Finite Group Databasebeta

← #366490

Subgroups — #366490

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1950C211yesmaximal, characteristic, complemented
2C2×#1929C212yesmaximal
3C2×#1806C211yesmaximal, characteristic
4#20645C211yesmaximal, characteristic
5#20437C211yesmaximal, characteristic, complemented
6C4×#408C211yesmaximal, characteristic
7#11258C212yesmaximal, complemented
8#11191C212yesmaximal
9#10851C211yesmaximal, characteristic, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10809C212yesmaximal, complemented
12C2×C4×#41C2211yescharacteristic
13C22×#108C2212yes—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#104C2212yes—
17C2×#416C2212yes—
18C2×#408C2211yescharacteristic
19C2×#408C2211yescharacteristic
20C2×#405C2212yes—
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1950—24no—
26#1929—28no—
27#1806—24no—
28#1805C412yescomplemented
29#1804C412yescomplemented
30#1803C411yescharacteristic, complemented
31#1803C411yescharacteristic, complemented
32#1802C411yescharacteristic, complemented
33#1802C411yescharacteristic, complemented
34#1770—21no—
35#1770—21no—
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38#1750—21no—
39#1750—21no—
40C4×C22⋊C8—21no—
41C4×C22⋊C8—21no—
42C2×#341C2212yes—
43#1735C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1729C2212yes—
50C2×#336C2212yes—
51C2×#336C2212yes—
52C22×#41C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×#117—21no—
70C2×#117C2312yes—
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#108—24no—
77C2×#108—24no—
78C2×#108—42no—
79#423—22no—
80#423D414yes—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2×C412yes—
88C2×#104—24no—
89C2×#104—24no—
90C2×#104—42no—
91#416—24no—
92#416—24no—
93C2×#103C2×C412yes—
94C2×#103C2×C412yes—
95C2×#102C2×C412yes—
96C2×#102C2×C412yes—
97#408—22no—
98#408—22no—
99#408—22no—
100#408D414yes—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105#405—24no—
106#405—24no—
107C4×M16—22no—
108C4×M16—22no—
109C2×#97—22no—
110C2×#97—21no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—21no—
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8D411yescharacteristic
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#382C2×C412yes—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C2×#95—21no—
129#341—24no—
130#340C2×C412yes—
131#340C2×C412yes—
132#340C2×C412yes—
133#340C2×C412yes—
134#339—24no—
135#339—24no—
136#339—24no—
137#336—24no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—22no—
143#336—22no—
144#336—42no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41C2×D412yes—
156C2×#41#4112yes—
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4#4111yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—22no—
182C2×M16C2×D412yes—
183C2×M16C2×D414yes—
184C2×M16#4112yes—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8C22×C412yes—
205C22×C8C22×C412yes—
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D412yes—
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8#4112yes—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224#117—21no—
225#117—22no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4112yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271#108—48no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—24no—
277#104—48no—
278#103—24no—
279#103—24no—
280#102—24no—
281#102—24no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286#97—44no—
287#97—42no—
288C4×C8—22no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—44no—
293C4×C8—42no—
294#95—22no—
295#95—22no—
296#95—42no—
297C24—21no—
298C24C2×#4111yescharacteristic
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C4×D412yes—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4C4×D412yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
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×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4#12412yes—
366C22×C4#12412yes—
367C22×C4#12612yes—
368C22×C4#12612yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C22×C42−1+411yescharacteristic
383M16—24no—
384M16—22no—
385M16—22no—
386M16—24no—
387M16—22no—
388M16—22no—
389M16—42no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—42no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—22no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8#12112yes—
433C2×C8#12112yes—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C42—42no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42#11712yes—
466C42#11712yes—
467C42C4×D412yes—
468C42C4×D412yes—
469C42#12112yes—
470C42#12112yes—
471C42#12312yes—
472C42#12312yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C2×C4×D411yescharacteristic
485C23C2×#12411yescharacteristic
486C23C2×#12611yescharacteristic
487C23#52911yescharacteristic
488C23#53211yescharacteristic
489C23#54111yescharacteristic
490C23#54211yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—41no—
494Q8—41no—
495D4—22no—
496D4—22no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4#38812yes—
542C2×C4#38812yes—
543C2×C4#38812yes—
544C2×C4#38812yes—
545C2×C4#44312yes—
546C2×C4#44412yes—
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C4×#4111yescharacteristic
551C2×C4#51911yescharacteristic
552C2×C4#52011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4#52511yescharacteristic
557C2×C4#53611yescharacteristic
558C2×C4#53711yescharacteristic
559C2×C4D4211yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54711yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4#55211yescharacteristic
567C2×C4#55211yescharacteristic
568C2×C4#55211yescharacteristic
569C2×C4#55211yescharacteristic
570C2×C4#55211yescharacteristic
571C2×C4#55311yescharacteristic
572C2×C4#55511yescharacteristic
573C2×C4#55511yescharacteristic
574C2×C4#55611yescharacteristic
575C8—22no—
576C8—22no—
577C8—42no—
578C8—42no—
579C8—41no—
580C8—41no—
581C8—42no—
582C8—42no—
583C8—44no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×#38811yescharacteristic
604C22C4×#12111yescharacteristic
605C22#232311yescharacteristic
606C22#248911yescharacteristic
607C22#249111yescharacteristic
608C22#249511yescharacteristic
609C22#249611yescharacteristic
610C22C2×#44311yescharacteristic
611C22C2×#44411yescharacteristic
612C22#295511yescharacteristic
613C22#295511yescharacteristic
614C4—22no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—84no—
619C4—82no—
620C4—82no—
621C4—81no—
622C4—81no—
623C4#192914yescomplemented
624C4#234411yescharacteristic
625C4#234611yescharacteristic
626C4#236111yescharacteristic
627C4#236211yescharacteristic
628C4#248511yescharacteristic
629C4#248611yescharacteristic
630C4#253611yescharacteristic
631C4#253711yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—82no—
637C2#1399411yescharacteristic
638C2C2×#192911yescharacteristic
639C2#2107011yescharacteristic
640C2#2151711yescharacteristic
641C2#2151811yescharacteristic
642C2#2213011yescharacteristic
643C2#2213111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1950C2semidirect
5. #20437C2semidirect
7. #11258C2semidirect
9. #10851C2semidirect
10. #10834C2semidirect
11. #10809C2semidirect
28. #1805C4semidirect
29. #1804C4semidirect
30. #1803C4semidirect
31. #1803C4semidirect
32. #1802C4semidirect
33. #1802C4semidirect
623. C4#1929semidirect

Maximal subgroups

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

subgroupindex
C2×#19502
C2×#19292
C2×#18062
#206452
#204372
C4×#4082
#112582
#111912
#108512
#108342
#108092