Finite Group Databasebeta

← #6510716

Subgroups — #6510716

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3#29118C214yesmaximal, complemented
4#27971C214yesmaximal, complemented
5#27721C214yesmaximal, complemented
6#27714C214yesmaximal, complemented
7#27705C214yesmaximal, complemented
8#27698C214yesmaximal, complemented
9#21768C211yesmaximal, characteristic
10#21762C211yesmaximal, characteristic
11#20701C211yesmaximal, characteristic
12#20699C211yesmaximal, characteristic
13#20527C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2855—22no—
19#2842—21no—
20#2840—21no—
21#2665—22no—
22#2665—22no—
23#2628C2214yes—
24#2625—22no—
25#2607—22no—
26#2503C2212yes—
27#2502C2212yes—
28#2499C2214yes—
29#2484—22no—
30#2483—21no—
31#2483—21no—
32#2439C2214yes—
33#2439C2214yes—
34#2317C2212yes—
35#2310C2212yes—
36#2280C2211yescharacteristic
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39#2101C2214yescomplemented
40#2101C2214yescomplemented
41#2100C2214yescomplemented
42#2099C2214yescomplemented
43#2074C2214yescomplemented
44#2074C2214yescomplemented
45#2024C2214yescomplemented
46#2024C2214yescomplemented
47#1993C2214yescomplemented
48#1990C2214yescomplemented
49#1935C2214yescomplemented
50#1935C2214yescomplemented
51#1933C2214yescomplemented
52#1933C2214yescomplemented
53#1894C2214yes—
54#1892C2212yes—
55#1892C2212yes—
56#1873C2214yes—
57#1873C2214yes—
58#1836C2214yescomplemented
59#1835C2214yescomplemented
60#1825C2214yes—
61#1825C2214yes—
62#1825C2214yes—
63#1824C2212yes—
64#1824C2212yes—
65#1823C2212yes—
66#1823C2212yes—
67#1787C2212yes—
68#1787C2212yes—
69#1753C2214yes—
70#1752C2212yes—
71C8×C22⋊C4C2212yes—
72#1750C2214yes—
73#1736C2212yes—
74#1736C2212yes—
75#1735C2212yes—
76#1735C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×M16C2312yes—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×C4⋊Q8C2314yes—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2312yes—
88C2×C22⋊Q8C2312yes—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C2×#117C2312yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C42—21no—
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109#407—24no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×#97C2314yes—
116C2×C4×C8C2314yes—
117#400—24no—
118#400—24no—
119#399—24no—
120#398—24no—
121#397—24no—
122#397—24no—
123#396—24no—
124#396—24no—
125#395—24no—
126#393—24no—
127#393—24no—
128#393—24no—
129#391—24no—
130#391—24no—
131#391—24no—
132#391C2314yes—
133#391C2314yes—
134#389—24no—
135#389C2314yes—
136#389C2314yes—
137#387—24no—
138#387—24no—
139#387—24no—
140#387—24no—
141#385—22no—
142#385—22no—
143#385—22no—
144#385—22no—
145#384C2314yes—
146#382C2314yes—
147#382C2314yes—
148#381—21no—
149#381—21no—
150#381—22no—
151#381—22no—
152#381—22no—
153#381—22no—
154#381D412yes—
155#380D412yes—
156#380D412yes—
157#380C2312yes—
158#380C2312yes—
159#380C2312yes—
160#380C2312yes—
161#379—21no—
162#379—21no—
163#379D411yescharacteristic
164#379D411yescharacteristic
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4C2312yes—
170C4×C22⋊C4C2312yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2314yes—
185C25C2×D411yescharacteristic
186C22×Q8—21no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8C2412yes—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4#4114yes—
203C23×C4#4114yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×M16—24no—
208C2×M16—42no—
209C2×M16—42no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8C2412yes—
215C22×C8C2412yes—
216C22×C8C2414yes—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—44no—
224C22⋊Q8—44no—
225#117—24no—
226#117—24no—
227#117—24no—
228#117—24no—
229#117—42no—
230#117—42no—
231#117—44no—
232#117—44no—
233#117—42no—
234#117—42no—
235#117—42no—
236#117—42no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
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—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—44no—
283C2×C42C2414yes—
284#103—44no—
285#103—44no—
286#103—44no—
287#103—44no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292#97—44no—
293C4×C8—44no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95C2×D414yes—
309#95C2×D414yes—
310C24—42no—
311C24—42no—
312C24—42no—
313C24#12414yes—
314C24C22×D411yescharacteristic
315C24C22×D411yescharacteristic
316C24C22×D411yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—84no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4#12414yes—
356C22×C4#12414yes—
357C22×C4#12414yes—
358C22×C4C22×D414yes—
359C22×C4C22×D414yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4C2511yescharacteristic
370M16—84no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—82no—
379C2×C8—82no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C42—24no—
418C42—24no—
419C42—44no—
420C42—44no—
421C42—82no—
422C42—82no—
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×#12412yes—
437C23C2×#12412yes—
438C23C2×#12412yes—
439C23C23×D411yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—44no—
443Q8—84no—
444Q8—168no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—44no—
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—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4C2×#12412yes—
485C2×C4C2×#12412yes—
486C2×C4C2×#12412yes—
487C2×C4C22×#4111yescharacteristic
488C2×C4C22×#4111yescharacteristic
489C2×C4#58512yes—
490C2×C4#58512yes—
491C8—42no—
492C8—42no—
493C8—44no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—88no—
507C22C22×#12411yescharacteristic
508C22#348512yes—
509C22#358412yes—
510C22#358612yes—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—44no—
515C4—44no—
516C4—84no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4#349111yescharacteristic
523C4#349111yescharacteristic
524C4#349112yes—
525C2—42no—
526C2—42no—
527C2—42no—
528C2#3545511yescharacteristic
529C2#6061211yescharacteristic
530C2#6062011yescharacteristic
531C2#6176014yes—
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. #29118C2semidirect
4. #27971C2semidirect
5. #27721C2semidirect
6. #27714C2semidirect
7. #27705C2semidirect
8. #27698C2semidirect
39. #2101C22semidirect
40. #2101C22semidirect
41. #2100C22semidirect
42. #2099C22semidirect
43. #2074C22semidirect
44. #2074C22semidirect
45. #2024C22semidirect
46. #2024C22semidirect
47. #1993C22semidirect
48. #1990C22semidirect
49. #1935C22semidirect
50. #1935C22semidirect
51. #1933C22semidirect
52. #1933C22semidirect
58. #1836C22semidirect
59. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#353112
#291182
#279712
#277212
#277142
#277052
#276982
#217682
#217622
#207012
#206992
#205272