Finite Group Databasebeta

← #365760

Subgroups — #365760

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1836C211yesmaximal, characteristic, complemented
5#13905C211yesmaximal, characteristic
6#11096C212yesmaximal, complemented
7#11095C212yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10731C212yesmaximal, complemented
10#10731C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20#2354—24no—
21#2354—24no—
22C2×#390—21no—
23C2×#388—22no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1955C418yescomplemented
33#1949—24no—
34#1836—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1445—21no—
44#1438—21no—
45#1437—22no—
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C414yes—
56C2×C4×D4Q814yes—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—42no—
62C2×#117—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D412yes—
77C22×C42—22no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—42no—
83C2×#106—22no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8C2×C414yes—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8C2×C418yes—
93C2×C22⋊C8C2×C418yes—
94C2×#97—22no—
95C2×#97—22no—
96C2×#97C2×C414yes—
97#390—28no—
98#388—216no—
99#384—22no—
100#384D414yes—
101#382—22no—
102#382—22no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—42no—
108#380—42no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—24no—
127#340—42no—
128#340—42no—
129C25C2×D412yes—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4C4⋊C414yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C22×C414yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×Q814yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C22×C414yes—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180#117—48no—
181#117—48no—
182#117—48no—
183#117—48no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
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—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4C4⋊C414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4C4⋊C418yes—
235C2×C42—28no—
236C2×C42—24no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—81no—
242C2×C42—81no—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249#107—44no—
250#106—44no—
251C4⋊C8—28no—
252C22⋊C8—48no—
253C22⋊C8—48no—
254#97—28no—
255#95—28no—
256#95—24no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—42no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C2×C4⋊C414yes—
277C24C22×D411yescharacteristic
278C2×D4—28no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
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—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4C2×C4⋊C414yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12312yes—
363C22×C4#12312yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C2×C8—24no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—84no—
413C4⋊C4#12414yes—
414C22⋊C4—216no—
415C22⋊C4—416no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C42—48no—
420C42—416no—
421C42—84no—
422C42C4×D414yes—
423C42C4×D414yes—
424C42C22⋊Q814yes—
425C42C22⋊Q814yes—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23C22×C4⋊C411yescharacteristic
462C23C2×#12011yescharacteristic
463C23C2×#12311yescharacteristic
464C23#54612yes—
465C23#54612yes—
466C23#57512yes—
467C23#57512yes—
468C23#57611yescharacteristic
469C23#57711yescharacteristic
470C23#57711yescharacteristic
471C23#57811yescharacteristic
472D4—416no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4#38514yes—
533C2×C4#38514yes—
534C2×C4C2×C4×D411yescharacteristic
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C4×#4112yes—
537C2×C4#51812yes—
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4#52912yes—
542C2×C4#53211yescharacteristic
543C2×C4#53511yescharacteristic
544C2×C4#53511yescharacteristic
545C2×C4#54012yes—
546C2×C4#54612yes—
547C2×C4#55012yes—
548C2×C4#55312yes—
549C2×C4#55412yes—
550C2×C4#55412yes—
551C2×C4#55512yes—
552C8—42no—
553C8—42no—
554C8—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22C2×#38511yescharacteristic
571C22#234912yes—
572C22#242012yes—
573C22#243111yescharacteristic
574C22#243511yescharacteristic
575C22#291112yes—
576C22#291211yescharacteristic
577C22#291311yescharacteristic
578C22#300511yescharacteristic
579C22#300511yescharacteristic
580C22#301211yescharacteristic
581C22#301311yescharacteristic
582C22#309312yes—
583C22#309611yescharacteristic
584C22#309711yescharacteristic
585C22#330712yes—
586C22#330912yes—
587C22#331112yes—
588C22#331312yes—
589C22#331512yes—
590C22#331712yes—
591C22#331912yes—
592C22#332112yes—
593C4—24no—
594C4—24no—
595C4—44no—
596C4—44no—
597C4—84no—
598C4—84no—
599C4—82no—
600C4—161no—
601C4—161no—
602C4C4×#12411yescharacteristic
603C4#232811yescharacteristic
604C4#236312yes—
605C4#245412yes—
606C4#256311yescharacteristic
607C4#256611yescharacteristic
608C2—84no—
609C2#1397911yescharacteristic
610C2#2084511yescharacteristic
611C2#2084911yescharacteristic
612C2#2271012yes—
613C2#2271212yes—
614C2#2497812yes—
615C2#2498012yes—
616C2#2557811yescharacteristic
617C2#2558011yescharacteristic
618C2#2589211yescharacteristic
619C2#2589311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1949C2semidirect
4. C2×#1836C2semidirect
6. #11096C2semidirect
7. #11095C2semidirect
8. #10849C2semidirect
9. #10731C2semidirect
10. #10731C2semidirect
11. #10703C2semidirect
32. #1955C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#19552
C2×#19492
C2×#18362
#139052
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