Finite Group Databasebeta

← #387059

Subgroups — #387059

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#13923C212yesmaximal, complemented
3#11353C212yesmaximal, complemented
4#11203C212yesmaximal, complemented
5#11043C212yesmaximal
6#10960C212yesmaximal
7#10912C212yesmaximal, complemented
8#10761C211yesmaximal, characteristic
9#10761C211yesmaximal, characteristic
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2212yes—
13C22×#99C2211yescharacteristic
14C22×#99C2211yescharacteristic
15#2487—24no—
16#2487—24no—
17C2×#400C412yescomplemented
18C2×#399C412yescomplemented
19C2×#389C412yescomplemented
20C2×#386C2212yes—
21C2×#385C2212yes—
22C2×#380C412yescomplemented
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2212yes—
26C2×#342C2212yes—
27C2×#342C2212yes—
28C2×#342C2211yescharacteristic
29C2×#342C2212yes—
30C2×#342C2211yescharacteristic
31C2×#342C2212yes—
32C2×#342C2212yes—
33#1441—24no—
34#1436—24no—
35C22⋊#95C2212yes—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C22⋊#95C2212yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2312yes—
43C2×C4×D4—24no—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—42no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—41no—
49C22×C4⋊C4—42no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4D414yes—
57C22×C4⋊C4D414yes—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C414yes—
60C22×C22⋊C4C2312yes—
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2312yes—
64C22×C22⋊C4C2312yes—
65C22×C22⋊C4C2312yes—
66C22×C42—22no—
67C22×C42C2×C411yescharacteristic
68C2×#99—24no—
69C2×#99—22no—
70C2×#99—24no—
71C2×#99—41no—
72C2×#99—41no—
73C2×#99—42no—
74C2×#99—41no—
75C2×#99—41no—
76C2×#99Q814yes—
77#400—22no—
78#400—24no—
79#400—22no—
80#399—22no—
81#399—24no—
82#399—22no—
83#389—24no—
84#389—22no—
85#389—22no—
86#386—22no—
87#386—24no—
88#386—22no—
89#385—22no—
90#385—24no—
91#385—22no—
92#380—24no—
93#380—22no—
94#380—22no—
95#380—24no—
96#380—42no—
97#380—42no—
98C4×C22⋊C4—42no—
99C4×C22⋊C4—42no—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—24no—
105C2×#95—24no—
106C2×#95C2×C412yes—
107C2×#95C2×C414yes—
108C2×#95C2×C414yes—
109#342—24no—
110#342—24no—
111#342—41no—
112#342—42no—
113#342—42no—
114#342—41no—
115#342—42no—
116C25C22×C411yescharacteristic
117C25C2411yescharacteristic
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4C2×Q812yes—
127C22×D4#4112yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—44no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—44no—
144C23×C4—84no—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C414yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C414yes—
152C23×C4C2×D414yes—
153C23×C4C2×D412yes—
154C23×C4C2×D414yes—
155C23×C4C2×D412yes—
156C4×D4—48no—
157C4×D4—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4—84no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C414yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C414yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—84no—
221C2×C22⋊C4C2×Q814yes—
222C2×C22⋊C4#4114yes—
223C2×C42—22no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—44no—
227C2×C42—84no—
228#99—44no—
229#99—48no—
230#99—44no—
231#95—24no—
232#95—24no—
233#95—22no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95—48no—
238#95—48no—
239#95—44no—
240#95—44no—
241#95—84no—
242#95—84no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24C4×Q812yes—
254C24C23×C411yescharacteristic
255C24C22×D412yes—
256C24C22×Q811yescharacteristic
257C24C2×#4111yescharacteristic
258C24C2×#4112yes—
259C24H(5,𝔽2)11yescharacteristic
260C24H(5,𝔽2)11yescharacteristic
261C24H(5,𝔽2)12yes—
262C24H(5,𝔽2)11yescharacteristic
263C242−1+412yes—
264C242−1+411yescharacteristic
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—28no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C414yes—
358C22×C4C2×C22⋊C414yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C4×Q812yes—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—88no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—44no—
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—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—44no—
399C42—88no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—81no—
428C23—81no—
429C23—82no—
430C23—82no—
431C23—81no—
432C23—81no—
433C23C22×C22⋊C411yescharacteristic
434C23C2×C4×Q811yescharacteristic
435C23C4×#4111yescharacteristic
436C23#51811yescharacteristic
437C23#51811yescharacteristic
438C23#51911yescharacteristic
439C23#52011yescharacteristic
440C23#53412yes—
441C23#53412yes—
442C23#53811yescharacteristic
443C23#53911yescharacteristic
444C23#53912yes—
445C23#54212yes—
446C23#54311yescharacteristic
447C23#54411yescharacteristic
448C23#54812yes—
449C23#54812yes—
450C23D4×Q812yes—
451C23#55012yes—
452C23#55112yes—
453C23#55412yes—
454C23#56212yes—
455C23#56312yes—
456D4—44no—
457D4—44no—
458D4—44no—
459D4—44no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—88no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#41014yes—
526C22—24no—
527C22—24no—
528C22—22no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—21no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22#230811yescharacteristic
562C22#232911yescharacteristic
563C22#233211yescharacteristic
564C22Q8×C22⋊C411yescharacteristic
565C22#234111yescharacteristic
566C22#236811yescharacteristic
567C22#237811yescharacteristic
568C22#272812yes—
569C22#272912yes—
570C22#273012yes—
571C22#273212yes—
572C22C2×#41012yes—
573C22#288312yes—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—44no—
578C4—44no—
579C4—84no—
580C4—84no—
581C4—84no—
582C4—81no—
583C4—81no—
584C4—162no—
585C4—162no—
586C2—22no—
587C2—22no—
588C2—22no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—22no—
594C2—82no—
595C2#1444911yescharacteristic
596C2#2174711yescharacteristic
597C2#2175011yescharacteristic
598C2#2251312yes—
599C2#2253512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13923C2semidirect
3. #11353C2semidirect
4. #11203C2semidirect
7. #10912C2semidirect
17. C2×#400C4semidirect
18. C2×#399C4semidirect
19. C2×#389C4semidirect
22. C2×#380C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#139232
#113532
#112032
#110432
#109602
#109122
#107612