Finite Group Databasebeta

← #362111

Subgroups — #362111

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1940C211yesmaximal, characteristic, complemented
3C2×#1802C212yesmaximal
4C2×#1802C211yesmaximal, characteristic
5C2×#1762C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic, complemented
7#11228C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2349—24no—
21#2349C418yescomplemented
22C2×#389—22no—
23C2×#389C2212yes—
24C2×#385—22no—
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27C2×#382C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C22×#95—21no—
32#1940—24no—
33#1802—28no—
34#1802—24no—
35#1762—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—22no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4C2×C418yes—
62C2×C4×D4C2×C414yes—
63C2×C4×D4D414yes—
64C2×C4×D4Q814yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—42no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102D414yes—
95C2×#102Q814yes—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99#389—24no—
100#389—48no—
101#389C2×C418yes—
102#385—216no—
103#384—24no—
104#384—22no—
105#384—22no—
106#384—42no—
107#382—22no—
108#382C2×C414yes—
109#380—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C414yes—
113C4×C4⋊C4D414yes—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—28no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—42no—
129#340—42no—
130#336—24no—
131C25—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C22×D4C2×D412yes—
141C22×D4C2×D412yes—
142C22×D4C2×Q812yes—
143C22×D4C2×Q812yes—
144C22×D4#4112yes—
145C22×D4#4112yes—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—28no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D412yes—
188C22×C8C2×Q812yes—
189C4×D4—216no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4C4⋊C4116yescomplemented
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
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—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C418yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×Q812yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42C22×C414yes—
256C2×C42C22×C412yes—
257C2×C42C2×D414yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C4⋊C8—48no—
264C4⋊C8—44no—
265#102—28no—
266#102—28no—
267#102—48no—
268#97—44no—
269#95—24no—
270#95—28no—
271#95—24no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95C22⋊C414yes—
276#95C22⋊C414yes—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24C22×D411yescharacteristic
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—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4C2×C4⋊C414yes—
311C2×D4C4×D414yes—
312C2×D4C4×Q814yes—
313C2×D4#12114yes—
314C2×D4C22⋊Q814yes—
315C2×D4#12514yes—
316C2×D4C4⋊Q814yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
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—216no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—416no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
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—416no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4C2×C22⋊C414yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C414yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4C22⋊Q812yes—
379C22×C4C22⋊Q812yes—
380C22×C4AGL(1,ℤ/8)14yes—
381C22×C4#13714yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—216no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—88no—
442C4⋊C4C2×C4⋊C414yes—
443C22⋊C4—216no—
444C22⋊C4—416no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—84no—
450C42C2×C4⋊C414yes—
451C42#11714yes—
452C42#12114yes—
453C42C22⋊Q814yes—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23C22×C22⋊C411yescharacteristic
478C23C2×#12111yescharacteristic
479C23C2×C22⋊Q811yescharacteristic
480C23D4211yescharacteristic
481C23#54611yescharacteristic
482C23#54811yescharacteristic
483C23D4×Q811yescharacteristic
484C23C2×AGL(1,ℤ/8)12yes—
485C23C2×#13712yes—
486C23#57612yes—
487C23#57712yes—
488D4—24no—
489D4—24no—
490D4—24no—
491D4—24no—
492D4—24no—
493D4—24no—
494D4—24no—
495D4#38918yescomplemented
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—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—816no—
530C2×C4—832no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#38014yes—
549C2×C4#38014yes—
550C2×C4#41718yes—
551C2×C4C22×C4⋊C411yescharacteristic
552C2×C4C2×#11711yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×Q811yescharacteristic
555C2×C4#51811yescharacteristic
556C2×C4#51911yescharacteristic
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×#12111yescharacteristic
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×C22⋊Q811yescharacteristic
561C2×C4C2×#12511yescharacteristic
562C2×C4C2×C4⋊Q811yescharacteristic
563C2×C4#53711yescharacteristic
564C2×C4#53711yescharacteristic
565C2×C4#54411yescharacteristic
566C2×C4#54411yescharacteristic
567C2×C4D4211yescharacteristic
568C2×C4#54611yescharacteristic
569C2×C4#54711yescharacteristic
570C2×C4#54711yescharacteristic
571C2×C4#54811yescharacteristic
572C2×C4D4×Q811yescharacteristic
573C2×C4#55211yescharacteristic
574C2×C4#55211yescharacteristic
575C2×C4#55311yescharacteristic
576C2×C4#55311yescharacteristic
577C2×C4#55411yescharacteristic
578C2×C4#55411yescharacteristic
579C8—42no—
580C8—42no—
581C8—82no—
582C22—28no—
583C22—44no—
584C22—48no—
585C22—48no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—44no—
593C22C2×#38011yescharacteristic
594C22D4×C22⋊C411yescharacteristic
595C22Q8×C22⋊C411yescharacteristic
596C22#241611yescharacteristic
597C22#241711yescharacteristic
598C22#244011yescharacteristic
599C22#244211yescharacteristic
600C22C2×#41712yes—
601C22#289712yes—
602C22#303811yescharacteristic
603C22#303911yescharacteristic
604C22#304111yescharacteristic
605C22#304211yescharacteristic
606C22#307912yes—
607C22#308112yes—
608C22#326412yes—
609C22#326612yes—
610C22#330612yes—
611C22#331412yes—
612C22#336612yes—
613C22#337212yes—
614C22#338412yes—
615C22#338512yes—
616C4—28no—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—162no—
625C4C2×#38911yescharacteristic
626C4#231611yescharacteristic
627C4#234411yescharacteristic
628C4#234511yescharacteristic
629C4#241611yescharacteristic
630C4#241711yescharacteristic
631C4#244011yescharacteristic
632C4#244211yescharacteristic
633C2—48no—
634C2#1390711yescharacteristic
635C2#2064811yescharacteristic
636C2#2065011yescharacteristic
637C2#2241912yes—
638C2#2252212yes—
639C2#2474411yescharacteristic
640C2#2474811yescharacteristic
641C2#2482911yescharacteristic
642C2#2483311yescharacteristic
643C2#2605812yes—
644C2#2612412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1940C2semidirect
6. #13910C2semidirect
7. #11228C2semidirect
8. #11207C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
21. #2349C4semidirect
196. C4×D4C4⋊C4semidirect
495. D4#389semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#19402
C2×#18022
C2×#17622
#139102
#112282
#112072
#108502
#108432
#107882