Finite Group Databasebeta

← #404162

Subgroups — #404162

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2319C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic, complemented
5#11293C212yesmaximal, complemented
6#11208C211yesmaximal, characteristic, complemented
7#11123C212yesmaximal
8#11051C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2319—24no—
18#2319—24no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×#382C2211yescharacteristic
26C2×#381—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2057—24no—
31#2023—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1446—21no—
42#1446—21no—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4Q814yescomplemented
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390D414yes—
92#389—22no—
93#389—22no—
94#389—22no—
95#389D414yes—
96#388—28no—
97#388—28no—
98#382—42no—
99#381—28no—
100#380—28no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103#376—22no—
104#376—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—24no—
127C25—21no—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4C2×Q812yes—
142C22×D4#4112yes—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8#4114yes—
174C4⋊Q8—48no—
175C4⋊Q8—48no—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
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—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4#4114yes—
212C2×C4⋊C4#4112yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—82no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240#107—48no—
241#102—48no—
242#102—48no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—42no—
251#95—42no—
252#95—82no—
253#95—82no—
254#95—82no—
255#95—82no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24C22×D411yescharacteristic
268C2×Q8—48no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4C22⋊Q814yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—216no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4C22⋊Q812yes—
352C22×C4C22⋊Q812yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4C22⋊Q814yes—
396C22⋊C4—216no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—24no—
407C42—42no—
408C42—42no—
409C42—84no—
410C23—24no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C2×#12011yescharacteristic
434C23C2×C22⋊Q811yescharacteristic
435C23#52511yescharacteristic
436C23#52511yescharacteristic
437C23#53211yescharacteristic
438C23#53511yescharacteristic
439C23#53611yescharacteristic
440C23#57612yes—
441C23#57612yes—
442C23#57712yes—
443C23#57812yes—
444Q8—816no—
445D4—28no—
446D4—44no—
447D4—44no—
448D4—44no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—816no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4#52511yescharacteristic
513C2×C4#52912yes—
514C2×C4#53312yes—
515C2×C4#54212yes—
516C2×C4#54412yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4#54712yes—
520C2×C4#54812yes—
521C2×C4#55112yes—
522C2×C4#55212yes—
523C2×C4#55512yes—
524C2×C4#55612yes—
525C8—44no—
526C8—82no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—84no—
531C22—82no—
532C22—82no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22#239611yescharacteristic
539C22#240411yescharacteristic
540C22#243211yescharacteristic
541C22#259911yescharacteristic
542C22#261011yescharacteristic
543C22#262811yescharacteristic
544C22#284811yescharacteristic
545C22#300811yescharacteristic
546C22#300811yescharacteristic
547C22#300911yescharacteristic
548C22#301111yescharacteristic
549C22#308112yes—
550C22#308312yes—
551C22#311812yes—
552C22#312112yes—
553C22#312212yes—
554C22#312412yes—
555C22#317512yes—
556C22#317912yes—
557C22#322012yes—
558C22#322512yes—
559C22#324112yes—
560C22#324312yes—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4—162no—
572C4#243812yes—
573C4#243912yes—
574C4#247412yes—
575C4#247712yes—
576C2—84no—
577C2#1488711yescharacteristic
578C2#2378712yes—
579C2#2393911yescharacteristic
580C2#2394611yescharacteristic
581C2#2568111yescharacteristic
582C2#2568511yescharacteristic
583C2#2940112yes—
584C2#2956612yes—
585C2#2999112yes—
586C2#3296211yescharacteristic
587C2#3296311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2057C2semidirect
4. #14282C2semidirect
5. #11293C2semidirect
6. #11208C2semidirect
9. #10907C2semidirect
10. #10796C2semidirect
61. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23192
C2×#20572
C2×#20232
#142822
#112932
#112082
#111232
#110512
#109072
#107962
#107882