Finite Group Databasebeta

← #615773

Subgroups — #615773

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15225C212yesmaximal
2#15225C212yesmaximal
3#15028C214yesmaximal
4#14957C214yesmaximal
5#14712C211yesmaximal, characteristic
6#14707C212yesmaximal
7#14699C212yesmaximal
8#14151C211yesmaximal, characteristic
9#14132C212yesmaximal
10#14048C212yesmaximal
11#14041C212yesmaximal
12#13911C211yesmaximal, characteristic
13#13911C211yesmaximal, characteristic
14#13910C211yesmaximal, characteristic
15#13884C212yesmaximal
16C4×#389C211yesmaximal, characteristic
17C42×C4⋊C4C211yesmaximal, characteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2211yescharacteristic
27C2×#387C2214yes—
28C2×#384C2212yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2212yes—
32C2×#382C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C43C2211yescharacteristic
43C2×C43C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78#1434C2214yes—
79C4×#95C2212yes—
80C4×#95C2212yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#389—24no—
119#389—28no—
120#389—28no—
121#389—44no—
122#389—44no—
123#389—44no—
124#389—44no—
125#389—44no—
126#389Q818yes—
127#387—48no—
128#384—28no—
129#384—24no—
130#384—24no—
131#382—28no—
132#382—44no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4Q818yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C43—24no—
173C43—44no—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×Q814yes—
244C2×C4⋊C4C2×Q814yes—
245C2×C4⋊C4C2×Q814yes—
246C2×C4⋊C4C2×Q818yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4#4114yes—
249C2×C4⋊C4#4114yes—
250C2×C4⋊C4#4114yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—41no—
286C2×C42—41no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C2×Q818yes—
290C2×C42C2×Q814yes—
291C2×C42C2×Q814yes—
292C2×C42#4114yes—
293C2×C42#4114yes—
294C2×C42#4114yes—
295C2×C42#4114yes—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×Q818yes—
325#95C2×Q818yes—
326C24C2511yescharacteristic
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—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
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—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4C22×Q814yes—
415C22×C4C22×Q814yes—
416C22×C4C22×Q814yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q812yes—
419C22×C4C22×Q812yes—
420C22×C4C22×Q812yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q812yes—
423C22×C4C22×Q812yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—84no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C42—28no—
467C42—28no—
468C42—28no—
469C42—216no—
470C42—44no—
471C42—44no—
472C42—48no—
473C42—48no—
474C42—48no—
475C42—48no—
476C42—48no—
477C42—48no—
478C42—48no—
479C42—44no—
480C42#125116yes—
481C23C23×Q811yescharacteristic
482C23C23×Q811yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23#58511yescharacteristic
495C23#58511yescharacteristic
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
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—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
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—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4C2×#12514yes—
599C2×C4C2×#12514yes—
600C2×C4C2×#12514yes—
601C2×C4#52914yes—
602C2×C4#52914yes—
603C2×C4#52914yes—
604C2×C4#52914yes—
605C2×C4#53314yes—
606C2×C4#53314yes—
607C2×C4#53314yes—
608C2×C4#54014yes—
609C2×C4#55314yes—
610C2×C4#55414yes—
611C2×C4#55414yes—
612C2×C4#55414yes—
613C2×C4#55714yes—
614C2×C4Q8214yes—
615C22C22×#12511yescharacteristic
616C22C2×#52911yescharacteristic
617C22C2×#52911yescharacteristic
618C22C2×#52911yescharacteristic
619C22C2×#52911yescharacteristic
620C22C2×#53311yescharacteristic
621C22C2×#53311yescharacteristic
622C22C2×#53311yescharacteristic
623C22C2×#54011yescharacteristic
624C22#346011yescharacteristic
625C22#346111yescharacteristic
626C22C2×#55311yescharacteristic
627C22C2×#55411yescharacteristic
628C22C2×#55411yescharacteristic
629C22C2×#55411yescharacteristic
630C22#347511yescharacteristic
631C22C2×#55711yescharacteristic
632C22C2×Q8211yescharacteristic
633C22Q8×#4111yescharacteristic
634C22Q8×#4111yescharacteristic
635C22#348011yescharacteristic
636C22#348211yescharacteristic
637C22#348311yescharacteristic
638C22#349411yescharacteristic
639C22#349511yescharacteristic
640C22#350311yescharacteristic
641C22#350411yescharacteristic
642C22#350611yescharacteristic
643C22#350611yescharacteristic
644C22#350811yescharacteristic
645C22#351111yescharacteristic
646C22#351411yescharacteristic
647C22#351411yescharacteristic
648C22#351911yescharacteristic
649C22#352011yescharacteristic
650C4—24no—
651C4—24no—
652C4—24no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—84no—
663C4—82no—
664C4—84no—
665C4#250718yes—
666C4#286418yes—
667C2C2×#250711yescharacteristic
668C2#3494111yescharacteristic
669C2#3495111yescharacteristic
670C2#3519611yescharacteristic
671C2#3519711yescharacteristic
672C2#3526511yescharacteristic
673C2#3640711yescharacteristic
674C2#3641611yescharacteristic
675C2#3644911yescharacteristic
676C2#3646011yescharacteristic
677C2#3655111yescharacteristic
678C2#3655211yescharacteristic
679C2#3656311yescharacteristic
680C2#3656511yescharacteristic
681C2C2×#286411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
#152252
#150282
#149572
#147122
#147072
#146992
#141512
#141322
#140482
#140412
#139112
#139102
#138842
C4×#3892
C42×C4⋊C42