Finite Group Databasebeta

← #6499416

Subgroups — #6499416

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35341C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic, complemented
3#33282C212yesmaximal, complemented
4#33282C212yesmaximal, complemented
5#30217C212yesmaximal, complemented
6#30216C212yesmaximal, complemented
7#29361C212yesmaximal, complemented
8#29360C212yesmaximal, complemented
9#29359C212yesmaximal, complemented
10#29358C212yesmaximal, complemented
11#29120C214yesmaximal, complemented
12#29119C214yesmaximal, complemented
13#21838C212yesmaximal, complemented
14#21809C212yesmaximal
15#20467C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#432C2212yes—
19#2868C2211yescharacteristic
20#2861C2212yes—
21#2861C2212yes—
22#2859C2212yes—
23#2859C2211yescharacteristic
24#2642C2211yescharacteristic
25#2642C2211yescharacteristic
26#2641C2211yescharacteristic
27#2641C2211yescharacteristic
28#2629C2212yes—
29#2628C2212yes—
30#2591—22no—
31#2591—22no—
32#2570—22no—
33#2562—22no—
34#2562—22no—
35#2550—22no—
36#2441—22no—
37#2439—22no—
38#2331C2212yes—
39#2330C2212yes—
40#2311C2212yes—
41#2311C2212yes—
42#2285C2212yes—
43C4×#117C2211yescharacteristic
44#2074C2214yes—
45#2073C2214yes—
46#2072C2214yes—
47#2071C2214yes—
48#2026C2214yes—
49#2025C2214yes—
50#2024C2214yes—
51#2023C2214yes—
52#1992C2212yes—
53#1992C2212yes—
54#1992C2212yes—
55#1992C2212yes—
56#1989C2214yes—
57#1989C2214yes—
58#1982C2214yes—
59#1981C2214yes—
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1959C2212yes—
64#1936C2214yes—
65#1936C2214yes—
66#1935C2214yes—
67#1935C2214yes—
68#1934C2214yes—
69#1934C2214yes—
70#1933C2214yes—
71#1933C2214yes—
72#1824C2214yes—
73#1823C2214yes—
74#1788C2214yes—
75#1788C2214yes—
76#1768C2212yes—
77#1768C2212yes—
78#1750C2212yes—
79#1750C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C22×#41C2312yes—
84#532—28no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#125—22no—
90C2×#125—22no—
91C2×#125—28no—
92C2×#125C2312yes—
93C2×#125C2311yescharacteristic
94C2×#125C2311yescharacteristic
95C2×#125C2312yes—
96C2×#124C2312yes—
97C2×#121—22no—
98C2×C4×D4—22no—
99C2×#117—22no—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105#432—28no—
106C2×C4⋊C8C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×#97C2312yes—
119C2×#97C2312yes—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122#400—22no—
123#400—22no—
124#400—22no—
125#400—22no—
126#400—22no—
127#399—22no—
128#398—24no—
129#396—24no—
130#395—22no—
131#394—22no—
132#394—22no—
133#394—22no—
134#394—22no—
135#393—22no—
136#391—24no—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#391C2312yes—
141#390C2312yes—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—24no—
150#389C2314yes—
151#389C2314yes—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—24no—
161#387—24no—
162#386—22no—
163#386—22no—
164#386—22no—
165#386—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#382—22no—
172#382C2312yes—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185#376C2311yescharacteristic
186#376C2311yescharacteristic
187C43C2311yescharacteristic
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C2×#41—28no—
200C2×#41—44no—
201C2×#41—44no—
202C22×Q8—22no—
203C22×Q8C2412yes—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4C2412yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2411yescharacteristic
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—24no—
214C4⋊Q8—44no—
215C4⋊Q8—84no—
216#127—24no—
217#125—28no—
218#125—24no—
219#125—24no—
220#125—44no—
221#125—48no—
222#125—84no—
223#124—28no—
224#121—48no—
225C4×D4—48no—
226#117—42no—
227#117—42no—
228#117—84no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
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—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
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×C22⋊C4—22no—
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D411yescharacteristic
267C2×C42—24no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42#4118yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C4⋊C8—28no—
281#103—44no—
282#103—44no—
283#103—44no—
284#103—44no—
285#102—44no—
286#102—44no—
287#102—44no—
288#102—44no—
289C22⋊C8—28no—
290#97—24no—
291#97—24no—
292#97—42no—
293#97—42no—
294C4×C8—42no—
295C4×C8—42no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95C2×D412yes—
311#95C2×D412yes—
312#95C2×D412yes—
313#95C2×D412yes—
314C24—22no—
315C24C22×D411yescharacteristic
316#41—816no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—44no—
321C2×Q8—84no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—82no—
330C2×D4—82no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4#12418yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C2×#4114yes—
357C22×C4C2511yescharacteristic
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—84no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—84no—
411C42#12418yes—
412C42C2×#4114yes—
413C42C2×#4114yes—
414C23—22no—
415C23—22no—
416C23—21no—
417C23—21no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23C2×#12412yes—
426C23C23×D411yescharacteristic
427Q8—48no—
428Q8—164no—
429D4—48no—
430D4—88no—
431D4—88no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—28no—
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—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—48no—
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—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—88no—
470C2×C4—88no—
471C2×C4C2×#12412yes—
472C2×C4C2×#12414yes—
473C2×C4C2×#12414yes—
474C2×C4C22×#4112yes—
475C2×C4#58512yes—
476C2×C4#58512yes—
477C8—44no—
478C8—44no—
479C22—22no—
480C22—24no—
481C22—41no—
482C22—41no—
483C22—42no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22C22×#12411yescharacteristic
492C22#344911yescharacteristic
493C22#344911yescharacteristic
494C22#358512yes—
495C22#358612yes—
496C4—22no—
497C4—24no—
498C4—24no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—42no—
504C4—42no—
505C4—82no—
506C4—84no—
507C4—84no—
508C4—82no—
509C4—82no—
510C4#351014yes—
511C2—42no—
512C2—162no—
513C2#3533811yescharacteristic
514C2#6062112yes—
515C2#6101911yescharacteristic
516C2#6102011yescharacteristic
517C2#6102112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35294C2semidirect
3. #33282C2semidirect
4. #33282C2semidirect
5. #30217C2semidirect
6. #30216C2semidirect
7. #29361C2semidirect
8. #29360C2semidirect
9. #29359C2semidirect
10. #29358C2semidirect
11. #29120C2semidirect
12. #29119C2semidirect
13. #21838C2semidirect
15. #20467C2semidirect

Maximal subgroups

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

subgroupindex
#353412
#352942
#332822
#302172
#302162
#293612
#293602
#293592
#293582
#291202
#291192
#218382
#218092
#204672