Finite Group Databasebeta

← #418102

Subgroups — #418102

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3C2×#2043C212yesmaximal, complemented
4C2×#1872C212yesmaximal, complemented
5#10970C212yesmaximal, complemented
6#10970C214yesmaximal, complemented
7#10964C211yesmaximal, characteristic
8#10884C212yesmaximal, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C22×#123—21no—
11C22×C22⋊Q8C2212yes—
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#103C2214yes—
16C22×#103C2214yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19#2542—28no—
20#2486—28no—
21C2×#399—21no—
22C2×#399—21no—
23C2×#398—21no—
24C2×#397—21no—
25C2×#390—21no—
26C2×#389C2212yes—
27C2×#384—21no—
28C2×#381—21no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2043—28no—
33#1872—28no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C23×Q8C2312yes—
38C23×D4D411yescharacteristic
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×#123—28no—
47C2×#123—44no—
48C2×#123—44no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—28no—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8—44no—
55C2×C22⋊Q8—44no—
56C2×C22⋊Q8D418yes—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#103—28no—
82C2×#103—28no—
83C2×#103—28no—
84C2×#103—44no—
85C2×#103—44no—
86C2×#103—44no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—44no—
89C2×C22⋊C8—44no—
90C2×C22⋊C8—42no—
91#399—28no—
92#399—28no—
93#398—28no—
94#397—28no—
95#390—28no—
96#389—24no—
97#389D418yes—
98#384—28no—
99#381—28no—
100#380—24no—
101#380—24no—
102#380—24no—
103C4×C4⋊C4—24no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108#340—28no—
109#340—28no—
110#336—24no—
111C25C2×D411yescharacteristic
112C25C2×D411yescharacteristic
113C22×Q8—22no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8C2×D414yes—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4SD1614yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8—24no—
159C22×C8—24no—
160C22×C8—24no—
161C4⋊Q8—416no—
162#123—432no—
163C22⋊Q8—216no—
164C22⋊Q8—48no—
165C22⋊Q8—416no—
166C22⋊Q8—416no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C4×D4—416no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—416no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4#4118yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42C2×D414yes—
224#103—416no—
225#103—416no—
226C22⋊C8—48no—
227C22⋊C8—48no—
228#95—24no—
229#95—28no—
230#95—24no—
231#95—28no—
232C24—21no—
233C24—21no—
234C24—24no—
235C24—21no—
236C24—21no—
237C24—24no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—24no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—24no—
246C24#12012yes—
247C24C2×SD1612yes—
248C24C2×SD1612yes—
249C24#13712yes—
250C24C22×D411yescharacteristic
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—28no—
254C2×Q8—28no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×D4—28no—
268C2×D4—28no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—44no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4#12012yes—
338C22×C4#12012yes—
339C22×C4#12012yes—
340C22×C4C2×SD1612yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C2×#4114yes—
345C22×C4H(5,𝔽2)14yes—
346C22×C4H(5,𝔽2)12yes—
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—44no—
354C4⋊C4—24no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—44no—
360C4⋊C4—416no—
361C4⋊C4—48no—
362C4⋊C4—416no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—416no—
368C4⋊C4#12318yescomplemented
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—416no—
372C22⋊C4—416no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C42—28no—
376C42—48no—
377C23—22no—
378C23—24no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—28no—
386C23—28no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—28no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—44no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23#44814yes—
411C23C2×#12011yescharacteristic
412C23C2×#12011yescharacteristic
413C23#53411yescharacteristic
414C23#54614yes—
415C23C22×SD1611yescharacteristic
416C23C2×AGL(1,ℤ/8)11yescharacteristic
417C23C2×#13711yescharacteristic
418C23#57511yescharacteristic
419C23#57814yes—
420Q8—48no—
421Q8—48no—
422Q8—84no—
423Q8—84no—
424D4—416no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—816no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4#45014yes—
471C2×C4C2×#12312yes—
472C2×C4#53212yes—
473C2×C4#53412yes—
474C2×C4#53512yes—
475C2×C4#54014yes—
476C2×C4D4214yes—
477C2×C4#54614yes—
478C2×C4#55014yes—
479C2×C4#55912yes—
480C2×C4#56012yes—
481C8—84no—
482C22—24no—
483C22—28no—
484C22—44no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—44no—
494C22—44no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—82no—
499C22—82no—
500C22#240311yescharacteristic
501C22#240512yes—
502C22#243112yes—
503C22#270912yes—
504C22C2×#45011yescharacteristic
505C22C2×#44811yescharacteristic
506C22#300611yescharacteristic
507C22#300711yescharacteristic
508C22#301112yes—
509C22#301112yes—
510C22#318711yescharacteristic
511C22#319311yescharacteristic
512C22#320212yes—
513C22#329814yes—
514C22#331214yes—
515C22#331514yes—
516C22#333014yes—
517C4—24no—
518C4—42no—
519C4—42no—
520C4—48no—
521C4—84no—
522C4—81no—
523C4—84no—
524C4—84no—
525C4#244514yes—
526C4#267314yes—
527C2—44no—
528C2—82no—
529C2#1526111yescharacteristic
530C2#2391911yescharacteristic
531C2#2392811yescharacteristic
532C2#2404112yes—
533C2#2405412yes—
534C2#2557112yes—
535C2#2558412yes—
536C2#3202712yes—
537C2#3203912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2043C2semidirect
4. C2×#1872C2semidirect
5. #10970C2semidirect
6. #10970C2semidirect
8. #10884C2semidirect
368. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24862
C2×#20432
C2×#18722
#109702
#109642
#108842