Finite Group Databasebeta

← #360342

Subgroups — #360342

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1826C211yesmaximal, characteristic
2#20685C211yesmaximal, characteristic
3C2×#1810C211yesmaximal, characteristic
4#20379C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6#11317C212yesmaximal, complemented
7#11258C212yesmaximal, complemented
8#10910C212yesmaximal, complemented
9#10809C212yesmaximal, complemented
10#10733C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#108C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#405C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1826—24no—
22#1818—22no—
23#1817—21no—
24#1817—21no—
25#1810—24no—
26#1750—22no—
27#1749—21no—
28C4×C22⋊C8—21no—
29C2×#341C2212yes—
30#1735C411yescharacteristic, complemented
31#1735C411yescharacteristic, complemented
32#1735C411yescharacteristic, complemented
33#1735C411yescharacteristic, complemented
34#1735C2211yescharacteristic
35#1735C2211yescharacteristic
36#1735C2211yescharacteristic
37#1735C2211yescharacteristic
38C2×#340C411yescharacteristic, complemented
39C2×#340C411yescharacteristic, complemented
40C2×#340C411yescharacteristic, complemented
41C2×#340C411yescharacteristic, complemented
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46#1729C2212yes—
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—42no—
58C23×C8C2×C411yescharacteristic
59C23×C8C2×C411yescharacteristic
60C23×C8C2×C411yescharacteristic
61C23×C8C2×C411yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—24no—
74C2×#117—42no—
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2312yes—
93C2×#108—24no—
94C2×#108—24no—
95C2×#108—42no—
96#407—22no—
97#407—22no—
98#407—22no—
99#407D414yes—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8D411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D412yes—
109#405—24no—
110#405—42no—
111C2×#97—22no—
112C2×C4×C8—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—21no—
116#341—24no—
117#340C2×C411yescharacteristic
118#340C2×C411yescharacteristic
119#340C2×C411yescharacteristic
120#340C2×C411yescharacteristic
121#340C2×C411yescharacteristic
122#340C2×C411yescharacteristic
123#340C2×C411yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#339—24no—
134#339—24no—
135#339—24no—
136#339—42no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—24no—
144#336—42no—
145C25#4111yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4111yescharacteristic
164C23×C4C2411yescharacteristic
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16C2×D414yes—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—41no—
201C22×C8—41no—
202C22×C8—41no—
203C22×C8—41no—
204C22×C8—48no—
205C22×C8—48no—
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8#4114yes—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—44no—
231#117—44no—
232#117—41no—
233#117—41no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117#4114yes—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
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—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—44no—
288C2×C42—42no—
289C2×C42—42no—
290#108—48no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—24no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298#97—44no—
299C4×C8—44no—
300#95—42no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—41no—
306C24—41no—
307C24C4×D411yescharacteristic
308C24C4×D411yescharacteristic
309C24#12412yes—
310C24#12612yes—
311C24C2×#4111yescharacteristic
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
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—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4#11712yes—
362C22×C4#11712yes—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4#12412yes—
372C22×C4#12612yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383M16—24no—
384M16—24no—
385M16—42no—
386M16—42no—
387M16—42no—
388M16—42no—
389M16—42no—
390M16—42no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8#11712yes—
426C2×C8#11712yes—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4#11712yes—
446C4⋊C4#11712yes—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—48no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C42—24no—
459C42—42no—
460C42—42no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—24no—
483C23—24no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23#38812yes—
497C23#38812yes—
498C23#44312yes—
499C23#44412yes—
500C23C2×C4×D411yescharacteristic
501C23C2×#12411yescharacteristic
502C23C2×#12611yescharacteristic
503C23#55112yes—
504C23#55212yes—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—84no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—82no—
558C2×C4#38812yes—
559C2×C4#38812yes—
560C2×C4C2×#11711yescharacteristic
561C2×C4C2×#11711yescharacteristic
562C2×C4C2×#11711yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4#51811yescharacteristic
565C2×C4#51911yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C2×C4#52912yes—
569C2×C4#52912yes—
570C2×C4#53912yes—
571C2×C4#53912yes—
572C2×C4#54612yes—
573C2×C4#54612yes—
574C2×C4#54712yes—
575C2×C4#54712yes—
576C2×C4#55112yes—
577C2×C4#55212yes—
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C8—42no—
583C8—42no—
584C8—44no—
585C8—82no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—48no—
604C22—48no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22#192914yes—
610C22C2×#38811yescharacteristic
611C22#234211yescharacteristic
612C22#234411yescharacteristic
613C22#248812yes—
614C22#249312yes—
615C22C2×#44311yescharacteristic
616C22C2×#44411yescharacteristic
617C22#295711yescharacteristic
618C22#295711yescharacteristic
619C4—22no—
620C4—22no—
621C4—42no—
622C4—42no—
623C4—84no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4#228311yescharacteristic
631C4#231311yescharacteristic
632C4#234211yescharacteristic
633C4#234411yescharacteristic
634C4#241912yes—
635C4#242512yes—
636C2—22no—
637C2—24no—
638C2—22no—
639C2—42no—
640C2#1387411yescharacteristic
641C2C2×#192911yescharacteristic
642C2#2107211yescharacteristic
643C2#2258611yescharacteristic
644C2#2258711yescharacteristic
645C2#2260611yescharacteristic
646C2#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
6. #11317C2semidirect
7. #11258C2semidirect
8. #10910C2semidirect
9. #10809C2semidirect
30. #1735C4semidirect
31. #1735C4semidirect
32. #1735C4semidirect
33. #1735C4semidirect
38. C2×#340C4semidirect
39. C2×#340C4semidirect
40. C2×#340C4semidirect
41. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18262
#206852
C2×#18102
#203792
C2×#17352
#113172
#112582
#109102
#108092
#107332
#107142