Finite Group Databasebeta

← #440817

Subgroups — #440817

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2074C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4#15223C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#10970C212yesmaximal
9#10928C211yesmaximal, characteristic, complemented
10#10922C211yesmaximal, characteristic
11#10731C212yesmaximal
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18#2499—24no—
19#2499—24no—
20C2×#397—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#2074—24no—
31#1836—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—22no—
41#1439—22no—
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—42no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—24no—
54C2×#117—42no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79C2×#97—24no—
80C2×#97—22no—
81#397—216no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391D414yes—
86#389—24no—
87#389—42no—
88#387—216no—
89#384—22no—
90#384D414yes—
91#382—22no—
92#382—22no—
93#382—22no—
94#382—22no—
95#380—24no—
96#380—42no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—22no—
114#336—42no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C2×D412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4#4114yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
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—22no—
155C22×C8#4114yes—
156C22⋊Q8—48no—
157C22⋊Q8—48no—
158#117—48no—
159#117—48no—
160C2×C4⋊C4—22no—
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—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4#4114yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—82no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216#103—48no—
217#103—48no—
218C22⋊C8—48no—
219#97—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—24no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—82no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24#12412yes—
237C24C22×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C22×C4—21no—
255C22×C4—21no—
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—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12014yes—
322C22×C4#12412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—416no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—88no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—88no—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—48no—
370C42—41no—
371C42—41no—
372C42—42no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23C2×#12011yescharacteristic
395C23C2×#12411yescharacteristic
396C23#53511yescharacteristic
397C23#54612yes—
398C23#54712yes—
399C23#57512yes—
400C23#57512yes—
401C23#57712yes—
402C23#57812yes—
403Q8—44no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4#52511yescharacteristic
465C2×C4#52912yes—
466C2×C4#53211yescharacteristic
467C2×C4#53511yescharacteristic
468C2×C4#53611yescharacteristic
469C2×C4#53912yes—
470C2×C4#53912yes—
471C2×C4#54012yes—
472C2×C4#54612yes—
473C2×C4#54612yes—
474C2×C4#54712yes—
475C2×C4#55012yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55212yes—
479C2×C4#56212yes—
480C8—44no—
481C8—82no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—44no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22#240611yescharacteristic
492C22#243111yescharacteristic
493C22#243511yescharacteristic
494C22#248812yes—
495C22#270612yes—
496C22#300512yes—
497C22#301312yes—
498C22#313511yescharacteristic
499C22#313712yes—
500C22#314211yescharacteristic
501C22#321912yes—
502C22#322111yescharacteristic
503C22#322311yescharacteristic
504C22#331112yes—
505C22#331112yes—
506C22#331312yes—
507C22#331312yes—
508C22#334812yes—
509C22#334912yes—
510C22#335112yes—
511C22#335212yes—
512C4—44no—
513C4—42no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—84no—
519C4—82no—
520C4—162no—
521C4#247112yes—
522C4#255312yes—
523C4#269412yes—
524C4#270612yes—
525C2—82no—
526C2#1566911yescharacteristic
527C2#2410612yes—
528C2#2558012yes—
529C2#2586211yescharacteristic
530C2#2586811yescharacteristic
531C2#2771412yes—
532C2#2772412yes—
533C2#3195512yes—
534C2#3195712yes—
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
2. C2×#2074C2semidirect
3. C2×#1836C2semidirect
5. #11371C2semidirect
6. #11208C2semidirect
7. #11067C2semidirect
9. #10928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#20742
C2×#18362
#152232
#113712
#112082
#110672
#109702
#109282
#109222
#107312