Finite Group Databasebeta

← #7623120

Subgroups — #7623120

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60707C211yesmaximal, characteristic
2C2×#2950C211yesmaximal, characteristic, complemented
3#33686C214yesmaximal, complemented
4#33681C214yesmaximal, complemented
5#33677C212yesmaximal, complemented
6#33672C212yesmaximal, complemented
7#33668C212yesmaximal, complemented
8#33663C212yesmaximal, complemented
9#33648C211yesmaximal, characteristic, complemented
10#33645C211yesmaximal, characteristic, complemented
11#33632C212yesmaximal, complemented
12#33587C212yesmaximal, complemented
13#33518C211yesmaximal, characteristic, complemented
14C2×#2185C211yesmaximal, characteristic, complemented
15#33457C212yesmaximal, complemented
16#33455C212yesmaximal, complemented
17#33422C211yesmaximal, characteristic, complemented
18C23×Q16C2211yescharacteristic
19#3494—21no—
20C2×#519—21no—
21C2×#510C2212yes—
22C22×SD32C2212yes—
23#3180—22no—
24#3179—22no—
25#3176C2212yes—
26#3175C2212yes—
27#3169C2212yes—
28#3168C2212yes—
29#3164C2211yescharacteristic
30#3155C2211yescharacteristic
31#3136—21no—
32#3055—21no—
33#3050C2211yescharacteristic
34C2×#466C2211yescharacteristic
35#2951C2211yescharacteristic
36#2950—21no—
37#2950—21no—
38#2950—22no—
39#2950C2212yes—
40#2950C2212yes—
41#2950C2214yescomplemented
42#2948C2211yescharacteristic
43#2946—21no—
44C2×#441—22no—
45C2×C4×Q16C2212yescomplemented
46C2×C4×Q16C2212yescomplemented
47C2×#427C2211yescharacteristic
48C2×#419—21no—
49C2×#405C2211yescharacteristic
50#2237C2212yescomplemented
51#2236C2211yescharacteristic, complemented
52#2235C2211yescharacteristic, complemented
53#2234C2214yes—
54#2234C2212yescomplemented
55#2233C2212yes—
56#2233C2211yescharacteristic, complemented
57#2232C2212yes—
58#2232C2211yescharacteristic, complemented
59#2229C2214yescomplemented
60#2229C2214yescomplemented
61#2227C2214yescomplemented
62#2225C2214yescomplemented
63#2211C2214yescomplemented
64#2211C2214yescomplemented
65#2207C2214yescomplemented
66#2207C2214yescomplemented
67#2188C2212yescomplemented
68#2188C2212yescomplemented
69#2185C2212yes—
70#2185C2212yes—
71#2185C2214yes—
72#2185C2212yescomplemented
73#2185C2212yescomplemented
74#2176C2214yescomplemented
75#2175C2214yescomplemented
76#2175C2214yescomplemented
77#2175C2214yescomplemented
78C4×Q32C2214yescomplemented
79C4×SD32C2214yescomplemented
80C4×SD32C2214yescomplemented
81C4×SD32C2214yescomplemented
82#2164C2214yescomplemented
83#2163C2212yescomplemented
84#2162C2212yescomplemented
85#2159C2212yes—
86#2158C2212yes—
87#2151C2212yes—
88#2141C2212yes—
89#2140C2212yes—
90#2138C2212yes—
91C2×#358C2212yes—
92#2112C2211yescharacteristic
93C2×C22⋊C16C2211yescharacteristic
94#2108C2212yes—
95C23×Q8—21no—
96C2×AGL(1,ℤ/8)—21no—
97C2×#135C2311yescharacteristic
98C22×Q16—28no—
99C22×Q16—21no—
100C22×Q16—21no—
101C22×Q16—21no—
102C22×Q16C2312yes—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×D8C2311yescharacteristic
106C22×M16D411yescharacteristic
107C23×C8C2311yescharacteristic
108#555—41no—
109#553—41no—
110#553—41no—
111#551—42no—
112#547—41no—
113#546—42no—
114#544—21no—
115#542—22no—
116#540—22no—
117#535—21no—
118#532—22no—
119#529—22no—
120#525—21no—
121C2×#121—21no—
122#520—21no—
123#519—22no—
124#519—22no—
125#519—24no—
126#519—42no—
127#519—41no—
128#519—41no—
129#518—21no—
130C2×C4×Q8—22no—
131C2×C4×Q8—22no—
132C2×C4×Q8—41no—
133C2×C4×Q8—41no—
134C2×#117—21no—
135C2×#117—42no—
136C2×#117D411yescharacteristic
137#510—28no—
138C2×Q32C2314yes—
139C2×SD32—28no—
140C2×SD32C2314yes—
141C2×SD32C2314yes—
142C2×SD32C2314yes—
143C2×M32C2312yes—
144C22×C16C2312yes—
145#501—22no—
146#500C2312yes—
147#499C2312yes—
148#496—22no—
149#495C2312yes—
150#493C2312yes—
151#490—22no—
152#489—22no—
153#488—22no—
154#486—22no—
155#469—22no—
156#468—22no—
157#467C2311yescharacteristic
158#467C2311yescharacteristic
159#466—22no—
160#466—21no—
161#466—21no—
162#466C2312yes—
163#466C2311yescharacteristic
164#466C2312yes—
165#466C2311yescharacteristic
166#465—22no—
167#465—21no—
168#465—21no—
169#442—22no—
170#442—22no—
171#441—24no—
172#441—24no—
173#441—24no—
174#441—24no—
175#440—22no—
176#440—22no—
177C4×Q16—22no—
178C4×Q16—22no—
179C4×Q16—22no—
180C4×Q16—22no—
181C4×Q16C2312yes—
182C4×Q16C2314yes—
183C4×Q16C2314yes—
184C4×Q16C2314yes—
185C4×Q16C2314yes—
186C4×SD16—22no—
187C4×D8—22no—
188C4×D8C2312yes—
189C4×D8C2312yes—
190C4×D8C2312yes—
191#428—21no—
192#427—21no—
193#427—21no—
194#427C2312yes—
195#427C2311yescharacteristic
196#427C2312yes—
197C2×#107C2311yescharacteristic
198C2×#107C2311yescharacteristic
199C2×#107C2311yescharacteristic
200C2×#106—21no—
201#419—22no—
202#419—22no—
203#419—24no—
204#418—21no—
205#417—21no—
206#416—21no—
207C2×#103—22no—
208C2×#103—22no—
209C2×#102—21no—
210#405D411yescharacteristic
211#405D412yes—
212#405D411yescharacteristic
213#405C2311yescharacteristic
214#405C2311yescharacteristic
215#405C2312yes—
216C2×#97D412yes—
217C2×C4×C8C2312yes—
218#367C2312yes—
219#367C2312yes—
220#367C2314yes—
221#366C2312yes—
222#366C2312yes—
223#366C2314yes—
224C4⋊C16C2314yes—
225C4⋊C16C2314yes—
226#358C2314yes—
227#358C2314yes—
228#358C2312yes—
229#358C2312yes—
230#358C2314yes—
231#358C2314yes—
232#357C2312yes—
233#357C2312yes—
234#357C2314yes—
235#357C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2311yescharacteristic
238C22⋊C16C2312yes—
239C22⋊C16C2312yes—
240C22⋊C16C2312yes—
241#346C2314yes—
242C4×C16C2314yes—
243C2×#41—21no—
244C2×#41—42no—
245C22×Q8—28no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—44no—
250C22×Q8—41no—
251C22×Q8—41no—
252C22×Q8—41no—
253C22×D4—21no—
254C22×D4—41no—
255C23×C4—41no—
256C23×C4C2×D411yescharacteristic
257AGL(1,ℤ/8)—44no—
258#135—24no—
259C2×Q16—28no—
260C2×Q16—28no—
261C2×Q16—28no—
262C2×Q16—28no—
263C2×Q16—28no—
264C2×Q16—22no—
265C2×Q16—22no—
266C2×Q16—22no—
267C2×Q16—24no—
268C2×Q16#4118yescomplemented
269C2×Q16C2411yescharacteristic
270C2×Q16C2414yes—
271C2×SD16—21no—
272C2×SD16—22no—
273C2×D8—21no—
274C2×D8—22no—
275C2×D8—24no—
276C2×D8C2412yes—
277C2×D8C2411yescharacteristic
278C2×M16—42no—
279C2×M16C2×D411yescharacteristic
280C2×M16C2×D411yescharacteristic
281C2×M16C2×D412yes—
282C22×C8—24no—
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D412yes—
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289C22×C8C2411yescharacteristic
290C4⋊Q8—22no—
291C4⋊Q8—41no—
292#127—22no—
293#126—42no—
294#126—42no—
295#125—22no—
296#125—41no—
297#124—22no—
298#124—42no—
299#124—42no—
300#123—41no—
301#123—42no—
302#123—42no—
303#123—42no—
304#123—42no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—42no—
308C22⋊Q8—42no—
309#121—22no—
310#121—22no—
311#121—21no—
312#121—21no—
313#121—41no—
314#121—41no—
315#121—42no—
316#121—41no—
317#121—41no—
318#121—42no—
319#121—42no—
320#120—41no—
321#120—42no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—22no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×Q8—42no—
330C4×Q8—42no—
331C4×Q8—44no—
332C4×Q8—44no—
333C4×Q8—42no—
334C4×Q8—42no—
335C4×Q8—42no—
336C4×Q8—42no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—41no—
341C4×D4—41no—
342C4×D4—42no—
343C4×D4—41no—
344C4×D4—41no—
345C4×D4—42no—
346C4×D4—42no—
347#117—22no—
348#117—22no—
349#117—21no—
350#117—41no—
351#117—41no—
352#117—41no—
353#117—44no—
354#117—44no—
355#117—44no—
356#117—44no—
357#117—42no—
358#117D812yes—
359#117D811yescharacteristic
360#117D811yescharacteristic
361#117C2×D411yescharacteristic
362#117C2×D411yescharacteristic
363#117C2×D412yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—42no—
368C2×C4⋊C4—42no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4D811yescharacteristic
371C2×C4⋊C4D811yescharacteristic
372C2×C22⋊C4—41no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4D812yes—
376C2×C42—41no—
377C2×C42—42no—
378C2×C42—42no—
379C2×C42C2×D412yes—
380Q32—28no—
381SD32—28no—
382SD32—28no—
383SD32—28no—
384M32—24no—
385C2×C16—24no—
386C2×C16C2412yes—
387C2×C16C2414yes—
388C2×C16C2412yes—
389#107—21no—
390#107—21no—
391#107—21no—
392#107—22no—
393#107C2412yes—
394#107C2412yes—
395#107C2412yes—
396#107C2411yescharacteristic
397#107C2411yescharacteristic
398#106—22no—
399#106—22no—
400#106—21no—
401#106—22no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—21no—
407#103—21no—
408#103—22no—
409#102—22no—
410#102—22no—
411#102—21no—
412#102—21no—
413#102—22no—
414#102—22no—
415#102—22no—
416#97C2×D412yes—
417#97C2×D412yes—
418#97C2×D412yes—
419#97C2×D412yes—
420C4×C8C2×D412yes—
421C4×C8C2×D412yes—
422C4×C8C2412yes—
423C4×C8C2412yes—
424C24—41no—
425C24C2×D811yescharacteristic
426#41—42no—
427#41—42no—
428#41—164no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—24no—
433C2×Q8—21no—
434C2×Q8—48no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—42no—
449C2×D4—41no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—41no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—162no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4C2×D812yes—
471C22×C4C2×D811yescharacteristic
472C22×C4C2×D811yescharacteristic
473C22×C4C2×D811yescharacteristic
474C22×C4C2×D812yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478Q16—28no—
479Q16—28no—
480Q16—28no—
481Q16—24no—
482Q16—22no—
483Q16C2×#4118yes—
484SD16—44no—
485SD16—42no—
486D8—24no—
487D8—22no—
488D8—44no—
489D8—42no—
490M16—22no—
491M16—22no—
492M16—42no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—42no—
499C2×C8C22×D411yescharacteristic
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C2×C8C22×D411yescharacteristic
503C2×C8C22×D411yescharacteristic
504C2×C8C2×#4114yes—
505C2×C8C2511yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—41no—
524C4⋊C4C2×D811yescharacteristic
525C4⋊C4C2×D811yescharacteristic
526C4⋊C4C2×D811yescharacteristic
527C4⋊C4C2×D811yescharacteristic
528C4⋊C4C2×D812yes—
529C4⋊C4C2×D812yes—
530C22⋊C4—41no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—41no—
542C22⋊C4—42no—
543C22⋊C4—41no—
544C22⋊C4C2×D812yes—
545C22⋊C4C2×D812yes—
546C22⋊C4C2×D812yes—
547C22⋊C4C2×D812yes—
548C42—42no—
549C42—42no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—41no—
555C42—42no—
556C42C2×D812yes—
557C42C2×D812yes—
558C42C22×D412yes—
559C42C22×D412yes—
560C16—22no—
561C16—22no—
562C16—24no—
563C23—22no—
564C23—22no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—82no—
569C23—161no—
570C23—161no—
571C23C22×D811yescharacteristic
572C23C22×D811yescharacteristic
573C23C22×D811yescharacteristic
574Q8—28no—
575Q8—48no—
576Q8—48no—
577Q8—48no—
578Q8—44no—
579Q8—41no—
580Q8—41no—
581Q8—162no—
582D4—42no—
583D4—41no—
584D4—41no—
585D4—42no—
586D4—82no—
587D4—84no—
588D4—82no—
589D4—82no—
590D4—162no—
591D4—162no—
592D4—161no—
593D4—162no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—162no—
617C2×C4—162no—
618C2×C4—162no—
619C2×C4—161no—
620C2×C4#52512yes—
621C2×C4C22×D812yes—
622C2×C4C22×D812yes—
623C2×C4C22×D812yes—
624C2×C4C22×D812yes—
625C2×C4C22×D811yescharacteristic
626C2×C4C22×D811yescharacteristic
627C2×C4C22×D811yescharacteristic
628C2×C4#57512yes—
629C2×C4C23×D411yescharacteristic
630C8—21no—
631C8—21no—
632C8—22no—
633C8—22no—
634C8#52512yes—
635C8C22×#4112yes—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—44no—
644C22—81no—
645C22—82no—
646C22—81no—
647C22—82no—
648C22—82no—
649C22—81no—
650C22—322no—
651C22#310012yes—
652C22#341612yes—
653C22C23×D811yescharacteristic
654C4—21no—
655C4—21no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—48no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4—81no—
667C4#310012yes—
668C4C2×#52511yescharacteristic
669C4C2×#57511yescharacteristic
670C2—22no—
671C2—22no—
672C2—41no—
673C2—161no—
674C2—162no—
675C2C2×#310011yescharacteristic
676C2C2×#341611yescharacteristic
677C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60707C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2950C2semidirect
3. #33686C2semidirect
4. #33681C2semidirect
5. #33677C2semidirect
6. #33672C2semidirect
7. #33668C2semidirect
8. #33663C2semidirect
9. #33648C2semidirect
10. #33645C2semidirect
11. #33632C2semidirect
12. #33587C2semidirect
13. #33518C2semidirect
14. C2×#2185C2semidirect
15. #33457C2semidirect
16. #33455C2semidirect
17. #33422C2semidirect
41. #2950C22semidirect
45. C2×C4×Q16C22semidirect
46. C2×C4×Q16C22semidirect
50. #2237C22semidirect
51. #2236C22semidirect
52. #2235C22semidirect
54. #2234C22semidirect
56. #2233C22semidirect
58. #2232C22semidirect
59. #2229C22semidirect
60. #2229C22semidirect
61. #2227C22semidirect
62. #2225C22semidirect
63. #2211C22semidirect
64. #2211C22semidirect
65. #2207C22semidirect
66. #2207C22semidirect
67. #2188C22semidirect
68. #2188C22semidirect
72. #2185C22semidirect
73. #2185C22semidirect
74. #2176C22semidirect
75. #2175C22semidirect
76. #2175C22semidirect
77. #2175C22semidirect
78. C4×Q32C22semidirect
79. C4×SD32C22semidirect
80. C4×SD32C22semidirect
81. C4×SD32C22semidirect
82. #2164C22semidirect
83. #2163C22semidirect
84. #2162C22semidirect
268. C2×Q16#41semidirect

Maximal subgroups

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

subgroupindex
#607072
C2×#29502
#336862
#336812
#336772
#336722
#336682
#336632
#336482
#336452
#336322
#335872
#335182
C2×#21852
#334572
#334552
#334222