Finite Group Databasebeta

← #412523

Subgroups — #412523

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2068C212yesmaximal
2C2×#1947C211yesmaximal, characteristic
3C2×#1843C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6#14276C211yesmaximal, characteristic
7#13786C211yesmaximal, characteristic
8#11232C212yesmaximal
9#11227C212yesmaximal
10#11077C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#2068—28no—
28#1947—24no—
29#1843—24no—
30#1805—24no—
31#1803—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1441—22no—
38#1441—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—22no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8Q814yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8D414yes—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88#391—22no—
89#391—22no—
90#391D414yes—
91#391Q814yes—
92#389—24no—
93#389—24no—
94#384—22no—
95#384D414yes—
96#382—22no—
97#382—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100#376—22no—
101#376—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—42no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8C2×D412yes—
131C22×Q8C2×Q812yes—
132C22×Q8C2×Q812yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
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×C8C2×D414yes—
158C4×Q8—28no—
159C4×Q8—28no—
160C4×Q8—44no—
161C4×Q8—44no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4C2×Q812yes—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—24no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—82no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×Q812yes—
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225#107—48no—
226C4⋊C8—28no—
227C4⋊C8—48no—
228C4⋊C8—48no—
229C4⋊C8—44no—
230#103—48no—
231#103—48no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—28no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—82no—
241#95—82no—
242#95—82no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8C4⋊Q814yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
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—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
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#12114yes—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4C22⋊Q814yes—
331C22×C4C22⋊Q812yes—
332C22×C4C22⋊Q812yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
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—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—48no—
394C42—84no—
395C42#12114yes—
396C42C22⋊Q814yes—
397C42C22⋊Q814yes—
398C42#12614yes—
399C23C2×#12111yescharacteristic
400C23C2×#12111yescharacteristic
401C23C2×#12111yescharacteristic
402C23C2×C22⋊Q811yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23#53711yescharacteristic
405C23#53711yescharacteristic
406C23#57612yes—
407C23#57712yes—
408C23#57812yes—
409C23#57812yes—
410Q8—24no—
411Q8—24no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
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#39614yes—
477C2×C4#39614yes—
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×C22⋊Q811yescharacteristic
481C2×C4C2×#12611yescharacteristic
482C2×C4C2×C4⋊Q811yescharacteristic
483C2×C4#53211yescharacteristic
484C2×C4#53311yescharacteristic
485C2×C4#53311yescharacteristic
486C2×C4#54011yescharacteristic
487C2×C4#54111yescharacteristic
488C2×C4#54411yescharacteristic
489C2×C4#54411yescharacteristic
490C2×C4D4211yescharacteristic
491C2×C4D4211yescharacteristic
492C2×C4#54611yescharacteristic
493C2×C4#54711yescharacteristic
494C2×C4#54711yescharacteristic
495C2×C4#54711yescharacteristic
496C2×C4#54811yescharacteristic
497C2×C4D4×Q811yescharacteristic
498C2×C4D4×Q811yescharacteristic
499C2×C4#55011yescharacteristic
500C2×C4#55211yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55411yescharacteristic
503C2×C4#55411yescharacteristic
504C2×C4#55511yescharacteristic
505C2×C4#55611yescharacteristic
506C8—44no—
507C8—82no—
508C22C2×#39611yescharacteristic
509C22#241611yescharacteristic
510C22#241711yescharacteristic
511C22#253711yescharacteristic
512C22#253811yescharacteristic
513C22#262911yescharacteristic
514C22#285711yescharacteristic
515C22#304111yescharacteristic
516C22#304211yescharacteristic
517C22#304411yescharacteristic
518C22#304411yescharacteristic
519C22#304412yes—
520C22#304512yes—
521C22#306112yes—
522C22#306312yes—
523C22#308212yes—
524C22#308212yes—
525C22#308312yes—
526C22#308312yes—
527C22#325712yes—
528C22#325812yes—
529C22#326112yes—
530C22#326212yes—
531C4—24no—
532C4—24no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—82no—
537C4—84no—
538C4—82no—
539C4—82no—
540C4—162no—
541C4#242811yescharacteristic
542C4#242911yescharacteristic
543C4#243911yescharacteristic
544C4#244011yescharacteristic
545C4#244011yescharacteristic
546C4#244111yescharacteristic
547C4#249611yescharacteristic
548C4#249711yescharacteristic
549C2#1513111yescharacteristic
550C2#2361111yescharacteristic
551C2#2361511yescharacteristic
552C2#2476512yes—
553C2#2485012yes—
554C2#2825711yescharacteristic
555C2#2826111yescharacteristic
556C2#2828811yescharacteristic
557C2#2829111yescharacteristic
558C2#3000512yes—
559C2#3320612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#20682
C2×#19472
C2×#18432
C2×#18052
C2×#18032
#142762
#137862
#112322
#112272
#110772
#108502