Finite Group Databasebeta

← #6631953

Subgroups — #6631953

Order 512.

1267 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#35268C211yesmaximal, characteristic
3#32853C212yesmaximal, complemented
4#32836C212yesmaximal, complemented
5#32835C212yesmaximal, complemented
6#32819C212yesmaximal, complemented
7#32794C212yesmaximal, complemented
8#32779C212yesmaximal
9#32756C214yesmaximal, complemented
10#23845C212yesmaximal, complemented
11#23830C212yesmaximal, complemented
12#23786C214yesmaximal, complemented
13#23760C212yesmaximal, complemented
14#23755C212yesmaximal, complemented
15#20784C211yesmaximal, characteristic, complemented
16#2868C2211yescharacteristic
17#2864C2211yescharacteristic
18#2860C2212yes—
19#2859C2212yes—
20#2859C2211yescharacteristic
21#2858C2211yescharacteristic
22#2629C2212yes—
23#2628C2212yes—
24#2628C2211yescharacteristic
25#2624C2212yes—
26#2619C2211yescharacteristic
27#2570—22no—
28#2550—22no—
29#2521—22no—
30#2516—22no—
31#2467—22no—
32#2459—22no—
33#2441—22no—
34#2439—22no—
35#2331C2211yescharacteristic
36#2331C2212yes—
37#2330C2212yes—
38#2328C2211yescharacteristic
39#2319C2212yes—
40C4×#121C2212yes—
41#2285C2212yes—
42C4×#117C2211yescharacteristic
43#2101C2214yes—
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2094C2212yes—
48#2094C2212yes—
49#2092C2214yes—
50#2088C2214yes—
51#2086C2214yes—
52#2085C2214yes—
53#2084C2214yes—
54#2084C2212yes—
55#2083C2212yes—
56#2077C2214yes—
57#2077C2214yes—
58#2071C2214yes—
59#2071C2212yes—
60#2071C2212yes—
61#2065C2214yes—
62#2065C2214yes—
63#2059C2214yes—
64#2058C2214yes—
65#2058C2214yes—
66#2041C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#2025C2214yes—
70#2023C2214yes—
71#1839C2212yes—
72#1831C2212yes—
73#1818C2212yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1768C2212yes—
78#1762C2214yes—
79C4×#102C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#126—22no—
88C2×#125—22no—
89C2×#125C2312yes—
90C2×#124—22no—
91C2×#121C2312yes—
92C2×C4×D4—22no—
93C2×C4×D4C2312yes—
94C2×#117—22no—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×#106C2314yes—
102C2×C4⋊C8C2314yes—
103C2×#102C2314yes—
104C2×#102C2314yes—
105C2×#102C2314yes—
106C2×#102C2314yes—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109C2×#97C2312yes—
110C2×C4×C8C2312yes—
111#400—22no—
112#400—21no—
113#399—22no—
114#399—22no—
115#399—21no—
116#398—24no—
117#397—22no—
118#396—24no—
119#395—21no—
120#395—22no—
121#394—21no—
122#394—21no—
123#394—22no—
124#394—22no—
125#393—22no—
126#393—21no—
127#392—21no—
128#392—21no—
129#391—24no—
130#391—22no—
131#390—22no—
132#390C2312yes—
133#390C2312yes—
134#390C2312yes—
135#390C2312yes—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—24no—
141#389C2312yes—
142#389C2314yes—
143#389C2312yes—
144#389C2312yes—
145#389C2314yes—
146#389C2312yes—
147#388—22no—
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#384—22no—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382—22no—
166#382—22no—
167#382C2311yescharacteristic
168#382C2311yescharacteristic
169#382C2312yes—
170#381—22no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174#380C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4D412yes—
182C4×C22⋊C4D412yes—
183#376C2311yescharacteristic
184#376C2311yescharacteristic
185C43C2311yescharacteristic
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
200C22×Q8—21no—
201C22×Q8—22no—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4C2412yes—
206C23×C4—22no—
207C23×C4C2×D412yes—
208C23×C4C2411yescharacteristic
209C22×C8C2414yes—
210C22×C8C2414yes—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214C4⋊Q8—84no—
215#127—42no—
216#126—84no—
217#125—44no—
218#125—84no—
219#124—84no—
220#121—42no—
221#121—42no—
222C4×D4—28no—
223C4×D4—84no—
224#117—42no—
225#117—42no—
226#117—84no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
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⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D412yes—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#107—44no—
279#106—44no—
280#106—44no—
281#106—44no—
282C4⋊C8—44no—
283#102—44no—
284#102—44no—
285#102—44no—
286#102—44no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289#97—24no—
290C4×C8—24no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—22no—
304#95—21no—
305#95—22no—
306#95—22no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311C24—22no—
312C24C22×D411yescharacteristic
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—82no—
316C2×Q8—84no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—44no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C2511yescharacteristic
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—82no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—84no—
383C22⋊C4—41no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—84no—
397C22⋊C4—82no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—42no—
408C42—84no—
409C42—82no—
410C422−1+414yes—
411C23—22no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C23×D411yescharacteristic
424Q8—162no—
425Q8—164no—
426D4—44no—
427D4—44no—
428D4—88no—
429D4—162no—
430D4—162no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—164no—
470C2×C4#53614yes—
471C2×C4#57714yes—
472C2×C4C2×2−1+412yes—
473C2×C4#58512yes—
474C2×C4#58512yes—
475C8—44no—
476C8—44no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22C2×#53611yescharacteristic
490C22#344911yescharacteristic
491C22#344911yescharacteristic
492C22C2×#57712yes—
493C22#358512yes—
494C4—24no—
495C4—24no—
496C4—42no—
497C4—42no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—84no—
504C4—82no—
505C4—82no—
506C4—84no—
507C4—82no—
508C4#353114yes—
509C2—81no—
510C2—162no—
511C2#3669911yescharacteristic
512C2#6087312yes—
513C2#6098912yes—
514C2#6101211yescharacteristic
515C2#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
1. #35294C2semidirect
3. #32853C2semidirect
4. #32836C2semidirect
5. #32835C2semidirect
6. #32819C2semidirect
7. #32794C2semidirect
9. #32756C2semidirect
10. #23845C2semidirect
11. #23830C2semidirect
12. #23786C2semidirect
13. #23760C2semidirect
14. #23755C2semidirect
15. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#352682
#328532
#328362
#328352
#328192
#327942
#327792
#327562
#238452
#238302
#237862
#237602
#237552
#207842