Finite Group Databasebeta

← #403891

Subgroups — #403891

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11293C212yesmaximal
7#11228C211yesmaximal, characteristic
8#11202C211yesmaximal, characteristic
9#10907C212yesmaximal
10#10907C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C2×#400—21no—
16#2369—24no—
17#2369—24no—
18C2×#391—21no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100—24no—
30#1947—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1445—21no—
42#1442—21no—
43#1441—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
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⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80#400—44no—
81#391—28no—
82#391—28no—
83#389—28no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389D414yes—
88#384—22no—
89#384—22no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#336—24no—
120#336—22no—
121#336—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—82no—
126C22×Q8—82no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
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#4114yes—
154C4⋊Q8—28no—
155C4⋊Q8—28no—
156C4⋊Q8—28no—
157C4⋊Q8—28no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
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—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4Q1614yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C4⋊C4#4112yes—
206C2×C4⋊C4#4112yes—
207C2×C4⋊C4#4114yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—82no—
224C2×C42—82no—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229#107—48no—
230C4⋊C8—28no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—42no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—82no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—216no—
246C2×Q8—88no—
247C2×Q8—88no—
248C2×Q8—84no—
249C2×Q8—84no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4#12012yes—
308C22×C4#12012yes—
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4C2×Q1612yes—
312C22×C4C2×Q1612yes—
313C22×C4C2×Q1612yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C42−1+412yes—
325C22×C42−1+412yes—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—216no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—44no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4C22⋊Q814yes—
367C4⋊C4C22⋊Q814yes—
368C4⋊C4#12414yes—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—24no—
376C42—48no—
377C42—41no—
378C42—41no—
379C42—42no—
380C42—84no—
381C42—84no—
382C23C2×#12011yescharacteristic
383C23C2×C22⋊Q811yescharacteristic
384C23#52511yescharacteristic
385C23#54712yes—
386C23#54812yes—
387C23C22×Q1611yescharacteristic
388C23C2×AGL(1,ℤ/8)11yescharacteristic
389C23C2×#13711yescharacteristic
390C23#57511yescharacteristic
391C23#57612yes—
392C23#57712yes—
393Q8—816no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#45114yes—
459C2×C4#47714yes—
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4C2×C22⋊Q811yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4#52511yescharacteristic
464C2×C4#52511yescharacteristic
465C2×C4C2×#12411yescharacteristic
466C2×C4#52912yes—
467C2×C4#52912yes—
468C2×C4#53611yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#54112yes—
471C2×C4#54112yes—
472C2×C4#54712yes—
473C2×C4#54812yes—
474C2×C4#55612yes—
475C2×C4#55612yes—
476C2×C4#55712yes—
477C2×C4#55712yes—
478C8—44no—
479C8—82no—
480C22#240611yescharacteristic
481C22#243611yescharacteristic
482C22#245312yes—
483C22#246912yes—
484C22#284811yescharacteristic
485C22C2×#45111yescharacteristic
486C22#300611yescharacteristic
487C22#300811yescharacteristic
488C22#300911yescharacteristic
489C22C2×#47711yescharacteristic
490C22#307811yescharacteristic
491C22#308112yes—
492C22#310411yescharacteristic
493C22#311211yescharacteristic
494C22#311812yes—
495C22#333812yes—
496C22#334612yes—
497C22#334812yes—
498C22#335412yes—
499C22#336312yes—
500C22#336612yes—
501C22#336912yes—
502C22#337012yes—
503C4—24no—
504C4—24no—
505C4—42no—
506C4—42no—
507C4—44no—
508C4—42no—
509C4—44no—
510C4—44no—
511C4—81no—
512C4—82no—
513C4—162no—
514C4—162no—
515C4#239811yescharacteristic
516C4#245312yes—
517C4#246912yes—
518C4#256811yescharacteristic
519C4#258111yescharacteristic
520C4#263411yescharacteristic
521C2#1488411yescharacteristic
522C2#2409811yescharacteristic
523C2#2410411yescharacteristic
524C2#2593611yescharacteristic
525C2#2594611yescharacteristic
526C2#2651212yes—
527C2#2651912yes—
528C2#2713212yes—
529C2#2714712yes—
530C2#3295711yescharacteristic
531C2#3296211yescharacteristic
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×#23692
C2×#21002
C2×#19472
#151522
#113842
#112932
#112282
#112022
#109072
#108072