Finite Group Databasebeta

← #364819

Subgroups — #364819

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C2×#2960C211yesmaximal, characteristic, complemented
3#21312C211yesmaximal, characteristic, complemented
4C2×#1800C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#11216C212yesmaximal, complemented
7#11061C212yesmaximal
8#10943C212yesmaximal
9#10896C212yesmaximal, complemented
10#10847C211yesmaximal, characteristic, complemented
11#10844C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14#3077—24no—
15#3077C418yescomplemented
16C2×#478—22no—
17C2×#476—21no—
18C2×#474—21no—
19#2960—24no—
20#2960—24no—
21C2×#446—21no—
22C2×#446—21no—
23C2×C8×Q8—21no—
24C2×C8×Q8—21no—
25C2×#428C2212yes—
26C2×#424C2212yes—
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#418C2212yes—
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#100C2212yes—
34C2×#405C2212yes—
35C22×C4×C8C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1989—22no—
38#1988—21no—
39#1987—21no—
40#1800—24no—
41C4×#102—21no—
42C4×#102—21no—
43C4×#102—21no—
44C4×#102—21no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2212yes—
48C2×#341C2212yes—
49C2×#341C2212yes—
50#1736C2212yes—
51#1736C2212yes—
52C2×#340C2211yescharacteristic
53C2×#340C2211yescharacteristic
54C23×D4C2311yescharacteristic
55C24×C4—21no—
56C22×M16C2312yes—
57C22×M16C2312yes—
58C22×M16C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61#533—22no—
62#533—22no—
63#533C2×C414yes—
64#533D414yes—
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8D411yescharacteristic
67C2×#125D412yes—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4C2×C414yes—
80C2×#117C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#478—216no—
88#476—28no—
89#474—28no—
90#446—28no—
91#446—28no—
92C8×Q8—28no—
93C8×Q8—28no—
94#428—24no—
95#428C2×C418yes—
96C2×#107—22no—
97C2×#106—22no—
98#424—24no—
99#424—24no—
100#423—22no—
101#423—22no—
102#423—22no—
103#423—22no—
104#423C2×C414yes—
105#423D414yes—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114#418—24no—
115#418C2×C418yes—
116C2×#102—22no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102Q814yescomplemented
126C2×#100—24no—
127C2×#100—24no—
128C2×#100—42no—
129#405—24no—
130#405—24no—
131C2×#97—22no—
132C2×C4×C8—22no—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8D411yescharacteristic
136C2×C4×C8D411yescharacteristic
137C2×C4×C8D411yescharacteristic
138C2×C4×C8D411yescharacteristic
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C2×#95—21no—
144#343—24no—
145#343—24no—
146#343—42no—
147#341—24no—
148#341—42no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157C25—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C22×C412yes—
172C22×D4C2×Q812yes—
173C22×D4#4112yes—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—41no—
179C23×C4—41no—
180C23×C4—42no—
181C23×C4—82no—
182C23×C4—82no—
183C23×C4#4111yescharacteristic
184C23×C4#4111yescharacteristic
185C23×C4C2411yescharacteristic
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16C22×C414yes—
203C2×M16C2×D414yes—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×Q812yes—
222C22×C8#4112yes—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8C22⋊C414yes—
226C4⋊Q8C22⋊C414yes—
227#125—24no—
228#125C22⋊C418yes—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245#117—22no—
246#117—22no—
247#117C22×C414yes—
248#117C2×D414yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×Q812yes—
272C2×C4⋊C4#4112yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—42no—
278C2×C42—22no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—82no—
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42#4112yes—
290#107—28no—
291#106—28no—
292C4⋊C8—28no—
293C4⋊C8—28no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300#102—24no—
301#102—24no—
302#102—24no—
303#102—24no—
304#102—28no—
305#102—28no—
306#100—48no—
307#97—28no—
308C4×C8—28no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—22no—
313#95—41no—
314#95—41no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—81no—
323C24—81no—
324C24—81no—
325C24—81no—
326C24C2×#4111yescharacteristic
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4C4×Q814yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12312yes—
397C22×C4#12312yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C42−1+411yescharacteristic
410C22×C42−1+411yescharacteristic
411C22×C42−1+411yescharacteristic
412M16—24no—
413M16—24no—
414M16—24no—
415M16—24no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421M16—42no—
422M16—42no—
423M16—42no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—24no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8C22⋊Q812yes—
452C2×C8C22⋊Q812yes—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4C2×C22⋊C414yes—
472C4⋊C4C2×C22⋊C414yes—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C42—24no—
478C42—24no—
479C42—82no—
480C42—82no—
481C42C2×C22⋊C412yes—
482C42C2×C22⋊C412yes—
483C42C4×D412yes—
484C42C4×D412yes—
485C42#12012yes—
486C42#12012yes—
487C42#12112yes—
488C42#12112yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23C2×C4×D411yescharacteristic
517C23C2×#12111yescharacteristic
518C23C2×#12311yescharacteristic
519C23#55211yescharacteristic
520C23#55211yescharacteristic
521C23#55411yescharacteristic
522C23#55611yescharacteristic
523Q8—48no—
524Q8—48no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—42no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—88no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—88no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4#38612yes—
567C2×C4#38612yes—
568C2×C4#38612yes—
569C2×C4#38612yes—
570C2×C4#47112yes—
571C2×C4#47212yes—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×C4×Q811yescharacteristic
575C2×C4C4×#4111yescharacteristic
576C2×C4#51911yescharacteristic
577C2×C4#52011yescharacteristic
578C2×C4C2×#12011yescharacteristic
579C2×C4C2×#12111yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4#52511yescharacteristic
582C2×C4#52911yescharacteristic
583C2×C4#53311yescharacteristic
584C2×C4#53611yescharacteristic
585C2×C4#53711yescharacteristic
586C2×C4#54111yescharacteristic
587C2×C4#54211yescharacteristic
588C2×C4#54611yescharacteristic
589C2×C4#54711yescharacteristic
590C2×C4#54811yescharacteristic
591C2×C4#54811yescharacteristic
592C2×C4#54811yescharacteristic
593C2×C4#54811yescharacteristic
594C2×C4#54811yescharacteristic
595C2×C4D4×Q811yescharacteristic
596C2×C4D4×Q811yescharacteristic
597C2×C4D4×Q811yescharacteristic
598C2×C4#55011yescharacteristic
599C2×C4#55011yescharacteristic
600C8—22no—
601C8—22no—
602C8—44no—
603C8—41no—
604C8—41no—
605C8—44no—
606C8—82no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—44no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×#38611yescharacteristic
631C22#236011yescharacteristic
632C22#236211yescharacteristic
633C22#242111yescharacteristic
634C22#242311yescharacteristic
635C22#253611yescharacteristic
636C22#253811yescharacteristic
637C22C2×#47111yescharacteristic
638C22C2×#47211yescharacteristic
639C22#306711yescharacteristic
640C22#306711yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#190714yes—
650C4C4×C22⋊Q811yescharacteristic
651C4#232311yescharacteristic
652C4Q8×C22⋊C411yescharacteristic
653C4#234111yescharacteristic
654C4#240711yescharacteristic
655C4#240911yescharacteristic
656C4#242611yescharacteristic
657C4#242911yescharacteristic
658C2—21no—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—84no—
663C2#1396911yescharacteristic
664C2C2×#190711yescharacteristic
665C2#2100311yescharacteristic
666C2#2821711yescharacteristic
667C2#2821811yescharacteristic
668C2#2829511yescharacteristic
669C2#2829611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2960C2semidirect
3. #21312C2semidirect
6. #11216C2semidirect
9. #10896C2semidirect
10. #10847C2semidirect
11. #10844C2semidirect
15. #3077C4semidirect
125. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C2×#29602
#213122
C2×#18002
C2×C4×#1022
#112162
#110612
#109432
#108962
#108472
#108442