Finite Group Databasebeta

← #6503099

Subgroups — #6503099

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37481C211yesmaximal, characteristic
2#34370C211yesmaximal, characteristic
3#29172C214yesmaximal, complemented
4#26417C214yesmaximal, complemented
5#26351C214yesmaximal, complemented
6#26018C214yesmaximal, complemented
7#25973C214yesmaximal, complemented
8#24850C214yesmaximal, complemented
9#21939C211yesmaximal, characteristic
10#21762C211yesmaximal, characteristic
11#20972C211yesmaximal, characteristic
12#20836C211yesmaximal, characteristic
13#20660C211yesmaximal, characteristic
14C23×M16C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#479C2214yescomplemented
17C2×#427C2212yes—
18C2×#424C2212yes—
19C2×#419C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21#2842—21no—
22#2842—21no—
23#2667—22no—
24#2667—22no—
25#2629C2214yes—
26#2608—22no—
27#2441C2214yes—
28#2441C2214yes—
29#2378—22no—
30#2362C2214yes—
31#2345—22no—
32#2345C2212yes—
33Q8×C22⋊C4C2212yes—
34#2332—22no—
35#2323C2214yes—
36#2312—21no—
37#2312—21no—
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41#2280C2211yescharacteristic
42#2100C2214yescomplemented
43#2096C2214yescomplemented
44#2088C2214yescomplemented
45#2086C2214yescomplemented
46#2068C2214yescomplemented
47#2068C2214yescomplemented
48#2059C2214yescomplemented
49#2026C2214yescomplemented
50#1989C2214yescomplemented
51#1947C2214yescomplemented
52#1920C2214yescomplemented
53#1919C2214yescomplemented
54#1894C2214yes—
55#1892C2212yes—
56#1892C2212yes—
57#1892C2212yes—
58#1892C2212yes—
59#1873C2214yes—
60#1865C2214yescomplemented
61#1863C2214yescomplemented
62#1854C2214yes—
63#1854C2214yes—
64#1851C2214yes—
65#1850C2212yes—
66#1850C2212yes—
67#1823C2212yes—
68#1809C2214yes—
69#1809C2214yes—
70#1808C2212yes—
71#1807C2212yes—
72#1803C2214yescomplemented
73#1787C2212yes—
74#1736C2212yes—
75#1735C2212yes—
76#1729C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×M16—24no—
80C22×M16—44no—
81C22×M16C2312yes—
82C22×M16C2312yes—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#125C2314yes—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8Q818yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8C2314yes—
95C2×#117C2312yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102#479—48no—
103#427—44no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108#424—44no—
109C2×C4⋊C8C2314yes—
110C2×C4⋊C8C2314yes—
111#419—44no—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×C22⋊C8—24no—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119#399—24no—
120#399—24no—
121#398—24no—
122#398—24no—
123#396—24no—
124#396—24no—
125#396—24no—
126#396—24no—
127#391—24no—
128#391C2314yes—
129#389—24no—
130#389—24no—
131#389—24no—
132#389C2314yes—
133#389C2314yes—
134#389C2314yes—
135#388—24no—
136#388—24no—
137#388—24no—
138#388—24no—
139#387—24no—
140#387—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#384C2314yes—
145#381—21no—
146#381—21no—
147#381—21no—
148#381D412yes—
149#380—22no—
150#380—22no—
151#380D412yes—
152#380D412yes—
153#380C2312yes—
154#380C2312yes—
155#380C2312yes—
156#380C2312yes—
157#380C2312yes—
158#380C2312yes—
159#379—21no—
160#379—21no—
161#379—21no—
162#379D411yescharacteristic
163#379D411yescharacteristic
164C4×C4⋊C4—24no—
165C4×C4⋊C4C2314yes—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—22no—
173C4×C22⋊C4—22no—
174C4×C22⋊C4C2312yes—
175C4×C22⋊C4C2312yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#336C2314yes—
185#336C2314yes—
186C25C2×D411yescharacteristic
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8C2×Q814yes—
192C22×Q8C2412yes—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×Q814yes—
201C23×C4#4114yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C2×M16—24no—
206C2×M16—24no—
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—42no—
210C2×M16—42no—
211C2×M16—416no—
212C22×C8—22no—
213C22×C8—24no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224#125—44no—
225#125—44no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8C2×Q818yes—
231C22⋊Q8C2×Q818yes—
232C4×Q8—44no—
233C4×Q8—44no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117—42no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42C2414yes—
285#107—44no—
286#107—44no—
287#106—28no—
288#106—44no—
289C4⋊C8—28no—
290C4⋊C8—44no—
291#103—28no—
292#103—44no—
293#103—44no—
294#103—44no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95C2×D414yes—
310#95C2×D414yes—
311C24—42no—
312C24—42no—
313C24—42no—
314C24C22⋊Q814yes—
315C24C22×D411yescharacteristic
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—48no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×Q8C22×Q814yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—44no—
341C22×C4—48no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4C22⋊Q814yes—
351C22×C4C22×D414yes—
352C22×C4C22×D414yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C22×Q812yes—
358C22×C4C22×Q812yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C4C2511yescharacteristic
363M16—48no—
364M16—48no—
365C2×C8—28no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—44no—
376C2×C8—82no—
377C2×C8—82no—
378C2×C8H(5,𝔽2)14yes—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4C22×Q814yes—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C42—24no—
417C42—24no—
418C42—84no—
419C42—82no—
420C42—82no—
421C42—82no—
422C42—84no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—88no—
436C23C2×C22⋊Q812yes—
437C23C2×C22⋊Q812yes—
438C23C2×C22⋊Q812yes—
439C23C23×D411yescharacteristic
440Q8—28no—
441Q8—42no—
442Q8—42no—
443Q8—84no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4C2×C22⋊Q812yes—
481C2×C4C2×C22⋊Q812yes—
482C2×C4C2×C22⋊Q812yes—
483C2×C4C23×Q811yescharacteristic
484C2×C4C22×#4111yescharacteristic
485C2×C4C2×H(5,𝔽2)12yes—
486C2×C4#58512yes—
487C8—24no—
488C8—42no—
489C8—42no—
490C8—42no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—88no—
504C22C22×C22⋊Q811yescharacteristic
505C22#349612yes—
506C22#358412yes—
507C22#358612yes—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—44no—
512C4—44no—
513C4—44no—
514C4—44no—
515C4—84no—
516C4—84no—
517C4—84no—
518C4#349011yescharacteristic
519C4#349111yescharacteristic
520C4#350112yes—
521C2—42no—
522C2—42no—
523C2—42no—
524C2#3535911yescharacteristic
525C2#6048911yescharacteristic
526C2#6049111yescharacteristic
527C2#6231414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29172C2semidirect
4. #26417C2semidirect
5. #26351C2semidirect
6. #26018C2semidirect
7. #25973C2semidirect
8. #24850C2semidirect
16. C2×#479C22semidirect
42. #2100C22semidirect
43. #2096C22semidirect
44. #2088C22semidirect
45. #2086C22semidirect
46. #2068C22semidirect
47. #2068C22semidirect
48. #2059C22semidirect
49. #2026C22semidirect
50. #1989C22semidirect
51. #1947C22semidirect
52. #1920C22semidirect
53. #1919C22semidirect
60. #1865C22semidirect
61. #1863C22semidirect
72. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#374812
#343702
#291722
#264172
#263512
#260182
#259732
#248502
#219392
#217622
#209722
#208362
#206602