Finite Group Databasebeta

← #406809

Subgroups — #406809

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1852C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3C2×#1807C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#13881C211yesmaximal, characteristic
7#11378C212yesmaximal, complemented
8#11377C212yesmaximal, complemented
9#11372C212yesmaximal, complemented
10#11325C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#382C2212yes—
20C2×#381—22no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1852—24no—
25#1823—24no—
26#1807—24no—
27C2×#340—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1445—22no—
38#1442—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—42no—
50C23×C8D414yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4Q814yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42C2312yes—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—24no—
82C2×#106—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8D414yes—
86#382—24no—
87#382—42no—
88#381—48no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—24no—
101C2×#95D412yes—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—24no—
110#336—42no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—44no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—82no—
131C23×C4C2×D414yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×Q812yes—
136C23×C4#4112yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—28no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—28no—
155C22×C8—44no—
156C22×C8—48no—
157C22×C8—48no—
158C22×C8C2×D412yes—
159C22×C8C2×D414yes—
160C22×C8C2×D412yes—
161C22×C8#4114yes—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
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—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×Q814yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215#107—48no—
216#106—44no—
217#106—44no—
218C22⋊C8—28no—
219#95—24no—
220#95—48no—
221#95—48no—
222#95—48no—
223#95—42no—
224#95—42no—
225#95—84no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24C22⋊Q812yes—
231C24#12312yes—
232C24#12712yes—
233C24#13514yes—
234C24AGL(1,ℤ/8)14yes—
235C24C22×D411yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
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—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—48no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—416no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
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—816no—
289C22×C4—84no—
290C22×C4—88no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C22⋊Q814yes—
295C22×C4C22⋊Q814yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12314yes—
298C22×C4#12314yes—
299C22×C4#12312yes—
300C22×C4#12712yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4112yes—
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+412yes—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—416no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8#12314yes—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—48no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C22⋊C4—48no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C23—28no—
359C23—24no—
360C23—28no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—28no—
368C23—28no—
369C23—22no—
370C23#39914yes—
371C23#47818yes—
372C23#48218yes—
373C23#49514yes—
374C23#49614yes—
375C23C2×C22⋊Q811yescharacteristic
376C23C2×#12311yescharacteristic
377C23C2×#12711yescharacteristic
378C23#54612yes—
379C23#54712yes—
380C23C2×#13512yes—
381C23C2×AGL(1,ℤ/8)12yes—
382C23#57712yes—
383C23#57812yes—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—816no—
406C2×C4—84no—
407C2×C4—816no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#39914yes—
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4#52511yescharacteristic
436C2×C4#52912yes—
437C2×C4#53312yes—
438C2×C4#53411yescharacteristic
439C2×C4#53611yescharacteristic
440C2×C4#53912yes—
441C2×C4#54412yes—
442C2×C4#54612yes—
443C2×C4#54712yes—
444C2×C4#55112yes—
445C2×C4#55112yes—
446C2×C4#55212yes—
447C2×C4#55212yes—
448C8—24no—
449C8—44no—
450C8—42no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22—216no—
455C22—216no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22#209218yes—
460C22C2×#39911yescharacteristic
461C22#241912yes—
462C22#243812yes—
463C22#250111yescharacteristic
464C22#253911yescharacteristic
465C22C2×#47812yes—
466C22#308212yes—
467C22C2×#48212yes—
468C22#309712yes—
469C22C2×#49511yescharacteristic
470C22C2×#49611yescharacteristic
471C22#315411yescharacteristic
472C22#315611yescharacteristic
473C22#330014yes—
474C22#330614yes—
475C22#333714yes—
476C22#334314yes—
477C4—24no—
478C4—84no—
479C4—84no—
480C4—84no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4#241912yes—
486C4#243812yes—
487C4#248311yescharacteristic
488C4#248411yescharacteristic
489C4#262711yescharacteristic
490C4#285511yescharacteristic
491C2—28no—
492C2#1493611yescharacteristic
493C2C2×#209211yescharacteristic
494C2#2373611yescharacteristic
495C2#2494514yes—
496C2#2599814yes—
497C2#2792611yescharacteristic
498C2#2793911yescharacteristic
499C2#2833511yescharacteristic
500C2#2834311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11378C2semidirect
8. #11377C2semidirect
9. #11372C2semidirect
10. #11325C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18522
C2×#18232
C2×#18072
C22×#3402
#139042
#138812
#113782
#113772
#113722
#113252
#107142