Finite Group Databasebeta

← #398313

Subgroups — #398313

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#2016C212yesmaximal
4C2×#1981C212yesmaximal
5C2×#1878C211yesmaximal, characteristic
6#15190C211yesmaximal, characteristic
7#13817C211yesmaximal, characteristic
8#11322C211yesmaximal, characteristic
9#11077C211yesmaximal, characteristic
10#11010C211yesmaximal, characteristic
11#10932C212yesmaximal
12C23×Q16C2211yescharacteristic
13C23×Q16C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C2×#462—28no—
18C2×#462—24no—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×C4×C4⋊C4C2212yes—
31#2056—28no—
32#2016—28no—
33#1981—28no—
34#1878—24no—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444—21no—
40#1444—22no—
41#1444—21no—
42#1436—22no—
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C4×#95—21no—
46C4×#95C2211yescharacteristic
47C23×Q8C2312yes—
48C23×Q8C2311yescharacteristic
49C23×Q8C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—22no—
54C22×Q16—24no—
55C22×Q16—24no—
56C22×Q16—24no—
57C22×Q16—24no—
58C22×Q16—22no—
59C22×Q16—42no—
60C22×Q16—42no—
61C22×Q16—42no—
62C23×C8C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—24no—
70C2×C4⋊Q8—24no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—42no—
73C2×C4⋊Q8—42no—
74C2×C4⋊Q8—42no—
75C2×C4×Q8—22no—
76C2×C4×Q8—24no—
77C2×C4×Q8—42no—
78C2×C4×Q8D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C42—21no—
88C22×C42—22no—
89C22×C42C2311yescharacteristic
90C22×C42C2312yes—
91#462—232no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—42no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—22no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—42no—
110C2×#103—42no—
111C2×#103—42no—
112C2×#103D418yes—
113#391—24no—
114#391—24no—
115#391—22no—
116#391—22no—
117#391—24no—
118#391D418yes—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—42no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95D411yescharacteristic
131C2×#95D412yes—
132C2×#95D411yescharacteristic
133#340—24no—
134#340—22no—
135#336—22no—
136#336—24no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—21no—
167C22×Q8C2×D414yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×D414yes—
170C22×Q8#4112yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2411yescharacteristic
182C2×Q16—28no—
183C2×Q16—216no—
184C2×Q16—48no—
185C2×Q16—48no—
186C2×Q16—48no—
187C2×Q16—48no—
188C2×Q16—48no—
189C2×Q16—48no—
190C2×Q16—44no—
191C2×Q16—48no—
192C2×Q16—44no—
193C2×Q16—48no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8C2×D414yes—
210C22×C8#4112yes—
211C4⋊Q8—216no—
212C4⋊Q8—28no—
213C4⋊Q8—48no—
214C4⋊Q8—48no—
215C4⋊Q8—48no—
216C4⋊Q8—44no—
217C4⋊Q8—48no—
218C4×Q8—28no—
219C4×Q8—44no—
220C4×Q8—48no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—82no—
258C2×C42C2×D412yes—
259C2×C42C2×D414yes—
260C2×C42#4112yes—
261C2×C42#4114yes—
262C4⋊C8—28no—
263C4⋊C8—48no—
264#103—216no—
265#103—48no—
266#103—48no—
267#103—48no—
268#103—44no—
269#103—44no—
270#95—28no—
271#95—28no—
272#95—22no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—42no—
278#95—44no—
279#95—42no—
280#95—81no—
281#95Q1614yescomplemented
282C24C22×D411yescharacteristic
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8#12714yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4#12114yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12114yes—
401C22×C4C2×Q1612yes—
402C22×C4C2×Q1614yes—
403C22×C4#13712yes—
404C22×C4#13714yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415Q16—432no—
416Q16—416no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—42no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—48no—
442C4⋊C4—416no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4#12014yes—
449C42—22no—
450C42—24no—
451C42—22no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—48no—
457C42—88no—
458C23C2×#12111yescharacteristic
459C23C2×#12112yes—
460C23D4212yes—
461C23#54611yescharacteristic
462C23#55011yescharacteristic
463C23C22×Q1611yescharacteristic
464C23C2×#13711yescharacteristic
465C23C2×#13712yes—
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468C23#57812yes—
469Q8—24no—
470Q8—24no—
471Q8—42no—
472Q8—42no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—42no—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—42no—
481Q8—44no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485Q8—82no—
486Q8—82no—
487Q8—82no—
488Q8—82no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4#46214yes—
546C2×C4#46218yescomplemented
547C2×C4C2×#12011yescharacteristic
548C2×C4#52512yes—
549C2×C4#52511yescharacteristic
550C2×C4C2×#12711yescharacteristic
551C2×C4#53211yescharacteristic
552C2×C4#53212yes—
553C2×C4#53212yes—
554C2×C4#53411yescharacteristic
555C2×C4#53511yescharacteristic
556C2×C4#53812yes—
557C2×C4#54012yes—
558C2×C4D4212yes—
559C2×C4D4212yes—
560C2×C4#54612yes—
561C2×C4#54612yes—
562C2×C4#54612yes—
563C2×C4#55311yescharacteristic
564C2×C4#55511yescharacteristic
565C8—44no—
566C8—42no—
567C8—81no—
568C22#241011yescharacteristic
569C22#241411yescharacteristic
570C22#242412yes—
571C22#242812yes—
572C22#285611yescharacteristic
573C22C2×#46212yes—
574C22C2×#46211yescharacteristic
575C22#303912yes—
576C22#303911yescharacteristic
577C22#304412yes—
578C22#304411yescharacteristic
579C22#304612yes—
580C22#304611yescharacteristic
581C22#326612yes—
582C22#327012yes—
583C22#327612yes—
584C22D4×Q1612yes—
585C22#329911yescharacteristic
586C22#331212yes—
587C22#332811yescharacteristic
588C22#339511yescharacteristic
589C22#339812yes—
590C22#340011yescharacteristic
591C4—24no—
592C4—28no—
593C4—44no—
594C4—42no—
595C4—44no—
596C4—44no—
597C4—84no—
598C4—84no—
599C4—82no—
600C4—161no—
601C4#238811yescharacteristic
602C4#243311yescharacteristic
603C4#246412yes—
604C4#253911yescharacteristic
605C4#255512yes—
606C4#266111yescharacteristic
607C2#1476211yescharacteristic
608C2#2436011yescharacteristic
609C2#2437611yescharacteristic
610C2#2463211yescharacteristic
611C2#2464811yescharacteristic
612C2#2508212yes—
613C2#2509512yes—
614C2#2537012yes—
615C2#2538212yes—
616C2#3316311yescharacteristic
617C2#3316911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
281. #95Q16semidirect
546. C2×C4#462semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#20562
C2×#20162
C2×#19812
C2×#18782
#151902
#138172
#113222
#110772
#110102
#109322