Finite Group Databasebeta

← #6640540

Subgroups — #6640540

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic, complemented
3#32978C212yesmaximal, complemented
4#32258C214yesmaximal, complemented
5#32026C214yesmaximal, complemented
6#31994C218yesmaximal, complemented
7#29890C212yesmaximal, complemented
8#27610C212yesmaximal, complemented
9#24550C214yesmaximal, complemented
10#24060C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C2×#535—21no—
13C2×#532C2211yescharacteristic
14C22×#123—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#480C2214yes—
18C22×C22⋊C8C2212yes—
19#2847C2212yes—
20#2847C2212yes—
21#2617—21no—
22#2617—21no—
23#2547C2212yes—
24#2544—22no—
25#2541C2212yes—
26#2537C2214yes—
27#2535—22no—
28#2486—22no—
29#2486—22no—
30#2486C2214yes—
31#2486C2214yes—
32C2×#398—21no—
33C2×#397—21no—
34#2311—21no—
35C2×#389C2212yes—
36#2285C2211yescharacteristic
37C2×#381—21no—
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40#2099C2214yescomplemented
41#2083C2214yescomplemented
42#2075C2218yescomplemented
43#2075C2218yescomplemented
44#2057C2214yescomplemented
45#2041C2214yescomplemented
46#2033C2218yescomplemented
47#2033C2218yescomplemented
48#1918C2218yescomplemented
49#1893C2218yescomplemented
50#1874C2214yes—
51#1874C2218yescomplemented
52#1869C2214yes—
53#1869C2214yes—
54#1862C2218yescomplemented
55#1852C2214yes—
56#1852C2218yescomplemented
57#1823C2214yes—
58#1817C2212yes—
59#1808C2218yescomplemented
60#1779C2212yes—
61C22×#41—21no—
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67#535—48no—
68#532—44no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127—22no—
71C2×#127C2311yescharacteristic
72C2×#124—22no—
73C2×#124—21no—
74C2×#123—24no—
75C2×#123—24no—
76C2×#123—48no—
77C2×#121—24no—
78C2×#121—21no—
79C2×#121C2312yes—
80C2×#121C2314yes—
81C2×#121C2314yes—
82C2×#120—22no—
83C2×#120—22no—
84C2×C4×D4—21no—
85C2×C4×D4—44no—
86C2×C4×D4D412yes—
87C2×C4×D4D412yes—
88C2×C4×D4C2312yes—
89C2×C4×D4C2312yes—
90C2×C4×D4C2312yes—
91C2×#117—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4—22no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4D411yescharacteristic
103C22×C22⋊C4D411yescharacteristic
104C22×C42C2311yescharacteristic
105#480—48no—
106C2×#107C2318yes—
107C2×#107C2318yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×C22⋊C8—44no—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2318yes—
114#398—24no—
115#398—24no—
116#397—22no—
117#397—22no—
118#397—24no—
119#397—24no—
120#394—24no—
121#394—24no—
122#392—22no—
123#392—22no—
124#391—22no—
125#390—22no—
126#389—22no—
127#389—22no—
128#389C2314yes—
129#389C2314yes—
130#386—22no—
131#386—22no—
132#386—24no—
133#386—24no—
134#384C2312yes—
135#382—21no—
136#381—24no—
137#381—24no—
138#380—21no—
139#380—24no—
140#380—22no—
141#380—24no—
142#380C2314yes—
143#380C2318yes—
144#380C2314yes—
145#380C2314yes—
146#380C2312yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2312yes—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#336C2314yes—
157#336C2314yes—
158C25C2×D411yescharacteristic
159C25C2×D411yescharacteristic
160C2×#41—44no—
161C2×#41—44no—
162C2×#41—44no—
163C22×Q8—21no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4D818yes—
172C22×D4C2×D412yes—
173C22×D4C2×D412yes—
174C22×D4C2412yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C22×C8—48no—
196C22×C8—44no—
197C22×C8C2418yes—
198C4⋊Q8—42no—
199#127—42no—
200#127—48no—
201#124—44no—
202#124—48no—
203#123—416no—
204#123—416no—
205#121—28no—
206#121—44no—
207#121—416no—
208#121—48no—
209#121—48no—
210#120—48no—
211#120—48no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—44no—
220#117—44no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2414yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2414yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×D412yes—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266#107—48no—
267#107—48no—
268#102—48no—
269#102—48no—
270C22⋊C8—28no—
271C22⋊C8—44no—
272C22⋊C8—48no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—22no—
278#95—22no—
279#95—24no—
280#95—24no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24C2×D814yes—
291C24C2×D814yes—
292C24C22×D412yes—
293C24C22×D412yes—
294C24C22×D411yescharacteristic
295#41—816no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×D4—24no—
300C2×D4—28no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—416no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—42no—
315C2×D4—82no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—84no—
320C2×D4C2×D818yes—
321C2×D4C2×D818yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—416no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—88no—
359C22×C4—816no—
360C22×C4C2×D814yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4H(5,𝔽2)14yes—
369C22×C4C2511yescharacteristic
370C2×C8—28no—
371C2×C8—28no—
372C2×C8—28no—
373C2×C8—88no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—82no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—48no—
395C22⋊C4—416no—
396C22⋊C4—44no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—44no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—44no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—28no—
419C23—48no—
420C23—48no—
421C23—44no—
422C23—48no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—82no—
430C23—84no—
431C23—82no—
432C23#53412yes—
433C23C22×D814yes—
434C23C22×D814yes—
435C23C22×D812yes—
436C23#57512yes—
437C23C23×D411yescharacteristic
438Q8—84no—
439D4—48no—
440D4—48no—
441D4—48no—
442D4—84no—
443D4—84no—
444D4—88no—
445D4—168no—
446D4—168no—
447C2×C4—28no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—84no—
481C2×C4—88no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—88no—
485C2×C4—816no—
486C2×C4—88no—
487C2×C4#53412yes—
488C2×C4C22×D812yes—
489C2×C4C22×D812yes—
490C2×C4C2×H(5,𝔽2)12yes—
491C2×C4#58514yes—
492C8—48no—
493C22—24no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—28no—
500C22—28no—
501C22—416no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—88no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—84no—
514C22#318614yes—
515C22C2×#53411yescharacteristic
516C22#348512yes—
517C22C23×D811yescharacteristic
518C22C2×#57511yescharacteristic
519C22#358612yes—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—41no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—84no—
528C4—82no—
529C4—84no—
530C4—84no—
531C4#353314yes—
532C2—24no—
533C2—44no—
534C2—44no—
535C2—162no—
536C2#3695511yescharacteristic
537C2C2×#318611yescharacteristic
538C2#6075411yescharacteristic
539C2#6169412yes—
540C2#6171112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2486C2semidirect
3. #32978C2semidirect
4. #32258C2semidirect
5. #32026C2semidirect
6. #31994C2semidirect
7. #29890C2semidirect
8. #27610C2semidirect
9. #24550C2semidirect
10. #24060C2semidirect
11. #21738C2semidirect
40. #2099C22semidirect
41. #2083C22semidirect
42. #2075C22semidirect
43. #2075C22semidirect
44. #2057C22semidirect
45. #2041C22semidirect
46. #2033C22semidirect
47. #2033C22semidirect
48. #1918C22semidirect
49. #1893C22semidirect
51. #1874C22semidirect
54. #1862C22semidirect
56. #1852C22semidirect
59. #1808C22semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#24862
#329782
#322582
#320262
#319942
#298902
#276102
#245502
#240602
#217382