Finite Group Databasebeta

← #524574

Subgroups — #524574

Order 512.

2674 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3C2×#2295C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic
5#14465C214yesmaximal
6#14441C214yesmaximal
7#14276C212yesmaximal
8#14151C212yesmaximal
9#14151C211yesmaximal, characteristic
10#14045C212yesmaximal
11#14005C212yesmaximal
12#13954C212yesmaximal
13#13935C212yesmaximal
14#13926C212yesmaximal
15#13910C211yesmaximal, characteristic
16#13783C212yesmaximal
17C4×#389C211yesmaximal, characteristic
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20#2355—28no—
21Q8×C4⋊C4C4116yescomplemented
22#2295—28no—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#387C2214yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×#376C2212yes—
57C2×C43C2211yescharacteristic
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1436C2212yes—
69#1436C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2211yescharacteristic
84#1434C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C2×C4×Q8—44no—
88C2×C4×Q8—44no—
89C2×C4×Q8C2×C4116yes—
90C2×C4×Q8C2×C418yes—
91C2×C4×Q8D418yes—
92C2×C4×Q8Q818yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#391—28no—
131#391—24no—
132#391—24no—
133#389—24no—
134#389—24no—
135#389—24no—
136#389—28no—
137#389—48no—
138#389—44no—
139#389—44no—
140#389—44no—
141#389—44no—
142#389—44no—
143#389—44no—
144#389C2×C4116yes—
145#389C2×C4132yes—
146#389Q818yescomplemented
147#387—216no—
148#387—28no—
149#387—28no—
150#384—24no—
151#384—44no—
152#384C2×C418yes—
153#384C2×C4116yes—
154#382—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—44no—
161C4×C4⋊C4C2×C418yes—
162C4×C4⋊C4C2×C4116yes—
163#376—28no—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C43—24no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8C22×C414yes—
197C22×Q8C2×D414yes—
198C22×Q8C2×Q814yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4×Q8—432no—
217C4×Q8C4⋊C4164yescomplemented
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C22×C418yes—
274C2×C4⋊C4C22×C414yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C4116yes—
277C2×C4⋊C4C22×C4116yes—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C414yes—
281C2×C4⋊C4C22×C414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4C2×Q818yes—
286C2×C4⋊C4C2×Q818yes—
287C2×C4⋊C4C2×Q814yes—
288C2×C4⋊C4C2×Q814yes—
289C2×C4⋊C4#4114yes—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—41no—
317C2×C42—41no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C22×C418yes—
321C2×C42C22×C4116yes—
322C2×C42C22×C418yes—
323C2×C42C22×C414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326C2×C42C2×Q814yes—
327C2×C42C2×Q818yes—
328C2×C42C2×Q814yes—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95C22×C418yes—
354#95C22×C4116yes—
355#95C2×Q818yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8C2×C4⋊C4116yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4C23×C414yes—
451C22×C4C23×C418yes—
452C22×C4C23×C412yes—
453C22×C4C23×C414yes—
454C22×C4C23×C414yes—
455C22×C4C23×C414yes—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C22×C42−1+412yes—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—84no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C4⋊C4C2×C4⋊C4132yes—
505C4⋊C4C2×C4⋊C4116yes—
506C4⋊C4C4×Q8116yes—
507C42—216no—
508C42—28no—
509C42—416no—
510C42—44no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—48no—
515C42—44no—
516C42—48no—
517C42—88no—
518C42C2×C4⋊C4132yes—
519C42C2×C4⋊C4116yes—
520C23C24×C411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C23×Q811yescharacteristic
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23#58511yescharacteristic
534C23#58511yescharacteristic
535Q8—232no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4C22×C4⋊C414yes—
638C2×C4C22×C4⋊C418yes—
639C2×C4C22×C4⋊C418yes—
640C2×C4C22×C4⋊C414yes—
641C2×C4C22×C4⋊C414yes—
642C2×C4C2×C4×Q818yes—
643C2×C4C2×C4×Q814yes—
644C2×C4#51814yes—
645C2×C4#51914yes—
646C2×C4#52014yes—
647C2×C4#53714yes—
648C2×C4#53714yes—
649C2×C4#54414yes—
650C2×C4#54414yes—
651C2×C4D4×Q814yes—
652C2×C4#55014yes—
653C2×C4#55714yes—
654C2×C4Q8214yes—
655C22C23×C4⋊C411yescharacteristic
656C22C22×C4×Q811yescharacteristic
657C22C2×#51811yescharacteristic
658C22C2×#51911yescharacteristic
659C22C2×#52011yescharacteristic
660C22C4×H(5,𝔽2)11yescharacteristic
661C22C4×2−1+411yescharacteristic
662C22C2×#53711yescharacteristic
663C22C2×#53711yescharacteristic
664C22C2×#54411yescharacteristic
665C22C2×#54411yescharacteristic
666C22C2×D4×Q811yescharacteristic
667C22C2×#55011yescharacteristic
668C22C2×#55711yescharacteristic
669C22C2×Q8211yescharacteristic
670C22#348811yescharacteristic
671C22#348911yescharacteristic
672C22#350111yescharacteristic
673C22#350211yescharacteristic
674C22#350311yescharacteristic
675C22#350311yescharacteristic
676C22#350311yescharacteristic
677C22#350411yescharacteristic
678C22#351411yescharacteristic
679C22#351811yescharacteristic
680C22#352011yescharacteristic
681C22#352211yescharacteristic
682C22#354311yescharacteristic
683C22#354611yescharacteristic
684C22#354811yescharacteristic
685C22#355211yescharacteristic
686C22#355411yescharacteristic
687C22#355411yescharacteristic
688C22#355511yescharacteristic
689C22#355611yescharacteristic
690C4—28no—
691C4—28no—
692C4—24no—
693C4—28no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—44no—
698C4—48no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—84no—
705C4#231618yes—
706C4Q8×C4⋊C418yes—
707C2C2×#231611yescharacteristic
708C2C2×Q8×C4⋊C411yescharacteristic
709C2#3425011yescharacteristic
710C2#3433411yescharacteristic
711C2#3433511yescharacteristic
712C2#3448811yescharacteristic
713C2#3449411yescharacteristic
714C2#3913611yescharacteristic
715C2#3913711yescharacteristic
716C2#3918011yescharacteristic
717C2#3918211yescharacteristic
718C2#3959811yescharacteristic
719C2#3959911yescharacteristic
720C2#3962511yescharacteristic
721C2#3962611yescharacteristic
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
21. Q8×C4⋊C4C4semidirect
146. #389Q8semidirect
217. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×Q8×C4⋊C42
C2×#22952
#147122
#144652
#144412
#142762
#141512
#140452
#140052
#139542
#139352
#139262
#139102
#137832
C4×#3892