Finite Group Databasebeta

← #368367

Subgroups — #368367

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal, complemented
5C2×#1776C211yesmaximal, characteristic, complemented
6#13905C211yesmaximal, characteristic
7#11095C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21#2354—24no—
22#2354—24no—
23C2×#390—21no—
24C2×#388—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1955C418yescomplemented
34#1948—24no—
35#1854—28no—
36#1776—24no—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1445—21no—
45#1438—21no—
46#1437—22no—
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C414yes—
57C2×C4×D4Q814yes—
58C2×#117—24no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C2×#107—22no—
83C2×#107—22no—
84C2×#107D414yescomplemented
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—42no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yes—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8C2×C418yes—
99C2×C22⋊C8C2×C418yes—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97C2×C414yes—
103#390—28no—
104#388—216no—
105#384—22no—
106#384D414yes—
107#382—22no—
108#382—22no—
109#380—24no—
110#380—24no—
111#380—24no—
112#380—24no—
113#380—42no—
114#380—42no—
115C4×C4⋊C4—28no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D414yes—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—42no—
133#340—42no—
134C25C2×D412yes—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C4⋊C414yes—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C22×C414yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×Q814yes—
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8C22×C412yes—
181C22×C8C22×C414yes—
182C22×C8C22×C412yes—
183C22×C8C2×D412yes—
184C22×C8#4112yes—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—28no—
189#117—48no—
190#117—48no—
191#117—48no—
192#117—48no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C4⋊C414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4#4112yes—
235C2×C4⋊C4#4112yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4C4⋊C418yes—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—42no—
257C2×C42—82no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264#107—28no—
265#106—28no—
266#106—48no—
267C4⋊C8—28no—
268C22⋊C8—216no—
269C22⋊C8—48no—
270#97—28no—
271#95—28no—
272#95—24no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—42no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24C2×C4⋊C414yes—
293C24C22×D411yescharacteristic
294C2×D4—28no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—44no—
298C2×D4—44no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4C2×C4⋊C414yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C2×C4⋊C412yes—
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C42−1+411yescharacteristic
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8C4×D414yes—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—84no—
439C4⋊C4#12114yes—
440C22⋊C4—216no—
441C22⋊C4—416no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—48no—
446C42—416no—
447C42—84no—
448C42C4×D414yes—
449C42C4×D414yes—
450C42C22⋊Q814yes—
451C42C22⋊Q814yes—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23C22×C4⋊C411yescharacteristic
488C23C2×#12011yescharacteristic
489C23C2×#12311yescharacteristic
490C23D4211yescharacteristic
491C23#54611yescharacteristic
492C23#54611yescharacteristic
493C23#54711yescharacteristic
494C23#57512yes—
495C23#57512yes—
496C23#57712yes—
497C23#57712yes—
498D4—416no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#38514yes—
557C2×C4#38514yes—
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C4×#4111yescharacteristic
562C2×C4#51811yescharacteristic
563C2×C4#52011yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4#52511yescharacteristic
568C2×C4#52911yescharacteristic
569C2×C4#53211yescharacteristic
570C2×C4#53511yescharacteristic
571C2×C4#53711yescharacteristic
572C2×C4#54011yescharacteristic
573C2×C4#54211yescharacteristic
574C2×C4D4211yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54811yescharacteristic
577C2×C4D4×Q811yescharacteristic
578C2×C4#55011yescharacteristic
579C2×C4#55011yescharacteristic
580C2×C4#55411yescharacteristic
581C2×C4#55411yescharacteristic
582C2×C4#55411yescharacteristic
583C2×C4#55611yescharacteristic
584C2×C4#55911yescharacteristic
585C2×C4#55911yescharacteristic
586C8—24no—
587C8—42no—
588C8—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×#38511yescharacteristic
605C22D4×C4⋊C411yescharacteristic
606C22#234911yescharacteristic
607C22#240711yescharacteristic
608C22#240811yescharacteristic
609C22#244511yescharacteristic
610C22#244711yescharacteristic
611C22#291112yes—
612C22#291212yes—
613C22#300511yescharacteristic
614C22#300511yescharacteristic
615C22#300911yescharacteristic
616C22#301011yescharacteristic
617C22#309312yes—
618C22#309912yes—
619C22#326712yes—
620C22#327112yes—
621C22#330712yes—
622C22#331112yes—
623C22#331712yes—
624C22#332112yes—
625C22#334812yes—
626C22#335012yes—
627C4—24no—
628C4—24no—
629C4—44no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—84no—
635C4—162no—
636C4C4×#12111yescharacteristic
637C4#232411yescharacteristic
638C4#236611yescharacteristic
639C4#236711yescharacteristic
640C4#245211yescharacteristic
641C4#245411yescharacteristic
642C4#253811yescharacteristic
643C4#258011yescharacteristic
644C2—84no—
645C2#1401811yescharacteristic
646C2#2084511yescharacteristic
647C2#2084811yescharacteristic
648C2#2266212yes—
649C2#2271012yes—
650C2#2413111yescharacteristic
651C2#2413311yescharacteristic
652C2#2424911yescharacteristic
653C2#2425411yescharacteristic
654C2#2624512yes—
655C2#2630912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1948C2semidirect
4. C2×#1854C2semidirect
5. C2×#1776C2semidirect
7. #11095C2semidirect
8. #11095C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10731C2semidirect
33. #1955C4semidirect
84. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#19552
C2×#19482
C2×#18542
C2×#17762
#139052
#110952
#108502
#108432
#107312