Finite Group Databasebeta

← #416716

Subgroups — #416716

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1960C211yesmaximal, characteristic
4#14155C211yesmaximal, characteristic
5#11067C212yesmaximal
6#11067C212yesmaximal
7#11052C212yesmaximal
8#11050C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#97C2211yescharacteristic
19#2570—24no—
20#2570—24no—
21C2×#400—21no—
22C2×#398—21no—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1981—24no—
36#1960—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C23×Q8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92#400—28no—
93#398—28no—
94#391—24no—
95#391—24no—
96#391—42no—
97#391—42no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#387—28no—
103#387—28no—
104#384—22no—
105#384—22no—
106#382—22no—
107#382—22no—
108#382—22no—
109#382D414yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4D414yes—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134#336—42no—
135#336—42no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C2×Q814yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8#4114yes—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179C4⋊Q8—44no—
180C4⋊Q8—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
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—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4#4114yes—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—82no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C4⋊C8—44no—
245#103—48no—
246#103—48no—
247#103—48no—
248#103—48no—
249#97—44no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—42no—
264C24C22×D411yescharacteristic
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4C22⋊Q812yes—
354C22×C4C22⋊Q812yes—
355C22×C4#12412yes—
356C22×C4#12412yes—
357C22×C4#13712yes—
358C22×C4#13712yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)12yes—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—42no—
412C42—42no—
413C42—84no—
414C42—84no—
415C42C22⋊Q814yes—
416C42C22⋊Q814yes—
417C42#12414yes—
418C42#12414yes—
419C23C2×#12011yescharacteristic
420C23C2×C22⋊Q811yescharacteristic
421C23C2×#12411yescharacteristic
422C23#54612yes—
423C23#55012yes—
424C23C2×#13711yescharacteristic
425C23C2×#13711yescharacteristic
426C23#57511yescharacteristic
427C23#57511yescharacteristic
428C23#57712yes—
429C23#57812yes—
430Q8—44no—
431Q8—44no—
432Q8—44no—
433Q8—44no—
434Q8—82no—
435Q8—82no—
436Q8—82no—
437Q8—82no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4#39314yes—
503C2×C4#39314yes—
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4C2×#12411yescharacteristic
507C2×C4C2×#12411yescharacteristic
508C2×C4#52912yes—
509C2×C4#52912yes—
510C2×C4#53211yescharacteristic
511C2×C4#53211yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53511yescharacteristic
514C2×C4#54012yes—
515C2×C4#54012yes—
516C2×C4#54612yes—
517C2×C4#55012yes—
518C2×C4#55312yes—
519C2×C4#55312yes—
520C2×C4#55412yes—
521C2×C4#55412yes—
522C8—44no—
523C8—82no—
524C22C2×#39311yescharacteristic
525C22#243111yescharacteristic
526C22#243111yescharacteristic
527C22#245412yes—
528C22#249012yes—
529C22#300711yescharacteristic
530C22#300711yescharacteristic
531C22#300911yescharacteristic
532C22#301111yescharacteristic
533C22#307811yescharacteristic
534C22#307811yescharacteristic
535C22#308212yes—
536C22#313712yes—
537C22#313911yescharacteristic
538C22#314011yescharacteristic
539C22#330712yes—
540C22#331212yes—
541C22#331512yes—
542C22#332012yes—
543C22#339812yes—
544C22#340012yes—
545C22#340112yes—
546C22#340212yes—
547C4—24no—
548C4—24no—
549C4—42no—
550C4—42no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—84no—
556C4—81no—
557C4—82no—
558C4—81no—
559C4#245412yes—
560C4#249012yes—
561C4#256611yescharacteristic
562C4#256611yescharacteristic
563C4#264111yescharacteristic
564C4#286111yescharacteristic
565C2#1522911yescharacteristic
566C2#2345511yescharacteristic
567C2#2345711yescharacteristic
568C2#2558211yescharacteristic
569C2#2558411yescharacteristic
570C2#2564111yescharacteristic
571C2#2564311yescharacteristic
572C2#2653912yes—
573C2#2654012yes—
574C2#2776212yes—
575C2#2776412yes—
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×#25702
C2×#19812
C2×#19602
#141552
#110672
#110522
#110502
#108502
#108352
#107032