Finite Group Databasebeta

← #6630248

Subgroups — #6630248

Order 512.

2322 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic, complemented
4C4×#518C211yesmaximal, characteristic
5#33201C212yesmaximal, complemented
6#33162C212yesmaximal, complemented
7#33119C214yesmaximal, complemented
8#33032C212yesmaximal, complemented
9#33026C212yesmaximal, complemented
10#32914C214yesmaximal
11#21514C218yesmaximal, complemented
12#21392C211yesmaximal, characteristic, complemented
13#21389C211yesmaximal, characteristic, complemented
14#20815C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C2×#533C2211yescharacteristic
17C2×#518C2212yes—
18C22×C4×D4C2211yescharacteristic
19#3110—28no—
20#3101—28no—
21C2×#484C2212yes—
22C2×#483C2212yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×#461C2212yes—
26C2×#461C2212yes—
27C2×#461C2218yes—
28C2×#450C2212yes—
29C2×#450C2212yes—
30C2×#440C2214yes—
31C2×C4×SD16C2214yes—
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34#2868C2211yescharacteristic, complemented
35#2862C2211yescharacteristic, complemented
36#2862C2211yescharacteristic, complemented
37#2862C2211yescharacteristic, complemented
38#2621C2214yes—
39#2565—22no—
40#2565—22no—
41#2499—22no—
42#2499—22no—
43#2327C2214yescomplemented
44#2275—22no—
45#2275C2214yes—
46#2274C2212yes—
47#2273—22no—
48C42×D4—22no—
49C42×D4C2212yes—
50C42×D4C2212yes—
51C4×#117C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C2×C4×C22⋊C4—22no—
54#1988C2212yescomplemented
55#1987C2212yescomplemented
56#1981C2212yescomplemented
57#1981C2212yescomplemented
58#1981C2218yescomplemented
59#1949C2218yescomplemented
60#1928C2218yescomplemented
61#1927C2218yescomplemented
62#1877C2218yes—
63#1877C2218yes—
64#1873C2214yes—
65#1873C2214yes—
66#1842C2212yescomplemented
67#1831C2212yescomplemented
68#1804C2218yes—
69#1798C2214yes—
70#1768C2214yescomplemented
71#1763C2214yescomplemented
72C4×#103C2214yescomplemented
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×SD16C2314yes—
76C22×SD16C2314yes—
77#533—24no—
78#533—24no—
79#533—24no—
80#533—24no—
81C2×C4⋊Q8C2314yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#125—21no—
87C2×#125—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92#518—216no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×D4—24no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—24no—
101C2×C4×D4D418yes—
102C2×C4×D4C2314yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—24no—
113C22×C42—22no—
114C22×C42—22no—
115C22×C42C2311yescharacteristic
116#484—28no—
117#483—28no—
118#476—28no—
119#474—28no—
120#461—28no—
121#461—28no—
122#461—232no—
123#450—28no—
124#450—28no—
125#440—216no—
126C4×SD16—216no—
127#432—24no—
128#431—24no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2318yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2318yes—
137C2×#103C2318yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144#398—28no—
145#397—28no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391C2314yes—
151#391C2314yes—
152#387—28no—
153#387—28no—
154#384—22no—
155#384—22no—
156#382—22no—
157#382—22no—
158#382C2314yes—
159C4×C4⋊C4—24no—
160C4×C4⋊C4D412yes—
161C4×C4⋊C4D412yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2312yes—
165C4×C4⋊C4C2312yes—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—22no—
170C4×C22⋊C4—22no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—24no—
173C4×C22⋊C4—24no—
174C4×C22⋊C4—24no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4D412yes—
178#376—21no—
179#376C2312yes—
180C2×#95—24no—
181C43—21no—
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184#340C2318yes—
185#340C2318yes—
186#336C2318yes—
187C25—21no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8C2412yes—
191C22×Q8C2412yes—
192C22×D4—28no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4C2×D414yes—
196C22×D4C2412yes—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—84no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C2×SD16—28no—
208C2×SD16—28no—
209C2×SD16—216no—
210C2×SD16—216no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—216no—
222#125—44no—
223#125—44no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—44no—
235C4×Q8—44no—
236C4×D4—232no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—28no—
244C4×D4C2×D418yes—
245C4×D4C2×D418yes—
246C4×D4#41116yes—
247#117—28no—
248#117—28no—
249#117—44no—
250#117—44no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C22⋊C4—232no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4C2×D414yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—84no—
288C2×C42—84no—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297C2×C42C2412yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2412yes—
301#107—28no—
302#106—28no—
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—216no—
306#103—28no—
307#103—28no—
308#103—216no—
309#103—216no—
310#102—28no—
311#102—28no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314#97—24no—
315C4×C8—24no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—28no—
326C24—22no—
327C24—24no—
328C24—82no—
329C24—82no—
330C24C22×D411yescharacteristic
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—44no—
341C2×D4—28no—
342C2×D4—216no—
343C2×D4—216no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4C22×D414yes—
348C2×D4C2×#4118yes—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—216no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—816no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4#12114yes—
374C22×C4#12114yes—
375C22×C4C22×D414yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D414yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4112yes—
384C22×C4C2511yescharacteristic
385SD16—416no—
386SD16—416no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—28no—
390C2×C8—28no—
391C4⋊C4—24no—
392C4⋊C4—28no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—416no—
410C4⋊C4C22×D414yes—
411C4⋊C4C2×#4118yes—
412C22⋊C4—28no—
413C22⋊C4—216no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—216no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C42—28no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—44no—
428C42—44no—
429C42—82no—
430C42—84no—
431C42—84no—
432C42#12114yes—
433C42#12114yes—
434C42C22×D412yes—
435C42C22×D412yes—
436C42C2×#4112yes—
437C42C2×#4112yes—
438C42C2×#4118yes—
439C23—216no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—28no—
444C23—22no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—84no—
450C23—816no—
451C23C2×#12112yes—
452C23C23×D411yescharacteristic
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457Q8—84no—
458Q8—84no—
459D4—216no—
460D4—24no—
461D4—28no—
462D4#53218yes—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—416no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—816no—
492C2×C4—88no—
493C2×C4—84no—
494C2×C4—816no—
495C2×C4C2×#12112yes—
496C2×C4C2×#12112yes—
497C2×C4C2×#12112yes—
498C2×C4C2×#12112yes—
499C2×C4C2×#12112yes—
500C2×C4#52518yes—
501C2×C4#57714yes—
502C2×C4C23×D411yescharacteristic
503C2×C4C22×#4114yes—
504C2×C4C22×#4111yescharacteristic
505C8—44no—
506C8—44no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—28no—
511C22—44no—
512C22—44no—
513C22—48no—
514C22—48no—
515C22—44no—
516C22—816no—
517C22C22×#12111yescharacteristic
518C22C2×#52512yes—
519C22C2×#57712yes—
520C22#358612yes—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—28no—
526C4—44no—
527C4—44no—
528C4—48no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4#304214yes—
534C4C2×#53212yes—
535C4#344412yes—
536C2—42no—
537C2—48no—
538C2#3662411yescharacteristic
539C2C2×#304211yescharacteristic
540C2#6042711yescharacteristic
541C2#6059114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
3. #35322C2semidirect
5. #33201C2semidirect
6. #33162C2semidirect
7. #33119C2semidirect
8. #33032C2semidirect
9. #33026C2semidirect
11. #21514C2semidirect
12. #21392C2semidirect
13. #21389C2semidirect
14. #20815C2semidirect
34. #2868C22semidirect
35. #2862C22semidirect
36. #2862C22semidirect
37. #2862C22semidirect
43. #2327C22semidirect
54. #1988C22semidirect
55. #1987C22semidirect
56. #1981C22semidirect
57. #1981C22semidirect
58. #1981C22semidirect
59. #1949C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
66. #1842C22semidirect
67. #1831C22semidirect
70. #1768C22semidirect
71. #1763C22semidirect
72. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#31012
#353222
C4×#5182
#332012
#331622
#331192
#330322
#330262
#329142
#215142
#213922
#213892
#208152