Finite Group Databasebeta

← #414133

Subgroups — #414133

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3C2×#2043C211yesmaximal, characteristic
4C2×#1927C212yesmaximal
5#14714C211yesmaximal, characteristic
6C4×#389C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic
8#11077C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10832C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#2084—24no—
29#2066—28no—
30#2043—24no—
31#1927—28no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1436—21no—
39#1436—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×Q8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—42no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92#391—24no—
93#391—24no—
94#391—24no—
95#391—24no—
96#389—22no—
97#389—22no—
98#389D414yes—
99#389D414yes—
100#384—22no—
101#384—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—42no—
104C4×C4⋊C4—44no—
105C4×C4⋊C4D414yes—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C43—42no—
119#340—24no—
120#340—24no—
121#340—24no—
122#336—22no—
123#336—24no—
124#336—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8#4114yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8#4114yes—
166C4⋊Q8—48no—
167C4⋊Q8—48no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42SD1614yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240#106—48no—
241C4⋊C8—48no—
242#103—216no—
243#103—48no—
244#103—48no—
245#103—48no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—42no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—42no—
258C24C22×D411yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
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#12112yes—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4C2×SD1612yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×SD1612yes—
366C22×C4#13512yes—
367C22×C4#13712yes—
368C22×C4#13712yes—
369C22×C4#13712yes—
370C22×C4#13712yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)12yes—
380C22×C42−1+412yes—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4#12714yes—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—44no—
422C42—44no—
423C42—44no—
424C42—44no—
425C42—48no—
426C42—84no—
427C23C2×#12111yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×#12111yescharacteristic
430C23#52511yescharacteristic
431C23#53211yescharacteristic
432C23#53411yescharacteristic
433C23#53611yescharacteristic
434C23C22×SD1611yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×#13711yescharacteristic
437C23C2×#13711yescharacteristic
438C23C2×#13711yescharacteristic
439C23C2×#13711yescharacteristic
440C23#57511yescharacteristic
441C23#57511yescharacteristic
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449Q8—82no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#46114yes—
531C2×C4#46114yes—
532C2×C4#46114yes—
533C2×C4#46414yes—
534C2×C4#52512yes—
535C2×C4#52912yes—
536C2×C4C2×#12711yescharacteristic
537C2×C4#53211yescharacteristic
538C2×C4#53211yescharacteristic
539C2×C4#53211yescharacteristic
540C2×C4#53712yes—
541C2×C4#53912yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54812yes—
545C2×C4#54812yes—
546C2×C4#55112yes—
547C2×C4#55212yes—
548C2×C4#55312yes—
549C2×C4#55612yes—
550C8—44no—
551C8—82no—
552C22#239211yescharacteristic
553C22#253411yescharacteristic
554C22#253611yescharacteristic
555C22#254311yescharacteristic
556C22#254511yescharacteristic
557C22#261811yescharacteristic
558C22#285611yescharacteristic
559C22C2×#46111yescharacteristic
560C22C2×#46111yescharacteristic
561C22C2×#46111yescharacteristic
562C22C2×#46411yescharacteristic
563C22#303511yescharacteristic
564C22#303511yescharacteristic
565C22#303711yescharacteristic
566C22#303711yescharacteristic
567C22#303911yescharacteristic
568C22#303911yescharacteristic
569C22#303911yescharacteristic
570C22#304011yescharacteristic
571C22#310311yescharacteristic
572C22#310811yescharacteristic
573C22#311311yescharacteristic
574C22#311711yescharacteristic
575C22#316711yescharacteristic
576C22#317211yescharacteristic
577C22#318111yescharacteristic
578C22#318511yescharacteristic
579C22#318711yescharacteristic
580C22#319011yescharacteristic
581C22#319511yescharacteristic
582C22#319611yescharacteristic
583C22#322911yescharacteristic
584C22#323111yescharacteristic
585C22#323611yescharacteristic
586C22#323711yescharacteristic
587C4—24no—
588C4—24no—
589C4—24no—
590C4—24no—
591C4—42no—
592C4—44no—
593C4—42no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—162no—
599C4#254012yes—
600C4#255812yes—
601C4#256012yes—
602C4#260112yes—
603C2#1516311yescharacteristic
604C2#2367711yescharacteristic
605C2#2368511yescharacteristic
606C2#2811511yescharacteristic
607C2#2812411yescharacteristic
608C2#2820511yescharacteristic
609C2#2820911yescharacteristic
610C2#2857111yescharacteristic
611C2#2858111yescharacteristic
612C2#2865311yescharacteristic
613C2#2866011yescharacteristic
614C2#2975611yescharacteristic
615C2#2975911yescharacteristic
616C2#3316211yescharacteristic
617C2#3316511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#20662
C2×#20432
C2×#19272
#147142
C4×#3892
#113572
#110772
#110102
#109322
#108322