Finite Group Databasebeta

← #379583

Subgroups — #379583

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2031C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic, complemented
3C2×#1759C211yesmaximal, characteristic
4#11387C211yesmaximal, characteristic, complemented
5#11261C211yesmaximal, characteristic, complemented
6#11082C211yesmaximal, characteristic
7#10946C211yesmaximal, characteristic
8#10945C211yesmaximal, characteristic
9#10826C211yesmaximal, characteristic, complemented
10#10816C211yesmaximal, characteristic, complemented
11#10813C211yesmaximal, characteristic, complemented
12#10805C211yesmaximal, characteristic, complemented
13#10772C211yesmaximal, characteristic
14#10758C211yesmaximal, characteristic
15#10701C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22#2896C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2031C418yescomplemented
30#1995—24no—
31#1759—24no—
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1742C2211yescharacteristic
37#1740C2211yescharacteristic
38#1738C2211yescharacteristic
39#1733C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#338C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C22×#41C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61#533—22no—
62#533—41no—
63#533—41no—
64#533C2×C414yescomplemented
65C2×C4⋊Q8—21no—
66C2×#125—21no—
67C2×C22⋊Q8—41no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—41no—
72C2×C4×Q8—21no—
73C2×C4×Q8—41no—
74C2×C4×D4—21no—
75C2×C4×D4—41no—
76C2×#117—21no—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#423—22no—
87#423—41no—
88C2×C4⋊C8—41no—
89#421—41no—
90#421C2×C414yes—
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—21no—
94C2×#104—21no—
95C2×#104—21no—
96C2×#104—21no—
97C2×#104—41no—
98C2×#104C2×C414yes—
99C2×#104Q814yescomplemented
100#419—21no—
101#418—21no—
102#417—22no—
103#416—41no—
104#413—22no—
105#413—41no—
106C2×#101—21no—
107C2×#100—21no—
108#410—41no—
109#410C2×C414yes—
110C2×#99—41no—
111C4×M16—22no—
112C4×M16—22no—
113C4×M16—22no—
114C4×M16—41no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C4×C4⋊C4—22no—
120C4×C22⋊C4—41no—
121#376—21no—
122#376—41no—
123#376D414yes—
124C2×#95—41no—
125C43—21no—
126#344—22no—
127#344—41no—
128#344—41no—
129#344C2×C414yescomplemented
130#343—22no—
131#342—41no—
132#341—41no—
133#339—22no—
134#339—22no—
135#339—22no—
136#339—22no—
137#339—22no—
138#339—22no—
139#339—21no—
140#339—21no—
141#339—21no—
142#339—22no—
143#339—21no—
144#339—41no—
145#339C2×C414yescomplemented
146#338—22no—
147H(3,ℤ/4)—22no—
148H(3,ℤ/4)—41no—
149H(3,ℤ/4)C2×C414yescomplemented
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41C22×C412yes—
161C2×#41C2×Q812yes—
162C22×Q8—41no—
163C22×Q8#4111yescharacteristic
164C22×D4—21no—
165C22×D4#4111yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—41no—
171C23×C4C2411yescharacteristic
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C2×Q812yes—
207C2×M16#4112yes—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C4⋊Q8—24no—
215C4⋊Q8—42no—
216#125—24no—
217#125—42no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220C4×Q8—24no—
221C4×Q8—44no—
222C4×Q8—42no—
223C4×Q8—44no—
224C4×D4—24no—
225C4×D4—42no—
226C4×D4—44no—
227C4×D4—44no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—42no—
245#117—42no—
246#117C22×C412yes—
247#117C22×C412yes—
248#117C22×C412yes—
249#117#4112yes—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4#4112yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—41no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—42no—
279C2×C42—41no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42C22×C412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×Q812yes—
287C2×C42#4111yescharacteristic
288C2×C42#4111yescharacteristic
289C2×C42#4111yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4112yes—
293C2×C42#4111yescharacteristic
294C4⋊C8—44no—
295#104—24no—
296#104—24no—
297#104—24no—
298#104—24no—
299#104—44no—
300#103—42no—
301#102—42no—
302#101—42no—
303#100—42no—
304#99—44no—
305#97—42no—
306#97—42no—
307C4×C8—42no—
308C4×C8—42no—
309#95—44no—
310#95—42no—
311C24—21no—
312C24H(5,𝔽2)11yescharacteristic
313#41—22no—
314#41—22no—
315#41—22no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—44no—
335C2×Q8—84no—
336C2×Q8C4×Q812yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C4×Q812yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—84no—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×Q811yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392M16—22no—
393M16—22no—
394M16—22no—
395M16—22no—
396M16—22no—
397M16—22no—
398M16—22no—
399M16—22no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—44no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—44no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—84no—
482C42C4×D412yes—
483C42C4×D412yes—
484C42C22⋊Q812yes—
485C42C22⋊Q812yes—
486C42#12412yes—
487C42#12412yes—
488C23—22no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—82no—
493C23#51811yescharacteristic
494C23#53411yescharacteristic
495C23#53811yescharacteristic
496C23#55111yescharacteristic
497C23#55111yescharacteristic
498C23#55111yescharacteristic
499C23#55411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—41no—
503Q8—41no—
504Q8—88no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—41no—
512D4—41no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—88no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×Q811yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4C4×#4111yescharacteristic
553C2×C4C4×#4111yescharacteristic
554C2×C4#51811yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4#52511yescharacteristic
557C2×C4C2×#12411yescharacteristic
558C2×C4#52911yescharacteristic
559C2×C4#52911yescharacteristic
560C2×C4#52911yescharacteristic
561C2×C4#53211yescharacteristic
562C2×C4#53311yescharacteristic
563C2×C4#53511yescharacteristic
564C2×C4#53811yescharacteristic
565C2×C4#55111yescharacteristic
566C2×C4#55111yescharacteristic
567C2×C4#55111yescharacteristic
568C2×C4#55111yescharacteristic
569C2×C4#55111yescharacteristic
570C2×C4#55411yescharacteristic
571C2×C4#55411yescharacteristic
572C2×C4#55411yescharacteristic
573C2×C4#55511yescharacteristic
574C2×C4#55511yescharacteristic
575C2×C4#55511yescharacteristic
576C2×C4#55511yescharacteristic
577C8—42no—
578C8—42no—
579C8—42no—
580C8—42no—
581C8—81no—
582C8—81no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22#232111yescharacteristic
595C22#236811yescharacteristic
596C22#237211yescharacteristic
597C22#256111yescharacteristic
598C22#256411yescharacteristic
599C22#257211yescharacteristic
600C22#257411yescharacteristic
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4C4×C22⋊Q811yescharacteristic
614C4C4×#12411yescharacteristic
615C4#236811yescharacteristic
616C4#237211yescharacteristic
617C4#256311yescharacteristic
618C4#257211yescharacteristic
619C4#257411yescharacteristic
620C4#261011yescharacteristic
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—21no—
627C2—82no—
628C2#1424311yescharacteristic
629C2#2923312yes—
subgroupG/H
centerC2×C4#555
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2572
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1995C2semidirect
4. #11387C2semidirect
5. #11261C2semidirect
9. #10826C2semidirect
10. #10816C2semidirect
11. #10813C2semidirect
12. #10805C2semidirect
15. #10701C2semidirect
29. #2031C4semidirect
64. #533C2×C4semidirect
99. C2×#104Q8semidirect
129. #344C2×C4semidirect
145. #339C2×C4semidirect
149. H(3,ℤ/4)C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20312
C2×#19952
C2×#17592
#113872
#112612
#110822
#109462
#109452
#108262
#108162
#108132
#108052
#107722
#107582
#107012