Finite Group Databasebeta

← #355086

Subgroups — #355086

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C2×#1788C212yesmaximal
3C2×#1750C211yesmaximal, characteristic, complemented
4#13777C211yesmaximal, characteristic, complemented
5#10932C212yesmaximal, complemented
6#10931C212yesmaximal, complemented
7#10928C211yesmaximal, characteristic, complemented
8#10927C211yesmaximal, characteristic, complemented
9#10926C211yesmaximal, characteristic, complemented
10#10925C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C23×#41C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2311—24no—
23#2311C418yescomplemented
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#386—22no—
29C2×#380—22no—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32#1788C4116yescomplemented
33#1750—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1437—22no—
40#1436—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C22×#41—22no—
44C22×#41—24no—
45C22×#41—42no—
46C22×#41—42no—
47C22×#41—42no—
48C22×#41—42no—
49C22×#41C2×C414yes—
50C22×#41C2×C418yes—
51C23×Q8C2311yescharacteristic
52C23×D4—21no—
53C23×D4C2311yescharacteristic
54C24×C4—21no—
55C24×C4—21no—
56C24×C4C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×#117—42no—
60C2×#117—42no—
61C2×#117C2×C414yes—
62C2×#117D414yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D412yes—
77C22×C42—22no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103C2×C418yescomplemented
85C2×#103C2×C418yescomplemented
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102C2×C418yescomplemented
91C2×#102C2×C418yescomplemented
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8C2×C418yes—
95C2×#97—22no—
96C2×#97—42no—
97C2×C4×C8—22no—
98C2×C4×C8—42no—
99#391—22no—
100#391—22no—
101#391C2×C414yes—
102#391C2×C414yes—
103#390—22no—
104#390—22no—
105#390C2×C414yes—
106#390C2×C414yes—
107#386—216no—
108#380—24no—
109#380—216no—
110#380C2×C418yes—
111C4×C4⋊C4—42no—
112C4×C4⋊C4—42no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119#340—42no—
120#340—42no—
121#340—42no—
122#340—42no—
123#336—24no—
124C25—21no—
125C25C2×D411yescharacteristic
126C2×#41—28no—
127C2×#41—216no—
128C2×#41—48no—
129C2×#41—48no—
130C2×#41—48no—
131C2×#41—48no—
132C2×#41—48no—
133C2×#41—48no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—48no—
137C2×#41—48no—
138C2×#41—48no—
139C2×#41—48no—
140C2×#41—48no—
141C2×#41C42116yescomplemented
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C22×Q8C22×C414yes—
150C22×D4—24no—
151C22×D4—28no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—44no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4C22×C412yes—
162C22×D4C22×C414yes—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—82no—
181C23×C4—82no—
182C23×C4C22×C412yes—
183C23×C4C22×C414yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4#4114yes—
189C23×C4C2411yescharacteristic
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
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×C8C22×C414yes—
203C22×C8C22×C414yes—
204#117—48no—
205#117—44no—
206#117—44no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4C22⋊C418yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C22×C414yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4C22⋊C418yes—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—44no—
252C2×C42—82no—
253C2×C42—82no—
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258#103—48no—
259#103—48no—
260#102—48no—
261#102—48no—
262C22⋊C8—216no—
263#97—44no—
264C4×C8—44no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—42no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C2×C22⋊C412yes—
288C24C4×D414yes—
289C24C22×D411yescharacteristic
290#41—432no—
291#41—432no—
292#41—432no—
293#41—432no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8C2×C4214yes—
310C2×D4—216no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4C2×C4214yes—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
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—22no—
348C22×C4—24no—
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—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—216no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—416no—
369C22×C4—416no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—48no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—88no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—88no—
398C22×C4C2×C4214yes—
399C22×C4C2×C22⋊C414yes—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4#11714yes—
403C22×C4C4×D418yes—
404C22×C4C4×D414yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C23×C412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C22⋊C4—48no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—88no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—82no—
492C23—82no—
493C23C4×C22⋊C414yes—
494C23C22×C22⋊C411yescharacteristic
495C23C2×C4×D412yes—
496C23#53412yes—
497C23#53511yescharacteristic
498C23#53511yescharacteristic
499C23#57612yes—
500C23#57712yes—
501C23#57712yes—
502C23#57812yes—
503Q8—28no—
504Q8—44no—
505Q8—44no—
506Q8—44no—
507D4—28no—
508D4—432no—
509D4—44no—
510D4—44no—
511D4—44no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—432no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—816no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4C4×C22⋊C414yes—
566C2×C4C22×C4211yescharacteristic
567C2×C4C2×#11711yescharacteristic
568C2×C4C2×C4×D412yes—
569C2×C4#51812yes—
570C2×C4#51812yes—
571C2×C4#51812yes—
572C2×C4#51812yes—
573C2×C4#51812yes—
574C2×C4#51812yes—
575C2×C4#53411yescharacteristic
576C2×C4#53411yescharacteristic
577C2×C4#53512yes—
578C2×C4#53812yes—
579C2×C4#53812yes—
580C2×C4#53812yes—
581C2×C4#53812yes—
582C8—44no—
583C8—44no—
584C22—28no—
585C22—24no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—48no—
594C22—44no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—82no—
601C22—82no—
602C22C2×C4×C22⋊C411yescharacteristic
603C22#230811yescharacteristic
604C22#230911yescharacteristic
605C22#232112yes—
606C22#232112yes—
607C22#289712yes—
608C22#289812yes—
609C22#294612yes—
610C22#294712yes—
611C22#295112yes—
612C22#295212yes—
613C22#319812yes—
614C22#320112yes—
615C22#320312yes—
616C22#320412yes—
617C22#322011yescharacteristic
618C22#322111yescharacteristic
619C22#322211yescharacteristic
620C22#322311yescharacteristic
621C22#322412yes—
622C22#322512yes—
623C4—24no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—84no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4#227312yes—
632C4#231212yes—
633C4#232112yes—
634C4#232112yes—
635C2—44no—
636C2—84no—
637C2#1377511yescharacteristic
638C2#2041512yes—
639C2#2177212yes—
640C2#2181311yescharacteristic
641C2#2181411yescharacteristic
642C2#2205312yes—
643C2#2205512yes—
644C2#2205712yes—
645C2#2205912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1750C2semidirect
4. #13777C2semidirect
5. #10932C2semidirect
6. #10931C2semidirect
7. #10928C2semidirect
8. #10927C2semidirect
9. #10926C2semidirect
10. #10925C2semidirect
11. #10730C2semidirect
23. #2311C4semidirect
32. #1788C4semidirect
84. C2×#103C2×C4semidirect
85. C2×#103C2×C4semidirect
90. C2×#102C2×C4semidirect
91. C2×#102C2×C4semidirect
141. C2×#41C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#17882
C2×#17502
#137772
#109322
#109312
#109282
#109272
#109262
#109252
#107302