Finite Group Databasebeta

← #534550

Subgroups — #534550

Order 512.

2838 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#14502C214yesmaximal
5#14113C214yesmaximal
6#14045C212yesmaximal
7#14017C212yesmaximal
8#14006C212yesmaximal
9#13978C212yesmaximal
10#13942C212yesmaximal
11#13933C212yesmaximal
12#13911C211yesmaximal, characteristic
13#13911C211yesmaximal, characteristic
14Q8×#95C212yesmaximal
15#13804C212yesmaximal
16C4×#389C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22#2365—28no—
23#2365C4116yescomplemented
24C4×C4⋊Q8—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#1444C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2214yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81#1434C2214yes—
82#1434C2214yes—
83C4×#95C2214yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C2×C4⋊Q8C2×C418yes—
88C2×C4⋊Q8D418yes—
89C2×C4⋊Q8Q818yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—44no—
95C2×C4×Q8—44no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×Q8C2×C4116yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133#391—28no—
134#391—28no—
135#391C2×C4116yes—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#389C2×C4116yes—
144#387—216no—
145#387C2×C4132yes—
146#384—24no—
147#382—28no—
148#382—44no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—44no—
156C4×C4⋊C4C2×C418yes—
157C4×C4⋊C4C2×C418yes—
158C4×C4⋊C4C2×C4116yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C43—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8C22×C418yes—
195C22×Q8C2×D418yes—
196C22×Q8C2×Q818yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8C4⋊C4164yescomplemented
215C4×Q8—432no—
216C4×Q8—432no—
217C4×Q8—432no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C4116yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×Q818yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C4⋊C4#4114yes—
277C2×C4⋊C4#4114yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—41no—
311C2×C42—41no—
312C2×C42C22×C418yes—
313C2×C42C22×C418yes—
314C2×C42C22×C4116yes—
315C2×C42C22×C414yes—
316C2×C42C2×D414yes—
317C2×C42C2×Q814yes—
318C2×C42#4114yes—
319C2×C42#4114yes—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—42no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95C22×C418yes—
347#95C22×C418yes—
348#95C22×C4116yes—
349#95C22×C4116yes—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—216no—
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8C2×C4⋊C4132yes—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4C23×C414yes—
446C22×C4C23×C414yes—
447C22×C4C23×C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C23×C418yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D418yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q818yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C22×C42−1+412yes—
476C22×C42−1+412yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4C2×C4⋊C4132yes—
496C4⋊C4C2×C4⋊C4132yes—
497C42—28no—
498C42—216no—
499C42—216no—
500C42—28no—
501C42—28no—
502C42—416no—
503C42—416no—
504C42—416no—
505C42—44no—
506C42—44no—
507C42—48no—
508C42—44no—
509C42—48no—
510C42C2×C4⋊C4116yes—
511C23C24×C411yescharacteristic
512C23C23×D411yescharacteristic
513C23C23×Q811yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23#58511yescharacteristic
525C23#58511yescharacteristic
526Q8—264no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4C22×C4⋊C414yes—
630C2×C4C22×C4⋊C4116yes—
631C2×C4C22×C4⋊C414yes—
632C2×C4C22×C4⋊C414yes—
633C2×C4C4×#4114yes—
634C2×C4C4×#4114yes—
635C2×C4#51814yes—
636C2×C4#51914yes—
637C2×C4#54614yes—
638C2×C4#54614yes—
639C2×C4#54814yes—
640C2×C4#54814yes—
641C2×C4#55414yes—
642C2×C4#55414yes—
643C2×C4#55614yes—
644C2×C4#55714yes—
645C2×C4#55714yes—
646C22C23×C4⋊C411yescharacteristic
647C22C2×C4×#4111yescharacteristic
648C22C2×C4×#4111yescharacteristic
649C22C2×#51811yescharacteristic
650C22C2×#51911yescharacteristic
651C22C4×2−1+411yescharacteristic
652C22C4×2−1+411yescharacteristic
653C22C2×#54611yescharacteristic
654C22C2×#54611yescharacteristic
655C22C2×#54811yescharacteristic
656C22C2×#54811yescharacteristic
657C22C2×#55411yescharacteristic
658C22C2×#55411yescharacteristic
659C22C2×#55611yescharacteristic
660C22#347511yescharacteristic
661C22C2×#55711yescharacteristic
662C22C2×#55711yescharacteristic
663C22#348011yescharacteristic
664C22#348211yescharacteristic
665C22#348611yescharacteristic
666C22#348711yescharacteristic
667C22#350211yescharacteristic
668C22#350211yescharacteristic
669C22#350611yescharacteristic
670C22#350711yescharacteristic
671C22#350911yescharacteristic
672C22#351111yescharacteristic
673C22#351511yescharacteristic
674C22#351711yescharacteristic
675C22#352011yescharacteristic
676C22#352311yescharacteristic
677C22#354711yescharacteristic
678C22#355011yescharacteristic
679C22#355211yescharacteristic
680C22#355411yescharacteristic
681C4—216no—
682C4—24no—
683C4—24no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4#234918yes—
697C4#235118yes—
698C2C2×#234911yescharacteristic
699C2C2×#235111yescharacteristic
700C2C4×#55611yescharacteristic
701C2#3424911yescharacteristic
702C2#3431611yescharacteristic
703C2#3432811yescharacteristic
704C2#3448711yescharacteristic
705C2#5903411yescharacteristic
706C2#5904211yescharacteristic
707C2#5907811yescharacteristic
708C2#5913411yescharacteristic
709C2#5945611yescharacteristic
710C2#5946711yescharacteristic
711C2#5966611yescharacteristic
712C2#5987011yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #2365C4semidirect
214. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×C4×C4⋊Q82
#145022
#141132
#140452
#140172
#140062
#139782
#139422
#139332
#139112
Q8×#952
#138042
C4×#3892