Finite Group Databasebeta

← #406800

Subgroups — #406800

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C212yesmaximal
2C2×#1825C212yesmaximal
3#14686C211yesmaximal, characteristic
4#13756C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11377C212yesmaximal, complemented
7#11373C211yesmaximal, characteristic, complemented
8#11373C211yesmaximal, characteristic, complemented
9#11324C211yesmaximal, characteristic, complemented
10#11323C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#380—22no—
21C2×#380C2212yes—
22#1854—28no—
23#1825—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C4⋊#95—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C22⋊#95—22no—
40C4×#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—24no—
49C2×#117—24no—
50C2×#117—24no—
51C2×#117—42no—
52C2×#117Q814yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42—24no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—42no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8D418yes—
82#384—22no—
83#384—22no—
84#384—42no—
85#380—24no—
86#380—24no—
87#380—48no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#336—22no—
101#336—22no—
102#336—42no—
103C25C2×D411yescharacteristic
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D414yes—
126C23×C4C2×D412yes—
127C23×C4C2×Q812yes—
128C23×C4#4114yes—
129C23×C4#4112yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C2×D414yes—
144C22×C8C2×D414yes—
145#117—216no—
146#117—48no—
147#117—48no—
148#117—48no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C2×Q812yes—
180C2×C4⋊C4C2×Q812yes—
181C2×C4⋊C4#4112yes—
182C2×C4⋊C4#4112yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—44no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—48no—
202C2×C42—48no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42#4112yes—
209C2×C42#4112yes—
210#107—48no—
211#106—48no—
212C22⋊C8—216no—
213#95—44no—
214#95—48no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24C22⋊Q812yes—
228C24#12312yes—
229C24#12712yes—
230C24C22×D411yescharacteristic
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—216no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4C22⋊Q818yes—
297C22×C4C22⋊Q812yes—
298C22×C4#12318yes—
299C22×C4#12312yes—
300C22×C4#12314yes—
301C22×C4#12712yes—
302C22×C4#13514yes—
303C22×C4AGL(1,ℤ/8)14yes—
304C22×C4#13714yes—
305C22×C4C22×D412yes—
306C22×C4C22×Q811yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)12yes—
314C22×C42−1+412yes—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—88no—
346C4⋊C4—88no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—88no—
350C42—24no—
351C42—24no—
352C42—48no—
353C42—48no—
354C42—42no—
355C42—42no—
356C42—48no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23#39914yes—
380C23C2×C22⋊Q811yescharacteristic
381C23C2×#12311yescharacteristic
382C23C2×#12311yescharacteristic
383C23#52511yescharacteristic
384C23C2×#12711yescharacteristic
385C23#53411yescharacteristic
386C23#53611yescharacteristic
387C23C2×#13512yes—
388C23C2×AGL(1,ℤ/8)12yes—
389C23C2×#13712yes—
390C23#57512yes—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—432no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#39914yes—
446C2×C4#48218yes—
447C2×C4C2×C22⋊Q812yes—
448C2×C4C2×#12312yes—
449C2×C4#52911yescharacteristic
450C2×C4#52911yescharacteristic
451C2×C4#53311yescharacteristic
452C2×C4#53311yescharacteristic
453C2×C4#53912yes—
454C2×C4#54412yes—
455C2×C4#54612yes—
456C2×C4#54612yes—
457C2×C4#54712yes—
458C2×C4#54712yes—
459C2×C4#55112yes—
460C2×C4#55112yes—
461C2×C4#55212yes—
462C2×C4#55212yes—
463C8—44no—
464C8—44no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—42no—
469C22—42no—
470C22—44no—
471C22—44no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22C2×#39911yescharacteristic
477C22#248311yescharacteristic
478C22#248411yescharacteristic
479C22#250111yescharacteristic
480C22#253911yescharacteristic
481C22#262711yescharacteristic
482C22#285511yescharacteristic
483C22#307612yes—
484C22#307912yes—
485C22C2×#48212yes—
486C22#309112yes—
487C22#309212yes—
488C22#309512yes—
489C22#310712yes—
490C22#311512yes—
491C22#314711yescharacteristic
492C22#315011yescharacteristic
493C22#315111yescharacteristic
494C22#315211yescharacteristic
495C22#319012yes—
496C22#319412yes—
497C22#322812yes—
498C22#323712yes—
499C4—24no—
500C4—28no—
501C4—44no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—162no—
507C4—162no—
508C4#241912yes—
509C4#241912yes—
510C4#243812yes—
511C4#243812yes—
512C2—44no—
513C2#1493611yescharacteristic
514C2#2373011yescharacteristic
515C2#2373411yescharacteristic
516C2#2760212yes—
517C2#2761512yes—
518C2#2790711yescharacteristic
519C2#2791611yescharacteristic
520C2#2830811yescharacteristic
521C2#2831611yescharacteristic
522C2#2997112yes—
523C2#3315412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #11378C2semidirect
6. #11377C2semidirect
7. #11373C2semidirect
8. #11373C2semidirect
9. #11324C2semidirect
10. #11323C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18252
#146862
#137562
#113782
#113772
#113732
#113242
#113232
#107122