Finite Group Databasebeta

← #375389

Subgroups — #375389

Order 512.

1304 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
1C22×#501C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1776C211yesmaximal, characteristic
5C2×#1771C211yesmaximal, characteristic
6#14161C211yesmaximal, characteristic
7#14160C211yesmaximal, characteristic
8#11379C211yesmaximal, characteristic
9#11378C211yesmaximal, characteristic
10#11378C211yesmaximal, characteristic
11#11377C211yesmaximal, characteristic
12#10863C211yesmaximal, characteristic
13#10863C211yesmaximal, characteristic
14#10862C211yesmaximal, characteristic
15#10861C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#501—24no—
19C2×#501—24no—
20C2×#501C418yescomplemented
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1989—24no—
35#1988—24no—
36#1776—24no—
37#1771—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50#1446—22no—
51#1441—22no—
52#1436—22no—
53C4⋊#95—22no—
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C23×Q8—21no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8C2×C414yes—
73C2×#125—22no—
74C2×#125—22no—
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125C2×C414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D412yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#501—232no—
97C2×#107—22no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107C2×C414yes—
104C2×#107C2×C414yes—
105C2×#107Q814yescomplemented
106C2×#106—22no—
107C2×#106—22no—
108C2×#106—22no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—22no—
112C2×#106C2×C414yes—
113C2×#106C2×C414yes—
114C2×#106Q814yescomplemented
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8D414yes—
118C2×#97—22no—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97C2×C414yes—
122C2×#97D414yes—
123C2×#97D414yes—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95D412yes—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#336—22no—
155#336—22no—
156C22×Q8—24no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—82no—
160C22×Q8—82no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C22×C412yes—
191C22×C8C22×C412yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C22×C8C2×D412yes—
198C22×C8C2×Q812yes—
199C22×C8C2×Q812yes—
200C22×C8#4112yes—
201C22×C8#4112yes—
202C4⋊Q8—28no—
203C4⋊Q8—28no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206#125—28no—
207#125—28no—
208#125—44no—
209#125—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C2×Q812yes—
267C2×C4⋊C4C2×Q812yes—
268C2×C4⋊C4C2×Q812yes—
269C2×C4⋊C4C2×Q812yes—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—82no—
286C2×C42C22×C412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292#107—28no—
293#107—28no—
294#107—28no—
295#106—28no—
296#106—28no—
297#106—28no—
298C4⋊C8—28no—
299#97—44no—
300#97C22⋊C4116yes—
301C4×C8—44no—
302#95—24no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—82no—
308#95—82no—
309#95—82no—
310#95—82no—
311#95—82no—
312#95—82no—
313C24C22×D411yescharacteristic
314C2×Q8—48no—
315C2×Q8—88no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—88no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
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×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12712yes—
381C22×C4#12712yes—
382C22×C4#13714yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C42−1+411yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396C22×C42−1+411yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8C2×C22⋊C414yes—
418C2×C8C2×C22⋊C414yes—
419C2×C8C4×Q814yes—
420C2×C8C4×Q814yes—
421C2×C8C22⋊Q814yes—
422C2×C8C22⋊Q814yes—
423C2×C8#12414yes—
424C2×C8C4⋊Q814yes—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—48no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—84no—
470C42—84no—
471C42C2×C22⋊C414yes—
472C42C4×D414yes—
473C42#12114yes—
474C42#12414yes—
475C23C2×C4×D411yescharacteristic
476C23C2×#12111yescharacteristic
477C23C2×#12711yescharacteristic
478C23#54711yescharacteristic
479C23#54711yescharacteristic
480C23D4×Q811yescharacteristic
481C23D4×Q811yescharacteristic
482C23C2×#13712yes—
483C23#57512yes—
484C23#57612yes—
485C23#57712yes—
486Q8—816no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4#39014yes—
550C2×C4#39014yes—
551C2×C4C22×C22⋊C411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×Q811yescharacteristic
554C2×C4C2×C4×Q811yescharacteristic
555C2×C4#51911yescharacteristic
556C2×C4#51911yescharacteristic
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×#12411yescharacteristic
561C2×C4C2×#12411yescharacteristic
562C2×C4C2×C4⋊Q811yescharacteristic
563C2×C4#53611yescharacteristic
564C2×C4#53611yescharacteristic
565C2×C4#53611yescharacteristic
566C2×C4#53611yescharacteristic
567C2×C4#54711yescharacteristic
568C2×C4#54711yescharacteristic
569C2×C4D4×Q811yescharacteristic
570C2×C4D4×Q811yescharacteristic
571C2×C4#55611yescharacteristic
572C2×C4#55611yescharacteristic
573C2×C4#55611yescharacteristic
574C2×C4#55611yescharacteristic
575C2×C4#55711yescharacteristic
576C2×C4#55711yescharacteristic
577C2×C4#55711yescharacteristic
578C2×C4#55711yescharacteristic
579C8—24no—
580C8—42no—
581C8—42no—
582C8#39118yes—
583C22C2×#39011yescharacteristic
584C22#236511yescharacteristic
585C22#236511yescharacteristic
586C22#254211yescharacteristic
587C22#254211yescharacteristic
588C22#254611yescharacteristic
589C22#254711yescharacteristic
590C22C4×#13712yes—
591C22#294612yes—
592C22#305612yes—
593C22#306212yes—
594C22#315211yescharacteristic
595C22#315311yescharacteristic
596C22#315411yescharacteristic
597C22#315511yescharacteristic
598C22#334412yes—
599C22#334612yes—
600C22#334712yes—
601C22#334812yes—
602C22#338512yes—
603C22#338712yes—
604C22#338812yes—
605C22#338912yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4C2×#39111yescharacteristic
619C4#231011yescharacteristic
620C4#236511yescharacteristic
621C4#236511yescharacteristic
622C4#254611yescharacteristic
623C4#254711yescharacteristic
624C4#256811yescharacteristic
625C4#256811yescharacteristic
626C2#1415511yescharacteristic
627C2#2123511yescharacteristic
628C2#2123711yescharacteristic
629C2#2316712yes—
630C2#2316912yes—
631C2#2854311yescharacteristic
632C2#2854411yescharacteristic
633C2#2854911yescharacteristic
634C2#2855011yescharacteristic
635C2#2868612yes—
636C2#2871512yes—
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
20. C2×#501C4semidirect
105. C2×#107Q8semidirect
114. C2×#106Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#19892
C2×#19882
C2×#17762
C2×#17712
#141612
#141602
#113792
#113782
#113772
#108632
#108622
#108612