Finite Group Databasebeta

← #365108

Subgroups — #365108

Order 512.

2084 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#419C211yesmaximal, characteristic
2C2×#1927C212yesmaximal, complemented
3C2×#1854C211yesmaximal, characteristic
4C2×#1787C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6#13801C211yesmaximal, characteristic
7#11190C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#10932C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#419—28no—
18C2×#419—24no—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1927—28no—
31#1854—24no—
32#1787C418yescomplemented
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1446—21no—
40#1446—21no—
41#1437—22no—
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—22no—
45C4×#95—21no—
46C23×Q8—21no—
47C23×Q8—22no—
48C23×Q8—24no—
49C23×Q8—24no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—42no—
55C23×Q8—42no—
56C23×Q8C2×C414yes—
57C23×Q8C2312yes—
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—24no—
64C22×M16—42no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2312yes—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117D414yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42C2312yes—
84C2×#107—22no—
85C2×#107—42no—
86C2×#106—22no—
87C2×#106—24no—
88#419—232no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103C2×C418yes—
99C2×#103C2×C418yes—
100C2×#103D418yes—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2×C414yes—
104#391—24no—
105#391—24no—
106#391—42no—
107#391—42no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380C2×C414yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—42no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—22no—
129#340—22no—
130#340—42no—
131#336—24no—
132#336—42no—
133C25C2×D411yescharacteristic
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—28no—
139C22×Q8—216no—
140C22×Q8—28no—
141C22×Q8—216no—
142C22×Q8—28no—
143C22×Q8—28no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—48no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8C22×C414yes—
169C22×Q8C2×D414yes—
170C22×Q8#4114yes—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C22×C412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C2×M16—28no—
197C2×M16—48no—
198C2×M16—48no—
199C2×M16—44no—
200C2×M16—44no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D414yes—
218#117—28no—
219#117—44no—
220#117—48no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—81no—
252C2×C22⋊C4—81no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—82no—
264#107—44no—
265#106—44no—
266#103—216no—
267#103—216no—
268#103—216no—
269#103—48no—
270C22⋊C8—28no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—82no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24C4×D412yes—
291C24#12112yes—
292C24#12312yes—
293C24C22×D411yescharacteristic
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—232no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—416no—
311C2×Q8—432no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—416no—
315C2×Q8—416no—
316C2×Q8—416no—
317C2×Q8—416no—
318C2×Q8—432no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—48no—
332C2×Q8C4×D418yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—416no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—416no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4#11714yes—
402C22×C4C4×D414yes—
403C22×C4C4×D412yes—
404C22×C4#12112yes—
405C22×C4#12312yes—
406C22×C4#12414yes—
407C22×C4#12614yes—
408C22×C4#13714yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4111yescharacteristic
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)12yes—
419M16—416no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4#12014yes—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C42—48no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—88no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23#38814yes—
486C23C2×C4×D411yescharacteristic
487C23C2×#12111yescharacteristic
488C23C2×#12311yescharacteristic
489C23D4212yes—
490C23#54612yes—
491C23C2×#13712yes—
492C23#57512yes—
493C23#57711yescharacteristic
494C23#57711yescharacteristic
495C23#57811yescharacteristic
496C23#57811yescharacteristic
497Q8—28no—
498Q8—28no—
499Q8—432no—
500Q8—44no—
501Q8—44no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—432no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4#38814yes—
566C2×C4C2×#11711yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×C4×D412yes—
569C2×C4#51812yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52512yes—
573C2×C4C2×#12411yescharacteristic
574C2×C4C2×#12611yescharacteristic
575C2×C4#53411yescharacteristic
576C2×C4#53511yescharacteristic
577C2×C4#53812yes—
578C2×C4#54612yes—
579C2×C4#54612yes—
580C2×C4#54612yes—
581C2×C4#55112yes—
582C2×C4#55112yes—
583C2×C4#55112yes—
584C8—42no—
585C8—44no—
586C8—42no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22C2×#38811yescharacteristic
599C22#235812yes—
600C22#241011yescharacteristic
601C22#241111yescharacteristic
602C22#241812yes—
603C22C4×#13712yes—
604C22#294911yescharacteristic
605C22#295011yescharacteristic
606C22#303911yescharacteristic
607C22#304011yescharacteristic
608C22#304611yescharacteristic
609C22#304611yescharacteristic
610C22#309512yes—
611C22#309711yescharacteristic
612C22#309711yescharacteristic
613C22#326612yes—
614C22#327112yes—
615C22#327612yes—
616C22#327712yes—
617C22#331212yes—
618C22#331312yes—
619C22#332612yes—
620C22#332912yes—
621C4—24no—
622C4—44no—
623C4—44no—
624C4—48no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4—161no—
629C4—161no—
630C4C4×#12011yescharacteristic
631C4#232111yescharacteristic
632C4#234212yes—
633C4#243511yescharacteristic
634C4#248012yes—
635C4#256111yescharacteristic
636C2—44no—
637C2#1397111yescharacteristic
638C2#2106111yescharacteristic
639C2#2106611yescharacteristic
640C2#2286912yes—
641C2#2288512yes—
642C2#2437511yescharacteristic
643C2#2437611yescharacteristic
644C2#2449711yescharacteristic
645C2#2449811yescharacteristic
646C2#2488412yes—
647C2#2488812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1927C2semidirect
7. #11190C2semidirect
8. #11124C2semidirect
9. #10932C2semidirect
32. #1787C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4192
C2×#19272
C2×#18542
C2×#17872
#139222
#138012
#111902
#111242
#109322
#107222
#107152