Finite Group Databasebeta

← #367045

Subgroups — #367045

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1981C212yesmaximal
4C2×#1960C211yesmaximal, characteristic
5C2×#1940C211yesmaximal, characteristic
6C2×#1805C211yesmaximal, characteristic
7C2×#1763C211yesmaximal, characteristic
8#13804C211yesmaximal, characteristic
9#11077C212yesmaximal
10#11010C212yesmaximal
11#10932C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#97C2211yescharacteristic
25#2546—24no—
26#2546C418yescomplemented
27C2×#398—21no—
28C2×#398—21no—
29C2×#391C2212yes—
30C2×#389—22no—
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1981—28no—
36#1981—24no—
37#1960—24no—
38#1940—24no—
39#1805—24no—
40#1763—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95C2212yes—
47#1434—21no—
48C4×#95—21no—
49C23×Q8C2312yes—
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4⋊Q8C2×C418yes—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2×C414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—42no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—42no—
109C2×#103—42no—
110C2×#103—42no—
111C2×#103—42no—
112C2×#103—42no—
113C2×#103D414yes—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117#398—28no—
118#398—28no—
119#391—24no—
120#391C2×C418yes—
121#389—216no—
122#382—22no—
123#382D414yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—42no—
127C4×C4⋊C4C2×C414yes—
128C4×C4⋊C4D414yes—
129C4×C4⋊C4D414yes—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140#340—24no—
141#340—22no—
142#340—22no—
143#336—24no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—21no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—21no—
167C22×Q8—22no—
168C22×Q8C22×C414yes—
169C22×Q8C22×C412yes—
170C22×Q8C2×D412yes—
171C22×Q8#4112yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8C2×D412yes—
203C22×C8#4112yes—
204C4⋊Q8—216no—
205C4⋊Q8—28no—
206C4⋊Q8—28no—
207C4⋊Q8—48no—
208C4⋊Q8—48no—
209C4⋊Q8—48no—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—44no—
215C4×Q8—44no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C22⋊C418yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4#4112yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—82no—
269C2×C42—82no—
270C2×C42C22×C414yes—
271C2×C42C22×C412yes—
272C2×C42C2×D414yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278C4⋊C8—48no—
279C4⋊C8—44no—
280#103—28no—
281#103—28no—
282#103—48no—
283#103—48no—
284#103—48no—
285#103—48no—
286#103—48no—
287#97—44no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—82no—
296#95C22⋊C414yes—
297#95C22⋊C414yes—
298C24C22×D411yescharacteristic
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—41no—
330C2×Q8—42no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8C4×D414yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
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—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C2×C22⋊C414yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C414yes—
409C22×C4#12112yes—
410C22×C4#12112yes—
411C22×C4#12112yes—
412C22×C4#12112yes—
413C22×C4#13714yes—
414C22×C4#13714yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—216no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4#12114yes—
472C42—24no—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—48no—
477C42—84no—
478C42—88no—
479C42C4×D414yes—
480C42C4×D414yes—
481C42#12414yes—
482C42#12714yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×#12111yescharacteristic
485C23C2×#12111yescharacteristic
486C23D4211yescharacteristic
487C23#54611yescharacteristic
488C23#54811yescharacteristic
489C23#55011yescharacteristic
490C23C2×#13712yes—
491C23C2×#13712yes—
492C23#57712yes—
493C23#57812yes—
494Q8—24no—
495Q8—24no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509Q8—82no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4#39014yes—
568C2×C4#39014yes—
569C2×C4#41918yes—
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C4×#4111yescharacteristic
574C2×C4#51811yescharacteristic
575C2×C4#52011yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4#52511yescharacteristic
578C2×C4C2×#12411yescharacteristic
579C2×C4#52911yescharacteristic
580C2×C4C2×#12711yescharacteristic
581C2×C4#53211yescharacteristic
582C2×C4#53511yescharacteristic
583C2×C4#53711yescharacteristic
584C2×C4#54011yescharacteristic
585C2×C4#54211yescharacteristic
586C2×C4D4211yescharacteristic
587C2×C4D4211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54611yescharacteristic
590C2×C4#54711yescharacteristic
591C2×C4#54811yescharacteristic
592C2×C4#54811yescharacteristic
593C2×C4#55011yescharacteristic
594C2×C4#55011yescharacteristic
595C2×C4#55311yescharacteristic
596C2×C4#55311yescharacteristic
597C2×C4#55511yescharacteristic
598C8—42no—
599C8—42no—
600C8—82no—
601C22C2×#39011yescharacteristic
602C22D4×C22⋊C411yescharacteristic
603C22#234111yescharacteristic
604C22#241411yescharacteristic
605C22#241611yescharacteristic
606C22#242611yescharacteristic
607C22#242811yescharacteristic
608C22C2×#41912yes—
609C22#289812yes—
610C22#303912yes—
611C22#303911yescharacteristic
612C22#303911yescharacteristic
613C22#304211yescharacteristic
614C22#304411yescharacteristic
615C22#304612yes—
616C22#326612yes—
617C22#327012yes—
618C22#331212yes—
619C22#332012yes—
620C22#337212yes—
621C22#337412yes—
622C22#339812yes—
623C22#340012yes—
624C4—24no—
625C4—24no—
626C4—28no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—84no—
634C4C4×#12111yescharacteristic
635C4#232411yescharacteristic
636C4#236111yescharacteristic
637C4#236311yescharacteristic
638C4#249011yescharacteristic
639C4#249111yescharacteristic
640C4#254011yescharacteristic
641C4#254111yescharacteristic
642C2#1399811yescharacteristic
643C2#2125911yescharacteristic
644C2#2126211yescharacteristic
645C2#2242512yes—
646C2#2254712yes—
647C2#2464611yescharacteristic
648C2#2464811yescharacteristic
649C2#2474611yescharacteristic
650C2#2474811yescharacteristic
651C2#2529012yes—
652C2#2538212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
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
26. #2546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#19812
C2×#19602
C2×#19402
C2×#18052
C2×#17632
#138042
#110772
#110102
#109322