Finite Group Databasebeta

← #877283

Subgroups — #877283

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C212yesmaximal
2#17078C212yesmaximal
3#17075C212yesmaximal
4#15852C214yesmaximal
5#15676C214yesmaximal
6#15228C212yesmaximal
7#15218C212yesmaximal
8#15133C212yesmaximal
9#14714C211yesmaximal, characteristic
10#14712C211yesmaximal, characteristic
11#14467C212yesmaximal
12#14441C212yesmaximal
13#14029C212yesmaximal
14#13885C212yesmaximal
15C4×#389C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2212yes—
18#2550—216no—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#395C2214yes—
22C2×#391C2214yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C43C2211yescharacteristic
43#1445C2214yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74#1434C2214yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8D418yes—
80C2×#125—28no—
81C2×#125—44no—
82C2×#125—44no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#398—216no—
116#398—48no—
117#395—216no—
118#391—216no—
119#391—28no—
120#391—44no—
121#389—216no—
122#389—28no—
123#389—28no—
124#389—24no—
125#389—28no—
126#389—24no—
127#389—44no—
128#389—44no—
129#387—48no—
130#384—42no—
131#384—42no—
132#384Q818yescomplemented
133#382—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—42no—
137C4×C4⋊C4Q818yescomplemented
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C43—42no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8C2×D418yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C4⋊Q8—416no—
188#125—432no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4C2×Q814yes—
251C2×C4⋊C4C2×Q818yes—
252C2×C4⋊C4C2×Q818yes—
253C2×C4⋊C4C2×Q814yes—
254C2×C4⋊C4C2×Q818yes—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—28no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C2×D414yes—
280C2×C42C2×Q814yes—
281C2×C42C2×Q818yes—
282C2×C42C2×Q814yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—44no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C2×Q818yes—
311C24C2511yescharacteristic
312C2×Q8—216no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—84no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4C22×D418yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×Q812yes—
402C22×C4C22×Q814yes—
403C22×C4C22×Q814yes—
404C22×C4C22×Q818yes—
405C22×C4C22×Q812yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4C22×Q812yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+412yes—
421C22×C42−1+412yes—
422C22×C42−1+412yes—
423C22×C42−1+412yes—
424C4⋊C4—28no—
425C4⋊C4—216no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C42—28no—
447C42—28no—
448C42—28no—
449C42—216no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—48no—
455C42—88no—
456C23C23×D411yescharacteristic
457C23C23×Q811yescharacteristic
458C23C23×Q811yescharacteristic
459C23C22×#4111yescharacteristic
460C23C22×#4111yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×2−1+411yescharacteristic
466C23C2×2−1+411yescharacteristic
467C23C2×2−1+411yescharacteristic
468C23C2×2−1+411yescharacteristic
469C23#58511yescharacteristic
470C23#58511yescharacteristic
471Q8—816no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4#53314yes—
566C2×C4#53414yes—
567C2×C4#54414yes—
568C2×C4#54414yes—
569C2×C4#54714yes—
570C2×C4D4×Q814yes—
571C2×C4D4×Q814yes—
572C2×C4#55014yes—
573C2×C4#55414yes—
574C2×C4#55714yes—
575C2×C4Q8214yes—
576C2×C4#56314yes—
577C22C2×#53311yescharacteristic
578C22C2×#53411yescharacteristic
579C22C2×#54411yescharacteristic
580C22C2×#54411yescharacteristic
581C22#345811yescharacteristic
582C22C2×#54711yescharacteristic
583C22C2×D4×Q811yescharacteristic
584C22C2×D4×Q811yescharacteristic
585C22C2×#55011yescharacteristic
586C22C2×#55411yescharacteristic
587C22C2×#55711yescharacteristic
588C22C2×Q8211yescharacteristic
589C22#349611yescharacteristic
590C22#349711yescharacteristic
591C22#350111yescharacteristic
592C22#350211yescharacteristic
593C22#350311yescharacteristic
594C22#350311yescharacteristic
595C22#350511yescharacteristic
596C22#350511yescharacteristic
597C22#350711yescharacteristic
598C22#350711yescharacteristic
599C22#350811yescharacteristic
600C22#350911yescharacteristic
601C22#351111yescharacteristic
602C22#351111yescharacteristic
603C22C2×#56311yescharacteristic
604C22#353611yescharacteristic
605C22#353611yescharacteristic
606C22#354811yescharacteristic
607C22#355611yescharacteristic
608C22#355811yescharacteristic
609C22#355911yescharacteristic
610C22#356111yescharacteristic
611C22#356311yescharacteristic
612C4—24no—
613C4—24no—
614C4—24no—
615C4—24no—
616C4—24no—
617C4—44no—
618C4—48no—
619C4—48no—
620C4—42no—
621C4—44no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—84no—
626C4#273418yes—
627C2#3510411yescharacteristic
628C2#3613411yescharacteristic
629C2#3656311yescharacteristic
630C2#3730911yescharacteristic
631C2#3936511yescharacteristic
632C2#3948611yescharacteristic
633C2#3949611yescharacteristic
634C2#3962111yescharacteristic
635C2#3974311yescharacteristic
636C2#3975111yescharacteristic
637C2#4601411yescharacteristic
638C2#4602211yescharacteristic
639C2C2×#273411yescharacteristic
640C2#5982311yescharacteristic
641C2#5982811yescharacteristic
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
132. #384Q8semidirect
137. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
#170782
#170752
#158522
#156762
#152282
#152182
#151332
#147142
#147122
#144672
#144412
#140292
#138852
C4×#3892