Finite Group Databasebeta

← #385159

Subgroups — #385159

Order 512.

1630 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×#1869C211yesmaximal, characteristic
2C2×#1852C212yesmaximal, complemented
3C2×#1808C212yesmaximal, complemented
4#13906C211yesmaximal, characteristic
5#13872C211yesmaximal, characteristic
6#11324C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C2×#390—21no—
19C2×#386—21no—
20C2×#384C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1869C418yescomplemented
27#1852—28no—
28#1808—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1441—21no—
36#1439—21no—
37#1439—21no—
38#1439—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—24no—
47C23×C8—24no—
48C23×C8—41no—
49C23×C8—41no—
50C23×C8C2×C414yescomplemented
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4Q814yes—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#107—42no—
86C2×#107—42no—
87C2×#107—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8C2×C418yes—
93C2×C22⋊C8C2×C414yescomplemented
94#390—44no—
95#386—44no—
96#384—41no—
97#384—41no—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—42no—
102#380—42no—
103#380—42no—
104#380—41no—
105#380—41no—
106C4×C4⋊C4—41no—
107C4×C4⋊C4—41no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340D418yescomplemented
120#336C2×C418yescomplemented
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×D4—41no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4C4⋊C414yes—
128C22×D4D814yes—
129C22×D4Q1614yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×Q812yes—
162C23×C4C2×Q812yes—
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—216no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8—48no—
178C22×C8—44no—
179C22×C8C22×C414yes—
180C22×C8C22×C414yes—
181C22×C8C2×D414yes—
182C22×C8#4114yes—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—44no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C4⋊C414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4C4⋊C414yes—
241C2×C22⋊C4C4⋊C414yes—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C22×C412yes—
250C2×C42C2×Q812yes—
251#107—48no—
252#107—48no—
253#107—48no—
254C22⋊C8—216no—
255C22⋊C8—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—82no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24C2×C4⋊C412yes—
280C24C2×C4⋊C412yes—
281C24C2×D812yes—
282C24C2×D812yes—
283C24C2×Q1612yes—
284C24C2×Q1612yes—
285C24C22×D411yescharacteristic
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4#107116yescomplemented
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
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—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—88no—
375C22×C4—88no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4C2×C4⋊C412yes—
379C22×C4C2×C4⋊C412yes—
380C22×C4C2×C4⋊C412yes—
381C22×C4C2×C4⋊C412yes—
382C22×C4C2×D812yes—
383C22×C4C2×Q1612yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)12yes—
393C22×C42−1+412yes—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—416no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8C4×D418yes—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—84no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—84no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C2×#10714yes—
485C23C2×#10714yes—
486C23C22×C4⋊C411yescharacteristic
487C23#53411yescharacteristic
488C23#53411yescharacteristic
489C23D4212yes—
490C23#54712yes—
491C23C22×D811yescharacteristic
492C23C22×Q1611yescharacteristic
493C23#57511yescharacteristic
494C23#57511yescharacteristic
495C23#57612yes—
496C23#57812yes—
497D4—416no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—816no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4C2×#10714yes—
561C2×C4C2×C4×D412yes—
562C2×C4#51811yescharacteristic
563C2×C4#51811yescharacteristic
564C2×C4#52012yes—
565C2×C4#53512yes—
566C2×C4#53712yes—
567C2×C4#54012yes—
568C2×C4#54212yes—
569C2×C4#54311yescharacteristic
570C2×C4#54311yescharacteristic
571C2×C4D4211yescharacteristic
572C2×C4#54612yes—
573C2×C4D4×Q812yes—
574C2×C4#55111yescharacteristic
575C2×C4#55412yes—
576C2×C4#56011yescharacteristic
577C2×C4#56212yes—
578C2×C4#56311yescharacteristic
579C8—28no—
580C8—44no—
581C22—28no—
582C22—28no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22#231511yescharacteristic
604C22D4×C4⋊C412yes—
605C22#266911yescharacteristic
606C22#267112yes—
607C22#271711yescharacteristic
608C22C22×#10711yescharacteristic
609C22#291111yescharacteristic
610C22#291312yes—
611C22#318611yescharacteristic
612C22#318611yescharacteristic
613C22#318911yescharacteristic
614C22#318911yescharacteristic
615C22#319911yescharacteristic
616C22#320211yescharacteristic
617C22#320312yes—
618C22#326812yes—
619C22#326912yes—
620C22D4×D812yes—
621C22D4×Q1612yes—
622C22#333412yes—
623C22#333812yes—
624C22#334712yes—
625C22#334912yes—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—84no—
632C4—162no—
633C4—162no—
634C4#232412yes—
635C4#235811yescharacteristic
636C4#237611yescharacteristic
637C4#268311yescharacteristic
638C4#269212yes—
639C4#274811yescharacteristic
640C2—44no—
641C2—44no—
642C2#1440111yescharacteristic
643C2#2194011yescharacteristic
644C2#2194711yescharacteristic
645C2D4×#10712yes—
646C2#2266012yes—
647C2#3049911yescharacteristic
648C2#3050711yescharacteristic
649C2#3055712yes—
650C2#3056312yes—
651C2#3225811yescharacteristic
652C2#3226111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1852C2semidirect
3. C2×#1808C2semidirect
6. #11324C2semidirect
7. #11096C2semidirect
8. #11096C2semidirect
9. #10907C2semidirect
26. #1869C4semidirect
50. C23×C8C2×C4semidirect
93. C2×C22⋊C8C2×C4semidirect
119. #340D4semidirect
120. #336C2×C4semidirect
292. C2×D4#107semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C2×#18522
C2×#18082
#139062
#138722
#113242
#110962
#109072