Finite Group Databasebeta

← #6479680

Subgroups — #6479680

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35287C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#32967C212yesmaximal, complemented
4#32912C212yesmaximal
5#29910C212yesmaximal, complemented
6#29624C212yesmaximal, complemented
7#29500C212yesmaximal, complemented
8#29494C212yesmaximal, complemented
9#29378C214yesmaximal, complemented
10#29138C212yesmaximal, complemented
11#28392C212yesmaximal, complemented
12#28209C212yesmaximal, complemented
13#28160C212yesmaximal, complemented
14#22058C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#479C2214yes—
19C2×#460C2214yes—
20C2×#448C2214yes—
21#2860C2212yes—
22#2854C2212yes—
23#2647C2211yescharacteristic
24#2641C2211yescharacteristic
25#2629C2211yescharacteristic
26#2619C2211yescharacteristic
27#2593—22no—
28#2554—22no—
29#2468—22no—
30#2457—22no—
31#2365—22no—
32#2365—22no—
33#2344—22no—
34#2344—22no—
35#2327C2212yes—
36#2327C2212yes—
37#2324C2212yes—
38#2319C2212yes—
39#2310C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#127C2211yescharacteristic
43C4×#125C2211yescharacteristic
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2092C2214yes—
48#2084C2212yes—
49#2083C2212yes—
50#2071C2212yes—
51#2071C2212yes—
52#2043C2212yes—
53#2043C2212yes—
54#2043C2214yes—
55#2013C2212yes—
56#2013C2212yes—
57#2013C2214yes—
58#1973C2214yes—
59#1968C2214yes—
60#1935C2214yes—
61#1933C2214yes—
62#1928C2214yes—
63#1927C2214yes—
64#1880C2214yes—
65#1880C2214yes—
66#1877C2214yes—
67#1877C2214yes—
68#1875C2214yes—
69#1865C2214yes—
70#1863C2214yes—
71#1854C2214yes—
72#1840C2212yes—
73#1831C2212yes—
74#1817C2212yes—
75#1805C2214yes—
76#1803C2214yes—
77#1798C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83C23×Q8C2312yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#126—22no—
92C2×#125C2311yescharacteristic
93C2×#123—22no—
94C2×#121C2312yes—
95#519—48no—
96C2×C4×Q8—24no—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×#117—24no—
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—22no—
106C22×C42C2311yescharacteristic
107#479—48no—
108#460—48no—
109#448—216no—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#400—22no—
125#398—22no—
126#398—22no—
127#396—22no—
128#394—22no—
129#394—22no—
130#394—22no—
131#392—22no—
132#391—22no—
133#391—22no—
134#391C2314yes—
135#391C2314yes—
136#390—22no—
137#390—22no—
138#390C2312yes—
139#390C2312yes—
140#390C2312yes—
141#390C2312yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389C2312yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#387—22no—
159#387—24no—
160#387—24no—
161#387—24no—
162#387—24no—
163#386—22no—
164#386—22no—
165#385—22no—
166#384C2311yescharacteristic
167#384C2312yes—
168#382C2311yescharacteristic
169#382C2312yes—
170#381—22no—
171#381—22no—
172#381—22no—
173#380—22no—
174#380—22no—
175#380C2312yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8#4118yes—
206C22×Q8C2414yes—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4C2412yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4C2×D412yes—
213C23×C4C2411yescharacteristic
214C2×SD16—28no—
215C2×SD16—28no—
216C2×SD16—44no—
217C2×SD16—44no—
218C2×SD16—44no—
219C2×SD16—44no—
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—216no—
223C4⋊Q8—28no—
224C4⋊Q8—28no—
225#127—24no—
226#126—84no—
227#125—24no—
228#123—84no—
229#121—28no—
230C4×Q8—28no—
231C4×Q8—28no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×Q8—416no—
235C4×D4—44no—
236C4×D4—84no—
237#117—24no—
238#117—42no—
239#117—416no—
240#117—44no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—44no—
294#106—44no—
295C4⋊C8—28no—
296#103—28no—
297#103—28no—
298#103—44no—
299#103—44no—
300#103—44no—
301#103—44no—
302#102—44no—
303#102—44no—
304C22⋊C8—28no—
305#97—42no—
306C4×C8—42no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323C24—22no—
324C24C22×D411yescharacteristic
325C2×Q8—28no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—416no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—416no—
335C2×Q8C2×#4118yes—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C2×#4114yes—
372C22×C4C2511yescharacteristic
373SD16—88no—
374SD16—88no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—216no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C22⋊C4—24no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—42no—
427C42—42no—
428C42—44no—
429C42—48no—
430C42—48no—
431C42—82no—
432C42H(5,𝔽2)14yes—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23#52512yes—
445C23C23×D411yescharacteristic
446Q8—28no—
447Q8—28no—
448Q8—416no—
449Q8—44no—
450Q8—44no—
451D4—44no—
452D4—44no—
453D4—88no—
454D4—164no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—48no—
478C2×C4—416no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—164no—
494C2×C4#52512yes—
495C2×C4#53514yes—
496C2×C4#57714yes—
497C2×C4C22×#4112yes—
498C2×C4C2×H(5,𝔽2)12yes—
499C2×C4#58512yes—
500C8—44no—
501C8—44no—
502C22—22no—
503C22—24no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22C2×#52511yescharacteristic
515C22C2×#53511yescharacteristic
516C22#344911yescharacteristic
517C22C2×#57712yes—
518C22#358612yes—
519C4—22no—
520C4—24no—
521C4—24no—
522C4—48no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4—84no—
532C4—82no—
533C4#350514yes—
534C2—42no—
535C2—162no—
536C2#3510211yescharacteristic
537C2#6058212yes—
538C2#6083612yes—
539C2#6101211yescharacteristic
540C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
3. #32967C2semidirect
5. #29910C2semidirect
6. #29624C2semidirect
7. #29500C2semidirect
8. #29494C2semidirect
9. #29378C2semidirect
10. #29138C2semidirect
11. #28392C2semidirect
12. #28209C2semidirect
13. #28160C2semidirect
14. #22058C2semidirect
15. #20780C2semidirect

Maximal subgroups

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

subgroupindex
#352872
#342092
#329672
#329122
#299102
#296242
#295002
#294942
#293782
#291382
#283922
#282092
#281602
#220582
#207802