Finite Group Databasebeta

← #6631985

Subgroups — #6631985

Order 512.

1269 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35295C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic, complemented
3#32857C212yesmaximal, complemented
4#32840C212yesmaximal, complemented
5#32835C212yesmaximal, complemented
6#32827C212yesmaximal, complemented
7#32787C212yesmaximal, complemented
8#32773C212yesmaximal
9#32751C214yesmaximal, complemented
10#23845C212yesmaximal, complemented
11#23828C212yesmaximal, complemented
12#23780C212yesmaximal, complemented
13#23775C212yesmaximal, complemented
14#23771C214yesmaximal, complemented
15#20781C211yesmaximal, characteristic, complemented
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18#2860C2212yes—
19#2857C2212yes—
20#2856C2211yescharacteristic
21#2851C2211yescharacteristic
22#2647C2212yes—
23#2647C2211yescharacteristic
24#2641C2211yescharacteristic
25#2628C2212yes—
26#2624C2212yes—
27#2597—22no—
28#2559—22no—
29#2511—22no—
30#2492—22no—
31#2479—22no—
32#2476—22no—
33#2451—22no—
34#2423—22no—
35#2331C2212yes—
36#2330C2211yescharacteristic
37#2328C2211yescharacteristic
38#2324C2212yes—
39#2314C2212yes—
40C4×C4⋊Q8C2212yes—
41#2293C2212yes—
42C4×#117C2211yescharacteristic
43#2101C2214yes—
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2094C2212yes—
48#2094C2212yes—
49#2092C2214yes—
50#2088C2214yes—
51#2088C2214yes—
52#2085C2214yes—
53#2084C2214yes—
54#2084C2214yes—
55#2077C2214yes—
56#2075C2214yes—
57#2073C2214yes—
58#2071C2214yes—
59#2067C2214yes—
60#2065C2214yes—
61#2059C2214yes—
62#2058C2212yes—
63#2058C2214yes—
64#2057C2212yes—
65#2041C2214yes—
66#2034C2214yes—
67#2034C2214yes—
68#2023C2212yes—
69#2023C2212yes—
70#2023C2214yes—
71#1839C2212yes—
72#1832C2212yes—
73#1817C2212yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1762C2214yes—
80C4×#102C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#126—22no—
88C2×#125C2311yescharacteristic
89C2×C22⋊Q8—22no—
90C2×#121—22no—
91C2×#121C2312yes—
92C2×C4×Q8—22no—
93C2×C4×D4C2312yes—
94C2×#117—22no—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×#106C2314yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104C2×#102C2314yes—
105C2×#102C2314yes—
106C2×#102C2314yes—
107C2×#102C2314yes—
108C2×C22⋊C8C2314yes—
109C2×#97C2312yes—
110C2×C4×C8C2312yes—
111#400—22no—
112#400—22no—
113#400—24no—
114#399—22no—
115#399—22no—
116#398—24no—
117#398—22no—
118#397—22no—
119#395—22no—
120#394—22no—
121#394—22no—
122#394—22no—
123#394—22no—
124#393—22no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391—24no—
130#390—22no—
131#390—22no—
132#390C2312yes—
133#390C2312yes—
134#390C2312yes—
135#390C2312yes—
136#389—22no—
137#389—22no—
138#389—22no—
139#389C2314yes—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—24no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#387—24no—
153#387—22no—
154#387—22no—
155#387—24no—
156#386—22no—
157#386—22no—
158#386—22no—
159#385—22no—
160#385—22no—
161#384—22no—
162#384C2312yes—
163#384C2312yes—
164#384C2311yescharacteristic
165#382C2312yes—
166#382C2311yescharacteristic
167#382C2312yes—
168#380—22no—
169#380C2314yes—
170#380C2312yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181C43C2311yescharacteristic
182C43C2311yescharacteristic
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2312yes—
193#336C2312yes—
194#336C2314yes—
195#336C2314yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4C2412yes—
201C23×C4—22no—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—84no—
209#127—42no—
210#126—84no—
211#125—42no—
212C22⋊Q8—84no—
213#121—44no—
214#121—84no—
215C4×Q8—84no—
216C4×D4—24no—
217C4×D4—24no—
218#117—22no—
219#117—22no—
220#117—42no—
221#117—84no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2414yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—42no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—44no—
273#106—44no—
274#106—44no—
275#106—44no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278#102—44no—
279#102—44no—
280#102—44no—
281#102—44no—
282C22⋊C8—44no—
283#97—24no—
284C4×C8—24no—
285#95—22no—
286#95—22no—
287#95—24no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—22no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95C2×D412yes—
300#95C2×D412yes—
301#95C2×D412yes—
302#95C2×D412yes—
303C24—22no—
304C24C22×D411yescharacteristic
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—84no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C2511yescharacteristic
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C22⋊C4—24no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—84no—
390C42—24no—
391C42—24no—
392C42—21no—
393C42—21no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—82no—
402C42—84no—
403C422−1+412yes—
404C422−1+412yes—
405C23—22no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23C23×D411yescharacteristic
418Q8—88no—
419Q8—164no—
420D4—44no—
421D4—44no—
422D4—164no—
423D4—164no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—84no—
463C2×C4—88no—
464C2×C4#53614yes—
465C2×C4#57714yes—
466C2×C4C2×2−1+411yescharacteristic
467C2×C4C2×2−1+411yescharacteristic
468C2×C4#58512yes—
469C2×C4#58512yes—
470C8—44no—
471C8—44no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22C2×#53611yescharacteristic
485C22#344911yescharacteristic
486C22#344911yescharacteristic
487C22C2×#57712yes—
488C22#358512yes—
489C4—24no—
490C4—24no—
491C4—42no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—42no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4—84no—
502C4—82no—
503C4#353112yes—
504C4#353112yes—
505C2—81no—
506C2—162no—
507C2#3669911yescharacteristic
508C2#6089212yes—
509C2#6098012yes—
510C2#6101211yescharacteristic
511C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. #35283C2semidirect
3. #32857C2semidirect
4. #32840C2semidirect
5. #32835C2semidirect
6. #32827C2semidirect
7. #32787C2semidirect
9. #32751C2semidirect
10. #23845C2semidirect
11. #23828C2semidirect
12. #23780C2semidirect
13. #23775C2semidirect
14. #23771C2semidirect
15. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#352952
#352832
#328572
#328402
#328352
#328272
#327872
#327732
#327512
#238452
#238282
#237802
#237752
#237712
#207812