Finite Group Databasebeta

← #364821

Subgroups — #364821

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic, complemented
3#21312C211yesmaximal, characteristic, complemented
4C2×#1800C211yesmaximal, characteristic
5C2×#1762C211yesmaximal, characteristic
6#11216C212yesmaximal, complemented
7#11061C212yesmaximal
8#10943C212yesmaximal
9#10896C212yesmaximal, complemented
10#10847C211yesmaximal, characteristic, complemented
11#10845C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14#3077—24no—
15#3077C418yescomplemented
16C2×#478—22no—
17C2×#476—21no—
18C2×#474—21no—
19#2961—24no—
20#2961—24no—
21C2×#446—22no—
22C2×#446—22no—
23C2×#428C2212yes—
24C2×#424C2212yes—
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#418C2212yes—
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#100C2212yes—
32C2×#405C2212yes—
33C22×#97C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1989—22no—
36#1988—21no—
37#1987—21no—
38#1800—24no—
39#1762—22no—
40#1762—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#341C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57#533—22no—
58#533—22no—
59#533C2×C414yes—
60#533D414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×#125D412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4C2×C414yes—
76C2×#117C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#478—216no—
84#476—28no—
85#474—28no—
86#446—216no—
87#446—216no—
88#428—24no—
89#428C2×C418yes—
90C2×#107—22no—
91C2×#106—22no—
92#424—24no—
93#424—24no—
94#423—22no—
95#423—22no—
96#423—22no—
97#423—22no—
98#423C2×C414yes—
99#423D414yes—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107#418—24no—
108#418C2×C418yes—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102Q814yescomplemented
117C2×#100—24no—
118C2×#100—24no—
119C2×#100—42no—
120#405—24no—
121#405—24no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—22no—
125C2×#97D412yes—
126C2×#97D412yes—
127C2×C4×C8—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C2×#95—21no—
133#343—24no—
134#343—24no—
135#343—42no—
136#341—24no—
137#341—42no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144C25—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22×C412yes—
159C22×D4C2×Q812yes—
160C22×D4#4112yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4—42no—
168C23×C4—82no—
169C23×C4—82no—
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4C2411yescharacteristic
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16C22×C414yes—
190C2×M16C2×D414yes—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×Q812yes—
207C22×C8#4112yes—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8C22⋊C414yes—
211C4⋊Q8C22⋊C414yes—
212#125—24no—
213#125C22⋊C418yes—
214C22⋊Q8—24no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—42no—
230#117—22no—
231#117—22no—
232#117C22×C414yes—
233#117C2×D414yes—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
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—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×Q812yes—
257C2×C4⋊C4#4112yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—42no—
263C2×C42—22no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—82no—
267C2×C42C22×C412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275#107—28no—
276#106—28no—
277C4⋊C8—216no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284#102—28no—
285#102—28no—
286#102—28no—
287#102—28no—
288#100—48no—
289#97—28no—
290#97—24no—
291#97—24no—
292C4×C8—28no—
293#95—41no—
294#95—41no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—81no—
303C24—81no—
304C24—81no—
305C24—81no—
306C24C2×#4111yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—81no—
310C2×Q8—81no—
311C2×Q8—81no—
312C2×Q8—81no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4C4×Q814yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391C22×C42−1+411yescharacteristic
392M16—24no—
393M16—24no—
394M16—24no—
395M16—24no—
396M16—42no—
397M16—42no—
398M16—42no—
399M16—42no—
400M16—42no—
401M16—42no—
402M16—42no—
403M16—42no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8C22⋊Q814yes—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4C2×C22⋊C414yes—
446C4⋊C4C2×C22⋊C414yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C42—24no—
452C42—24no—
453C42—82no—
454C42—82no—
455C42C2×C22⋊C412yes—
456C42C2×C22⋊C412yes—
457C42C4×D412yes—
458C42C4×D412yes—
459C42#12012yes—
460C42#12012yes—
461C42#12112yes—
462C42#12112yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×#12311yescharacteristic
493C23#55211yescharacteristic
494C23#55211yescharacteristic
495C23#55411yescharacteristic
496C23#55611yescharacteristic
497Q8—84no—
498Q8—84no—
499D4—24no—
500D4—24no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—88no—
529C2×C4—88no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#38612yes—
543C2×C4#38612yes—
544C2×C4#38612yes—
545C2×C4#38612yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×Q811yescharacteristic
549C2×C4C4×#4111yescharacteristic
550C2×C4#51911yescharacteristic
551C2×C4#52011yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4#52511yescharacteristic
556C2×C4#52911yescharacteristic
557C2×C4#53311yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53711yescharacteristic
560C2×C4#54111yescharacteristic
561C2×C4#54211yescharacteristic
562C2×C4#54611yescharacteristic
563C2×C4#54711yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4#54811yescharacteristic
567C2×C4#54811yescharacteristic
568C2×C4#54811yescharacteristic
569C2×C4D4×Q811yescharacteristic
570C2×C4D4×Q811yescharacteristic
571C2×C4D4×Q811yescharacteristic
572C2×C4#55011yescharacteristic
573C2×C4#55011yescharacteristic
574C8—24no—
575C8—44no—
576C8—42no—
577C8—44no—
578C8—82no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—44no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22C2×#38611yescharacteristic
603C22#236011yescharacteristic
604C22#236211yescharacteristic
605C22#242111yescharacteristic
606C22#242311yescharacteristic
607C22#253611yescharacteristic
608C22#253811yescharacteristic
609C22#306811yescharacteristic
610C22#306811yescharacteristic
611C22#306811yescharacteristic
612C22#306911yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—44no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4C4×C22⋊Q811yescharacteristic
623C4#232311yescharacteristic
624C4Q8×C22⋊C411yescharacteristic
625C4#234111yescharacteristic
626C4#240711yescharacteristic
627C4#240911yescharacteristic
628C4#242611yescharacteristic
629C4#242911yescharacteristic
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—84no—
635C2#1396911yescharacteristic
636C2#2100411yescharacteristic
637C2#2100511yescharacteristic
638C2#2822011yescharacteristic
639C2#2822111yescharacteristic
640C2#2829811yescharacteristic
641C2#2829911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2961C2semidirect
3. #21312C2semidirect
6. #11216C2semidirect
9. #10896C2semidirect
10. #10847C2semidirect
11. #10845C2semidirect
15. #3077C4semidirect
116. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C2×#29612
#213122
C2×#18002
C2×#17622
#112162
#110612
#109432
#108962
#108472
#108452