Finite Group Databasebeta

← #676310

Subgroups — #676310

Order 512.

2498 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#15964C214yesmaximal
4#15333C214yesmaximal
5#15202C212yesmaximal
6#15194C212yesmaximal
7#15137C212yesmaximal
8#15133C212yesmaximal
9#14714C211yesmaximal, characteristic
10#14161C211yesmaximal, characteristic
11#14151C211yesmaximal, characteristic
12#14053C212yesmaximal
13#14045C212yesmaximal
14#14041C212yesmaximal
15#13926C212yesmaximal
16#13910C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#398C2214yes—
22#2369—28no—
23C4×C4⋊Q8—28no—
24C2×#391C2212yes—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C43C2211yescharacteristic
52#1446C2214yes—
53#1446C2214yes—
54#1445C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2212yes—
69#1436C2214yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8Q818yescomplemented
92C2×C4⋊Q8Q818yescomplemented
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132#398—48no—
133#391—28no—
134#391—216no—
135#391—44no—
136#391—44no—
137#391—48no—
138#389—24no—
139#389—24no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—24no—
144#389—28no—
145#389—44no—
146#389—44no—
147#389—44no—
148#389Q818yescomplemented
149#384—24no—
150#384—24no—
151#384—44no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—44no—
159C4×C4⋊C4D418yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C43—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8C2×Q818yes—
201C22×Q8C2×Q818yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—232no—
220C4⋊Q8—232no—
221C4×Q8—432no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
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⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×Q814yes—
286C2×C4⋊C4C2×Q818yes—
287C2×C4⋊C4C2×Q818yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C4⋊C4C2×Q814yes—
291C2×C4⋊C4C2×Q818yes—
292C2×C4⋊C4C2×Q814yes—
293C2×C4⋊C4#4114yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—41no—
320C2×C42—41no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D418yes—
324C2×C42C2×D414yes—
325C2×C42C2×Q814yes—
326C2×C42C2×Q814yes—
327C2×C42C2×Q814yes—
328C2×C42#4114yes—
329C2×C42#4114yes—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95C2×D418yes—
356#95C2×Q818yes—
357C24C2511yescharacteristic
358C2×Q8—216no—
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—84no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8—84no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q812yes—
466C22×C4C22×Q814yes—
467C22×C4C22×Q818yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C22×Q812yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C42−1+412yes—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C4⋊C4C22⋊Q8116yes—
515C42—28no—
516C42—216no—
517C42—28no—
518C42—28no—
519C42—44no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42—48no—
524C42—48no—
525C42—88no—
526C42C22⋊Q8116yes—
527C23C23×D411yescharacteristic
528C23C23×Q811yescharacteristic
529C23C23×Q811yescharacteristic
530C23C23×Q811yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+411yescharacteristic
540C23#58511yescharacteristic
541C23#58511yescharacteristic
542Q8—432no—
543Q8—816no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4C2×C22⋊Q814yes—
646C2×C4C2×C22⋊Q818yes—
647C2×C4C2×C22⋊Q814yes—
648C2×C4C2×C22⋊Q814yes—
649C2×C4C2×C22⋊Q814yes—
650C2×C4#53314yes—
651C2×C4#53714yes—
652C2×C4#54214yes—
653C2×C4#54614yes—
654C2×C4D4×Q814yes—
655C2×C4#55414yes—
656C2×C4#55714yes—
657C2×C4#55714yes—
658C2×C4#55714yes—
659C2×C4Q8214yes—
660C2×C4Q8214yes—
661C2×C4Q8214yes—
662C2×C4#55914yes—
663C22C22×C22⋊Q811yescharacteristic
664C22C22×C22⋊Q811yescharacteristic
665C22C2×#53311yescharacteristic
666C22C2×#53711yescharacteristic
667C22C2×#54211yescharacteristic
668C22C2×#54611yescharacteristic
669C22C2×D4×Q811yescharacteristic
670C22C2×#55411yescharacteristic
671C22#347511yescharacteristic
672C22C2×#55711yescharacteristic
673C22C2×#55711yescharacteristic
674C22C2×#55711yescharacteristic
675C22C2×Q8211yescharacteristic
676C22C2×Q8211yescharacteristic
677C22C2×Q8211yescharacteristic
678C22Q8×#4111yescharacteristic
679C22C2×#55911yescharacteristic
680C22#348811yescharacteristic
681C22#348811yescharacteristic
682C22#350111yescharacteristic
683C22#350111yescharacteristic
684C22#350111yescharacteristic
685C22#350111yescharacteristic
686C22#350211yescharacteristic
687C22#350311yescharacteristic
688C22#350611yescharacteristic
689C22#350811yescharacteristic
690C22#351011yescharacteristic
691C22#351111yescharacteristic
692C22#351911yescharacteristic
693C22#352211yescharacteristic
694C22#354711yescharacteristic
695C22#354811yescharacteristic
696C22#354911yescharacteristic
697C22#355511yescharacteristic
698C4—28no—
699C4—24no—
700C4—24no—
701C4—24no—
702C4—44no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—42no—
707C4—44no—
708C4—48no—
709C4—82no—
710C4—84no—
711C4—82no—
712C4—82no—
713C4#245118yes—
714C4#258218yes—
715C2C2×#245111yescharacteristic
716C2C2×#258211yescharacteristic
717C2#3551511yescharacteristic
718C2#3615411yescharacteristic
719C2#3647611yescharacteristic
720C2Q8×C22⋊Q811yescharacteristic
721C2#3656311yescharacteristic
722C2#3920111yescharacteristic
723C2#3946911yescharacteristic
724C2#3959811yescharacteristic
725C2#3964311yescharacteristic
726C2#5966511yescharacteristic
727C2#5966811yescharacteristic
728C2#5987111yescharacteristic
729C2#5992411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
91. C2×C4⋊Q8Q8semidirect
92. C2×C4⋊Q8Q8semidirect
148. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×C4×C4⋊Q82
#159642
#153332
#152022
#151942
#151372
#151332
#147142
#141612
#141512
#140532
#140452
#140412
#139262
#139102
C4×#3892