Finite Group Databasebeta

← #355990

Subgroups — #355990

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#2314C211yesmaximal, characteristic
4#10850C212yesmaximal, complemented
5#10849C212yesmaximal, 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
15C2×#433—24no—
16C2×#433—24no—
17C2×#433C418yescomplemented
18C22×C4⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2317—24no—
23#2317C418yescomplemented
24#2314—24no—
25#2314C418yescomplemented
26C2×#389—22no—
27C2×#388—22no—
28C2×#387—22no—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#117—22no—
51C2×#117—42no—
52C2×#117—42no—
53C2×#117C2×C414yes—
54C2×#117C2×C414yes—
55C2×#117C2×C414yes—
56C2×#117C2×C414yes—
57C2×#117D414yes—
58C2×#117D414yes—
59C2×#117Q814yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2×C414yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42—22no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#433—232no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8C2×C418yescomplemented
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8C2×C418yes—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97C2×C414yescomplemented
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8C2×C414yescomplemented
95#389—216no—
96#388—216no—
97#387—216no—
98#385—216no—
99#384—22no—
100#384—22no—
101#384—22no—
102#384C2×C414yes—
103#384C2×C414yes—
104#384C2×C414yescomplemented
105#382—22no—
106#382—22no—
107#382—22no—
108#382C2×C414yes—
109#382C2×C414yes—
110#382C2×C414yescomplemented
111#380—24no—
112#380C2×C418yes—
113C4×C4⋊C4—24no—
114C4×C4⋊C4C2×C418yescomplemented
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—42no—
121#340—42no—
122#340—42no—
123#340—42no—
124#340—42no—
125#340—42no—
126#340—42no—
127#340—42no—
128C25C2×D411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×Q812yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C414yes—
167C22×C8C22×C414yes—
168#117—48no—
169#117—48no—
170#117C42116yes—
171#117C22⋊C4116yes—
172#117C4⋊C4116yes—
173#117C4⋊C4116yes—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
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—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22⋊C418yes—
211C2×C4⋊C4C22⋊C418yes—
212C2×C4⋊C4C4⋊C418yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C22×C414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4C22⋊C418yes—
226C2×C22⋊C4C22⋊C418yes—
227C2×C22⋊C4C4⋊C418yes—
228C2×C42—24no—
229C2×C42—28no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—82no—
233C2×C42—82no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×Q812yes—
245C2×C42C2×Q812yes—
246C4⋊C8—216no—
247C22⋊C8—216no—
248#97—28no—
249C4×C8—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24C2×C22⋊C412yes—
264C24C2×C22⋊C412yes—
265C24C2×C4⋊C412yes—
266C24C22×D411yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
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—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—48no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
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—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—88no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—88no—
330C22×C4C2×C4214yes—
331C22×C4C2×C22⋊C414yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C414yes—
335C22×C4C2×C22⋊C414yes—
336C22×C4C2×C22⋊C414yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C4⋊C414yes—
339C22×C4C2×C4⋊C414yes—
340C22×C4C2×C4⋊C414yes—
341C22×C4C2×C4⋊C414yes—
342C22×C4C2×C4⋊C412yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C23×C411yescharacteristic
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—88no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C4⋊C4#95132yes—
388C22⋊C4—48no—
389C22⋊C4—88no—
390C22⋊C4#95132yes—
391C42—48no—
392C42—88no—
393C42C2×C4214yes—
394C42C2×C4214yes—
395C42C2×C22⋊C414yes—
396C42C2×C22⋊C414yes—
397C42C2×C4⋊C414yes—
398C42C2×C4⋊C414yes—
399C42C2×C4⋊C414yes—
400C42C2×C4⋊C414yes—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C2×#9514yes—
424C23C22×C22⋊C411yescharacteristic
425C23C22×C22⋊C411yescharacteristic
426C23C22×C4⋊C411yescharacteristic
427C23#53511yescharacteristic
428C23#53511yescharacteristic
429C23#53611yescharacteristic
430C23#53611yescharacteristic
431C23#57612yes—
432C23#57712yes—
433C23#57712yes—
434C23#57812yes—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
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—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—816no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4C2×#9518yes—
493C2×C4C2×#9518yes—
494C2×C4C2×#9514yes—
495C2×C4C22×C4211yescharacteristic
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C22×C4⋊C411yescharacteristic
498C2×C4C22×C4⋊C411yescharacteristic
499C2×C4#51812yes—
500C2×C4#51912yes—
501C2×C4#52012yes—
502C2×C4#52012yes—
503C2×C4#52012yes—
504C2×C4#52012yes—
505C2×C4#53512yes—
506C2×C4#53612yes—
507C2×C4#53712yes—
508C2×C4#53712yes—
509C2×C4#54412yes—
510C2×C4#54412yes—
511C8—44no—
512C8—44no—
513C22—28no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22C22×#9511yescharacteristic
525C22#230911yescharacteristic
526C22#231011yescharacteristic
527C22#231111yescharacteristic
528C22#231111yescharacteristic
529C22#231711yescharacteristic
530C22#231711yescharacteristic
531C22#289712yes—
532C22#289712yes—
533C22#289812yes—
534C22#289812yes—
535C22#291212yes—
536C22#291312yes—
537C22#322011yescharacteristic
538C22#322111yescharacteristic
539C22#322211yescharacteristic
540C22#322311yescharacteristic
541C22#322412yes—
542C22#322512yes—
543C22#324112yes—
544C22#324212yes—
545C22#324312yes—
546C22#324412yes—
547C4—28no—
548C4—28no—
549C4—24no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—84no—
556C4#227412yes—
557C4#231112yes—
558C4#231612yes—
559C4#231712yes—
560C2—44no—
561C2#1379511yescharacteristic
562C2#2034812yes—
563C2#2181311yescharacteristic
564C2#2181411yescharacteristic
565C2#2184312yes—
566C2#2187112yes—
567C2#2187211yescharacteristic
568C2#2187311yescharacteristic
569C2#2199011yescharacteristic
570C2#2199111yescharacteristic
571C2#2199212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×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. #10850C2semidirect
5. #10849C2semidirect
6. #10843C2semidirect
7. #10843C2semidirect
8. #10842C2semidirect
9. #10841C2semidirect
17. C2×#433C4semidirect
23. #2317C4semidirect
25. #2314C4semidirect
85. C2×C4⋊C8C2×C4semidirect
91. C2×#97C2×C4semidirect
94. C2×C4×C8C2×C4semidirect
104. #384C2×C4semidirect
110. #382C2×C4semidirect
114. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#23172
C2×#23142
#108502
#108492
#108432
#108422
#108412
#107312
#107302