Finite Group Databasebeta

← #400434

Subgroups — #400434

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#1918C211yesmaximal, characteristic
4C2×#1863C212yesmaximal
5#15149C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11067C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C2×#398—21no—
20#2365—24no—
21#2365—24no—
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#387—22no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2043—24no—
31#1918—24no—
32#1863—28no—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—22no—
43#1441—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#107—22no—
77C2×#107—22no—
78C2×#107D414yes—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91#398—44no—
92#391—24no—
93#391—24no—
94#389—28no—
95#389—24no—
96#389—22no—
97#389—22no—
98#389—42no—
99#387—216no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95D412yes—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—24no—
120#336—24no—
121#336—22no—
122#336—22no—
123#336—42no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8C2×D414yes—
136C22×Q8#4114yes—
137C22×Q8#4114yes—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8C2×D412yes—
166C4⋊Q8—28no—
167C4⋊Q8—28no—
168C4⋊Q8—216no—
169C4×Q8—48no—
170C4×Q8—48no—
171C4×Q8—48no—
172C4×Q8—48no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4#4114yes—
216C2×C4⋊C4#4112yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—28no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233#107—28no—
234#103—216no—
235#103—48no—
236#103—48no—
237#103—48no—
238#95—28no—
239#95—28no—
240#95—24no—
241#95—24no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—81no—
248C24C22×D411yescharacteristic
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8#12318yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
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—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#13512yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13714yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C42−1+412yes—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—28no—
367C4⋊C4—216no—
368C4⋊C4—216no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—416no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—88no—
389C4⋊C4—84no—
390C4⋊C4#12114yes—
391C42—24no—
392C42—22no—
393C42—22no—
394C42—416no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—48no—
400C23C2×#12011yescharacteristic
401C23#52512yes—
402C23D4211yescharacteristic
403C23#54611yescharacteristic
404C23#54812yes—
405C23C2×#13511yescharacteristic
406C23C2×AGL(1,ℤ/8)11yescharacteristic
407C23C2×#13712yes—
408C23#57711yescharacteristic
409C23#57811yescharacteristic
410C23#57812yes—
411Q8—28no—
412Q8—28no—
413Q8—44no—
414Q8—44no—
415Q8—44no—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#44914yes—
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12312yes—
483C2×C4#52511yescharacteristic
484C2×C4#52511yescharacteristic
485C2×C4#52512yes—
486C2×C4#52912yes—
487C2×C4#53211yescharacteristic
488C2×C4#53511yescharacteristic
489C2×C4#53611yescharacteristic
490C2×C4#53912yes—
491C2×C4#53912yes—
492C2×C4D4211yescharacteristic
493C2×C4#54611yescharacteristic
494C2×C4#54712yes—
495C2×C4#54812yes—
496C2×C4#55112yes—
497C2×C4#55212yes—
498C2×C4#55312yes—
499C8—42no—
500C8—81no—
501C8—82no—
502C22#240811yescharacteristic
503C22#243211yescharacteristic
504C22#246012yes—
505C22#246312yes—
506C22#284911yescharacteristic
507C22C2×#44911yescharacteristic
508C22#300411yescharacteristic
509C22#301111yescharacteristic
510C22#301311yescharacteristic
511C22#310812yes—
512C22#311212yes—
513C22#312112yes—
514C22#312712yes—
515C22#327012yes—
516C22#327211yescharacteristic
517C22#328311yescharacteristic
518C22#330311yescharacteristic
519C22#331212yes—
520C22#333011yescharacteristic
521C22#336412yes—
522C22#336712yes—
523C22#337212yes—
524C22#337412yes—
525C4—24no—
526C4—44no—
527C4—44no—
528C4—48no—
529C4—42no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—161no—
535C4#239211yescharacteristic
536C4#241912yes—
537C4#244312yes—
538C4#253611yescharacteristic
539C4#260311yescharacteristic
540C4#263311yescharacteristic
541C2#1480111yescharacteristic
542C2#2423711yescharacteristic
543C2#2425511yescharacteristic
544C2#2565011yescharacteristic
545C2#2565711yescharacteristic
546C2#2675712yes—
547C2#2677312yes—
548C2#2689512yes—
549C2#2691912yes—
550C2#3300311yescharacteristic
551C2#3301911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20432
C2×#19182
C2×#18632
#151492
#113462
#111902
#110672
#110102
#108062
#107892