Finite Group Databasebeta

← #406779

Subgroups — #406779

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1855C211yesmaximal, characteristic
2#20836C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#20661C211yesmaximal, characteristic
6C22×#343C211yesmaximal, characteristic
7#11303C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#11242C212yesmaximal, complemented
10#11216C212yesmaximal, complemented
11#10739C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2211yescharacteristic
16C22×#108C2211yescharacteristic
17C2×#428C2212yes—
18C2×#424C2212yes—
19C2×#407C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2280C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1855—24no—
24#1851—22no—
25#1850—22no—
26#1824—22no—
27#1823—22no—
28#1823—22no—
29#1823—21no—
30#1823—21no—
31#1809—22no—
32#1808—21no—
33#1807—21no—
34C2×#343—24no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1729C2212yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—21no—
50C22×M16—22no—
51C22×M16—21no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—24no—
55C22×M16—22no—
56C22×M16—42no—
57C22×M16—42no—
58C22×M16D414yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2312yes—
63C22×M16C2311yescharacteristic
64C23×C8—21no—
65C23×C8—21no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#117C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4Q814yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C2×#108—22no—
82C2×#108—22no—
83C2×#108—24no—
84C2×#108—42no—
85#428—24no—
86#428—42no—
87C2×#107—22no—
88C2×#106—22no—
89#424—24no—
90#424—42no—
91C2×C4⋊C8—24no—
92#407—22no—
93#407—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8D412yes—
100#381—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380D411yescharacteristic
106#380D411yescharacteristic
107#380D412yes—
108#379—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C2×#95—21no—
112#343—24no—
113#343—24no—
114#343—24no—
115#343—42no—
116#341—22no—
117#341—22no—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#336—24no—
127C25#4111yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—42no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×Q812yes—
146C23×C4#4111yescharacteristic
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—21no—
151C2×M16—22no—
152C2×M16—24no—
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—28no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—28no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—22no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—44no—
172C2×M16—44no—
173C2×M16—44no—
174C2×M16—48no—
175C2×M16—48no—
176C2×M16—44no—
177C2×M16—42no—
178C2×M16C2×D412yes—
179C2×M16C2×D412yes—
180C2×M16#4114yes—
181C2×M16#4114yes—
182C22×C8—21no—
183C22×C8—24no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8—42no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×Q814yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239#108—44no—
240#108—44no—
241#107—44no—
242#106—44no—
243C4⋊C8—48no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—41no—
252#95—41no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—42no—
258C24#12111yescharacteristic
259C24#12111yescharacteristic
260C24C22⋊Q812yes—
261C24#12612yes—
262C24C2×#4111yescharacteristic
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—88no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—88no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12612yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C42−1+412yes—
325M16—22no—
326M16—24no—
327M16—22no—
328M16—24no—
329M16—22no—
330M16—24no—
331M16—48no—
332M16—48no—
333M16—42no—
334M16—42no—
335M16—42no—
336M16—41no—
337M16—41no—
338M16—41no—
339M16—41no—
340M16—42no—
341M16—42no—
342M16#12314yes—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—48no—
352C2×C8—48no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C42—42no—
388C42—42no—
389C23—22no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23#39612yes—
410C23#39612yes—
411C23#47112yes—
412C23#47212yes—
413C23C2×#12111yescharacteristic
414C23C2×C22⋊Q811yescharacteristic
415C23C2×#12611yescharacteristic
416C23#52912yes—
417C23#53912yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—84no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—162no—
452C2×C4—162no—
453C2×C4#39612yes—
454C2×C4#39612yes—
455C2×C4C2×#12111yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4C2×C22⋊Q811yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C2×C4#52511yescharacteristic
461C2×C4#53312yes—
462C2×C4#53411yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#54412yes—
465C2×C4#54612yes—
466C2×C4#54612yes—
467C2×C4#54712yes—
468C2×C4#54712yes—
469C2×C4#55112yes—
470C2×C4#55112yes—
471C2×C4#55212yes—
472C2×C4#55212yes—
473C8—24no—
474C8—42no—
475C8—42no—
476C8—42no—
477C8—44no—
478C8—81no—
479C8—81no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—24no—
489C22—22no—
490C22—22no—
491C22—24no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—44no—
496C22—44no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22#206314yes—
501C22C2×#39611yescharacteristic
502C22#242512yes—
503C22#249312yes—
504C22#262711yescharacteristic
505C22#285511yescharacteristic
506C22C2×#47111yescharacteristic
507C22C2×#47211yescharacteristic
508C22#306912yes—
509C4—22no—
510C4—22no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4—162no—
515C4#241211yescharacteristic
516C4#241311yescharacteristic
517C4#243812yes—
518C4#243812yes—
519C4#248611yescharacteristic
520C4#253411yescharacteristic
521C2—24no—
522C2—21no—
523C2—21no—
524C2—22no—
525C2—42no—
526C2#1493511yescharacteristic
527C2C2×#206311yescharacteristic
528C2#2360311yescharacteristic
529C2#2526412yes—
530C2#2526812yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11303C2semidirect
8. #11249C2semidirect
9. #11242C2semidirect
10. #11216C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18552
#208362
#207002
C2×#18232
#206612
C22×#3432
#113032
#112492
#112422
#112162
#107392