Finite Group Databasebeta

← #6434924

Subgroups — #6434924

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35271C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3#30158C214yesmaximal, complemented
4#30101C212yesmaximal, complemented
5#30069C212yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#29912C212yesmaximal, complemented
8#29754C212yesmaximal
9#29638C212yesmaximal, complemented
10#23844C212yesmaximal, complemented
11#23828C212yesmaximal, complemented
12#23787C214yesmaximal, complemented
13#21578C212yesmaximal, complemented
14C4×#480C212yesmaximal, complemented
15#20779C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#480C2212yes—
18C2×#480C2212yes—
19#2856C2211yescharacteristic
20#2854C2212yes—
21#2851C2211yescharacteristic
22#2647C2211yescharacteristic
23#2644C2212yes—
24#2641C2211yescharacteristic
25#2531—22no—
26#2526—22no—
27#2444—22no—
28#2425—22no—
29#2331—22no—
30#2331—22no—
31#2331C2212yes—
32#2331C2212yes—
33#2331C2212yes—
34#2331C2212yes—
35#2330C2211yescharacteristic
36#2328C2211yescharacteristic
37#2319C2212yes—
38C4×#121C2212yes—
39C2×#391—22no—
40C2×#389C2212yes—
41C2×#387—22no—
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×#376C2211yescharacteristic
45#2101C2214yes—
46#2101C2214yes—
47#2092C2214yes—
48#2088C2214yes—
49#2087C2214yes—
50#2085C2214yes—
51#2083C2214yes—
52#2073C2214yes—
53#2067C2214yes—
54#2067C2214yes—
55#2059C2214yes—
56#2057C2214yes—
57#2057C2214yes—
58#2041C2214yes—
59#2023C2214yes—
60#2023C2214yes—
61#1926C2214yes—
62#1925C2214yes—
63#1921C2214yes—
64#1918C2214yes—
65#1874C2212yes—
66#1874C2212yes—
67#1853C2212yes—
68#1852C2212yes—
69#1840C2212yes—
70#1832C2212yes—
71#1823C2212yes—
72#1823C2212yes—
73#1817C2212yes—
74#1797C2212yes—
75#1776C2214yes—
76C4×#107C2214yes—
77#1768C2212yes—
78#1762C2214yes—
79C4×#102C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#123—22no—
92C2×#121—22no—
93C2×#121C2312yes—
94C2×C4×D4—22no—
95C2×C4×D4C2312yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C42—22no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#480—28no—
105#480—28no—
106C2×#107C2314yes—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2312yes—
117C2×#97C2312yes—
118C2×C4×C8C2312yes—
119#400—22no—
120#398—22no—
121#397—22no—
122#397—22no—
123#397—22no—
124#396—22no—
125#394—22no—
126#394—22no—
127#391—24no—
128#391—24no—
129#391—24no—
130#391—24no—
131#390—22no—
132#390C2312yes—
133#390C2312yes—
134#390C2312yes—
135#390C2312yes—
136#389—22no—
137#389—22no—
138#389—22no—
139#389C2312yes—
140#389C2312yes—
141#389C2314yes—
142#389C2312yes—
143#389C2314yes—
144#389C2312yes—
145#388—22no—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—22no—
151#387—22no—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#386—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#385—22no—
161#384C2311yescharacteristic
162#382—21no—
163#382—21no—
164#382C2311yescharacteristic
165#382C2312yes—
166#382C2312yes—
167#381—22no—
168#380—22no—
169#380C2312yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—22no—
176#376D412yes—
177#376D412yes—
178#376C2311yescharacteristic
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C2×#95—22no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—22no—
186C2×#95D412yes—
187C2×#95D412yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201C22×Q8—28no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—84no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2412yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2411yescharacteristic
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—28no—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222#127—42no—
223#125—42no—
224#123—84no—
225#121—24no—
226#121—24no—
227#121—84no—
228C4×D4—28no—
229C4×D4—84no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
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—44no—
245C2×C4⋊C4—84no—
246C2×C4⋊C4#4118yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
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×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—84no—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280#107—28no—
281#107—28no—
282#107—44no—
283#106—44no—
284C4⋊C8—44no—
285#102—28no—
286#102—28no—
287#102—44no—
288#102—44no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291#97—24no—
292C4×C8—24no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313#95C2×D412yes—
314#95C2×D412yes—
315#95C2×D412yes—
316C24—22no—
317C24C22×D411yescharacteristic
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—816no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—82no—
332C2×D4—82no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C2×#4114yes—
369C22×C4C2511yescharacteristic
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4C2×#4118yes—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C42—22no—
408C42—22no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—42no—
416C42—84no—
417C42—84no—
418C42—82no—
419C422−1+414yes—
420C23—22no—
421C23—21no—
422C23—21no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#52512yes—
431C23#57614yes—
432C23C23×D411yescharacteristic
433Q8—88no—
434Q8—88no—
435D4—44no—
436D4—44no—
437D4—88no—
438D4—164no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—416no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—84no—
481C2×C4—88no—
482C2×C4—164no—
483C2×C4#52512yes—
484C2×C4#53614yes—
485C2×C4C22×#4112yes—
486C2×C4C2×2−1+412yes—
487C2×C4#58512yes—
488C8—44no—
489C8—44no—
490C22—22no—
491C22—24no—
492C22—22no—
493C22—24no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22#312214yes—
502C22C2×#52511yescharacteristic
503C22C2×#53611yescharacteristic
504C22#344911yescharacteristic
505C22C2×#57612yes—
506C22#358612yes—
507C4—22no—
508C4—24no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—41no—
513C4—48no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—82no—
518C4—84no—
519C4—82no—
520C4—82no—
521C4#346114yes—
522C2—24no—
523C2—162no—
524C2#3460211yescharacteristic
525C2C2×#312211yescharacteristic
526C2#6056611yescharacteristic
527C2#6087112yes—
528C2#6097612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2331C2semidirect
3. #30158C2semidirect
4. #30101C2semidirect
5. #30069C2semidirect
6. #29972C2semidirect
7. #29912C2semidirect
9. #29638C2semidirect
10. #23844C2semidirect
11. #23828C2semidirect
12. #23787C2semidirect
13. #21578C2semidirect
14. C4×#480C2semidirect
15. #20779C2semidirect

Maximal subgroups

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

subgroupindex
#352712
C2×#23312
#301582
#301012
#300692
#299722
#299122
#297542
#296382
#238442
#238282
#237872
#215782
C4×#4802
#207792