Finite Group Databasebeta

← #6595136

Subgroups — #6595136

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36196C211yesmaximal, characteristic, complemented
3#33295C211yesmaximal, characteristic, complemented
4#33058C212yesmaximal, complemented
5#29214C211yesmaximal, characteristic, complemented
6#28516C211yesmaximal, characteristic, complemented
7#28072C214yesmaximal, complemented
8#28071C214yesmaximal, complemented
9#27897C212yesmaximal, complemented
10#27880C212yesmaximal, complemented
11#25344C214yesmaximal, complemented
12#23065C214yesmaximal, complemented
13C4×#490C211yesmaximal, characteristic, complemented
14#20819C212yesmaximal, complemented
15#20494C211yesmaximal, characteristic, complemented
16C2×#490C2211yescharacteristic
17C2×#478C2214yes—
18C2×#436C2212yes—
19C2×C4×M16C2212yes—
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22#2856C2212yes—
23#2851C2212yes—
24#2573—22no—
25#2562C2212yes—
26#2535C2212yes—
27#2533—22no—
28#2513—22no—
29#2497C2212yes—
30#2496C2212yes—
31#2490C2214yes—
32#2466—22no—
33#2420—22no—
34#2414—22no—
35#2373—22no—
36#2370C2212yes—
37#2363—22no—
38#2357C2212yes—
39C4×#127C2211yescharacteristic
40C4×#125C2211yescharacteristic
41#2285C2212yes—
42#2283C2212yes—
43C42×D4C2211yescharacteristic
44#2094C2212yes—
45#2093C2212yes—
46#2077C2214yes—
47#2075C2214yes—
48#2059C2214yes—
49#2034C2214yes—
50#2033C2214yes—
51#1992C2212yes—
52#1992C2212yes—
53#1992C2212yes—
54#1992C2211yescharacteristic
55#1991C2214yes—
56#1961C2212yes—
57#1961C2211yescharacteristic
58#1949C2214yes—
59#1936C2214yes—
60#1934C2214yes—
61#1934C2214yes—
62#1926C2214yes—
63#1925C2214yes—
64#1921C2214yes—
65#1919C2214yes—
66#1895C2214yes—
67#1895C2214yes—
68#1847C2212yes—
69#1847C2212yes—
70#1836C2212yes—
71#1836C2212yes—
72#1836C2211yescharacteristic
73#1809C2214yes—
74#1809C2214yes—
75C4×#107C2214yes—
76C4×#106C2214yes—
77C8×C4⋊C4C2212yes—
78#1767C2211yescharacteristic
79C4×#102C2214yes—
80#1753C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C22×M16C2314yes—
84C2×#127C2311yescharacteristic
85C2×#126—22no—
86C2×#125C2312yes—
87C2×#125C2311yescharacteristic
88C2×#125C2312yes—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121C2314yes—
92C2×C4×D4—22no—
93C2×C4×D4D412yes—
94C2×C4×D4C2312yes—
95C2×C4×D4C2311yescharacteristic
96C2×#117—22no—
97C22×C42C2312yes—
98#490—24no—
99#478—216no—
100#436—28no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2312yes—
106C2×#102C2314yes—
107C2×#102C2314yes—
108C2×#102C2314yes—
109C2×#102C2314yes—
110C2×C22⋊C8C2314yes—
111C4×M16—28no—
112C2×#97C2311yescharacteristic
113C2×#97C2312yes—
114C2×#97C2312yes—
115C2×#97C2311yescharacteristic
116C2×C4×C8C2314yes—
117C2×C4×C8C2312yes—
118#400—22no—
119#400—22no—
120#400—22no—
121#397—22no—
122#397—22no—
123#397—24no—
124#396—22no—
125#394—22no—
126#394—24no—
127#392—22no—
128#390—24no—
129#390—22no—
130#390—22no—
131#390—22no—
132#390C2312yes—
133#390C2314yes—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#388—24no—
143#388—22no—
144#388—22no—
145#388—22no—
146#388—24no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#386—24no—
155#385—22no—
156#385—22no—
157#384C2311yescharacteristic
158#384C2312yes—
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#383—22no—
162#382—22no—
163#382C2311yescharacteristic
164#382C2312yes—
165#382C2312yes—
166#381—22no—
167#381—22no—
168#380—22no—
169#380C2312yes—
170#380C2312yes—
171#380C2312yes—
172#380C2312yes—
173#380C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2311yescharacteristic
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188#376—22no—
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2314yes—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4C2×D411yescharacteristic
203C22×D4C2412yes—
204C23×C4—22no—
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2412yes—
208C2×M16—28no—
209C2×M16—28no—
210C2×M16—44no—
211C22×C8C2412yes—
212C22×C8C2412yes—
213C22×C8C2414yes—
214#127—22no—
215#127—22no—
216#126—48no—
217#125—28no—
218#125—22no—
219#125—22no—
220#125—42no—
221#125—42no—
222#121—48no—
223#121—44no—
224#121—44no—
225#121—84no—
226C4×D4—24no—
227C4×D4—28no—
228C4×D4—44no—
229C4×D4—84no—
230#117—84no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C22⋊C4C2×D412yes—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2412yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#107—28no—
279#107—44no—
280#106—28no—
281#106—44no—
282C4⋊C8#4118yes—
283#102—28no—
284#102—28no—
285#102—28no—
286#102—44no—
287C22⋊C8—28no—
288#97—24no—
289#97—22no—
290#97—42no—
291#97#4114yes—
292C4×C8—28no—
293C4×C8—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95C2×D411yescharacteristic
312#95C2×D411yescharacteristic
313#95C2×D412yes—
314#95C2×D412yes—
315C24—22no—
316C24C22×D412yes—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—42no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4—84no—
328C2×D4—84no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—84no—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C2511yescharacteristic
355M16—416no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8C2×#4114yes—
364C2×C8C2×#4118yes—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—84no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—48no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—84no—
400C42—24no—
401C42—24no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—42no—
406C42—42no—
407C42—44no—
408C42—82no—
409C42—82no—
410C42—84no—
411C42C2×#4114yes—
412C42C2×#4112yes—
413C422−1+412yes—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23C23×D411yescharacteristic
427D4—44no—
428D4—44no—
429D4—44no—
430D4—88no—
431D4—168no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—84no—
468C2×C4—164no—
469C2×C4#53212yes—
470C2×C4#53212yes—
471C2×C4#54714yes—
472C2×C4#57514yes—
473C2×C4C22×#4111yescharacteristic
474C2×C4C22×#4112yes—
475C2×C4C2×2−1+411yescharacteristic
476C2×C4#58512yes—
477C8—28no—
478C8—24no—
479C8—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—82no—
488C22—82no—
489C22—84no—
490C22—82no—
491C22—84no—
492C22C2×#53211yescharacteristic
493C22C2×#54711yescharacteristic
494C22#350011yescharacteristic
495C22C2×#57512yes—
496C22#358512yes—
497C4—22no—
498C4—22no—
499C4—24no—
500C4—24no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4#347512yes—
511C4#351812yes—
512C2—82no—
513C2—162no—
514C2#3618511yescharacteristic
515C2#6067611yescharacteristic
516C2#6072511yescharacteristic
517C2#6125512yes—
518C2#6259012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
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
2. #36196C2semidirect
3. #33295C2semidirect
4. #33058C2semidirect
5. #29214C2semidirect
6. #28516C2semidirect
7. #28072C2semidirect
8. #28071C2semidirect
9. #27897C2semidirect
10. #27880C2semidirect
11. #25344C2semidirect
12. #23065C2semidirect
13. C4×#490C2semidirect
14. #20819C2semidirect
15. #20494C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361962
#332952
#330582
#292142
#285162
#280722
#280712
#278972
#278802
#253442
#230652
C4×#4902
#208192
#204942