Finite Group Databasebeta

← #405507

Subgroups — #405507

Order 512.

1492 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×#2628C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3#13795C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11325C212yesmaximal, complemented
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17#2628—28no—
18C2×#400—21no—
19C2×#399—21no—
20C2×#395—21no—
21C2×#393—21no—
22C2×#389C2212yes—
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25#1854—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C23×Q8—21no—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—41no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117—41no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4Q814yescomplemented
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—42no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8Q818yescomplemented
89#400—28no—
90#399—28no—
91#395—28no—
92#393—28no—
93#389—24no—
94#389—24no—
95#382—24no—
96#380—24no—
97#380D418yes—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—41no—
114#336—41no—
115C25C2×D411yescharacteristic
116C22×Q8—24no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—82no—
120C22×Q8—82no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D414yes—
142C23×C4C2×Q814yes—
143C23×C4C2×Q812yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
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×C8C2×Q814yes—
159C22×C8C2×Q814yes—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162#117—48no—
163#117—48no—
164#117—48no—
165#117—48no—
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—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—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4#4114yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42#4114yes—
232#107—48no—
233#106—48no—
234C22⋊C8—216no—
235#95—24no—
236#95—24no—
237#95—28no—
238#95—28no—
239#95—82no—
240#95—82no—
241#95—82no—
242#95—82no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24#12012yes—
252C24C22⋊Q812yes—
253C24C4⋊Q812yes—
254C24C22×D411yescharacteristic
255C2×Q8—48no—
256C2×Q8—88no—
257C2×Q8—88no—
258C2×Q8—84no—
259C2×Q8—84no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—48no—
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—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—88no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4#12012yes—
330C22×C4C22⋊Q818yes—
331C22×C4C22⋊Q812yes—
332C22×C4#12312yes—
333C22×C4#12312yes—
334C22×C4C4⋊Q818yes—
335C22×C4C4⋊Q812yes—
336C22×C4C22×D412yes—
337C22×C4C22×Q812yes—
338C22×C4C22×Q811yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—88no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—48no—
384C22⋊C4—88no—
385C42—24no—
386C42—24no—
387C42—48no—
388C42—42no—
389C42—42no—
390C23—24no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—24no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23#39514yes—
413C23C2×#12011yescharacteristic
414C23C2×C22⋊Q811yescharacteristic
415C23C2×#12311yescharacteristic
416C23#52511yescharacteristic
417C23C2×C4⋊Q811yescharacteristic
418C23#53411yescharacteristic
419C23#53511yescharacteristic
420C23#57612yes—
421C23#57712yes—
422C23#57712yes—
423C23#57812yes—
424Q8—816no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—816no—
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—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#39514yes—
485C2×C4C2×C22⋊Q812yes—
486C2×C4#52912yes—
487C2×C4C2×C4⋊Q812yes—
488C2×C4#53312yes—
489C2×C4#53811yescharacteristic
490C2×C4#53811yescharacteristic
491C2×C4#54311yescharacteristic
492C2×C4#54311yescharacteristic
493C2×C4#54612yes—
494C2×C4#54612yes—
495C2×C4#54612yes—
496C2×C4#54612yes—
497C2×C4#55412yes—
498C2×C4#55412yes—
499C2×C4#55412yes—
500C2×C4#55412yes—
501C8—44no—
502C8—44no—
503C22—28no—
504C22—24no—
505C22—42no—
506C22—42no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22C2×#39511yescharacteristic
515C22#240411yescharacteristic
516C22#240511yescharacteristic
517C22#250211yescharacteristic
518C22#258311yescharacteristic
519C22#262611yescharacteristic
520C22#285511yescharacteristic
521C22#301212yes—
522C22#301312yes—
523C22#308112yes—
524C22#308212yes—
525C22#309612yes—
526C22#309712yes—
527C22#312412yes—
528C22#312512yes—
529C22#316312yes—
530C22#316412yes—
531C22#319911yescharacteristic
532C22#320011yescharacteristic
533C22#320011yescharacteristic
534C22#320211yescharacteristic
535C22#322412yes—
536C22#322512yes—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—84no—
541C4—84no—
542C4—84no—
543C4—81no—
544C4—84no—
545C4—81no—
546C4#245012yes—
547C4#245012yes—
548C4#254912yes—
549C4#254912yes—
550C2—44no—
551C2#1491411yescharacteristic
552C2#2355712yes—
553C2#2394712yes—
554C2#2401511yescharacteristic
555C2#2401611yescharacteristic
556C2#2795912yes—
557C2#2931011yescharacteristic
558C2#2931111yescharacteristic
559C2#2995911yescharacteristic
560C2#2996011yescharacteristic
561C2#3315912yes—
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
4. #11384C2semidirect
5. #11325C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
68. C22×C4⋊C4Q8semidirect
88. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#18542
#137952
#113842
#113252
#112202
#112192
#107302
#107132