Finite Group Databasebeta

← #468044

Subgroups — #468044

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic, complemented
3C2×#2026C211yesmaximal, characteristic, complemented
4#14258C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11096C212yesmaximal
9#11067C212yesmaximal, complemented
10#10970C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19#2629—24no—
20#2629—24no—
21C2×#398—22no—
22C2×#396—22no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29#2095—24no—
30#2026—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1441—22no—
40#1440—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—22no—
48C2×#125—42no—
49C2×#125—42no—
50C2×#125D414yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—42no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8Q814yes—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—42no—
58C2×#117—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—42no—
87#398—216no—
88#396—216no—
89#391—22no—
90#391—22no—
91#391—42no—
92#389—24no—
93#389—24no—
94#384—22no—
95#384D414yes—
96#380—24no—
97#380—24no—
98#380—42no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—22no—
113#336—42no—
114C25C2×D411yescharacteristic
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8C2×Q812yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×Q812yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
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×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8#4114yes—
159#125—48no—
160#125—48no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163#117—48no—
164#117—48no—
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—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223#107—48no—
224#103—48no—
225#103—48no—
226C22⋊C8—48no—
227#95—28no—
228#95—28no—
229#95—42no—
230#95—44no—
231#95—44no—
232#95—82no—
233#95—82no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24C22⋊Q812yes—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
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—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
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—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4#12014yes—
334C22×C4C22⋊Q812yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—416no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4—88no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—88no—
381C22⋊C4—84no—
382C42—24no—
383C42—24no—
384C42—42no—
385C42—42no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C2×#12011yescharacteristic
410C23C2×C22⋊Q811yescharacteristic
411C23#53211yescharacteristic
412C23#53411yescharacteristic
413C23#53411yescharacteristic
414C23#53511yescharacteristic
415C23#53611yescharacteristic
416C23#57612yes—
417C23#57712yes—
418C23#57812yes—
419C23#57812yes—
420Q8—44no—
421Q8—44no—
422Q8—82no—
423Q8—82no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#53412yes—
486C2×C4#53512yes—
487C2×C4#53812yes—
488C2×C4#53912yes—
489C2×C4#54212yes—
490C2×C4#54312yes—
491C2×C4#54612yes—
492C2×C4#54811yescharacteristic
493C2×C4#54811yescharacteristic
494C2×C4D4×Q811yescharacteristic
495C2×C4D4×Q811yescharacteristic
496C2×C4#55012yes—
497C2×C4#55112yes—
498C2×C4#55412yes—
499C2×C4#56212yes—
500C2×C4#56212yes—
501C8—44no—
502C8—82no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—44no—
507C22—44no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22#240911yescharacteristic
515C22#243111yescharacteristic
516C22#256411yescharacteristic
517C22#261311yescharacteristic
518C22#262611yescharacteristic
519C22#266311yescharacteristic
520C22#281611yescharacteristic
521C22#301112yes—
522C22#301212yes—
523C22#308212yes—
524C22#308312yes—
525C22#317712yes—
526C22#317912yes—
527C22#320112yes—
528C22#320112yes—
529C22#320312yes—
530C22#320312yes—
531C22#322011yescharacteristic
532C22#322111yescharacteristic
533C22#322311yescharacteristic
534C22#322311yescharacteristic
535C22#324212yes—
536C22#324312yes—
537C4—44no—
538C4—42no—
539C4—84no—
540C4—82no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4—161no—
545C4—161no—
546C4#269212yes—
547C4#269612yes—
548C4#273012yes—
549C4#273212yes—
550C2—82no—
551C2#1660011yescharacteristic
552C2#2431112yes—
553C2#2560911yescharacteristic
554C2#2561111yescharacteristic
555C2#2917312yes—
556C2#2960812yes—
557C2#2996311yescharacteristic
558C2#2996411yescharacteristic
559C2#3037112yes—
560C2#3278311yescharacteristic
561C2#3278411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#2095C2semidirect
3. C2×#2026C2semidirect
5. #11385C2semidirect
6. #11373C2semidirect
7. #11219C2semidirect
9. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#20952
C2×#20262
#142582
#113852
#113732
#112192
#110962
#110672
#109702
#107132