Finite Group Databasebeta

← #360847

Subgroups — #360847

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C2×#1922C212yesmaximal, complemented
3#20628C211yesmaximal, characteristic
4#20379C211yesmaximal, characteristic
5#11376C212yesmaximal, complemented
6#11221C212yesmaximal, complemented
7#10909C212yesmaximal, complemented
8#10733C211yesmaximal, characteristic
9#10733C211yesmaximal, characteristic
10#10723C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#427—28no—
16C2×#427C411yescharacteristic, complemented
17C2×#427C411yescharacteristic, complemented
18C2×#427C411yescharacteristic, complemented
19C2×#427C411yescharacteristic, complemented
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C411yescharacteristic, complemented
25C22×#107C411yescharacteristic, complemented
26C22×#106C411yescharacteristic, complemented
27C22×#106C411yescharacteristic, complemented
28C2×#424C2212yes—
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C2×#405C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1922—28no—
35#1798—22no—
36#1797—21no—
37#1797—21no—
38#1750—22no—
39#1749—21no—
40C4×C22⋊C8—21no—
41C2×#341C2212yes—
42C2×#341C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1729C2212yes—
48#1729C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C24×C4C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—24no—
58C23×C8C2×C411yescharacteristic
59C23×C8C2×C411yescharacteristic
60C23×C8C2×C411yescharacteristic
61C23×C8C2×C411yescharacteristic
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117—42no—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2312yes—
91#427—24no—
92#427—24no—
93#427—24no—
94#427—24no—
95#427—24no—
96#427—24no—
97#427—24no—
98#427D418yes—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109C2×#106—24no—
110C2×#106—24no—
111C2×#106C2×C411yescharacteristic
112C2×#106C2×C411yescharacteristic
113C2×#106C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115C2×#106C2×C411yescharacteristic
116C2×#106C2×C411yescharacteristic
117C2×#106C2×C411yescharacteristic
118C2×#106C2×C411yescharacteristic
119#424—24no—
120#424—42no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123#407—22no—
124#407—22no—
125#407—22no—
126#407Q814yes—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D412yes—
136#405—24no—
137#405—42no—
138C2×#97—22no—
139C2×C4×C8—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C2×#95—21no—
143#341—24no—
144#341—24no—
145#339—24no—
146#339—24no—
147#339—42no—
148#339—42no—
149#336—22no—
150#336—22no—
151#336—24no—
152#336—22no—
153#336—22no—
154#336—24no—
155C25#4111yescharacteristic
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×Q812yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4#4111yescharacteristic
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16C2×Q814yes—
187C22×C8—216no—
188C22×C8—216no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D414yes—
222C22×C8#4114yes—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—44no—
234#117—44no—
235#117—44no—
236#117—44no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117C2×D414yes—
242#117C2×D414yes—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—44no—
291C2×C42—44no—
292#107—22no—
293#107—22no—
294#107—22no—
295#107—22no—
296#107—22no—
297#107—22no—
298#107C22⋊C414yescomplemented
299#107C22⋊C414yescomplemented
300#106—22no—
301#106—22no—
302#106—22no—
303#106—22no—
304#106—22no—
305#106—22no—
306#106C22⋊C414yescomplemented
307#106C22⋊C414yescomplemented
308C4⋊C8—44no—
309C4⋊C8—44no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—24no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317#97—44no—
318C4×C8—44no—
319#95—42no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—41no—
325C24—41no—
326C24C4×D411yescharacteristic
327C24C4×D411yescharacteristic
328C24#12512yes—
329C24#12612yes—
330C24C2×#4111yescharacteristic
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
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×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4#11712yes—
381C22×C4#11712yes—
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×Q812yes—
385C22×C4C4×Q812yes—
386C22×C4C22⋊Q812yes—
387C22×C4C22⋊Q812yes—
388C22×C4#12312yes—
389C22×C4#12312yes—
390C22×C4#12512yes—
391C22×C4#12612yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2×#4111yescharacteristic
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+412yes—
402M16—24no—
403M16—24no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409M16—42no—
410C2×C8—28no—
411C2×C8—28no—
412C2×C8—28no—
413C2×C8—28no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8C2×C22⋊C412yes—
443C2×C8C2×C22⋊C412yes—
444C2×C8#11712yes—
445C2×C8#11712yes—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4C2×C22⋊C412yes—
463C4⋊C4C2×C22⋊C412yes—
464C4⋊C4C2×C22⋊C412yes—
465C4⋊C4C2×C22⋊C412yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—48no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—24no—
476C42—24no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—24no—
498C23—24no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23#38712yes—
512C23#38712yes—
513C23#44312yes—
514C23#44412yes—
515C23C2×C4×D411yescharacteristic
516C23C2×#12511yescharacteristic
517C23C2×#12611yescharacteristic
518C23#54612yes—
519C23#55212yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
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—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4#38712yes—
574C2×C4#38712yes—
575C2×C4C22×C22⋊C411yescharacteristic
576C2×C4C2×#11711yescharacteristic
577C2×C4C2×#11711yescharacteristic
578C2×C4C2×C4×Q811yescharacteristic
579C2×C4#51811yescharacteristic
580C2×C4#51911yescharacteristic
581C2×C4C2×C22⋊Q811yescharacteristic
582C2×C4C2×#12311yescharacteristic
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4#53912yes—
586C2×C4#53912yes—
587C2×C4D4212yes—
588C2×C4#54712yes—
589C2×C4#54812yes—
590C2×C4D4×Q812yes—
591C2×C4#55312yes—
592C2×C4#55412yes—
593C8—22no—
594C8—22no—
595C8—22no—
596C8—22no—
597C8—42no—
598C8—42no—
599C8—44no—
600C8—82no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—48no—
619C22—48no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#192214yescomplemented
625C22C2×#38711yescharacteristic
626C22D4×C22⋊C411yescharacteristic
627C22#234411yescharacteristic
628C22#247312yes—
629C22#248112yes—
630C22C2×#44311yescharacteristic
631C22C2×#44411yescharacteristic
632C22#295711yescharacteristic
633C22#295711yescharacteristic
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—84no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4#228011yescharacteristic
646C4#231311yescharacteristic
647C4Q8×C22⋊C411yescharacteristic
648C4#234511yescharacteristic
649C4#244312yes—
650C4#244912yes—
651C2—22no—
652C2—24no—
653C2—22no—
654C2—42no—
655C2#1388211yescharacteristic
656C2C2×#192211yescharacteristic
657C2#2104311yescharacteristic
658C2#2245211yescharacteristic
659C2#2245411yescharacteristic
660C2#2260611yescharacteristic
661C2#2260711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1922C2semidirect
5. #11376C2semidirect
6. #11221C2semidirect
7. #10909C2semidirect
16. C2×#427C4semidirect
17. C2×#427C4semidirect
18. C2×#427C4semidirect
19. C2×#427C4semidirect
24. C22×#107C4semidirect
25. C22×#107C4semidirect
26. C22×#106C4semidirect
27. C22×#106C4semidirect
298. #107C22⋊C4semidirect
299. #107C22⋊C4semidirect
306. #106C22⋊C4semidirect
307. #106C22⋊C4semidirect
624. C22#1922semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#19222
#206282
#203792
#113762
#112212
#109092
#107332
#107232
#107142