Finite Group Databasebeta

← #464834

Subgroups — #464834

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#2093C211yesmaximal, characteristic
4C2×#2059C212yesmaximal, complemented
5C2×#2025C212yesmaximal
6#14700C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic, complemented
8#11371C211yesmaximal, characteristic, complemented
9#11325C212yesmaximal, complemented
10#11051C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#125C2212yes—
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2859—24no—
18#2859—24no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389—22no—
23C2×#389—22no—
24C2×#389C2211yescharacteristic
25C2×#388—22no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2212yes—
29C2×C43C2211yescharacteristic
30#2094—24no—
31#2093—24no—
32#2059—28no—
33#2025—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1437—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—24no—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—42no—
53C2×#125—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85#390—22no—
86#390—22no—
87#390—24no—
88#390—22no—
89#390Q814yes—
90#389—216no—
91#389—216no—
92#389—24no—
93#388—48no—
94#384—22no—
95#384—22no—
96#384D414yes—
97#384D414yes—
98#382—24no—
99#382—24no—
100C2×#95—24no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C43—22no—
105C43—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—22no—
111#336—22no—
112#336—24no—
113#336D418yes—
114C25—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C2×Q812yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2411yescharacteristic
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×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8C2×D414yes—
148C22×C8C2×D414yes—
149#125—48no—
150#125—48no—
151#125—48no—
152#125—48no—
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—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×Q812yes—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D414yes—
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221#107—48no—
222#106—48no—
223#102—48no—
224#102—48no—
225#95—216no—
226#95—44no—
227#95—42no—
228#95—44no—
229C24—22no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C22×D411yescharacteristic
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—42no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4#12014yes—
301C22×C4C22⋊Q814yes—
302C22×C4#12712yes—
303C22×C4#12712yes—
304C22×C4#13514yes—
305C22×C4AGL(1,ℤ/8)14yes—
306C22×C4C22×D412yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×Q811yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C42−1+412yes—
313C22×C42−1+412yes—
314C22×C42−1+411yescharacteristic
315C22×C42−1+411yescharacteristic
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—416no—
336C4⋊C4—416no—
337C4⋊C4—416no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—416no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—84no—
349C22⋊C4—416no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C22⋊C4—88no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—48no—
362C42—48no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23C2×#12011yescharacteristic
387C23C2×C22⋊Q811yescharacteristic
388C23C2×#12711yescharacteristic
389C23#53511yescharacteristic
390C23#53611yescharacteristic
391C23#53611yescharacteristic
392C23#53611yescharacteristic
393C23C2×#13512yes—
394C23C2×AGL(1,ℤ/8)12yes—
395C23#57512yes—
396C23#57512yes—
397D4—44no—
398D4—44no—
399D4—82no—
400D4—82no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4#44918yes—
450C2×C4#53612yes—
451C2×C4#53612yes—
452C2×C4#53612yes—
453C2×C4#53612yes—
454C2×C4#54212yes—
455C2×C4#54412yes—
456C2×C4D4212yes—
457C2×C4D4212yes—
458C2×C4#54712yes—
459C2×C4#54811yescharacteristic
460C2×C4#54811yescharacteristic
461C2×C4D4×Q811yescharacteristic
462C2×C4D4×Q811yescharacteristic
463C2×C4D4×Q812yes—
464C2×C4#56312yes—
465C2×C4#56312yes—
466C8—44no—
467C8—44no—
468C22—84no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—84no—
473C22—162no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22—162no—
479C22#240911yescharacteristic
480C22#243311yescharacteristic
481C22#254211yescharacteristic
482C22#261411yescharacteristic
483C22#262811yescharacteristic
484C22#265811yescharacteristic
485C22#282111yescharacteristic
486C22C2×#44912yes—
487C22#300512yes—
488C22#307612yes—
489C22#307712yes—
490C22#314711yescharacteristic
491C22#315011yescharacteristic
492C22#315111yescharacteristic
493C22#315311yescharacteristic
494C22#320712yes—
495C22#321912yes—
496C22#323212yes—
497C22#323212yes—
498C22#323312yes—
499C22#323412yes—
500C22#323412yes—
501C22#323512yes—
502C4—28no—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—82no—
507C4—84no—
508C4—84no—
509C4—161no—
510C4—161no—
511C4#267412yes—
512C4#267412yes—
513C4#267712yes—
514C4#267712yes—
515C2—162no—
516C2#1651111yescharacteristic
517C2#2430012yes—
518C2#2572011yescharacteristic
519C2#2572711yescharacteristic
520C2#2851311yescharacteristic
521C2#2851611yescharacteristic
522C2#2964012yes—
523C2#2998712yes—
524C2#3028211yescharacteristic
525C2#3028311yescharacteristic
526C2#3284112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2859C2semidirect
4. C2×#2059C2semidirect
7. #11373C2semidirect
8. #11371C2semidirect
9. #11325C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#20942
C2×#20932
C2×#20592
C2×#20252
#147002
#113732
#113712
#113252
#110512
#107952