Finite Group Databasebeta

← #360799

Subgroups — #360799

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34183C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic, complemented
3#20648C211yesmaximal, characteristic
4C2×C4×#104C211yesmaximal, characteristic
5#14687C211yesmaximal, characteristic, complemented
6#11276C214yesmaximal, complemented
7#10857C211yesmaximal, characteristic, complemented
8#10839C211yesmaximal, characteristic, complemented
9#10781C214yesmaximal
10C2×#520C2212yes—
11C2×C4×#41C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×#104C2212yes—
14C2×#417C2212yes—
15C2×C4×M16C2211yescharacteristic
16Q8×C4⋊C4—21no—
17Q8×C4⋊C4—21no—
18D4×C4⋊C4—21no—
19D4×C4⋊C4—21no—
20D4×C4⋊C4—22no—
21D4×C4⋊C4—22no—
22#2317—22no—
23#2317C2212yes—
24#2285—21no—
25#2285C2211yescharacteristic
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382—22no—
30C2×C43C2211yescharacteristic
31#1836—22no—
32#1834—21no—
33#1834—21no—
34#1803—21no—
35#1803—21no—
36#1802—21no—
37#1802—21no—
38C4×#104C418yescomplemented
39C2×#343C2214yes—
40#1736C2212yes—
41#1736C2214yes—
42C2×#339C2214yes—
43C2×#339C2212yes—
44C4⋊#95—22no—
45C4⋊#95C2214yes—
46C22×#41C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2312yes—
49#520—28no—
50#520—44no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41C2×C414yes—
54C4×#41Q814yes—
55C2×C4×Q8—22no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×#117—22no—
65C2×#117—21no—
66C2×#117C2312yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#423—22no—
83#423—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×#104—24no—
87C2×#104—24no—
88C2×#104C2×C418yes—
89#417—28no—
90C2×#103—22no—
91C2×#102—22no—
92C4×M16—22no—
93C4×M16C2×C414yescomplemented
94C2×#97—21no—
95C2×C4×C8—21no—
96#389—24no—
97#389—24no—
98#389—22no—
99#389—22no—
100#385—24no—
101#385—24no—
102#385—24no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384D411yescharacteristic
115#384D411yescharacteristic
116#382—22no—
117#382—48no—
118#382D412yes—
119#380—22no—
120#380—22no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C2×#95—28no—
128C43—22no—
129C43C2×C414yescomplemented
130#343—44no—
131#340—22no—
132#340—22no—
133#340—24no—
134#340—24no—
135#339—44no—
136#339C2×C418yescomplemented
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41C22×C412yes—
142C2×#41C2×Q812yes—
143C2×#41#4114yes—
144C2×#41#4114yes—
145C22×Q8C2×D411yescharacteristic
146C22×D4—21no—
147C22×D4C2×D411yescharacteristic
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D414yes—
157C23×C4C2411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—24no—
167C2×M16C22×C414yes—
168C22×C8—22no—
169C22×C8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×Q8C4⋊C414yes—
175C4×Q8C4⋊C414yes—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—416no—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4C4⋊C414yes—
186C4×D4C4⋊C414yes—
187#117—21no—
188#117—21no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—24no—
193#117—48no—
194#117C22×C412yes—
195#117C2×Q812yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—84no—
227C2×C4⋊C4—84no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22×C414yes—
256C2×C42C22×C412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C4⋊C8—42no—
265C4⋊C8—42no—
266#104—216no—
267#103—44no—
268#102—44no—
269#97—24no—
270C4×C8—24no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—22no—
275#95—416no—
276C24—21no—
277C24C22×D411yescharacteristic
278#41—24no—
279#41—24no—
280#41—24no—
281#41—24no—
282#41—24no—
283#41—24no—
284#41#11718yes—
285#41#12518yes—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C2×C4⋊C412yes—
290C2×Q8#12412yes—
291C2×Q8#12412yes—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4C2×C4⋊C412yes—
300C2×D4#12412yes—
301C2×D4#12412yes—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357M16—24no—
358M16—24no—
359M16—42no—
360M16—42no—
361M16—44no—
362M16—44no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4C2×C4⋊C412yes—
397C4⋊C4C2×C4⋊C412yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—48no—
403C22⋊C4—44no—
404C42—216no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—48no—
412C42—48no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42C2×C4⋊C412yes—
418C42C2×C4⋊C412yes—
419C42C4×D412yes—
420C42C4×D412yes—
421C42C22⋊Q812yes—
422C42C22⋊Q812yes—
423C42#12312yes—
424C42#12312yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C2×C4×D411yescharacteristic
437C23C2×#12011yescharacteristic
438C23C2×C22⋊Q811yescharacteristic
439C23#54612yes—
440C23#54812yes—
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8#38214yescomplemented
445D4—28no—
446D4—28no—
447D4—22no—
448D4—22no—
449D4—22no—
450D4#38214yescomplemented
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—88no—
489C2×C4—88no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—168no—
494C2×C4#38512yes—
495C2×C4#38512yes—
496C2×C4#38512yes—
497C2×C4#38512yes—
498C2×C4#45312yes—
499C2×C4#45412yes—
500C2×C4#45512yes—
501C2×C4#45612yes—
502C2×C4C22×C4⋊C411yescharacteristic
503C2×C4C2×#11712yes—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4#52012yes—
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4C2×#12411yescharacteristic
509C2×C4C2×#12411yescharacteristic
510C2×C4C2×#12512yes—
511C2×C4#53511yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#54412yes—
514C2×C4#54612yes—
515C2×C4#54812yes—
516C2×C4#55212yes—
517C2×C4#55212yes—
518C2×C4#55312yes—
519C2×C4#55312yes—
520C8—44no—
521C8—82no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22C2×#38511yescharacteristic
538C22#235212yes—
539C22#243511yescharacteristic
540C22#243611yescharacteristic
541C22#244212yes—
542C22C2×#45311yescharacteristic
543C22C2×#45511yescharacteristic
544C22C2×#45411yescharacteristic
545C22C2×#45611yescharacteristic
546C4—24no—
547C4—22no—
548C4—22no—
549C4—48no—
550C4—42no—
551C4—42no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—162no—
556C4#185914yes—
557C4#186014yes—
558C4C2×#38211yescharacteristic
559C4#231711yescharacteristic
560C4#235212yes—
561C4#244212yes—
562C4#250311yescharacteristic
563C4#250411yescharacteristic
564C2—22no—
565C2—22no—
566C2—44no—
567C2#1388111yescharacteristic
568C2C2×#185911yescharacteristic
569C2C2×#186011yescharacteristic
570C2#2587511yescharacteristic
571C2#2587611yescharacteristic
572C2#2592311yescharacteristic
573C2#2592411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20764C2semidirect
5. #14687C2semidirect
6. #11276C2semidirect
7. #10857C2semidirect
8. #10839C2semidirect
38. C4×#104C4semidirect
93. C4×M16C2×C4semidirect
129. C43C2×C4semidirect
136. #339C2×C4semidirect
444. Q8#382semidirect
450. D4#382semidirect

Maximal subgroups

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

subgroupindex
#341832
#207642
#206482
C2×C4×#1042
#146872
#112762
#108572
#108392
#107812