Finite Group Databasebeta

← #379581

Subgroups — #379581

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2031C211yesmaximal, characteristic
2#21321C211yesmaximal, characteristic, complemented
3C2×#1765C211yesmaximal, characteristic
4#11387C211yesmaximal, characteristic, complemented
5#11262C211yesmaximal, characteristic, complemented
6#11081C211yesmaximal, characteristic
7#10944C211yesmaximal, characteristic
8#10944C211yesmaximal, characteristic
9#10828C211yesmaximal, characteristic, complemented
10#10817C211yesmaximal, characteristic, complemented
11#10814C211yesmaximal, characteristic, complemented
12#10804C211yesmaximal, characteristic, complemented
13#10773C211yesmaximal, characteristic
14#10759C211yesmaximal, characteristic
15#10707C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#421C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#410C2211yescharacteristic
23C2×#410C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×#97C2211yescharacteristic
28#2285C2211yescharacteristic
29#2031C418yescomplemented
30#1993—22no—
31#1992—22no—
32#1765—22no—
33#1765—22no—
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38#1740C2211yescharacteristic
39#1740C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56C22×#41C2311yescharacteristic
57C2×#131—22no—
58C2×#131—22no—
59C2×#131—22no—
60C2×#131—22no—
61C2×#131—41no—
62C2×#131—41no—
63C22×M16—21no—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C23×C8—21no—
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71#533—22no—
72#533—41no—
73#533—41no—
74#533C2×C414yes—
75C2×C4⋊Q8—21no—
76C2×#125—21no—
77C2×C22⋊Q8—41no—
78C2×C4×Q8—41no—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#421—22no—
88#421—22no—
89#421—41no—
90#421C2×C414yes—
91C2×#104—21no—
92C2×#104—21no—
93C2×#104—22no—
94C2×#104—41no—
95C2×#104—41no—
96C2×#104C2×C414yes—
97#416—22no—
98#416—21no—
99#416—21no—
100C2×#103Q812yescomplemented
101C2×#102Q812yescomplemented
102#410—22no—
103#410—22no—
104#410—41no—
105#410C2×C414yes—
106C2×#99—41no—
107C2×#99—41no—
108#405—22no—
109#405—22no—
110#405—22no—
111#405—22no—
112#405—41no—
113#405—41no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—41no—
118C2×#97—41no—
119C2×#97D412yes—
120C2×#97D412yes—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124#384—21no—
125#382—21no—
126#380—41no—
127C4×C4⋊C4—41no—
128#344—22no—
129#344—22no—
130#344—41no—
131#344C2×C414yes—
132#342—41no—
133#342—41no—
134#341—22no—
135#341—21no—
136#341—21no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#339—21no—
143#339—21no—
144#339—22no—
145#339—41no—
146#339—41no—
147#339C2×C414yes—
148H(3,ℤ/4)—22no—
149H(3,ℤ/4)—22no—
150H(3,ℤ/4)—41no—
151H(3,ℤ/4)C2×C414yes—
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×#41—21no—
161C2×#41—21no—
162C2×#41C22×C412yes—
163C2×#41#4112yes—
164C22×Q8—41no—
165C22×Q8C2×Q811yescharacteristic
166C22×D4—21no—
167C22×D4C2×Q811yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C2411yescharacteristic
171#131—28no—
172#131—44no—
173#131—44no—
174#131—44no—
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—42no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16C22×C412yes—
199C2×M16C22×C412yes—
200C2×M16#4112yes—
201C2×M16#4112yes—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×Q812yes—
220C22×C8#4111yescharacteristic
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C22×C8#4111yescharacteristic
224C22×C8#4112yes—
225C4⋊Q8—24no—
226C4⋊Q8—42no—
227#125—24no—
228#125—42no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231C4×Q8—44no—
232C4×Q8—44no—
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—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117C22×C412yes—
258#117C22×C412yes—
259#117C22×C412yes—
260#117#4112yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×Q812yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42C22×C412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4112yes—
292#104—24no—
293#104—24no—
294#104—44no—
295#104—44no—
296#103—22no—
297#103—22no—
298#102—22no—
299#102—22no—
300#99—44no—
301#99—44no—
302#97—42no—
303#97—42no—
304#97—42no—
305#97—42no—
306#97—42no—
307C4×C8—42no—
308C4×C8—42no—
309C4×C8—42no—
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—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—82no—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390M16—48no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—48no—
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×C8C22⋊Q812yes—
429C2×C8C22⋊Q812yes—
430C2×C8#12412yes—
431C2×C8#12412yes—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—84no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—22no—
464C42—21no—
465C42—21no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42C4×D412yes—
482C42C4×D412yes—
483C23—22no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—82no—
488C23#51811yescharacteristic
489C23#53511yescharacteristic
490C23#53811yescharacteristic
491C23#55111yescharacteristic
492C23#55411yescharacteristic
493C23#55511yescharacteristic
494C23#55511yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—88no—
500D4—22no—
501D4—22no—
502D4—42no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—41no—
507D4—41no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4C2×C4×D411yescharacteristic
536C2×C4C2×C4×Q811yescharacteristic
537C2×C4C4×#4111yescharacteristic
538C2×C4C4×#4111yescharacteristic
539C2×C4C4×#4111yescharacteristic
540C2×C4#51811yescharacteristic
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4#52911yescharacteristic
545C2×C4#52911yescharacteristic
546C2×C4#52911yescharacteristic
547C2×C4#53211yescharacteristic
548C2×C4#53311yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53811yescharacteristic
551C2×C4#55111yescharacteristic
552C2×C4#55111yescharacteristic
553C2×C4#55111yescharacteristic
554C2×C4#55111yescharacteristic
555C2×C4#55111yescharacteristic
556C2×C4#55111yescharacteristic
557C2×C4#55111yescharacteristic
558C2×C4#55411yescharacteristic
559C2×C4#55411yescharacteristic
560C2×C4#55411yescharacteristic
561C2×C4#55511yescharacteristic
562C2×C4#55511yescharacteristic
563C8—22no—
564C8—22no—
565C8—22no—
566C8—22no—
567C8—42no—
568C8—42no—
569C8—42no—
570C8—42no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22#232111yescharacteristic
583C22#236811yescharacteristic
584C22#237211yescharacteristic
585C22#256311yescharacteristic
586C22#257211yescharacteristic
587C22#257411yescharacteristic
588C22#261011yescharacteristic
589C4—42no—
590C4—42no—
591C4—42no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—161no—
598C4—161no—
599C4C4×C22⋊Q811yescharacteristic
600C4C4×#12411yescharacteristic
601C4#236811yescharacteristic
602C4#237211yescharacteristic
603C4#256111yescharacteristic
604C4#256411yescharacteristic
605C4#257211yescharacteristic
606C4#257411yescharacteristic
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—82no—
614C2#1424311yescharacteristic
615C2#2915512yes—
subgroupG/H
centerC2×C4#555
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2563
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. #21321C2semidirect
4. #11387C2semidirect
5. #11262C2semidirect
9. #10828C2semidirect
10. #10817C2semidirect
11. #10814C2semidirect
12. #10804C2semidirect
15. #10707C2semidirect
29. #2031C4semidirect
100. C2×#103Q8semidirect
101. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20312
#213212
C2×#17652
#113872
#112622
#110812
#109442
#108282
#108172
#108142
#108042
#107732
#107592
#107072