Finite Group Databasebeta

← #531097

Subgroups — #531097

Order 512.

3524 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3C2×C4×#125C211yesmaximal, characteristic, complemented
4#14714C211yesmaximal, characteristic, complemented
5#14493C214yesmaximal
6#14474C214yesmaximal, complemented
7#14340C211yesmaximal, characteristic
8#14274C211yesmaximal, characteristic, complemented
9#14101C214yesmaximal
10#14054C214yesmaximal, complemented
11#13997C212yesmaximal
12#13997C212yesmaximal
13#13935C212yesmaximal, complemented
14#13817C212yesmaximal, complemented
15#13761C211yesmaximal, characteristic
16C2×#541—28no—
17C2×#541—28no—
18C2×#541C4116yescomplemented
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2214yes—
21C22×#125C2214yes—
22C22×#125C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×#117C2211yescharacteristic
26#2330—28no—
27C4×#125—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2211yescharacteristic
46C2×C43C2211yescharacteristic
47#1445C2214yes—
48#1445C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1437C2212yes—
59#1437C2214yes—
60#1437C2212yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C4×#95C2214yes—
74C4×#95C2211yescharacteristic
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C24×C4C2311yescharacteristic
78#541—2128no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8C2×C418yes—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126C2×C4132yes—
85C2×#125—216no—
86C2×#125—216no—
87C2×#125C2×C4132yes—
88C2×#125C2×C418yes—
89C2×#125D418yes—
90C2×#125D418yes—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8C2×C4116yes—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8C2×C4116yes—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117C2×C418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128#391—28no—
129#391—28no—
130#391—44no—
131#391—44no—
132#389—28no—
133#389—216no—
134#389—28no—
135#387—216no—
136#387—216no—
137#384—24no—
138#382—24no—
139#380—44no—
140C4×C4⋊C4—216no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145#376—24no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C43—24no—
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8C22×C418yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4C22×C414yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C4⋊Q8—232no—
202#126—2128no—
203#125—2128no—
204#125C22⋊C4164yescomplemented
205C22⋊Q8—264no—
206C4×Q8—264no—
207#117—232no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C418yes—
232C2×C4⋊C4C22×C4116yes—
233C2×C4⋊C4C22×C4116yes—
234C2×C4⋊C4C22×C4116yes—
235C2×C4⋊C4C22×C418yes—
236C2×C4⋊C4C22×C4116yes—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D4116yes—
240C2×C4⋊C4C2×D4116yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C22×C418yes—
256C2×C22⋊C4C22×C4116yes—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42C22×C414yes—
284C2×C42C22×C4116yes—
285C2×C42C22×C414yes—
286C2×C42C2×D414yes—
287C2×C42C2×D414yes—
288C2×C42#4114yes—
289C2×C42#4114yes—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24C23×C412yes—
326C24C2511yescharacteristic
327C2×Q8—216no—
328C2×Q8—216no—
329C2×Q8—216no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4C23×C414yes—
411C22×C4C23×C418yes—
412C22×C4C23×C412yes—
413C22×C4C23×C418yes—
414C22×C4C23×C414yes—
415C22×C4C23×C412yes—
416C22×C4C22×D414yes—
417C22×C4C22×D418yes—
418C22×C4C22×D418yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+414yes—
429C22×C42−1+414yes—
430C22×C42−1+412yes—
431C4⋊C4—216no—
432C4⋊C4—232no—
433C4⋊C4—232no—
434C4⋊C4—232no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4C2×C22⋊C4132yes—
441C4⋊C4C2×C22⋊C4164yes—
442C22⋊C4—216no—
443C22⋊C4—232no—
444C22⋊C4—48no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—88no—
448C42—28no—
449C42—232no—
450C42—28no—
451C42—28no—
452C42—48no—
453C42—416no—
454C42—48no—
455C42—44no—
456C42—44no—
457C42—88no—
458C42C2×C22⋊C4116yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C24×C411yescharacteristic
483C23C23×D411yescharacteristic
484C23C23×D411yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4111yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×2−1+411yescharacteristic
489C23C2×2−1+412yes—
490C23C2×2−1+412yes—
491C23#58512yes—
492C23#58512yes—
493Q8—432no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—416no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4C22×C22⋊C4116yes—
576C2×C4C22×C22⋊C418yes—
577C2×C4C22×C22⋊C414yes—
578C2×C4C4×#4114yes—
579C2×C4#52014yes—
580C2×C4#54114yescomplemented
581C2×C4#54814yes—
582C2×C4#54814yes—
583C2×C4#55014yes—
584C2×C4#55014yes—
585C22—42no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22C23×C22⋊C411yescharacteristic
597C22C2×C4×#4111yescharacteristic
598C22C2×#52011yescharacteristic
599C22C4×2−1+412yes—
600C22C4×2−1+412yes—
601C22C2×#54111yescharacteristic
602C22#345511yescharacteristic
603C22#346112yes—
604C22C2×#54811yescharacteristic
605C22C2×#54811yescharacteristic
606C22C2×#55011yescharacteristic
607C22C2×#55011yescharacteristic
608C22#348912yes—
609C22#348912yes—
610C22#349512yes—
611C22#349912yes—
612C22#350012yes—
613C22#350911yescharacteristic
614C22#351211yescharacteristic
615C22#355212yes—
616C22#355512yes—
617C22#356212yes—
618C22#356312yes—
619C4—24no—
620C4—28no—
621C4—216no—
622C4—44no—
623C4—42no—
624C4—48no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—84no—
629C4—81no—
630C4—84no—
631C4—81no—
632C4#234118yes—
633C2—82no—
634C2C4×#54111yescharacteristic
635C2C2×#234111yescharacteristic
636C2#3425912yes—
637C2#3428111yescharacteristic
638C2#3443012yes—
639C2#3770511yescharacteristic
640C2#3770911yescharacteristic
641C2#3777412yes—
642C2#3938211yescharacteristic
643C2#3938511yescharacteristic
644C2#3942712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2330C2semidirect
3. C2×C4×#125C2semidirect
4. #14714C2semidirect
6. #14474C2semidirect
8. #14274C2semidirect
10. #14054C2semidirect
13. #13935C2semidirect
14. #13817C2semidirect
18. C2×#541C4semidirect
204. #125C22⋊C4semidirect
580. C2×C4#541semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#23302
C2×C4×#1252
#147142
#144932
#144742
#143402
#142742
#141012
#140542
#139972
#139352
#138172
#137612