Finite Group Databasebeta

← #366931

Subgroups — #366931

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1940C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×#1762C211yesmaximal, characteristic
7#13804C211yesmaximal, characteristic, complemented
8#11358C212yesmaximal, complemented
9#11076C212yesmaximal
10#10931C212yesmaximal
11#10850C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2563—24no—
21#2563C418yescomplemented
22C2×#393—21no—
23C2×#393—21no—
24C2×#390C2212yes—
25C2×#382C2212yes—
26C2×#382C2211yescharacteristic
27C2×#380—22no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1993—24no—
32#1988—24no—
33#1940—24no—
34#1804—24no—
35#1762—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×Q8—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—42no—
94C2×#102Q814yescomplemented
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98#393—28no—
99#393—28no—
100#390—24no—
101#390C2×C418yes—
102#382—24no—
103#382—22no—
104#382C2×C418yes—
105#382D414yes—
106#380—216no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—42no—
110C4×C4⋊C4C2×C414yes—
111C4×C4⋊C4D414yes—
112C4×C4⋊C4D414yes—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#336—24no—
131C25—21no—
132C22×Q8—24no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—82no—
136C22×Q8—82no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C22×D4C2×Q812yes—
150C22×D4#4112yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
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—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C2×Q812yes—
186C22×C8#4112yes—
187C4⋊Q8—28no—
188C4⋊Q8—28no—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—44no—
194C4×D4—44no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C22⋊C418yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C4⋊C4#4112yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—82no—
248C2×C42—82no—
249C2×C42C22×C414yes—
250C2×C42C22×C412yes—
251C2×C42C2×D414yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—48no—
258C4⋊C8—44no—
259#102—28no—
260#102—28no—
261#102—48no—
262#97—44no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—44no—
269#95—44no—
270#95—82no—
271#95C22⋊C414yes—
272#95C22⋊C414yes—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24C22×D411yescharacteristic
285C2×Q8—48no—
286C2×Q8—88no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8—88no—
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—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4C4×Q814yes—
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—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—416no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
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—48no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×Q811yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C42−1+411yescharacteristic
388C22×C42−1+411yescharacteristic
389C22×C42−1+411yescharacteristic
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—48no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—48no—
432C4⋊C4C22⋊Q814yes—
433C22⋊C4—48no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—84no—
441C42—88no—
442C42C4×D414yes—
443C42C4×D414yes—
444C42#12414yes—
445C42#12714yes—
446C23—24no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—82no—
460C23—82no—
461C23—81no—
462C23—81no—
463C23—82no—
464C23—81no—
465C23—81no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23C22×C22⋊C411yescharacteristic
470C23C2×#12111yescharacteristic
471C23C2×#12111yescharacteristic
472C23#54611yescharacteristic
473C23#54711yescharacteristic
474C23#54811yescharacteristic
475C23D4×Q811yescharacteristic
476C23#57512yes—
477C23#57512yes—
478C23#57612yes—
479C23#57712yes—
480Q8—816no—
481D4—24no—
482D4—24no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
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—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—816no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4#39014yes—
542C2×C4#39014yes—
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×Q811yescharacteristic
546C2×C4C4×#4111yescharacteristic
547C2×C4#51911yescharacteristic
548C2×C4#52011yescharacteristic
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4C2×#12411yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4C2×#12711yescharacteristic
554C2×C4#53311yescharacteristic
555C2×C4#53611yescharacteristic
556C2×C4#53711yescharacteristic
557C2×C4#54111yescharacteristic
558C2×C4#54211yescharacteristic
559C2×C4#54611yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54811yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4D4×Q811yescharacteristic
566C2×C4D4×Q811yescharacteristic
567C2×C4#55211yescharacteristic
568C2×C4#55211yescharacteristic
569C2×C4#55411yescharacteristic
570C2×C4#55611yescharacteristic
571C8—42no—
572C8—42no—
573C8—82no—
574C22—44no—
575C22—48no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—82no—
584C22—82no—
585C22C2×#39011yescharacteristic
586C22#233911yescharacteristic
587C22Q8×C22⋊C411yescharacteristic
588C22#241511yescharacteristic
589C22#241711yescharacteristic
590C22#242711yescharacteristic
591C22#242911yescharacteristic
592C22#289612yes—
593C22#289712yes—
594C22#304011yescharacteristic
595C22#304011yescharacteristic
596C22#304012yes—
597C22#304111yescharacteristic
598C22#304311yescharacteristic
599C22#304512yes—
600C22#331912yes—
601C22#332112yes—
602C22#334712yes—
603C22#334812yes—
604C22#337012yes—
605C22#337612yes—
606C22#338812yes—
607C22#338912yes—
608C4—24no—
609C4—24no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—82no—
614C4—84no—
615C4—82no—
616C4—84no—
617C4C4×C22⋊Q811yescharacteristic
618C4#232311yescharacteristic
619C4#236011yescharacteristic
620C4#236211yescharacteristic
621C4#248911yescharacteristic
622C4#249211yescharacteristic
623C4#254011yescharacteristic
624C4#254211yescharacteristic
625C2—84no—
626C2#1399711yescharacteristic
627C2#2126111yescharacteristic
628C2#2126311yescharacteristic
629C2#2249212yes—
630C2#2252612yes—
631C2#2469311yescharacteristic
632C2#2469511yescharacteristic
633C2#2482811yescharacteristic
634C2#2483011yescharacteristic
635C2#2533512yes—
636C2#2546612yes—
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×#1993C2semidirect
3. C2×#1988C2semidirect
4. C2×#1940C2semidirect
7. #13804C2semidirect
8. #11358C2semidirect
11. #10850C2semidirect
21. #2563C4semidirect
94. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#19932
C2×#19882
C2×#19402
C2×#18042
C2×#17622
#138042
#113582
#110762
#109312
#108502