Finite Group Databasebeta

← #364834

Subgroups — #364834

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5#13785C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11066C212yesmaximal
8#10931C212yesmaximal
9#10908C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—24no—
15C2×#476—24no—
16C2×#476C418yescomplemented
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2212yes—
22#2331—24no—
23#2331—24no—
24C2×#391—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#387—22no—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1988—24no—
34#1804—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1437—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C22⋊#95—22no—
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8D414yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—22no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#476—232no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106C2×C418yes—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8C2×C414yes—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102C2×C418yes—
94#391—216no—
95#390—22no—
96#390—22no—
97#390—22no—
98#390Q814yescomplemented
99#389—24no—
100#389—24no—
101#387—216no—
102#382—22no—
103#382D414yes—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108#376—22no—
109#376—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#336—24no—
125#336—42no—
126C25—21no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—82no—
131C22×Q8—82no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C412yes—
144C22×D4C2×Q812yes—
145C22×D4#4112yes—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
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—22no—
174C22×C8C22×C414yes—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8C22⋊C4116yescomplemented
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—44no—
182C4×D4—44no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4112yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—28no—
239C2×C42—42no—
240C2×C42—82no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#106—216no—
252C4⋊C8—28no—
253C4⋊C8—44no—
254#102—216no—
255#102—48no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—42no—
263#95—82no—
264#95—82no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24C22×D411yescharacteristic
277C2×Q8—216no—
278C2×Q8—88no—
279C2×Q8—88no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4C4×Q814yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—88no—
358C22×C4—88no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12012yes—
364C22×C4#12012yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4C22⋊Q814yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C42−1+411yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
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—
395C2×C8—42no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—216no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—88no—
417C4⋊C4C2×C22⋊C418yes—
418C22⋊C4—48no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—84no—
430C42—88no—
431C42C2×C22⋊C414yes—
432C42C4×D414yes—
433C42#12114yes—
434C42#12314yes—
435C23—24no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23C2×C4×D411yescharacteristic
459C23C2×#12011yescharacteristic
460C23C2×#12111yescharacteristic
461C23C2×C22⋊Q811yescharacteristic
462C23#52511yescharacteristic
463C23#53611yescharacteristic
464C23#53711yescharacteristic
465C23#57612yes—
466C23#57712yes—
467C23#57712yes—
468C23#57712yes—
469Q8—816no—
470D4—24no—
471D4—24no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—816no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4#38614yes—
531C2×C4#38614yes—
532C2×C4C22×C22⋊C411yescharacteristic
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C2×C4×Q811yescharacteristic
535C2×C4C4×#4111yescharacteristic
536C2×C4#51911yescharacteristic
537C2×C4#52011yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4#52911yescharacteristic
541C2×C4#53311yescharacteristic
542C2×C4#54111yescharacteristic
543C2×C4#54211yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4#54711yescharacteristic
546C2×C4#54811yescharacteristic
547C2×C4#54811yescharacteristic
548C2×C4#54811yescharacteristic
549C2×C4#54811yescharacteristic
550C2×C4#54811yescharacteristic
551C2×C4D4×Q811yescharacteristic
552C2×C4D4×Q811yescharacteristic
553C2×C4D4×Q811yescharacteristic
554C2×C4#55011yescharacteristic
555C2×C4#55011yescharacteristic
556C2×C4#55211yescharacteristic
557C2×C4#55211yescharacteristic
558C2×C4#55411yescharacteristic
559C2×C4#55611yescharacteristic
560C8—44no—
561C8—82no—
562C22—48no—
563C22—44no—
564C22—82no—
565C22—82no—
566C22—84no—
567C22—84no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22C2×#38611yescharacteristic
574C22C4×C22⋊Q811yescharacteristic
575C22#232311yescharacteristic
576C22#240711yescharacteristic
577C22#240911yescharacteristic
578C22#253611yescharacteristic
579C22#253811yescharacteristic
580C22#294612yes—
581C22#294912yes—
582C22#301012yes—
583C22#301212yes—
584C22#304111yescharacteristic
585C22#304211yescharacteristic
586C22#304311yescharacteristic
587C22#304311yescharacteristic
588C22#308112yes—
589C22#308412yes—
590C22#311812yes—
591C22#312512yes—
592C22#324112yes—
593C22#324412yes—
594C22#325712yes—
595C22#326312yes—
596C4—24no—
597C4—24no—
598C4—48no—
599C4—44no—
600C4—82no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4Q8×C22⋊C411yescharacteristic
606C4#234111yescharacteristic
607C4#236011yescharacteristic
608C4#236211yescharacteristic
609C4#242111yescharacteristic
610C4#242311yescharacteristic
611C4#242611yescharacteristic
612C4#242911yescharacteristic
613C2—84no—
614C2#1396911yescharacteristic
615C2#2092911yescharacteristic
616C2#2093611yescharacteristic
617C2#2162312yes—
618C2#2212312yes—
619C2#2416612yes—
620C2#2430912yes—
621C2#2821011yescharacteristic
622C2#2821211yescharacteristic
623C2#2828911yescharacteristic
624C2#2829011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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×#2331C2semidirect
3. C2×#1988C2semidirect
6. #11220C2semidirect
9. #10908C2semidirect
10. #10850C2semidirect
11. #10833C2semidirect
16. C2×#476C4semidirect
98. #390Q8semidirect
178. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#23312
C2×#19882
C2×#18042
#137852
#112202
#110662
#109312
#109082
#108502
#108332