Finite Group Databasebeta

← #364264

Subgroups — #364264

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#13881C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11208C212yesmaximal, complemented
8#10850C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2347—24no—
18#2347C418yescomplemented
19C2×#389C2212yes—
20C2×#388—22no—
21C2×#383—21no—
22C2×#382C2212yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1851—24no—
29#1825—24no—
30#1736—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—42no—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117C2×C414yes—
60C2×#117D414yes—
61C2×#117Q814yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42—22no—
80C22×C42C2312yes—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—42no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87#389—24no—
88#389C2×C418yescomplemented
89#388—216no—
90#383—28no—
91#382—24no—
92#382—42no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380C2×C414yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—28no—
99C4×C4⋊C4C2×C418yescomplemented
100C4×C22⋊C4C2×C414yes—
101C4×C22⋊C4D414yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#340—42no—
122#336—24no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—44no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q812yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×M16—28no—
150C2×M16—48no—
151C2×M16—48no—
152C2×M16—44no—
153C2×M16—44no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8#4114yes—
171#117—28no—
172#117—28no—
173#117—28no—
174#117C4⋊C4116yes—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C4⋊C4C2×Q812yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4C22⋊C418yes—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—28no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42C22×C414yes—
237C2×C42C2×D414yes—
238C4⋊C8—48no—
239C22⋊C8—44no—
240#95—24no—
241#95—28no—
242#95—28no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—82no—
247#95—82no—
248#95C22⋊C414yes—
249#95C22⋊C414yes—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24C2×C22⋊C412yes—
259C24#12112yes—
260C24C22⋊Q812yes—
261C24C22×D411yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4C2×C22⋊C414yes—
332C22×C4C2×C22⋊C414yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C4⋊C414yes—
335C22×C4#11714yes—
336C22×C4#12114yes—
337C22×C4#12112yes—
338C22×C4C22⋊Q814yes—
339C22×C4C22⋊Q812yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×Q811yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4H(5,𝔽2)12yes—
349C22×C42−1+412yes—
350M16—416no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—416no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4C2×C4⋊C414yes—
391C4⋊C4C2×C4⋊C414yes—
392C22⋊C4—216no—
393C22⋊C4—48no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—216no—
401C42—88no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23#38014yes—
425C23C22×C22⋊C411yescharacteristic
426C23C2×#12111yescharacteristic
427C23C2×C22⋊Q811yescharacteristic
428C23#54612yes—
429C23#54712yes—
430C23#57512yes—
431C23#57512yes—
432C23#57611yescharacteristic
433C23#57711yescharacteristic
434C23#57711yescharacteristic
435C23#57811yescharacteristic
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—816no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#38014yes—
498C2×C4C22×C4⋊C411yescharacteristic
499C2×C4C2×#11711yescharacteristic
500C2×C4C4×#4112yes—
501C2×C4#52012yes—
502C2×C4C2×#12111yescharacteristic
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4#53312yes—
507C2×C4#53511yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#54412yes—
510C2×C4#54612yes—
511C2×C4#54712yes—
512C2×C4#55112yes—
513C2×C4#55112yes—
514C2×C4#55212yes—
515C2×C4#55212yes—
516C8—44no—
517C8—81no—
518C8—81no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—44no—
530C22C2×#38011yescharacteristic
531C22#233912yes—
532C22#241111yescharacteristic
533C22#241311yescharacteristic
534C22#243812yes—
535C22#289612yes—
536C22#289711yescharacteristic
537C22#289811yescharacteristic
538C22#304011yescharacteristic
539C22#304011yescharacteristic
540C22#304511yescharacteristic
541C22#304611yescharacteristic
542C22#308012yes—
543C22#308311yescharacteristic
544C22#308411yescharacteristic
545C22#331112yes—
546C22#331312yes—
547C22#332212yes—
548C22#332612yes—
549C22#334912yes—
550C22#335012yes—
551C22#335112yes—
552C22#335512yes—
553C4—24no—
554C4—44no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—84no—
559C4—84no—
560C4—161no—
561C4—161no—
562C4#229611yescharacteristic
563C4#231711yescharacteristic
564C4#234312yes—
565C4#241111yescharacteristic
566C4#241311yescharacteristic
567C4#243812yes—
568C2—44no—
569C2#1396211yescharacteristic
570C2#2064911yescharacteristic
571C2#2065211yescharacteristic
572C2#2249412yes—
573C2#2250112yes—
574C2#2449311yescharacteristic
575C2#2449711yescharacteristic
576C2#2459311yescharacteristic
577C2#2459711yescharacteristic
578C2#2601312yes—
579C2#2601512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11228C2semidirect
7. #11208C2semidirect
8. #10850C2semidirect
9. #10806C2semidirect
18. #2347C4semidirect
88. #389C2×C4semidirect
99. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#18512
C2×#18252
C2×#17362
#138812
#112282
#112082
#108502
#108062
#107302
#107222