Finite Group Databasebeta

← #674620

Subgroups — #674620

Order 512.

3534 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#2548C211yesmaximal, characteristic
3C2×#2545C212yesmaximal, complemented
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
6#15615C218yesmaximal, complemented
7#15226C212yesmaximal, complemented
8#15134C214yesmaximal, complemented
9#15134C212yesmaximal, complemented
10#14713C211yesmaximal, characteristic
11#14026C212yesmaximal, complemented
12#14022C214yesmaximal, complemented
13#13926C211yesmaximal, characteristic, complemented
14C4×#389C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#124C2212yes—
18C22×#124C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#2548—28no—
23#2548—28no—
24#2545—216no—
25C2×#396C2218yes—
26C2×#396C2218yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29Q8×C4⋊C4—28no—
30C4×C4⋊Q8—28no—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2214yes—
38C2×#390C2212yes—
39C2×#390C2214yes—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2214yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2214yes—
46C2×#389C2214yes—
47C2×#388C2218yes—
48C2×#384C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2214yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#1441C2214yes—
58#1440C2218yes—
59#1439C2218yes—
60#1439C2218yes—
61#1439C2218yes—
62#1437C2218yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4×#95C2212yes—
68C23×Q8C2311yescharacteristic
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8D418yescomplemented
76C2×#127—24no—
77C2×#127—24no—
78C2×#127—24no—
79C2×#124—28no—
80C2×#124—28no—
81C2×#124—28no—
82C2×#124—28no—
83C2×#124—28no—
84C2×#124—44no—
85C2×C4×Q8—28no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—28no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—44no—
93C2×C4×Q8D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2318yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#396—232no—
128#396—416no—
129#394—216no—
130#394—216no—
131#391—24no—
132#391—28no—
133#391—24no—
134#390—28no—
135#390—28no—
136#390—28no—
137#390—28no—
138#390—28no—
139#390—216no—
140#390—216no—
141#390—48no—
142#390—44no—
143#389—216no—
144#389—28no—
145#389—28no—
146#389—216no—
147#388—232no—
148#384—24no—
149#384—28no—
150#384—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—216no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4D418yes—
156C2×#95C2312yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C43—24no—
167C25C2414yes—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8C2×D414yes—
179C22×Q8C2×D414yes—
180C22×Q8C2×D414yes—
181C22×Q8#4114yes—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4—28no—
196C22×D4—24no—
197C22×D4—44no—
198C22×D4—44no—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4⋊Q8—232no—
213#127—416no—
214#124—432no—
215#124—432no—
216C4×Q8—232no—
217C4×Q8—432no—
218C4×Q8—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×D4116yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—44no—
303C2×C42—42no—
304C2×C42C2×D418yes—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4118yes—
310C2×C42#4114yes—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—216no—
317#95—28no—
318#95—44no—
319#95—48no—
320#95C2×D418yes—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—28no—
326C24—24no—
327C24—28no—
328C24—24no—
329C24—28no—
330C24—24no—
331C24—28no—
332C24C2511yescharacteristic
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8#121116yes—
347C2×D4—416no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—416no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—88no—
362C2×D4—84no—
363C2×D4—84no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
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—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C42−1+414yes—
440C22×C42−1+412yes—
441C4⋊C4—216no—
442C4⋊C4—28no—
443C4⋊C4—232no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—416no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—432no—
459C22⋊C4—432no—
460C22⋊C4—816no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—816no—
464C42—216no—
465C42—28no—
466C42—216no—
467C42—216no—
468C42—48no—
469C42—416no—
470C42—48no—
471C42—48no—
472C42—44no—
473C42#121116yes—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—44no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—48no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23C23×D411yescharacteristic
498C23C23×D411yescharacteristic
499C23C23×D411yescharacteristic
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4112yes—
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×2−1+412yes—
506C23C2×2−1+411yescharacteristic
507C23#58512yes—
508Q8—232no—
509Q8—416no—
510D4—816no—
511D4—816no—
512D4—816no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4C2×#12114yes—
576C2×C4C2×#12118yes—
577C2×C4C2×#12114yes—
578C2×C4C2×#12114yes—
579C2×C4C2×#12114yes—
580C2×C4#53214yes—
581C2×C4#53614yes—
582C2×C4#53714yes—
583C2×C4D4214yes—
584C2×C4#54714yes—
585C2×C4#54814yes—
586C2×C4#54818yes—
587C2×C4#55018yes—
588C2×C4#55314yes—
589C2×C4#55618yes—
590C22—84no—
591C22—88no—
592C22—84no—
593C22—88no—
594C22—84no—
595C22—84no—
596C22—88no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—88no—
601C22C22×#12111yescharacteristic
602C22C22×#12111yescharacteristic
603C22C2×#53211yescharacteristic
604C22C2×#53611yescharacteristic
605C22C2×#53711yescharacteristic
606C22C2×D4211yescharacteristic
607C22C2×#54711yescharacteristic
608C22C2×#54811yescharacteristic
609C22C2×#54812yes—
610C22C2×#55012yes—
611C22C2×#55311yescharacteristic
612C22C2×#55612yes—
613C22#348012yes—
614C22#348812yes—
615C22#349712yes—
616C22#349811yescharacteristic
617C22#350011yescharacteristic
618C22#350612yes—
619C22#350712yes—
620C22#350812yes—
621C22#352212yes—
622C22#355012yes—
623C22#355412yes—
624C4—28no—
625C4—24no—
626C4—28no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—84no—
633C4—82no—
634C4—84no—
635C4#241618yes—
636C4#254518yescomplemented
637C2—164no—
638C2C2×#241611yescharacteristic
639C2C2×#254511yescharacteristic
640C2#3553311yescharacteristic
641C2#3612811yescharacteristic
642C2#3653811yescharacteristic
643C2#3657612yes—
644C2#3916811yescharacteristic
645C2#3936311yescharacteristic
646C2#3972612yes—
647C2#5967912yes—
648C2#5982112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2545C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
5. C2×C4×C4⋊Q8C2semidirect
6. #15615C2semidirect
7. #15226C2semidirect
8. #15134C2semidirect
9. #15134C2semidirect
11. #14026C2semidirect
12. #14022C2semidirect
13. #13926C2semidirect
14. C4×#389C2semidirect
75. C2×C4⋊Q8D4semidirect
636. C4#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#25452
C2×Q8×C4⋊C42
C2×C4×C4⋊Q82
#156152
#152262
#151342
#147132
#140262
#140222
#139262
C4×#3892