405711yescharacteristic
687C2C2×#234511yescharacteristic
688C2#3410911yescharacteristic
689C2Q8×#11711yescharacteristic
690C2#3412611yescharacteristic
691C2#3484014yes—
692C2#3485614yes—
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13934C2semidirect
3. #13932C2semidirect
4. #13930C2semidirect
5. #13928C2semidirect
10. #13848C2semidirect
11. #13846C2semidirect
15. C4×#387C2semidirect
16. #13759C2semidirect
29. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
#139342
#139322
#139302
#139282
#139232
#139082
#138832
#138822
#138482
#138462
C4×#3802
C4×#3872
#137592
#137532
C42×C22⋊C42