Finite Group Databasebeta

← #363242

Subgroups — #363242

Order 512.

3822 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
1C22×#547C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic, complemented
6C2×#1762C211yesmaximal, characteristic, complemented
7#13905C211yesmaximal, characteristic, complemented
8#10969C214yesmaximal, complemented
9#10730C214yesmaximal, complemented
10C23×#41—21no—
11C2×#547—24no—
12C2×#547—24no—
13C2×#547—24no—
14C2×#547—24no—
15C2×#547—24no—
16C2×#547C418yescomplemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#123—22no—
19C22×C22⋊Q8—21no—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×C22⋊C8C2214yes—
28C22×#97C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2214yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1993—24no—
33#1987—24no—
34#1955—24no—
35#1804—24no—
36#1762—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1445—21no—
42#1438—21no—
43#1437—22no—
44C22×#41—28no—
45C22×#41—28no—
46C22×#41—28no—
47C22×#41—416no—
48C23×Q8—21no—
49C23×D4—22no—
50C23×D4D411yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—22no—
53C24×C4—21no—
54C24×C4C2312yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57#547—232no—
58#547—232no—
59#547—232no—
60#547—232no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4⋊Q8D414yes—
66C2×C4⋊Q8D414yes—
67C2×#123—216no—
68C2×#123—216no—
69C2×#123—216no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×#121—24no—
74C2×#121—24no—
75C2×#121—24no—
76C2×#121—24no—
77C2×#121—24no—
78C2×#121C2×C418yes—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4C2×C414yes—
87C2×C4×D4C2×C414yes—
88C2×C4×D4D414yes—
89C2×C4×D4D414yes—
90C2×C4×D4D414yes—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4C2×C418yes—
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4D412yes—
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102—44no—
118C2×C22⋊C8—28no—
119C2×C22⋊C8—28no—
120C2×C22⋊C8—44no—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97—22no—
124#382—22no—
125#382—22no—
126#380—28no—
127#380—44no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—21no—
131C2×#95—24no—
132C2×#95—21no—
133C2×#95—21no—
134#340—24no—
135#340—28no—
136#340—28no—
137#340—24no—
138C25—21no—
139C25C2×D412yes—
140C2×#41—264no—
141C2×#41—464no—
142C2×#41—464no—
143C2×#41—464no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—48no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—48no—
162C22×D4—48no—
163C22×D4C22⋊C414yes—
164C22×D4C22×C412yes—
165C22×D4C2×D412yes—
166C22×D4C2×D412yes—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—48no—
186C23×C4—48no—
187C23×C4—48no—
188C23×C4C22×C414yes—
189C23×C4C2×D412yes—
190C23×C4C2×D414yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2411yescharacteristic
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8#4114yescomplemented
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—28no—
209C4⋊Q8C22⋊C4116yescomplemented
210#123—2128no—
211C22⋊Q8—264no—
212#121—216no—
213#121—216no—
214#121—216no—
215#121—216no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4C22⋊C4116yescomplemented
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4C22⋊C414yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4114yescomplemented
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—84no—
270C2×C22⋊C4C22⋊C418yes—
271C2×C42—42no—
272C2×C42—82no—
273C2×C42C22×C412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C4⋊C8—44no—
280#102—216no—
281#102—48no—
282C22⋊C8—416no—
283#97—44no—
284#95—48no—
285#95—42no—
286#95—42no—
287#95—42no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—42no—
299C24—42no—
300C24—44no—
301C24—42no—
302C24—42no—
303C24C2×C22⋊C414yes—
304C24AGL(1,ℤ/8)18yes—
305C24C22×D411yescharacteristic
306#41—4256no—
307C2×Q8—216no—
308C2×Q8—416no—
309C2×Q8—416no—
310C2×Q8—416no—
311C2×D4—216no—
312C2×D4—216no—
313C2×D4—216no—
314C2×D4—232no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—28no—
327C2×D4—416no—
328C2×D4—432no—
329C2×D4—432no—
330C2×D4—416no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4C2×C22⋊C414yes—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—232no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—216no—
361C22×C4—24no—
362C22×C4—416no—
363C22×C4—416no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—432no—
367C22×C4—432no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—44no—
419C2×C8—44no—
420C4⋊C4—232no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—216no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4C2×C22⋊C414yes—
437C4⋊C4C2×C22⋊C418yes—
438C22⋊C4—232no—
439C22⋊C4—216no—
440C22⋊C4—216no—
441C22⋊C4—816no—
442C42—84no—
443C42C2×C22⋊C414yes—
444C42C2×C22⋊C414yes—
445C42#12014yes—
446C42#12014yes—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—24no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23#41818yes—
484C23C22×C22⋊C411yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×#12011yescharacteristic
487C23#54612yes—
488C23#54712yes—
489C23C2×AGL(1,ℤ/8)14yes—
490C23#57712yes—
491C23#57812yes—
492Q8—432no—
493D4—232no—
494D4—24no—
495D4—24no—
496D4—464no—
497D4—44no—
498D4—44no—
499D4—42no—
500D4—42no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—432no—
513C2×C4—464no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—84no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4#37914yes—
541C2×C4#37914yes—
542C2×C4C22×C22⋊C411yescharacteristic
543C2×C4C22×C22⋊C411yescharacteristic
544C2×C4C4×#4112yes—
545C2×C4#52012yes—
546C2×C4C2×#12011yescharacteristic
547C2×C4C2×#12011yescharacteristic
548C2×C4#52512yes—
549C2×C4#53211yescharacteristic
550C2×C4#53211yescharacteristic
551C2×C4#53511yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#53712yes—
554C2×C4#54612yes—
555C2×C4#54712yes—
556C2×C4#54812yes—
557C2×C4#54812yes—
558C2×C4#55012yes—
559C2×C4#55012yes—
560C8—44no—
561C8—82no—
562C22—28no—
563C22—216no—
564C22—48no—
565C22—48no—
566C22—48no—
567C22—44no—
568C22—44no—
569C22—48no—
570C22—48no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—88no—
578C22—84no—
579C22C2×#37911yescharacteristic
580C22#233912yes—
581C22#240712yes—
582C22#243111yescharacteristic
583C22#243211yescharacteristic
584C22C2×#41812yes—
585C22#289812yes—
586C22#300312yes—
587C22#300312yes—
588C22#301011yescharacteristic
589C22#301111yescharacteristic
590C22#301312yes—
591C22#330614yes—
592C22#330814yes—
593C22#334314yes—
594C22#334514yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—48no—
600C4—84no—
601C4—162no—
602C4#229711yescharacteristic
603C4#231111yescharacteristic
604C4#234112yes—
605C4#240712yes—
606C4#243111yescharacteristic
607C4#243211yescharacteristic
608C2—48no—
609C2—84no—
610C2#1393711yescharacteristic
611C2#2059811yescharacteristic
612C2#2060311yescharacteristic
613C2#2249514yes—
614C2#2412812yes—
615C2#2413212yes—
616C2#2557712yes—
617C2#2570912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#547C2
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
2. C2×#1993C2semidirect
3. C2×#1987C2semidirect
4. C2×#1955C2semidirect
5. C2×#1804C2semidirect
6. C2×#1762C2semidirect
7. #13905C2semidirect
8. #10969C2semidirect
9. #10730C2semidirect
16. C2×#547C4semidirect
205. C22×C8#41semidirect
209. C4⋊Q8C22⋊C4semidirect
223. C4×D4C22⋊C4semidirect
258. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#19932
C2×#19872
C2×#19552
C2×#18042
C2×#17622
#139052
#109692
#107302