Finite Group Databasebeta

← #534390

Subgroups — #534390

Order 512.

3142 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×#2351C212yesmaximal
4C2×C4×C4⋊Q8C211yesmaximal, characteristic
5#14404C214yesmaximal
6#14117C214yesmaximal
7#14046C212yesmaximal
8#14017C212yesmaximal
9#14005C212yesmaximal
10#13978C212yesmaximal
11#13935C212yesmaximal
12#13918C212yesmaximal
13#13911C211yesmaximal, characteristic
14#13910C211yesmaximal, characteristic
15C4⋊C42C212yesmaximal
16C4×#389C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22#2365—28no—
23#2365C4116yescomplemented
24#2351—216no—
25C4×C4⋊Q8—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2214yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#382C2214yes—
46C2×#382C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2214yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C43C2211yescharacteristic
59#1444C2214yes—
60#1444C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2212yes—
73#1436C2212yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84#1434C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8C2×C418yes—
89C2×C4⋊Q8D418yes—
90C2×C4⋊Q8Q818yes—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C2×C4×Q8C2×C4116yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#391—28no—
137#391—28no—
138#391C2×C4116yes—
139#389—28no—
140#389—28no—
141#389—216no—
142#389—28no—
143#389—28no—
144#389—24no—
145#389—24no—
146#389—24no—
147#389—216no—
148#389—44no—
149#389C2×C4116yes—
150#389D418yes—
151#387—216no—
152#387C2×C4132yes—
153#384—28no—
154#384—24no—
155#382—28no—
156#382—216no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—216no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4C2×C418yes—
166C4×C4⋊C4C2×C418yes—
167C4×C4⋊C4C2×C4116yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C43—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8C22×C418yes—
202C22×Q8C2×D418yes—
203C22×Q8C2×Q818yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8C4⋊C4164yescomplemented
222C4×Q8—264no—
223C4×Q8—432no—
224C4×Q8—432no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C22×C4116yes—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C22×C414yes—
278C2×C4⋊C4C22×C414yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C22×C414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×Q818yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4#4114yes—
291C2×C42—22no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—41no—
324C2×C42—41no—
325C2×C42C22×C4116yes—
326C2×C42C22×C418yes—
327C2×C42C22×C418yes—
328C2×C42C22×C414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×Q814yes—
332C2×C42#4114yes—
333#95—24no—
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—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—42no—
352#95—44no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95C22×C418yes—
357#95C22×C418yes—
358#95C22×C4116yes—
359#95C22×C4116yes—
360#95C2×D418yes—
361C24C2511yescharacteristic
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—216no—
367C2×Q8—216no—
368C2×Q8C2×C4⋊C4132yes—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4C23×C414yes—
455C22×C4C23×C414yes—
456C22×C4C23×C414yes—
457C22×C4C23×C412yes—
458C22×C4C23×C418yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D418yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q818yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C22×C42−1+412yes—
488C4⋊C4—232no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—88no—
507C4⋊C4C2×C4⋊C4132yes—
508C4⋊C4C2×C4⋊C4132yes—
509C4⋊C4C4×D4116yes—
510C42—28no—
511C42—232no—
512C42—216no—
513C42—216no—
514C42—28no—
515C42—28no—
516C42—416no—
517C42—416no—
518C42—44no—
519C42—48no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42C2×C4⋊C4116yes—
524C23C24×C411yescharacteristic
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C23×Q811yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+411yescharacteristic
534C23C2×2−1+411yescharacteristic
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+411yescharacteristic
537C23#58511yescharacteristic
538C23#58511yescharacteristic
539Q8—264no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
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—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4C22×C4⋊C414yes—
642C2×C4C22×C4⋊C4116yes—
643C2×C4C22×C4⋊C414yes—
644C2×C4C22×C4⋊C414yes—
645C2×C4C2×C4×D418yes—
646C2×C4C2×C4×D414yes—
647C2×C4C4×#4114yes—
648C2×C4#51914yes—
649C2×C4#52014yes—
650C2×C4D4214yes—
651C2×C4#54714yes—
652C2×C4#54714yes—
653C2×C4#54814yes—
654C2×C4#54814yes—
655C2×C4D4×Q814yes—
656C2×C4#55414yes—
657C2×C4#55714yes—
658C2×C4#55714yes—
659C22C23×C4⋊C411yescharacteristic
660C22C22×C4×D411yescharacteristic
661C22C2×C4×#4111yescharacteristic
662C22C2×#51911yescharacteristic
663C22C2×#52011yescharacteristic
664C22C4×2−1+411yescharacteristic
665C22C4×2−1+411yescharacteristic
666C22C2×D4211yescharacteristic
667C22C2×#54711yescharacteristic
668C22C2×#54711yescharacteristic
669C22C2×#54811yescharacteristic
670C22C2×#54811yescharacteristic
671C22C2×D4×Q811yescharacteristic
672C22D4×#4111yescharacteristic
673C22C2×#55411yescharacteristic
674C22#347511yescharacteristic
675C22C2×#55711yescharacteristic
676C22C2×#55711yescharacteristic
677C22#348211yescharacteristic
678C22#348611yescharacteristic
679C22#348711yescharacteristic
680C22#349711yescharacteristic
681C22#349911yescharacteristic
682C22#350211yescharacteristic
683C22#350211yescharacteristic
684C22#350711yescharacteristic
685C22#351111yescharacteristic
686C22#351411yescharacteristic
687C22#351811yescharacteristic
688C22#352311yescharacteristic
689C22#352311yescharacteristic
690C22#355211yescharacteristic
691C22#355411yescharacteristic
692C22#355511yescharacteristic
693C22#355611yescharacteristic
694C4—216no—
695C4—28no—
696C4—24no—
697C4—24no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4D4×C4⋊C418yes—
710C4#235118yescomplemented
711C2C2×D4×C4⋊C411yescharacteristic
712C2C2×#235111yescharacteristic
713C2C4×#54711yescharacteristic
714C2#3424911yescharacteristic
715C2#3431111yescharacteristic
716C2#3432211yescharacteristic
717C2#3448811yescharacteristic
718C2#5906511yescharacteristic
719C2#5907111yescharacteristic
720C2#5908011yescharacteristic
721C2#5912511yescharacteristic
722C2#5945611yescharacteristic
723C2#5946411yescharacteristic
724C2#5966711yescharacteristic
725C2#5982811yescharacteristic
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
221. C4⋊Q8C4⋊C4semidirect
710. C4#2351semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#23512
C2×C4×C4⋊Q82
#144042
#141172
#140462
#140172
#140052
#139782
#139352
#139182
#139112
#139102
C4⋊C422
C4×#3892