Finite Group Databasebeta

← #7625909

Subgroups — #7625909

Order 512.

10950 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#548C211yesmaximal, characteristic
2C2×#3428C211yesmaximal, characteristic, complemented
3C22×#519C211yesmaximal, characteristic, complemented
4#34312C218yesmaximal, complemented
5#34294C218yesmaximal, complemented
6#34293C218yesmaximal, complemented
7#34211C216yesmaximal, complemented
8#34204C216yesmaximal, complemented
9#34203C216yesmaximal, complemented
10#34182C211yesmaximal, characteristic, complemented
11#34174C211yesmaximal, characteristic, complemented
12#34173C211yesmaximal, characteristic, complemented
13C4×#548C218yesmaximal, complemented
14D4×#117C216yesmaximal, complemented
15C4×#519C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C2×#548C4132yescomplemented
19C2×#548C2214yes—
20C2×#548C2214yes—
21C2×#548C2214yes—
22C2×#548C2214yes—
23C22×#124C2213yes—
24C22×C22⋊Q8C2213yes—
25C22×#121C2213yes—
26#3428—216no—
27C2×#520C2218yescomplemented
28C2×#519—216no—
29C2×#519C2211yescharacteristic
30C2×#519C2211yescharacteristic
31C2×#519C2211yescharacteristic
32C2×#519C2216yes—
33C2×#519C2211yescharacteristic
34C2×#519C2218yescomplemented
35C2×#518C2216yes—
36C2×C4×#41C2218yescomplemented
37C22×C4×Q8C2211yescharacteristic
38C22×C4×Q8C2218yescomplemented
39C22×C4×D4C2213yes—
40C22×#117C2216yes—
41C22×#117C2216yes—
42#2367C22124yescomplemented
43#2367C22124yescomplemented
44#2363C22124yescomplemented
45#2362C22124yescomplemented
46#2361C22124yescomplemented
47#2360C22124yescomplemented
48#2354C22112yescomplemented
49#2352C22112yescomplemented
50#2352C22112yescomplemented
51#2351C2218yescomplemented
52Q8×C4⋊C4C2218yescomplemented
53D4×C4⋊C4C22112yescomplemented
54#2344C22112yes—
55#2342C2214yes—
56#2342C22112yes—
57#2342C22124yes—
58#2341C2218yescomplemented
59#2341C2218yescomplemented
60Q8×C22⋊C4C2218yescomplemented
61Q8×C22⋊C4C2218yescomplemented
62D4×C22⋊C4C22112yes—
63#2326C22124yescomplemented
64#2324C22124yescomplemented
65#2323C22124yescomplemented
66#2316C2216yes—
67#2310C2216yes—
68#2309C2216yes—
69C4×#124C22124yescomplemented
70C4×C22⋊Q8C22124yescomplemented
71C4×#121C22124yescomplemented
72#2293C22112yescomplemented
73#2290C22112yescomplemented
74#2290C22112yescomplemented
75#2285C2216yes—
76#2280C2216yes—
77#2280C2216yes—
78#2276C2218yescomplemented
79#2274C2216yes—
80C42×Q8C2218yescomplemented
81C42×D4C22112yescomplemented
82C4×#117C2216yes—
83C22×#41—28no—
84C22×#41—28no—
85C22×#41C2×C4116yescomplemented
86C22×#41C2314yes—
87C22×#41C2314yes—
88C23×Q8—28no—
89C23×Q8—28no—
90C23×Q8C2×C4116yescomplemented
91C23×Q8C2313yes—
92C23×Q8C2314yes—
93C23×Q8C2314yes—
94C23×Q8C2311yescharacteristic
95C23×D4C2313yes—
96C23×D4C2313yes—
97C24×C4C2313yes—
98C24×C4C2313yes—
99#548C2×C4164yescomplemented
100#548C2×C4164yescomplemented
101#548C2×C4164yescomplemented
102#548C2×C4164yescomplemented
103C2×#124C2×C4148yescomplemented
104C2×#124C23112yes—
105C2×#124C23112yes—
106C2×C22⋊Q8C2×C4148yescomplemented
107C2×C22⋊Q8C23112yes—
108C2×C22⋊Q8C23112yes—
109C2×#121C2×C4148yescomplemented
110C2×#121C23112yes—
111C2×#121C23112yes—
112#520—264no—
113#519—28no—
114#519—28no—
115#519—28no—
116#519—264no—
117#519—248no—
118#519D4116yescomplemented
119#518—248no—
120C4×#41—264no—
121C2×C4×Q8—264no—
122C2×C4×Q8C2×C4116yescomplemented
123C2×C4×Q8C23124yes—
124C2×C4×Q8C2318yes—
125C2×C4×Q8C2318yes—
126C2×C4×Q8C2318yes—
127C2×C4×Q8C23124yes—
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2318yes—
131C2×C4×D4C2×C4148yescomplemented
132C2×C4×D4C23124yes—
133C2×C4×D4C23124yes—
134C2×C4×D4C23124yes—
135C2×C4×D4C2316yes—
136C2×C4×D4C2316yes—
137C2×C4×D4C2316yes—
138C2×C4×D4C2316yes—
139C2×#117—248no—
140C2×#117—248no—
141C2×#117C2316yes—
142C2×#117C2316yes—
143C2×#117C2316yes—
144C2×#117C23112yes—
145C2×#117C23112yes—
146C2×#117C23112yes—
147C2×#117C2314yes—
148C2×#117C23112yes—
149C2×#117C23112yes—
150C2×#117C2316yes—
151C22×C4⋊C4C23112yes—
152C22×C4⋊C4C23112yes—
153C22×C4⋊C4C2313yes—
154C22×C4⋊C4C2313yes—
155C22×C22⋊C4C2314yes—
156C22×C22⋊C4C23112yes—
157C22×C22⋊C4C2311yescharacteristic
158C22×C22⋊C4C2313yes—
159C22×C22⋊C4C2316yes—
160C22×C42C23112yes—
161C22×C42C23112yes—
162C22×C42C2313yes—
163#391C23124yes—
164#391C23124yes—
165#390C23124yes—
166#390C23124yes—
167#390C23124yes—
168#389C23124yes—
169#389C23124yes—
170#389C23124yes—
171#388C23148yes—
172#388C23148yes—
173#388C23124yes—
174#388C2318yes—
175#388C23124yes—
176#388C2318yes—
177#387C23124yes—
178#387C23124yes—
179#386C23124yes—
180#386C23124yes—
181#385C23148yes—
182#385C23124yes—
183#385C2318yes—
184#384C23112yes—
185#382C23112yes—
186#381C23112yes—
187#381C2314yes—
188#380C23112yes—
189#380C23112yes—
190#380C23112yes—
191#380C23112yes—
192#380C23112yes—
193#380C2314yes—
194#379C23112yes—
195#379C2314yes—
196C4×C4⋊C4C23124yes—
197C4×C4⋊C4C23124yes—
198C4×C4⋊C4C23112yes—
199C4×C4⋊C4C23112yes—
200C4×C4⋊C4C23112yes—
201C4×C4⋊C4C23112yes—
202C4×C4⋊C4C23124yes—
203C4×C22⋊C4C23112yes—
204C4×C22⋊C4C23112yes—
205C4×C22⋊C4C23112yes—
206C4×C22⋊C4C23112yes—
207C4×C22⋊C4C23112yes—
208C4×C22⋊C4C2314yes—
209C4×C22⋊C4C2318yes—
210C4×C22⋊C4C23124yes—
211C4×C22⋊C4C23148yes—
212#376C23112yes—
213C43C23112yes—
214C25C2411yescharacteristic
215C25C2413yes—
216C2×#41—2128no—
217C2×#41—216no—
218C2×#41—216no—
219C2×#41—216no—
220C2×#41—216no—
221C2×#41C22×C4132yes—
222C2×#41C22×C4132yes—
223C22×Q8—2128no—
224C22×Q8—216no—
225C22×Q8—216no—
226C22×Q8—216no—
227C22×Q8—216no—
228C22×Q8—212no—
229C22×Q8—212no—
230C22×Q8C22×C4124yes—
231C22×Q8C22×C4132yes—
232C22×Q8C22×C4132yes—
233C22×Q8C22×C418yes—
234C22×Q8C2×D418yes—
235C22×Q8#4118yes—
236C22×Q8C24112yes—
237C22×Q8C2414yes—
238C22×D4—212no—
239C22×D4—212no—
240C22×D4—412no—
241C22×D4C22×C4124yes—
242C22×D4C22×C4124yes—
243C22×D4C2416yes—
244C22×D4C24112yes—
245C22×D4C2416yes—
246C23×C4—212no—
247C23×C4—212no—
248C23×C4—212no—
249C23×C4—212no—
250C23×C4C22×C4124yes—
251C23×C4C22×C4124yes—
252C23×C4C2418yes—
253C23×C4C2413yes—
254C23×C4C2411yescharacteristic
255C23×C4C2416yes—
256C23×C4C2416yes—
257C23×C4C2416yes—
258C23×C4C2416yes—
259C23×C4C24124yes—
260C23×C4C2411yescharacteristic
261C23×C4C2413yes—
262C23×C4C2413yes—
263#124C22×C4196yes—
264#124C22×C4196yes—
265C22⋊Q8C22×C4196yes—
266C22⋊Q8C22×C4196yes—
267#121C22×C4196yes—
268#121C22×C4196yes—
269C4×Q8—232no—
270C4×Q8—232no—
271C4×Q8—232no—
272C4×Q8—296no—
273C4×Q8—296no—
274C4×Q8C22×C4132yes—
275C4×Q8C22×C4132yes—
276C4×Q8C2×D4164yescomplemented
277C4×D4—296no—
278C4×D4—296no—
279C4×D4—448no—
280C4×D4C22×C4148yes—
281C4×D4C22×C4148yes—
282C4×D4C22×C4148yes—
283C4×D4C22×C4148yes—
284#117—224no—
285#117—224no—
286#117—224no—
287#117—248no—
288#117—248no—
289#117—216no—
290#117—248no—
291#117—248no—
292#117—248no—
293#117C2×D4148yes—
294C2×C4⋊C4—248no—
295C2×C4⋊C4—248no—
296C2×C4⋊C4C22×C4124yes—
297C2×C4⋊C4C22×C4124yes—
298C2×C4⋊C4C2418yes—
299C2×C4⋊C4C24112yes—
300C2×C4⋊C4C24112yes—
301C2×C4⋊C4C24124yes—
302C2×C4⋊C4C24112yes—
303C2×C4⋊C4C24124yes—
304C2×C4⋊C4C2418yes—
305C2×C4⋊C4C2416yes—
306C2×C4⋊C4C2416yes—
307C2×C4⋊C4C24112yes—
308C2×C4⋊C4C24112yes—
309C2×C22⋊C4—216no—
310C2×C22⋊C4—248no—
311C2×C22⋊C4C22×C418yes—
312C2×C22⋊C4C22×C4124yes—
313C2×C22⋊C4C22×C4148yes—
314C2×C22⋊C4C2414yes—
315C2×C22⋊C4C2414yes—
316C2×C22⋊C4C2418yes—
317C2×C22⋊C4C24124yes—
318C2×C22⋊C4C2414yes—
319C2×C22⋊C4C2418yes—
320C2×C22⋊C4C24124yes—
321C2×C22⋊C4C2418yes—
322C2×C22⋊C4C24124yes—
323C2×C22⋊C4C2411yescharacteristic
324C2×C22⋊C4C2411yescharacteristic
325C2×C22⋊C4C2411yescharacteristic
326C2×C22⋊C4C2411yescharacteristic
327C2×C22⋊C4C2416yes—
328C2×C22⋊C4C2416yes—
329C2×C22⋊C4C24124yes—
330C2×C22⋊C4C2414yes—
331C2×C42—248no—
332C2×C42—248no—
333C2×C42C22×C4124yes—
334C2×C42C2418yes—
335C2×C42C24112yes—
336C2×C42C24124yes—
337C2×C42C24112yes—
338C2×C42C24112yes—
339C2×C42C24124yes—
340C2×C42C2418yes—
341C2×C42C2416yes—
342C2×C42C2416yes—
343C2×C42C24112yes—
344#95C2418yes—
345#95C24124yes—
346#95C2418yes—
347#95C2418yes—
348#95C2418yes—
349#95C24124yes—
350#95C24124yes—
351#95C24124yes—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—26no—
356C24—26no—
357C24—42no—
358C24—46no—
359C24—46no—
360C24C23×C414yes—
361C24C23×C4112yes—
362C24C2511yescharacteristic
363C24C2516yes—
364C24C2511yescharacteristic
365C24C2511yescharacteristic
366#41—2128no—
367#41—2128no—
368C2×Q8—2128no—
369C2×Q8—2128no—
370C2×Q8—296no—
371C2×Q8—224no—
372C2×Q8—224no—
373C2×Q8C4×D4164yescomplemented
374C2×Q8C23×C4148yes—
375C2×Q8C23×C4116yes—
376C2×Q8C22×D4116yes—
377C2×Q8C2×#41116yes—
378C2×D4—224no—
379C2×D4—296no—
380C2×D4—224no—
381C2×D4—224no—
382C2×D4—412no—
383C2×D4—448no—
384C2×D4C23×C4124yes—
385C2×D4C23×C4148yes—
386C2×D4C23×C4124yes—
387C22×C4—22no—
388C22×C4—26no—
389C22×C4—22no—
390C22×C4—26no—
391C22×C4—296no—
392C22×C4—296no—
393C22×C4—212no—
394C22×C4—212no—
395C22×C4—212no—
396C22×C4—212no—
397C22×C4—212no—
398C22×C4—212no—
399C22×C4—212no—
400C22×C4—212no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—248no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—46no—
407C22×C4—424no—
408C22×C4—424no—
409C22×C4C23×C4112yes—
410C22×C4C23×C414yes—
411C22×C4C23×C4124yes—
412C22×C4C23×C4124yes—
413C22×C4C23×C4124yes—
414C22×C4C23×C4124yes—
415C22×C4C23×C414yes—
416C22×C4C23×C4112yes—
417C22×C4C23×C4112yes—
418C22×C4C22×D4112yes—
419C22×C4C2×#41112yes—
420C22×C4C2518yes—
421C22×C4C2518yes—
422C22×C4C2518yes—
423C22×C4C2511yescharacteristic
424C22×C4C2511yescharacteristic
425C22×C4C2511yescharacteristic
426C22×C4C2516yes—
427C22×C4C2516yes—
428C22×C4C2511yescharacteristic
429C22×C4C2518yes—
430C22×C4C2516yes—
431C4⋊C4—216no—
432C4⋊C4—224no—
433C4⋊C4—248no—
434C4⋊C4—224no—
435C4⋊C4—224no—
436C4⋊C4—48no—
437C4⋊C4—424no—
438C4⋊C4C23×C4124yes—
439C4⋊C4C23×C4124yes—
440C4⋊C4C23×C4148yes—
441C4⋊C4C22×D4148yes—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—248no—
448C22⋊C4—424no—
449C22⋊C4—48no—
450C22⋊C4—424no—
451C22⋊C4C23×C414yes—
452C22⋊C4C23×C414yes—
453C22⋊C4C23×C414yes—
454C22⋊C4C23×C414yes—
455C22⋊C4C23×C4124yes—
456C22⋊C4C23×C4124yes—
457C22⋊C4C23×C4196yes—
458C22⋊C4C22×D4116yes—
459C42—216no—
460C42—224no—
461C42—248no—
462C42—224no—
463C42—224no—
464C42—48no—
465C42—424no—
466C42C23×C4124yes—
467C42C23×C4124yes—
468C42C22×D4148yes—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—26no—
476C23—21no—
477C23—26no—
478C23—212no—
479C23—22no—
480C23—22no—
481C23—212no—
482C23—28no—
483C23—28no—
484C23—224no—
485C23—44no—
486C23—412no—
487C23—412no—
488C23—48no—
489C23—48no—
490C23—812no—
491C23—812no—
492C23C24×C4112yes—
493C23C24×C412yes—
494C23C24×C412yes—
495C23C24×C412yes—
496C23C23×D412yes—
497C23C22×#4112yes—
498C23#58514yes—
499C23C2611yescharacteristic
500Q8—296no—
501Q8C2×C4×D4164yes—
502D4—248no—
503D4—296no—
504D4—424no—
505C2×C4—21no—
506C2×C4—26no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—26no—
511C2×C4—28no—
512C2×C4—212no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—224no—
518C2×C4—248no—
519C2×C4—248no—
520C2×C4—248no—
521C2×C4—248no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—44no—
532C2×C4—412no—
533C2×C4—432no—
534C2×C4—848no—
535C2×C4C2×C4×D4148yes—
536C2×C4C24×C412yes—
537C2×C4C24×C412yes—
538C2×C4C24×C4112yes—
539C2×C4C24×C4112yes—
540C2×C4C24×C412yes—
541C2×C4C24×C412yes—
542C2×C4C24×C4112yes—
543C2×C4C23×D4116yes—
544C2×C4C23×D4112yes—
545C2×C4C22×#41112yes—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—212no—
556C22—212no—
557C22—22no—
558C22—22no—
559C22—212no—
560C22—416no—
561C22—412no—
562C22—42no—
563C22—42no—
564C22—412no—
565C22—824no—
566C22C22×C4×D414yes—
567C22#342814yescomplemented
568C22C25×C411yescharacteristic
569C22C24×D411yescharacteristic
570C22C23×#4111yescharacteristic
571C22C2×#58512yes—
572C222−1+612yes—
573C4—216no—
574C4—22no—
575C4—22no—
576C4—212no—
577C4—212no—
578C4—22no—
579C4—22no—
580C4—48no—
581C4—48no—
582C4—48no—
583C4C22×C4×D4124yes—
584C2—24no—
585C2—24no—
586C2—42no—
587C2—412no—
588C2C23×C4×D411yescharacteristic
589C2C2×#342811yescharacteristic
590C2#6265311yescharacteristic
591C2#6300212yes—
592C2#6300512yes—
subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC22×#548C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3428C2semidirect
3. C22×#519C2semidirect
4. #34312C2semidirect
5. #34294C2semidirect
6. #34293C2semidirect
7. #34211C2semidirect
8. #34204C2semidirect
9. #34203C2semidirect
10. #34182C2semidirect
11. #34174C2semidirect
12. #34173C2semidirect
13. C4×#548C2semidirect
14. D4×#117C2semidirect
15. C4×#519C2semidirect
18. C2×#548C4semidirect
27. C2×#520C22semidirect
34. C2×#519C22semidirect
36. C2×C4×#41C22semidirect
38. C22×C4×Q8C22semidirect
42. #2367C22semidirect
43. #2367C22semidirect
44. #2363C22semidirect
45. #2362C22semidirect
46. #2361C22semidirect
47. #2360C22semidirect
48. #2354C22semidirect
49. #2352C22semidirect
50. #2352C22semidirect
51. #2351C22semidirect
52. Q8×C4⋊C4C22semidirect
53. D4×C4⋊C4C22semidirect
58. #2341C22semidirect
59. #2341C22semidirect
60. Q8×C22⋊C4C22semidirect
61. Q8×C22⋊C4C22semidirect
63. #2326C22semidirect
64. #2324C22semidirect
65. #2323C22semidirect
69. C4×#124C22semidirect
70. C4×C22⋊Q8C22semidirect
71. C4×#121C22semidirect
72. #2293C22semidirect
73. #2290C22semidirect
74. #2290C22semidirect
78. #2276C22semidirect
80. C42×Q8C22semidirect
81. C42×D4C22semidirect
85. C22×#41C2×C4semidirect
90. C23×Q8C2×C4semidirect
99. #548C2×C4semidirect
100. #548C2×C4semidirect
101. #548C2×C4semidirect
102. #548C2×C4semidirect
103. C2×#124C2×C4semidirect
106. C2×C22⋊Q8C2×C4semidirect
109. C2×#121C2×C4semidirect
118. #519D4semidirect
122. C2×C4×Q8C2×C4semidirect
131. C2×C4×D4C2×C4semidirect
276. C4×Q8C2×D4semidirect
373. C2×Q8C4×D4semidirect
567. C22#3428semidirect

Maximal subgroups

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

subgroupindex
C22×#5482
C2×#34282
C22×#5192
#343122
#342942
#342932
#342112
#342042
#342032
#341822
#341742
#341732
C4×#5482
D4×#1172
C4×#5192