Finite Group Databasebeta

← #6468652

Subgroups — #6468652

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35332C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29361C212yesmaximal, complemented
4#29360C212yesmaximal, complemented
5#29345C214yesmaximal, complemented
6#29223C212yesmaximal, complemented
7#28754C212yesmaximal, complemented
8#27978C212yesmaximal, complemented
9#25909C212yesmaximal, complemented
10#23832C214yesmaximal, complemented
11#21655C212yesmaximal, complemented
12C4×#489C212yesmaximal, complemented
13#21399C212yesmaximal
14#21398C212yesmaximal, complemented
15#20465C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C2×#489C2212yes—
20C2×#489C2212yes—
21C2×#433C2212yes—
22#2859C2211yescharacteristic
23#2858C2211yescharacteristic
24#2850C2212yes—
25#2642C2211yescharacteristic
26#2641C2211yescharacteristic
27#2620C2212yes—
28#2559—22no—
29#2559—22no—
30#2558—22no—
31#2558—22no—
32#2360—22no—
33#2360—22no—
34#2355—22no—
35Q8×C4⋊C4—22no—
36#2330C2212yes—
37#2330C2212yes—
38#2326C2212yes—
39#2317C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#124C2212yes—
43C2×#384C2211yescharacteristic
44#2274C2212yes—
45C4×#117C2211yescharacteristic
46#2100C2214yes—
47#2099C2214yes—
48#2072C2214yes—
49#2071C2214yes—
50#2042C2214yes—
51#2042C2214yes—
52#2026C2214yes—
53#2025C2214yes—
54#1993C2212yes—
55#1993C2212yes—
56#1989C2214yes—
57#1989C2214yes—
58#1982C2214yes—
59#1981C2214yes—
60#1959C2212yes—
61#1959C2212yes—
62#1936C2214yes—
63#1935C2214yes—
64#1934C2214yes—
65#1933C2214yes—
66#1875C2214yes—
67#1874C2214yes—
68#1873C2214yes—
69#1872C2214yes—
70#1836C2212yes—
71#1836C2212yes—
72#1768C2212yes—
73#1767C2212yes—
74#1763C2214yes—
75#1762C2214yes—
76C4×#103C2214yes—
77C4×#102C2214yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#541—28no—
84#540—28no—
85#533—28no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#126—22no—
90C2×#125—22no—
91C2×#125C2312yes—
92C2×#125C2311yescharacteristic
93C2×#125C2311yescharacteristic
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×#123—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8C2312yes—
99C2×#121—22no—
100C2×#121C2312yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8C2312yes—
105C2×C4×D4C2312yes—
106C2×#117C2312yes—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2312yes—
111C22×C42C2311yescharacteristic
112#489—28no—
113#489—28no—
114#433—28no—
115C2×C4⋊C8C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2314yes—
125C2×#97C2312yes—
126C2×#97C2312yes—
127C2×#97C2312yes—
128C2×#97C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2312yes—
131#400—22no—
132#400—22no—
133#396—22no—
134#396—22no—
135#394—22no—
136#394—22no—
137#394—22no—
138#394—22no—
139#391—22no—
140#391—22no—
141#391C2312yes—
142#391C2312yes—
143#390—22no—
144#390—22no—
145#390C2312yes—
146#390C2312yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—24no—
152#389—24no—
153#389C2314yes—
154#389C2314yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#387—24no—
162#387—24no—
163#386—22no—
164#386—22no—
165#385—22no—
166#385—22no—
167#385—22no—
168#385—22no—
169#384—22no—
170#384—22no—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#382—22no—
175#382C2312yes—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4D412yes—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4—24no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4D412yes—
188#376C2311yescharacteristic
189#376C2311yescharacteristic
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8C2412yes—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4C2412yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2411yescharacteristic
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215C4⋊Q8#4118yes—
216#127—24no—
217#126—48no—
218#125—28no—
219#125—24no—
220#125—24no—
221#125—48no—
222#124—24no—
223#124—24no—
224#124—44no—
225#123—48no—
226C22⋊Q8—216no—
227C22⋊Q8—28no—
228#121—28no—
229#121—48no—
230C4×Q8—216no—
231C4×Q8—28no—
232C4×Q8—84no—
233C4×Q8—84no—
234C4×D4—28no—
235#117—28no—
236#117—42no—
237#117#4118yes—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—84no—
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
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×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D412yes—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
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
293C4⋊C8—28no—
294#103—28no—
295#103—28no—
296#103—44no—
297#103—44no—
298#102—28no—
299#102—28no—
300#102—44no—
301#102—44no—
302C22⋊C8—28no—
303#97—24no—
304#97—24no—
305#97—24no—
306#97—42no—
307C4×C8—24no—
308C4×C8—42no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
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—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—84no—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4114yes—
367C22×C4C2511yescharacteristic
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C4⋊C4—216no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4#12418yes—
397C4⋊C4C2×#4118yes—
398C22⋊C4—28no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4#12418yes—
413C42—28no—
414C42—24no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—44no—
423C42—84no—
424C42—84no—
425C42C2×#4114yes—
426C42C2×#4114yes—
427C42C2×#4114yes—
428C23—22no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23C2×#12412yes—
440C23C23×D411yescharacteristic
441Q8—416no—
442Q8—44no—
443Q8—44no—
444D4—44no—
445D4—44no—
446D4—164no—
447D4—164no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—164no—
485C2×C4—164no—
486C2×C4#52514yes—
487C2×C4C2×#12414yes—
488C2×C4C2×#12414yes—
489C2×C4C2×#12412yes—
490C2×C4C22×#4112yes—
491C2×C4C22×#4112yes—
492C2×C4#58512yes—
493C8—44no—
494C8—44no—
495C22—22no—
496C22—24no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22C2×#52511yescharacteristic
508C22C22×#12411yescharacteristic
509C22#344911yescharacteristic
510C22#358512yes—
511C22#358612yes—
512C4—22no—
513C4—24no—
514C4—24no—
515C4—24no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—48no—
522C4—84no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4#348014yes—
527C2—42no—
528C2—162no—
529C2#3495511yescharacteristic
530C2#6056512yes—
531C2#6062512yes—
532C2#6101911yescharacteristic
533C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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. #34214C2semidirect
3. #29361C2semidirect
4. #29360C2semidirect
5. #29345C2semidirect
6. #29223C2semidirect
7. #28754C2semidirect
8. #27978C2semidirect
9. #25909C2semidirect
10. #23832C2semidirect
11. #21655C2semidirect
12. C4×#489C2semidirect
14. #21398C2semidirect
15. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#353322
#342142
#293612
#293602
#293452
#292232
#287542
#279782
#259092
#238322
#216552
C4×#4892
#213992
#213982
#204652