Finite Group Databasebeta

← #6434948

Subgroups — #6434948

Order 512.

1458 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
1#35296C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30163C214yesmaximal, complemented
4#30093C212yesmaximal, complemented
5#30062C212yesmaximal, complemented
6#29977C212yesmaximal, complemented
7#29907C212yesmaximal, complemented
8#29765C212yesmaximal
9#29644C212yesmaximal, complemented
10#23845C212yesmaximal, complemented
11#23830C212yesmaximal, complemented
12#23780C212yesmaximal, complemented
13#23775C212yesmaximal, complemented
14#21588C214yesmaximal, complemented
15#20781C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#482C2214yes—
18#2864C2211yescharacteristic
19#2858C2211yescharacteristic
20#2854C2212yes—
21#2644C2212yes—
22#2628C2211yescharacteristic
23#2619C2211yescharacteristic
24#2595—22no—
25#2557—22no—
26#2474—22no—
27#2471—22no—
28#2360—22no—
29#2360—22no—
30#2355—22no—
31Q8×C4⋊C4—22no—
32#2331C2211yescharacteristic
33#2330C2212yes—
34#2330C2212yes—
35#2328C2211yescharacteristic
36#2324C2212yes—
37#2317C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40#2293C2212yes—
41C2×#384C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2094C2212yes—
48#2094C2212yes—
49#2088C2214yes—
50#2088C2214yes—
51#2085C2214yes—
52#2084C2214yes—
53#2071C2214yes—
54#2067C2214yes—
55#2065C2214yes—
56#2059C2214yes—
57#2058C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2041C2214yes—
61#2025C2214yes—
62#2023C2212yes—
63#2023C2212yes—
64#1926C2214yes—
65#1926C2214yes—
66#1920C2214yes—
67#1918C2214yes—
68#1874C2214yes—
69#1853C2214yes—
70#1839C2212yes—
71#1832C2212yes—
72#1825C2214yes—
73#1817C2212yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1762C2214yes—
80C4×#102C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83#533—28no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#125C2312yes—
88C2×#124—22no—
89C2×#123—22no—
90C2×C22⋊Q8—22no—
91C2×#121C2312yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×D4C2312yes—
96C2×#117—22no—
97C2×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2312yes—
102C22×C42C2311yescharacteristic
103#482—216no—
104C2×#107C2314yes—
105C2×#106C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2312yes—
110C2×#102C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2312yes—
116C2×C4×C8C2312yes—
117#400—21no—
118#399—21no—
119#397—22no—
120#397—22no—
121#397—22no—
122#396—22no—
123#395—21no—
124#394—21no—
125#394—21no—
126#393—21no—
127#392—21no—
128#392—21no—
129#391—22no—
130#391—22no—
131#391—22no—
132#390—22no—
133#390—22no—
134#390C2312yes—
135#390C2312yes—
136#390C2312yes—
137#390C2312yes—
138#389—22no—
139#389—22no—
140#389—24no—
141#389—24no—
142#389C2314yes—
143#389C2314yes—
144#389C2312yes—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#387—22no—
154#387—22no—
155#387—24no—
156#387—24no—
157#387—22no—
158#386—22no—
159#386—22no—
160#386—22no—
161#386—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#384—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#382—22no—
170#382C2311yescharacteristic
171#382C2311yescharacteristic
172#380C2312yes—
173C4×C4⋊C4—24no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—24no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185#376C2311yescharacteristic
186#376C2311yescharacteristic
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4C2412yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4C2×D412yes—
211C23×C4C2411yescharacteristic
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—28no—
215C4⋊Q8—42no—
216#127—42no—
217#125—28no—
218#124—84no—
219#123—84no—
220C22⋊Q8—216no—
221#121—28no—
222C4×Q8—216no—
223C4×Q8—84no—
224C4×Q8—84no—
225C4×D4—24no—
226C4×D4—24no—
227#117—24no—
228#117—22no—
229#117—22no—
230#117—42no—
231#117—84no—
232#117#4118yes—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
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⋊C4C2×D412yes—
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288#107—28no—
289#106—28no—
290#106—44no—
291#106—44no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294#102—28no—
295#102—28no—
296#102—44no—
297#102—44no—
298C22⋊C8—28no—
299#97—24no—
300C4×C8—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—22no—
314#95—22no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319C24—22no—
320C24C22×D411yescharacteristic
321C2×Q8—28no—
322C2×Q8—44no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—82no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—82no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C2×#4114yes—
363C22×C4C2511yescharacteristic
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C4⋊C4—216no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—84no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4C2×#4118yes—
394C22⋊C4—28no—
395C22⋊C4—24no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C42—24no—
410C42—28no—
411C42—24no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—21no—
416C42—21no—
417C42—22no—
418C42—22no—
419C42—42no—
420C42—42no—
421C42—82no—
422C42—84no—
423C42—84no—
424C422−1+412yes—
425C422−1+412yes—
426C23—22no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23#52512yes—
438C23C23×D411yescharacteristic
439Q8—416no—
440Q8—162no—
441D4—44no—
442D4—44no—
443D4—162no—
444D4—162no—
445D4—164no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
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—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—88no—
485C2×C4—164no—
486C2×C4#52512yes—
487C2×C4#53614yes—
488C2×C4#57714yes—
489C2×C4C22×#4112yes—
490C2×C4C2×2−1+411yescharacteristic
491C2×C4C2×2−1+411yescharacteristic
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
508C22C2×#53611yescharacteristic
509C22#344911yescharacteristic
510C22C2×#57712yes—
511C22#358512yes—
512C4—22no—
513C4—24no—
514C4—24no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—42no—
520C4—48no—
521C4—42no—
522C4—82no—
523C4—84no—
524C4—82no—
525C4—82no—
526C4#346112yes—
527C4#346112yes—
528C2—42no—
529C2—162no—
530C2#3460211yescharacteristic
531C2#6058812yes—
532C2#6089212yes—
533C2#6101211yescharacteristic
534C2#6101711yescharacteristic
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. #34214C2semidirect
3. #30163C2semidirect
4. #30093C2semidirect
5. #30062C2semidirect
6. #29977C2semidirect
7. #29907C2semidirect
9. #29644C2semidirect
10. #23845C2semidirect
11. #23830C2semidirect
12. #23780C2semidirect
13. #23775C2semidirect
14. #21588C2semidirect
15. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#352962
#342142
#301632
#300932
#300622
#299772
#299072
#297652
#296442
#238452
#238302
#237802
#237752
#215882
#207812