Finite Group Databasebeta

← #364850

Subgroups — #364850

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic
5C4×#419C211yesmaximal, characteristic
6#11214C212yesmaximal
7#11059C212yesmaximal
8#10952C212yesmaximal
9#10904C212yesmaximal
10#10856C211yesmaximal, characteristic
11C4⋊#338C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—22no—
15#3078—22no—
16#3078—22no—
17#3078—22no—
18C2×#479C414yescomplemented
19C2×#477C412yescomplemented
20C2×#475C412yescomplemented
21Q8×M16—22no—
22Q8×M16—22no—
23#2962—22no—
24#2962—22no—
25#2961—22no—
26#2960—22no—
27C2×#428C2212yes—
28#2905C2212yes—
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#419C2211yescharacteristic
34C2×#419C2212yes—
35C22×#101C2212yes—
36#2873C2212yes—
37C2×C4×M16C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1995—22no—
40#1994—22no—
41#1801—22no—
42#1801—22no—
43#1763—22no—
44C4×#103—22no—
45C2×#344C2212yes—
46C2×#344C2212yes—
47C2×#343C2212yes—
48#1736C2211yescharacteristic
49#1736C2211yescharacteristic
50#1733C2212yes—
51#1733C2212yes—
52C2×#338C2212yes—
53C2×#338C2212yes—
54C23×Q8C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2312yes—
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533D412yes—
65#533D412yes—
66#533D412yes—
67#533D412yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#125C2×C412yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8C2×C412yes—
86C2×C4×Q8C2×C412yes—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#479—28no—
95#479—28no—
96#477—24no—
97#477—24no—
98#475—24no—
99#475—24no—
100#446—24no—
101#446—24no—
102#446—28no—
103#446—28no—
104C8×Q8—24no—
105C8×Q8—24no—
106#428—24no—
107#428—24no—
108C2×#107C2×C414yes—
109C2×#106C2×C414yes—
110#424—24no—
111#424—24no—
112#423—24no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423—21no—
120#423—21no—
121#423—21no—
122#423—21no—
123#423D412yes—
124#423D412yes—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2×C412yes—
130C2×C4⋊C8C2×C412yes—
131#419—22no—
132#419—22no—
133#419—24no—
134#419—24no—
135#419—22no—
136#419Q814yescomplemented
137C2×#103—22no—
138C2×#103—22no—
139C2×#103C2×C414yes—
140C2×#103C2×C414yes—
141C2×#101—24no—
142C2×#101—24no—
143C2×#101—42no—
144#405—24no—
145#405—24no—
146C4×M16—24no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—21no—
150C4×M16—21no—
151C2×#97D412yes—
152C2×C4×C8D412yes—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157#376—21no—
158C43—21no—
159#344—24no—
160#344—42no—
161#343—24no—
162#340—22no—
163#340—22no—
164#339—24no—
165#339—24no—
166#338—24no—
167#338—42no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8C22×C412yes—
182C22×Q8C2×Q812yes—
183C22×Q8#4112yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4#4111yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—21no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16C2×D414yes—
217C2×M16C2×Q812yes—
218C2×M16#4112yes—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8C22×C414yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8C22⋊C412yes—
229C4⋊Q8C22⋊C412yes—
230C4⋊Q8C22⋊C412yes—
231C4⋊Q8C22⋊C412yes—
232#125—22no—
233#125—22no—
234#125C22⋊C414yes—
235#125C22⋊C414yes—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—41no—
261C4×Q8—41no—
262C4×Q8—41no—
263C4×Q8—41no—
264C4×Q8—41no—
265C4×Q8—41no—
266C4×Q8—41no—
267C4×Q8—41no—
268#117—22no—
269#117—22no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117C2×D414yes—
277#117C2×D414yes—
278#117C2×Q812yes—
279#117#4112yes—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—41no—
309C2×C42—41no—
310C2×C42—41no—
311C2×C42—41no—
312C2×C42—41no—
313C2×C42—41no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42#4111yescharacteristic
326#107—28no—
327#106—28no—
328C4⋊C8—28no—
329C4⋊C8—28no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—41no—
339C4⋊C8—41no—
340C4⋊C8—41no—
341C4⋊C8—41no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—28no—
346#101—48no—
347#97—28no—
348#97—22no—
349#97—22no—
350C4×C8—28no—
351C4×C8—22no—
352C4×C8—22no—
353#95—41no—
354#95—41no—
355C24C2×#4111yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8C4×Q814yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—81no—
406C22×C4—82no—
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12311yescharacteristic
416C22×C4#12311yescharacteristic
417C22×C4#12311yescharacteristic
418C22×C4#12311yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C42−1+411yescharacteristic
431C22×C42−1+411yescharacteristic
432C22×C42−1+411yescharacteristic
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—24no—
438M16—24no—
439M16—42no—
440M16—42no—
441M16—41no—
442M16—41no—
443M16—42no—
444M16—42no—
445M16—41no—
446M16—42no—
447M16—42no—
448M16—41no—
449M16—42no—
450M16—42no—
451M16—42no—
452M16—42no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8C22⋊Q814yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4C2×C22⋊C414yes—
491C4⋊C4C2×C22⋊C414yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—41no—
503C42—41no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—82no—
509C42—82no—
510C42C2×C22⋊C411yescharacteristic
511C42C2×C22⋊C411yescharacteristic
512C42C2×C22⋊C411yescharacteristic
513C42C2×C22⋊C411yescharacteristic
514C42C4×D411yescharacteristic
515C42C4×D411yescharacteristic
516C42C4×D411yescharacteristic
517C42C4×D411yescharacteristic
518C42#12011yescharacteristic
519C42#12011yescharacteristic
520C42#12011yescharacteristic
521C42#12011yescharacteristic
522C42#12111yescharacteristic
523C42#12111yescharacteristic
524C42#12111yescharacteristic
525C42#12111yescharacteristic
526C23—41no—
527C23—41no—
528C23C2×C4×D411yescharacteristic
529C23C2×#12111yescharacteristic
530C23C2×#12311yescharacteristic
531C23#55211yescharacteristic
532C23#55211yescharacteristic
533C23#55411yescharacteristic
534C23#55611yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—44no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—44no—
543Q8—42no—
544Q8—42no—
545Q8—82no—
546Q8—82no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#38611yescharacteristic
581C2×C4#38611yescharacteristic
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4#38611yescharacteristic
586C2×C4#38611yescharacteristic
587C2×C4#38611yescharacteristic
588C2×C4C22×C22⋊C411yescharacteristic
589C2×C4C2×C4×D411yescharacteristic
590C2×C4C2×C4×Q811yescharacteristic
591C2×C4C4×#4111yescharacteristic
592C2×C4#51911yescharacteristic
593C2×C4#52011yescharacteristic
594C2×C4C2×#12011yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4#52511yescharacteristic
598C2×C4#52911yescharacteristic
599C2×C4#53311yescharacteristic
600C2×C4#53611yescharacteristic
601C2×C4#53711yescharacteristic
602C2×C4#54111yescharacteristic
603C2×C4#54211yescharacteristic
604C2×C4#54611yescharacteristic
605C2×C4#54711yescharacteristic
606C2×C4#54811yescharacteristic
607C2×C4#54811yescharacteristic
608C2×C4#54811yescharacteristic
609C2×C4#54811yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4D4×Q811yescharacteristic
612C2×C4D4×Q811yescharacteristic
613C2×C4D4×Q811yescharacteristic
614C2×C4#55011yescharacteristic
615C2×C4#55011yescharacteristic
616C8—24no—
617C8—44no—
618C8—42no—
619C8—44no—
620C8—82no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22C2×#38611yescharacteristic
632C22#236011yescharacteristic
633C22#236211yescharacteristic
634C22#242111yescharacteristic
635C22#242311yescharacteristic
636C22#253611yescharacteristic
637C22#253811yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#190212yes—
652C4C4×C22⋊Q811yescharacteristic
653C4#232311yescharacteristic
654C4Q8×C22⋊C411yescharacteristic
655C4#234111yescharacteristic
656C4#240711yescharacteristic
657C4#240911yescharacteristic
658C4#242611yescharacteristic
659C4#242911yescharacteristic
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2#1396911yescharacteristic
667C2C2×#190211yescharacteristic
668C2#2099511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×#479C4semidirect
19. C2×#477C4semidirect
20. C2×#475C4semidirect
136. #419Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#601922
#213292
C2×#18012
C4×#4192
#112142
#110592
#109522
#109042
#108562
C4⋊#3382