Finite Group Databasebeta

← #416657

Subgroups — #416657

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11328C212yesmaximal
7#11208C212yesmaximal
8#11207C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#97C2211yescharacteristic
15#2570—24no—
16#2570—24no—
17C2×#400—21no—
18C2×#398—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#387—21no—
24C2×#387—21no—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1947—24no—
34#1836—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46C4⋊#95—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—42no—
72C2×#107—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8Q814yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×#97Q814yes—
79#400—28no—
80#398—28no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—24no—
86#389—24no—
87#389—42no—
88#389—42no—
89#387—28no—
90#387—28no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—44no—
96#382—22no—
97#382—22no—
98#382—22no—
99#382D414yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C4×C4⋊C4Q814yes—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#336—42no—
126#336—42no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
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×C8C2×Q814yes—
153C22×C8C2×Q814yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C4⋊C4#4114yes—
208C2×C4⋊C4#4114yes—
209C2×C42—24no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—82no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×Q812yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q812yes—
227C2×C42#4112yes—
228#107—48no—
229C4⋊C8—28no—
230#97—28no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—44no—
242#95—44no—
243C24C22×D411yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
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#12012yes—
303C22×C4#12012yes—
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4C4⋊Q812yes—
307C22×C4C4⋊Q812yes—
308C22×C4AGL(1,ℤ/8)12yes—
309C22×C4#13712yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q811yescharacteristic
313C22×C4C22×Q811yescharacteristic
314C22×C4C22×Q811yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4H(5,𝔽2)12yes—
319C22×C42−1+412yes—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—88no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—88no—
363C4⋊C4#12518yes—
364C42—48no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—42no—
370C42—42no—
371C42—84no—
372C42C22⋊Q814yes—
373C42C22⋊Q814yes—
374C42C4⋊Q814yes—
375C42C4⋊Q814yes—
376C23C2×#12011yescharacteristic
377C23C2×C22⋊Q811yescharacteristic
378C23C2×C4⋊Q811yescharacteristic
379C23#54612yes—
380C23#54812yes—
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×#13711yescharacteristic
383C23#57511yescharacteristic
384C23#57511yescharacteristic
385C23#57612yes—
386C23#57812yes—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
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#39514yes—
452C2×C4#39514yes—
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×#12512yes—
456C2×C4C2×C4⋊Q811yescharacteristic
457C2×C4C2×C4⋊Q811yescharacteristic
458C2×C4#53211yescharacteristic
459C2×C4#53211yescharacteristic
460C2×C4#53312yes—
461C2×C4#53511yescharacteristic
462C2×C4#53611yescharacteristic
463C2×C4#54412yes—
464C2×C4#54412yes—
465C2×C4#54712yes—
466C2×C4#55012yes—
467C2×C4#55212yes—
468C2×C4#55312yes—
469C2×C4#55412yes—
470C2×C4#55712yes—
471C8—44no—
472C8—44no—
473C22C2×#39511yescharacteristic
474C22#243111yescharacteristic
475C22#243211yescharacteristic
476C22#244212yes—
477C22#255012yes—
478C22#300611yescharacteristic
479C22#300711yescharacteristic
480C22#300811yescharacteristic
481C22#301111yescharacteristic
482C22#307711yescharacteristic
483C22#307811yescharacteristic
484C22#308312yes—
485C22#316412yes—
486C22#316511yescharacteristic
487C22#316611yescharacteristic
488C22#330912yes—
489C22#331212yes—
490C22#331512yes—
491C22#331812yes—
492C22#336712yes—
493C22#337012yes—
494C22#337512yes—
495C22#337712yes—
496C4—24no—
497C4—24no—
498C4—48no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—81no—
505C4—82no—
506C4—81no—
507C4—162no—
508C4#244112yes—
509C4#255212yes—
510C4#256511yescharacteristic
511C4#256611yescharacteristic
512C4#266011yescharacteristic
513C4#285911yescharacteristic
514C2#1522811yescharacteristic
515C2#2355411yescharacteristic
516C2#2355511yescharacteristic
517C2#2558111yescharacteristic
518C2#2558411yescharacteristic
519C2#2566011yescharacteristic
520C2#2566111yescharacteristic
521C2#2611912yes—
522C2#2612012yes—
523C2#2880112yes—
524C2#2880712yes—
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×#19472
C2×#18362
#139102
#113282
#112082
#112072
#108502
#108352
#107032