Finite Group Databasebeta

← #362872

Subgroups — #362872

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11358C212yesmaximal, complemented
5#11357C212yesmaximal, complemented
6#10843C211yesmaximal, characteristic, complemented
7#10843C211yesmaximal, characteristic, complemented
8#10842C211yesmaximal, characteristic, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2317—24no—
19#2317C418yescomplemented
20C2×#389—22no—
21C2×#385—22no—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29#1750—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1441—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—42no—
52C2×#117—42no—
53C2×#117C2×C414yes—
54C2×#117C2×C414yes—
55C2×#117D414yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—42no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8Q818yes—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86#389—216no—
87#385—216no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#384C2×C414yes—
93#384C2×C414yes—
94#382—22no—
95#382—22no—
96#382—22no—
97#382—22no—
98#382C2×C414yes—
99#382C2×C414yes—
100#380—24no—
101#380—24no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336—24no—
117C25C2×D411yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C22×C412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×Q814yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C2×Q814yes—
156C22×C8C2×Q814yes—
157#117—28no—
158#117—28no—
159#117—28no—
160#117—28no—
161#117—48no—
162#117—48no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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—28no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C22⋊C418yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4C22⋊C418yes—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C22⋊C8—216no—
233#97—44no—
234C4×C8—44no—
235#95—28no—
236#95—28no—
237#95—44no—
238#95—44no—
239#95—82no—
240#95—82no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24C2×C22⋊C412yes—
250C24#12412yes—
251C24C4⋊Q812yes—
252C24C22×D411yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—48no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
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—88no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—88no—
320C22×C4C2×C22⋊C414yes—
321C22×C4C2×C22⋊C414yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C4×Q818yes—
324C22×C4C22⋊Q818yes—
325C22×C4#12314yes—
326C22×C4#12314yes—
327C22×C4#12412yes—
328C22×C4C4⋊Q812yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q812yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C4H(5,𝔽2)12yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—216no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—88no—
372C4⋊C4—88no—
373C4⋊C4—88no—
374C22⋊C4—216no—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—48no—
382C42#11714yes—
383C42#11714yes—
384C42#12314yes—
385C42#12314yes—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23#39114yes—
409C23C22×C22⋊C411yescharacteristic
410C23C2×#12311yescharacteristic
411C23C2×#12311yescharacteristic
412C23C2×#12411yescharacteristic
413C23C2×C4⋊Q811yescharacteristic
414C23#53511yescharacteristic
415C23#53511yescharacteristic
416C23#57612yes—
417C23#57712yes—
418C23#57712yes—
419C23#57812yes—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—816no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#38118yes—
479C2×C4#39114yes—
480C2×C4C2×#11711yescharacteristic
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×C4×Q812yes—
483C2×C4#51812yes—
484C2×C4C2×C22⋊Q812yes—
485C2×C4C2×#12311yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4#53512yes—
488C2×C4#55312yes—
489C2×C4#55312yes—
490C2×C4#55312yes—
491C2×C4#55312yes—
492C2×C4#55412yes—
493C2×C4#55412yes—
494C2×C4#55412yes—
495C2×C4#55412yes—
496C8—44no—
497C8—44no—
498C22—28no—
499C22—24no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—44no—
505C22—44no—
506C22—42no—
507C22—42no—
508C22—44no—
509C22C2×#38111yescharacteristic
510C22C2×#39111yescharacteristic
511C22#230911yescharacteristic
512C22#250211yescharacteristic
513C22#250211yescharacteristic
514C22#250411yescharacteristic
515C22#250411yescharacteristic
516C22#289712yes—
517C22#289812yes—
518C22#309612yes—
519C22#309712yes—
520C22#309812yes—
521C22#309912yes—
522C22#313612yes—
523C22#313712yes—
524C22#316312yes—
525C22#316412yes—
526C22#322011yescharacteristic
527C22#322111yescharacteristic
528C22#322211yescharacteristic
529C22#322311yescharacteristic
530C22#322412yes—
531C22#322512yes—
532C4—24no—
533C4—28no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—162no—
540C4—162no—
541C4#234512yes—
542C4#234512yes—
543C4#248712yes—
544C4#248712yes—
545C2—44no—
546C2#1392411yescharacteristic
547C2#2070312yes—
548C2#2132612yes—
549C2#2181311yescharacteristic
550C2#2181411yescharacteristic
551C2#2795711yescharacteristic
552C2#2795811yescharacteristic
553C2#2795912yes—
554C2#2798412yes—
555C2#2798511yescharacteristic
556C2#2798611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11358C2semidirect
5. #11357C2semidirect
6. #10843C2semidirect
7. #10843C2semidirect
8. #10842C2semidirect
9. #10841C2semidirect
19. #2317C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#17502
#143402
#113582
#113572
#108432
#108422
#108412
#107312
#107302