Finite Group Databasebeta

← #413352

Subgroups — #413352

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C211yesmaximal, characteristic
2C2×#2085C211yesmaximal, characteristic
3C2×#2085C211yesmaximal, characteristic
4C2×#2083C211yesmaximal, characteristic
5C2×#1847C211yesmaximal, characteristic
6C2×#1846C211yesmaximal, characteristic
7#14354C211yesmaximal, characteristic
8#14277C211yesmaximal, characteristic
9#11325C211yesmaximal, characteristic
10#11325C211yesmaximal, characteristic
11#11324C211yesmaximal, characteristic
12#11323C211yesmaximal, characteristic
13#10843C211yesmaximal, characteristic
14#10841C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#2085—24no—
38#2085—24no—
39#2084—24no—
40#2083—24no—
41#1847—24no—
42#1846—24no—
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55#1444—22no—
56#1444—22no—
57#1444—22no—
58#1442—22no—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—41no—
92C2×#107—41no—
93C2×#107—41no—
94C2×#107—41no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—41no—
104C2×#106—41no—
105C2×#106—41no—
106C2×#106—41no—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97Q814yes—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8Q814yes—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#384—22no—
118#384—22no—
119#384—22no—
120#384—22no—
121#384—22no—
122#384—22no—
123#384—41no—
124#384—41no—
125#382—22no—
126#382—22no—
127#382—22no—
128#382—22no—
129#382—22no—
130#382—41no—
131#382—41no—
132#382D414yes—
133#376—22no—
134#376—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95D412yes—
145C2×#95D412yes—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#336D414yes—
155#336D414yes—
156#336Q814yes—
157#336Q814yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8C2×Q812yes—
192C22×C8C2×Q812yes—
193C22×C8C2×Q812yes—
194C22×C8C2×Q812yes—
195C22×C8C2×Q812yes—
196C22×C8C2×Q812yes—
197C22×C8C2×Q812yes—
198C22×C8C2×Q812yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×Q812yes—
276C2×C42C2×Q812yes—
277C2×C42C2×Q812yes—
278C2×C42C2×Q812yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281C2×C42#4112yes—
282#107—44no—
283#107—44no—
284#107—44no—
285#107—44no—
286#106—44no—
287#106—44no—
288#106—44no—
289#106—44no—
290#97—28no—
291C4×C8—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—82no—
302#95—82no—
303C24C22×D411yescharacteristic
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4#12114yes—
363C22×C4#12314yes—
364C22×C4#12712yes—
365C22×C4#12712yes—
366C22×C4C4⋊Q814yes—
367C22×C4C4⋊Q814yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C22×Q811yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8C4⋊Q814yes—
407C2×C8C4⋊Q814yes—
408C2×C8C4⋊Q814yes—
409C2×C8C4⋊Q814yes—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42C22⋊Q814yes—
458C42C22⋊Q814yes—
459C42#12514yes—
460C42#12514yes—
461C23C2×#12111yescharacteristic
462C23C2×#12311yescharacteristic
463C23C2×#12711yescharacteristic
464C23C2×C4⋊Q811yescharacteristic
465C23C2×C4⋊Q811yescharacteristic
466C23#53511yescharacteristic
467C23#53611yescharacteristic
468C23#57612yes—
469C23#57712yes—
470C23#57712yes—
471C23#57812yes—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#39818yes—
536C2×C4C2×C22⋊Q811yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×#12511yescharacteristic
539C2×C4C2×#12511yescharacteristic
540C2×C4C2×C4⋊Q811yescharacteristic
541C2×C4C2×C4⋊Q811yescharacteristic
542C2×C4C2×C4⋊Q811yescharacteristic
543C2×C4C2×C4⋊Q811yescharacteristic
544C2×C4#54411yescharacteristic
545C2×C4#54411yescharacteristic
546C2×C4#54411yescharacteristic
547C2×C4#54411yescharacteristic
548C2×C4D4211yescharacteristic
549C2×C4D4211yescharacteristic
550C2×C4#54711yescharacteristic
551C2×C4#54711yescharacteristic
552C2×C4#54711yescharacteristic
553C2×C4#54711yescharacteristic
554C2×C4D4×Q811yescharacteristic
555C2×C4D4×Q811yescharacteristic
556C2×C4#55011yescharacteristic
557C2×C4#55011yescharacteristic
558C2×C4#55411yescharacteristic
559C2×C4#55411yescharacteristic
560C2×C4#55411yescharacteristic
561C2×C4#55411yescharacteristic
562C2×C4#55711yescharacteristic
563C2×C4#55711yescharacteristic
564C8—24no—
565C8—24no—
566C8—24no—
567C8—24no—
568C22C2×#39811yescharacteristic
569C22#250111yescharacteristic
570C22#250211yescharacteristic
571C22#254111yescharacteristic
572C22#254611yescharacteristic
573C22#266011yescharacteristic
574C22#286811yescharacteristic
575C22#304512yes—
576C22#304612yes—
577C22#309612yes—
578C22#309712yes—
579C22#315411yescharacteristic
580C22#315411yescharacteristic
581C22#315511yescharacteristic
582C22#315611yescharacteristic
583C22#316312yes—
584C22#316312yes—
585C22#316412yes—
586C22#316412yes—
587C22#322412yes—
588C22#322512yes—
589C22#324112yes—
590C22#324212yes—
591C4—28no—
592C4—42no—
593C4—42no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4—161no—
598C4—161no—
599C4—161no—
600C4—161no—
601C4—161no—
602C4—161no—
603C4#244011yescharacteristic
604C4#244011yescharacteristic
605C4#244011yescharacteristic
606C4#244011yescharacteristic
607C4#255211yescharacteristic
608C4#255211yescharacteristic
609C4#255211yescharacteristic
610C4#255211yescharacteristic
611C2#1514511yescharacteristic
612C2#2366212yes—
613C2#2792211yescharacteristic
614C2#2792311yescharacteristic
615C2#2795912yes—
616C2#2845211yescharacteristic
617C2#2845311yescharacteristic
618C2#2869812yes—
619C2#3031512yes—
620C2#3331811yescharacteristic
621C2#3331911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20842
C2×#20852
C2×#20832
C2×#18472
C2×#18462
#143542
#142772
#113252
#113242
#113232
#108432
#108412
#106942