Finite Group Databasebeta

← #111876

Subgroups — #111876

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6C2×#1301C211yesmaximal, characteristic
7#7877C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#477—22no—
11C2×#477—22no—
12C2×#477—22no—
13C2×#477—22no—
14C2×#477—22no—
15C2×#477C414yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×#107—21no—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103—21no—
23C22×#101—21no—
24#1801—22no—
25#1801—22no—
26C2×#343—21no—
27#1549—22no—
28#1549—22no—
29#1548—22no—
30#1548—22no—
31C2×#333—22no—
32C2×#332—22no—
33#1479—22no—
34#1479—22no—
35C2×Q8⋊C8—22no—
36#1301—22no—
37#1277—22no—
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C23×C8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8D412yes—
54C2×C4⋊Q8D412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8D412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#477—28no—
74#477—28no—
75#477—28no—
76#477—28no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431D412yes—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88#423—21no—
89#423—21no—
90#423—21no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423D412yes—
95#423D412yes—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C412yes—
105C2×C4⋊C8D412yes—
106C2×C4⋊C8D412yes—
107C2×C4⋊C8D412yes—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#101—24no—
112C2×#101—42no—
113C2×#101—42no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118#343—24no—
119#333—28no—
120#332—28no—
121Q8⋊C8—28no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—84no—
140C22×Q8C22⋊C412yes—
141C22×Q8SD1612yes—
142C22×Q8Q1612yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—41no—
152C2×M16—41no—
153C2×M16—41no—
154C2×M16—41no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16—41no—
158C2×M16—41no—
159C2×M16—41no—
160C2×M16—41no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8C22⋊C414yes—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C412yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246#107—216no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8C22⋊C414yes—
263#103—216no—
264#101—48no—
265C22⋊C8—22no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269C4×C8—22no—
270C4×C8—41no—
271C4×C8—41no—
272C4×C8—41no—
273C24#12311yescharacteristic
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8#10314yes—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—84no—
328C22×C4—82no—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12311yescharacteristic
344C22×C4#12311yescharacteristic
345C22×C4#12411yescharacteristic
346C22×C4#12411yescharacteristic
347C22×C4C2×SD1611yescharacteristic
348C22×C4C2×Q1611yescharacteristic
349C22×C4#13511yescharacteristic
350C22×C4#13511yescharacteristic
351C22×C4AGL(1,ℤ/8)11yescharacteristic
352C22×C4#13711yescharacteristic
353C22×C4#13711yescharacteristic
354C22×C4#13711yescharacteristic
355M16—42no—
356M16—42no—
357M16—42no—
358M16—42no—
359M16—81no—
360M16—81no—
361M16—81no—
362M16—81no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—44no—
375C2×C8—82no—
376C2×C8—81no—
377C2×C8—81no—
378C2×C8—81no—
379C2×C8—81no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42C2×C22⋊C411yescharacteristic
401C42C2×C22⋊C411yescharacteristic
402C42C4×D411yescharacteristic
403C42C4×D411yescharacteristic
404C42#12011yescharacteristic
405C42#12011yescharacteristic
406C42#12011yescharacteristic
407C42#12011yescharacteristic
408C42#12011yescharacteristic
409C42#12111yescharacteristic
410C42#12111yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42#12311yescharacteristic
414C42#12711yescharacteristic
415C42#12711yescharacteristic
416C23—22no—
417C23—22no—
418C23#38611yescharacteristic
419C23#39411yescharacteristic
420C23#39411yescharacteristic
421C23#48311yescharacteristic
422C23#48411yescharacteristic
423C23#48511yescharacteristic
424C23#48511yescharacteristic
425Q8—24no—
426Q8—24no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433Q8—84no—
434Q8—84no—
435Q8—84no—
436Q8—84no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4—84no—
462C2×C4—82no—
463C2×C4#37911yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4#39011yescharacteristic
468C2×C4#39011yescharacteristic
469C2×C4#39211yescharacteristic
470C2×C4#39211yescharacteristic
471C2×C4#39211yescharacteristic
472C2×C4#39211yescharacteristic
473C2×C4#39411yescharacteristic
474C2×C4#39411yescharacteristic
475C2×C4C2×#10311yescharacteristic
476C2×C4#41611yescharacteristic
477C2×C4#41711yescharacteristic
478C2×C4#41911yescharacteristic
479C2×C4#44811yescharacteristic
480C2×C4#44811yescharacteristic
481C2×C4#44911yescharacteristic
482C2×C4#44911yescharacteristic
483C2×C4#45011yescharacteristic
484C2×C4#45111yescharacteristic
485C2×C4#45111yescharacteristic
486C2×C4#45111yescharacteristic
487C2×C4#45211yescharacteristic
488C2×C4#45211yescharacteristic
489C2×C4#45211yescharacteristic
490C2×C4#45211yescharacteristic
491C2×C4#46511yescharacteristic
492C2×C4#46511yescharacteristic
493C2×C4#46711yescharacteristic
494C2×C4#46711yescharacteristic
495C2×C4#46811yescharacteristic
496C2×C4#46811yescharacteristic
497C2×C4#47011yescharacteristic
498C2×C4#47011yescharacteristic
499C2×C4#47314yes—
500C8—44no—
501C8—82no—
502C8—82no—
503C8—82no—
504C8—82no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22#143811yescharacteristic
514C22#187211yescharacteristic
515C22#187311yescharacteristic
516C22#203511yescharacteristic
517C22#203511yescharacteristic
518C22#203611yescharacteristic
519C22#203611yescharacteristic
520C22#204712yes—
521C22#204812yes—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—44no—
529C4—82no—
530C4—84no—
531C4#178711yescharacteristic
532C4#178811yescharacteristic
533C4#189211yescharacteristic
534C4#189411yescharacteristic
535C4#197912yes—
536C4#200011yescharacteristic
537C4#200011yescharacteristic
538C4#200111yescharacteristic
539C4#200111yescharacteristic
540C4#202012yes—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2#1045412yes—
546C2#1045512yes—
547C2#1097011yescharacteristic
548C2#1102911yescharacteristic
549C2#1103411yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#18012
C2×#15492
C2×#15482
C2×#14792
C2×#13012
#78772