Finite Group Databasebeta

← #381358

Subgroups — #381358

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1817C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#11325C212yesmaximal, complemented
5#11268C212yesmaximal, complemented
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10C23×C42C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#382C2212yes—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C22⋊C4—21no—
20#1817C418yescomplemented
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C4⋊#95C2211yescharacteristic
30C4⋊#95C2211yescharacteristic
31C4⋊#95C2211yescharacteristic
32C4⋊#95C2211yescharacteristic
33#1434—21no—
34#1434—21no—
35#1434—21no—
36C22⋊#95—21no—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C4×#95—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C42—22no—
52C22×C42—42no—
53C22×C42—42no—
54C22×C42C2×C414yes—
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42Q814yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—24no—
61C2×#107—42no—
62C2×#107—42no—
63C2×#106—24no—
64C2×#106—42no—
65C2×#106—42no—
66C2×C22⋊C8—42no—
67C2×C22⋊C8—42no—
68C2×C22⋊C8C2×C414yes—
69C2×C22⋊C8C2×C414yes—
70#382—42no—
71#382—42no—
72#380—24no—
73#380—24no—
74#380—42no—
75#380—42no—
76C4×C4⋊C4—42no—
77C4×C4⋊C4—42no—
78C4×C22⋊C4—44no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336C2×C418yes—
99#336C2×C418yes—
100C25C2×D411yescharacteristic
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—42no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4C4⋊C414yes—
120C23×C4C4⋊C414yes—
121C23×C4C22×C412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×Q812yes—
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
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×C8C22×C414yes—
142C22×C8C22×C414yes—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
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—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#4114yes—
166C2×C4⋊C4#4114yes—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42—48no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42C4⋊C414yes—
186C2×C42C4⋊C414yes—
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C2×Q812yes—
190C2×C42C2×Q812yes—
191#107—48no—
192#106—48no—
193C22⋊C8—44no—
194C22⋊C8—44no—
195#95—28no—
196#95—42no—
197#95—42no—
198#95—42no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—42no—
207#95—42no—
208C24—22no—
209C24—22no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—21no—
216C24—21no—
217C24C2×C4⋊C412yes—
218C24#13514yes—
219C24C22×D411yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
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—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×C4C2×C4⋊C412yes—
293C22×C4C2×C4⋊C412yes—
294C22×C4C2×C4⋊C412yes—
295C22×C4C2×C4⋊C412yes—
296C22×C4C2×C4⋊C412yes—
297C22×C4AGL(1,ℤ/8)12yes—
298C22×C4#13712yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)12yes—
304C22×C4H(5,𝔽2)12yes—
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+412yes—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C22⋊C4—416no—
338C22⋊C4—88no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—24no—
344C42—24no—
345C42—44no—
346C42—44no—
347C42—88no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—22no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#42714yes—
371C23C22×C4⋊C411yescharacteristic
372C23#52512yes—
373C23#53412yes—
374C23#53511yescharacteristic
375C23#53611yescharacteristic
376C23C2×#13512yes—
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×#13711yescharacteristic
379C23#57511yescharacteristic
380C23#57511yescharacteristic
381C23#57512yes—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—416no—
399C2×C4—416no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4#42814yes—
441C2×C4C4×#4112yes—
442C2×C4#51811yescharacteristic
443C2×C4#51811yescharacteristic
444C2×C4#52012yes—
445C2×C4#53311yescharacteristic
446C2×C4#53311yescharacteristic
447C2×C4#53812yes—
448C2×C4#53812yes—
449C2×C4#53912yes—
450C2×C4#53912yes—
451C2×C4#54012yes—
452C2×C4#54012yes—
453C2×C4#54212yes—
454C2×C4#54212yes—
455C2×C4#54312yes—
456C2×C4#54412yes—
457C8—44no—
458C8—44no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—44no—
468C22—44no—
469C22—42no—
470C22#229611yescharacteristic
471C22#231511yescharacteristic
472C22#231711yescharacteristic
473C22#263512yes—
474C22#263812yes—
475C22C2×#42711yescharacteristic
476C22C2×#42811yescharacteristic
477C22#291111yescharacteristic
478C22#291111yescharacteristic
479C22#310512yes—
480C22#310812yes—
481C22#311112yes—
482C22#311112yes—
483C22#319112yes—
484C22#319112yes—
485C22#319412yes—
486C22#319612yes—
487C22#321112yes—
488C22#321411yescharacteristic
489C22#321611yescharacteristic
490C22#323212yes—
491C22#323511yescharacteristic
492C22#323711yescharacteristic
493C4—24no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—84no—
500C4—162no—
501C4—162no—
502C4#233512yes—
503C4#233512yes—
504C4#266512yes—
505C4#266712yes—
506C2—44no—
507C2#1429511yescharacteristic
508C2#2136111yescharacteristic
509C2#2136611yescharacteristic
510C2#2193911yescharacteristic
511C2#2194411yescharacteristic
512C2#2198311yescharacteristic
513C2#2198811yescharacteristic
514C2#3010712yes—
515C2#3011012yes—
516C2#3016712yes—
517C2#3017012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
4. #11325C2semidirect
5. #11268C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
20. #1817C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18172
#146822
C4×#3802
#113252
#112682
#110962
#110952
#109082
#109072