Finite Group Databasebeta

← #6850698

Subgroups — #6850698

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#32076C218yesmaximal, complemented
5#29807C212yesmaximal, complemented
6#29143C211yesmaximal, characteristic, complemented
7#28511C211yesmaximal, characteristic, complemented
8#27972C212yesmaximal, complemented
9#27891C212yesmaximal, complemented
10#24694C214yesmaximal, complemented
11#24132C214yesmaximal, complemented
12#21982C212yesmaximal, complemented
13#21692C211yesmaximal, characteristic, complemented
14#21265C211yesmaximal, characteristic, complemented
15C2×#547—22no—
16C2×#547C2212yes—
17C2×#492C2211yescharacteristic
18C2×#477C2214yes—
19C2×#440C2214yes—
20C2×#431C2212yes—
21#2868C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2212yes—
24#2856C2212yes—
25#2542—22no—
26#2542C2212yes—
27#2536C2212yes—
28#2508—22no—
29#2501—22no—
30#2501—22no—
31#2501C2214yes—
32#2495—22no—
33#2489C2214yes—
34#2485—22no—
35#2356C2212yes—
36#2346—22no—
37C4×C4⋊Q8C2211yescharacteristic
38C4×#127C2211yescharacteristic
39#2285C2212yes—
40C2×#384C2212yes—
41C42×D4C2211yescharacteristic
42#2095C2214yes—
43#2072C2218yes—
44#2055C2218yes—
45#2033C2218yes—
46#2025C2214yes—
47#2000C2218yes—
48#1993C2214yes—
49#1992C2214yes—
50#1973C2214yes—
51#1957C2212yes—
52#1957C2211yescharacteristic
53#1956C2211yescharacteristic
54#1956C2212yes—
55#1955C2212yes—
56#1947C2218yes—
57#1940C2212yes—
58#1925C2218yes—
59#1924C2214yes—
60#1872C2218yes—
61#1852C2218yes—
62#1845C2212yes—
63#1845C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1834C2212yes—
67#1804C2214yes—
68C4×C4⋊C8C2211yescharacteristic
69C4×C22⋊C8C2212yes—
70C4×#97C2211yescharacteristic
71C22×#41—22no—
72C22×SD16C2314yes—
73#547—48no—
74#547—416no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2312yes—
78C2×#127C2311yescharacteristic
79C2×#125C2312yes—
80C2×#123—24no—
81C2×#123—24no—
82C2×#123—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8C2314yes—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121C2314yes—
89C2×C4×Q8C2312yes—
90C2×C4×D4—22no—
91C2×C4×D4D412yes—
92C2×C4×D4C2312yes—
93C2×C4×D4C2311yescharacteristic
94C2×#117—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4C2314yes—
98C22×C42C2312yes—
99#492—24no—
100#477—48no—
101#440—48no—
102#431—28no—
103C2×#107C2318yes—
104C2×#107C2314yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2314yes—
107C2×#103C2318yes—
108C2×#103C2314yes—
109C2×#102C2318yes—
110C2×#102C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×#97C2312yes—
114C2×#97C2312yes—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2312yes—
117C2×C4×C8C2311yescharacteristic
118#400—24no—
119#399—22no—
120#399—22no—
121#399—24no—
122#399—24no—
123#398—22no—
124#397—24no—
125#394—22no—
126#391C2312yes—
127#391C2314yes—
128#390—22no—
129#390—22no—
130#390—22no—
131#390C2312yes—
132#390C2314yes—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#388—24no—
139#388—24no—
140#387—24no—
141#384—22no—
142#384—22no—
143#384—22no—
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2314yes—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#383—22no—
151#382—22no—
152#382—24no—
153#382C2312yes—
154#380—22no—
155#380—22no—
156#380—22no—
157#380C2314yes—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4D412yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2311yescharacteristic
163C4×C22⋊C4D412yes—
164C4×C22⋊C4D412yes—
165C4×C22⋊C4C2312yes—
166#376—22no—
167C43C2311yescharacteristic
168#340C2318yes—
169#340C2318yes—
170#340C2318yes—
171#336C2314yes—
172#336C2312yes—
173C2×#41—416no—
174C2×#41—48no—
175C22×Q8—22no—
176C22×Q8C2412yes—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4C2×D411yescharacteristic
180C22×D4C2412yes—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2412yes—
186C2×SD16—28no—
187C2×SD16—44no—
188C2×SD16—44no—
189C22×C8C2412yes—
190C22×C8C2414yes—
191C22×C8C2412yes—
192C4⋊Q8—22no—
193C4⋊Q8—28no—
194C4⋊Q8—22no—
195C4⋊Q8—48no—
196#127—22no—
197#127—22no—
198#125—44no—
199#123—416no—
200#123—416no—
201#123—416no—
202C22⋊Q8—48no—
203C22⋊Q8—416no—
204C22⋊Q8—48no—
205#121—48no—
206#121—48no—
207#121—48no—
208C4×Q8—28no—
209C4×D4—28no—
210C4×D4—24no—
211C4×D4—28no—
212C4×D4—48no—
213#117—48no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2411yescharacteristic
257C2×C42C2412yes—
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260#107—44no—
261#107—48no—
262C4⋊C8—28no—
263C4⋊C8—24no—
264#103—44no—
265#103—48no—
266#102—48no—
267#102—44no—
268C22⋊C8—28no—
269C22⋊C8—44no—
270#97—24no—
271#97—42no—
272C4×C8—24no—
273C4×C8—22no—
274C4×C8#4114yescomplemented
275#95—24no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95C2×D411yescharacteristic
283#95C2×D411yescharacteristic
284#95C2×D412yes—
285#95C2×D412yes—
286C24—22no—
287C24C22×D412yes—
288#41—832no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4AGL(1,ℤ/8)18yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C2511yescharacteristic
335SD16—416no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8C2×#4114yes—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—44no—
374C22⋊C4—48no—
375C22⋊C4—84no—
376C42—24no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—44no—
384C42—48no—
385C42C2×#4112yes—
386C42H(5,𝔽2)14yes—
387C422−1+412yes—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23C2×AGL(1,ℤ/8)14yes—
400C23C23×D411yescharacteristic
401Q8—44no—
402Q8—44no—
403Q8—88no—
404D4—28no—
405D4—44no—
406D4—44no—
407D4—816no—
408D4—88no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—216no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—88no—
440C2×C4—816no—
441C2×C4#53512yes—
442C2×C4#53512yes—
443C2×C4#54714yes—
444C2×C4C2×AGL(1,ℤ/8)14yes—
445C2×C4C2×AGL(1,ℤ/8)14yes—
446C2×C4#57814yes—
447C2×C4C22×#4111yescharacteristic
448C2×C4C2×H(5,𝔽2)12yes—
449C2×C4C2×2−1+411yescharacteristic
450C2×C4#58512yes—
451C8—24no—
452C8—44no—
453C8—42no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—84no—
462C22—82no—
463C22—84no—
464C22—82no—
465C22—82no—
466C22C2×#53511yescharacteristic
467C22C2×#54711yescharacteristic
468C22#350011yescharacteristic
469C22C22×AGL(1,ℤ/8)12yes—
470C22C2×#57812yes—
471C4—24no—
472C4—22no—
473C4—22no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—84no—
479C4—82no—
480C4—82no—
481C4—84no—
482C4#334318yes—
483C4#350612yes—
484C4#354712yes—
485C2—44no—
486C2—162no—
487C2#3949111yescharacteristic
488C2#6079811yescharacteristic
489C2#6082111yescharacteristic
490C2C2×#334312yes—
491C2#6253812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36209C2semidirect
3. #33297C2semidirect
4. #32076C2semidirect
5. #29807C2semidirect
6. #29143C2semidirect
7. #28511C2semidirect
8. #27972C2semidirect
9. #27891C2semidirect
10. #24694C2semidirect
11. #24132C2semidirect
12. #21982C2semidirect
13. #21692C2semidirect
14. #21265C2semidirect
274. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#362092
#361792
#332972
#320762
#298072
#291432
#285112
#279722
#278912
#246942
#241322
#219822
#216922
#212652