Finite Group Databasebeta

← #524654

Subgroups — #524654

Order 512.

4106 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3C2×#2293C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic, complemented
5#14364C218yesmaximal, complemented
6#14258C212yesmaximal, complemented
7#14161C211yesmaximal, characteristic, complemented
8#14138C212yesmaximal, complemented
9#14054C214yesmaximal, complemented
10#13926C211yesmaximal, characteristic, complemented
11#13916C214yesmaximal
12#13904C212yesmaximal, complemented
13#13795C212yesmaximal, complemented
14C4×#391C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554C4116yescomplemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2212yes—
24C22×C4×D4C2211yescharacteristic
25C22×#117C2212yes—
26C22×#117C2212yes—
27C23×C4⋊C4C2212yes—
28#2354—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2214yes—
32C2×#391C2212yes—
33#2293—28no—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#384C2211yescharacteristic
40C2×#381C2218yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C43C2211yescharacteristic
47#1446C2214yes—
48#1445C2218yes—
49#1441C2214yes—
50#1441C2218yes—
51#1440C2218yes—
52#1440C2218yes—
53#1436C2214yes—
54#1436C2218yes—
55C4⋊#95C2218yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C22⋊#95C2218yes—
67C4×#95C2214yes—
68C23×Q8C2311yescharacteristic
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2314yes—
73#554—2128no—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8C2×C418yes—
77C2×#125—28no—
78C2×#125—28no—
79C2×#125C2×C4116yes—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8C2×C4116yes—
85C2×C22⋊Q8C2×C4132yes—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8C2×C418yes—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4C2×C4116yes—
92C2×C4×D4C2×C418yes—
93C2×C4×D4D418yes—
94C2×C4×D4D418yes—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—44no—
100C2×#117—44no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4C2×C4116yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#391—24no—
134#391—24no—
135#391—48no—
136#391—44no—
137#390—28no—
138#390—28no—
139#389—28no—
140#388—216no—
141#388—216no—
142#384—24no—
143#381—416no—
144#380—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—28no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C43—24no—
164C25C2412yes—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8C22×C414yes—
176C22×Q8C22×C418yes—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4C22×C414yes—
180C22×D4C2×D414yes—
181C22×D4C2×D414yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—48no—
189C23×C4—48no—
190C23×C4C22×C418yes—
191C23×C4C22×C4116yes—
192C23×C4C2×D418yes—
193C23×C4C2×D418yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C4⋊Q8—232no—
208#125—264no—
209C22⋊Q8—264no—
210C22⋊Q8—2128no—
211C4×Q8—232no—
212C4×D4—264no—
213C4×D4C22⋊C4164yes—
214#117—432no—
215#117—432no—
216C2×C4⋊C4—264no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C22×C4116yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C4116yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4C22×C4116yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C2×D418yes—
273C2×C22⋊C4C2×D418yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—22no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42C22×C418yes—
296C2×C42C22×C414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42#4114yes—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—44no—
309#95—48no—
310#95—48no—
311#95—48no—
312#95—48no—
313#95—48no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24C23×C414yes—
323C24C22×D414yes—
324C24C22×D414yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—216no—
328C2×Q8—28no—
329C2×Q8—28no—
330C2×Q8—416no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—88no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×D4—216no—
339C2×D4—28no—
340C2×D4—216no—
341C2×D4—28no—
342C2×D4C2×C22⋊C4116yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—232no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—432no—
404C22×C4—816no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4C2×C22⋊C4132yes—
408C22×C4C23×C414yes—
409C22×C4C23×C414yes—
410C22×C4C23×C418yes—
411C22×C4C23×C412yes—
412C22×C4C23×C414yes—
413C22×C4C23×C412yes—
414C22×C4C23×C412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+414yes—
430C22×C42−1+414yes—
431C22×C42−1+412yes—
432C4⋊C4—28no—
433C4⋊C4—232no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—232no—
438C4⋊C4—28no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—84no—
445C4⋊C4C2×C22⋊C4116yes—
446C22⋊C4—232no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—88no—
451C22⋊C4C2×C22⋊C4132yes—
452C42—216no—
453C42—28no—
454C42—48no—
455C42—416no—
456C42—48no—
457C42—48no—
458C42—48no—
459C42—44no—
460C42C2×C22⋊C4116yes—
461C23—24no—
462C23—28no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23C22×C22⋊C418yes—
484C23C24×C411yescharacteristic
485C23C23×D411yescharacteristic
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×2−1+412yes—
492C23C2×2−1+412yes—
493C23C2×2−1+411yescharacteristic
494C23#58512yes—
495Q8—416no—
496Q8—816no—
497D4—232no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—264no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—216no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—832no—
563C2×C4C22×C22⋊C414yes—
564C2×C4C22×C22⋊C418yes—
565C2×C4C22×C22⋊C414yes—
566C2×C4C22×C22⋊C414yes—
567C2×C4C4×#4114yes—
568C2×C4#51918yes—
569C2×C4#52014yes—
570C2×C4#53614yes—
571C2×C4#53618yes—
572C2×C4#53614yes—
573C2×C4#54614yes—
574C2×C4#54614yes—
575C2×C4#54714yes—
576C2×C4#54714yes—
577C22—28no—
578C22—28no—
579C22—28no—
580C22—48no—
581C22—48no—
582C22—44no—
583C22—44no—
584C22—48no—
585C22—44no—
586C22—44no—
587C22—48no—
588C22C23×C22⋊C411yescharacteristic
589C22C2×C4×#4111yescharacteristic
590C22C2×#51912yes—
591C22C2×#52011yescharacteristic
592C22C4×H(5,𝔽2)12yes—
593C22C2×#53612yes—
594C22C2×#53611yescharacteristic
595C22C2×#53611yescharacteristic
596C22C2×#54611yescharacteristic
597C22C2×#54611yescharacteristic
598C22C2×#54711yescharacteristic
599C22C2×#54711yescharacteristic
600C22#348812yes—
601C22#349811yescharacteristic
602C22#350011yescharacteristic
603C22#350712yes—
604C22#351112yes—
605C22#351512yes—
606C22#352212yes—
607C22#354412yes—
608C22#354712yes—
609C22#354812yes—
610C22#355512yes—
611C4—28no—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—48no—
619C4—84no—
620C4—84no—
621C4—82no—
622C4#231018yes—
623C4#233918yes—
624C2—48no—
625C2C2×#231011yescharacteristic
626C2C2×#233911yescharacteristic
627C2#3425811yescharacteristic
628C2#3431712yes—
629C2#3448412yes—
630C2#3892112yes—
631C2#3895211yescharacteristic
632C2#3895511yescharacteristic
633C2#3947012yes—
634C2#3948911yescharacteristic
635C2#3949011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14714C2semidirect
5. #14364C2semidirect
6. #14258C2semidirect
7. #14161C2semidirect
8. #14138C2semidirect
9. #14054C2semidirect
10. #13926C2semidirect
12. #13904C2semidirect
13. #13795C2semidirect
14. C4×#391C2semidirect
17. C2×#554C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#23542
C2×#22932
#147142
#143642
#142582
#141612
#141382
#140542
#139262
#139162
#139042
#137952
C4×#3912