Finite Group Databasebeta

← #366989

Subgroups — #366989

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1969C212yesmaximal
4C2×#1947C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×C4×C4⋊C8C211yesmaximal, characteristic
7#14714C211yesmaximal, characteristic
8C4×#391C211yesmaximal, characteristic
9#11124C212yesmaximal
10#11024C212yesmaximal
11#10926C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#462—24no—
16C2×#462—24no—
17C2×#462C418yescomplemented
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×C4×C8C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1981—24no—
34#1969—28no—
35#1947—24no—
36#1803—24no—
37C4×C4⋊C8—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1436—22no—
42C4⋊#95—22no—
43C4⋊#95—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—22no—
47C23×Q8C2312yes—
48C23×Q8C2311yescharacteristic
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—42no—
54C22×Q16—42no—
55C22×Q16C2×C418yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42D412yes—
82C22×C42D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#462—232no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8D414yes—
94C2×C4⋊C8D414yes—
95C2×C4⋊C8D414yes—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103C2×C418yes—
108C2×C4×C8—24no—
109C2×C4×C8—24no—
110C2×C4×C8—42no—
111#391—22no—
112#391—22no—
113#391—22no—
114#391—24no—
115#391—24no—
116#391—42no—
117#391—42no—
118#391D414yes—
119#384—22no—
120#384—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95—21no—
130C43—22no—
131C43—22no—
132#340—24no—
133#340—24no—
134#336—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8C22×C414yes—
160C22×Q8C22×C412yes—
161C22×Q8C2×D412yes—
162C22×Q8#4112yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C2×Q16—216no—
175C2×Q16—216no—
176C2×Q16—48no—
177C2×Q16—48no—
178C2×Q16—48no—
179C2×Q16—48no—
180C2×Q16—48no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C22×C414yes—
192C22×C8C2×D414yes—
193C22×C8C2×D414yes—
194C22×C8C2×D414yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C4×Q8—28no—
200C4×Q8—28no—
201C4×Q8—44no—
202C4×Q8—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4#4112yes—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—44no—
260C2×C42Q1618yescomplemented
261C2×C42C22×C412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269#107—48no—
270C4⋊C8—28no—
271C4⋊C8C22⋊C4116yes—
272#103—216no—
273#103—48no—
274#103—48no—
275#103—48no—
276C4×C8—48no—
277#95—28no—
278#95—28no—
279#95—44no—
280#95—48no—
281#95—48no—
282#95—42no—
283#95—44no—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8C4×D414yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4#12112yes—
390C22×C4#12114yes—
391C22×C4#12112yes—
392C22×C4#12712yes—
393C22×C4#12712yes—
394C22×C4C2×Q1614yes—
395C22×C4C2×Q1614yes—
396C22×C4C2×Q1614yes—
397C22×C4#13514yes—
398C22×C4#13714yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413Q16—432no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8C2×C22⋊C418yes—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—216no—
454C42—48no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—48no—
460C42—84no—
461C42C2×C22⋊C414yes—
462C42C4×D414yes—
463C42#12114yes—
464C42#12414yes—
465C23C2×C4×D411yescharacteristic
466C23C2×#12111yescharacteristic
467C23C2×#12111yescharacteristic
468C23#52511yescharacteristic
469C23C2×#12711yescharacteristic
470C23#53511yescharacteristic
471C23#53711yescharacteristic
472C23C22×Q1612yes—
473C23C2×#13512yes—
474C23C2×#13712yes—
475C23#57512yes—
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—44no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—82no—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4#39014yes—
544C2×C4#39014yes—
545C2×C4C4×Q1618yes—
546C2×C4#46218yescomplemented
547C2×C4#46718yes—
548C2×C4#49414yes—
549C2×C4#49514yes—
550C2×C4C22×C22⋊C411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C4×#4111yescharacteristic
554C2×C4#51811yescharacteristic
555C2×C4#52011yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12411yescharacteristic
558C2×C4#52911yescharacteristic
559C2×C4#53211yescharacteristic
560C2×C4#54011yescharacteristic
561C2×C4#54211yescharacteristic
562C2×C4D4211yescharacteristic
563C2×C4D4211yescharacteristic
564C2×C4D4211yescharacteristic
565C2×C4#54611yescharacteristic
566C2×C4#54611yescharacteristic
567C2×C4#54611yescharacteristic
568C2×C4#54711yescharacteristic
569C2×C4#54811yescharacteristic
570C2×C4#54811yescharacteristic
571C2×C4#54811yescharacteristic
572C2×C4#55011yescharacteristic
573C2×C4#55011yescharacteristic
574C2×C4#55011yescharacteristic
575C2×C4#55311yescharacteristic
576C2×C4#55311yescharacteristic
577C2×C4#55511yescharacteristic
578C8—28no—
579C8—44no—
580C22C2×#39011yescharacteristic
581C22C4×#12111yescharacteristic
582C22#232411yescharacteristic
583C22#241411yescharacteristic
584C22#241611yescharacteristic
585C22#254011yescharacteristic
586C22#254111yescharacteristic
587C22C2×C4×Q1612yes—
588C22C4×#13712yes—
589C22C2×#46212yes—
590C22#303512yes—
591C22C2×#46712yes—
592C22#305612yes—
593C22#310412yes—
594C22#310712yes—
595C22C2×#49411yescharacteristic
596C22C2×#49511yescharacteristic
597C22#315211yescharacteristic
598C22#315311yescharacteristic
599C22#320912yes—
600C22#321212yes—
601C22#324812yes—
602C22#324912yes—
603C4—28no—
604C4—24no—
605C4—44no—
606C4—44no—
607C4—82no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—162no—
612C4#196918yescomplemented
613C4D4×C22⋊C411yescharacteristic
614C4#234111yescharacteristic
615C4#236111yescharacteristic
616C4#236311yescharacteristic
617C4#242611yescharacteristic
618C4#242811yescharacteristic
619C4#249011yescharacteristic
620C4#249111yescharacteristic
621C2#1399811yescharacteristic
622C2C2×#196911yescharacteristic
623C2#2123711yescharacteristic
624C2C4×#46212yes—
625C2#2214812yes—
626C2#2465212yes—
627C2#2475212yes—
628C2#2837211yescharacteristic
629C2#2837311yescharacteristic
630C2#2843911yescharacteristic
631C2#2844011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#462C4semidirect
260. C2×C42Q16semidirect
546. C2×C4#462semidirect
612. C4#1969semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#19812
C2×#19692
C2×#19472
C2×#18032
C2×C4×C4⋊C82
#147142
C4×#3912
#111242
#110242
#109262