Finite Group Databasebeta

← #7505183

Subgroups — #7505183

Order 512.

1537 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic
2#35339C211yesmaximal, characteristic, complemented
3#31596C214yesmaximal, complemented
4#31591C214yesmaximal, complemented
5#31580C214yesmaximal, complemented
6#31575C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23631C211yesmaximal, characteristic
11#23610C211yesmaximal, characteristic
12#23512C211yesmaximal, characteristic
13#21205C211yesmaximal, characteristic
14#21158C211yesmaximal, characteristic, complemented
15#21061C211yesmaximal, characteristic, complemented
16#21058C211yesmaximal, characteristic
17#21026C211yesmaximal, characteristic, complemented
18#20880C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#405C2211yescharacteristic
36#2864—22no—
37#2647—22no—
38#2570C2214yes—
39#2570C2214yes—
40#2569—24no—
41#2498—24no—
42#2448C2214yes—
43#2446C2214yes—
44#2423—22no—
45#2421—22no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2297C2211yescharacteristic
53C2×#391C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yes—
58#2097C2214yes—
59#2097C2214yes—
60#2095C2212yes—
61#2093C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2084C2214yes—
65#2074C2214yes—
66#2073C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2043C2214yes—
70#2042C2214yes—
71#1960C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1956C2212yes—
75#1943C2214yes—
76#1942C2214yes—
77#1928C2212yes—
78#1928C2214yes—
79#1927C2212yes—
80#1927C2214yes—
81#1926C2212yes—
82#1925C2212yes—
83#1917C2214yes—
84#1916C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1805C2214yes—
88#1803C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536—42no—
105C2×C4⋊Q8C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109#519—24no—
110#519—42no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2314yes—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#428—42no—
124#427—42no—
125C2×#107—24no—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106—42no—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134#424—42no—
135C2×C4⋊C8C2314yes—
136#419—24no—
137#419—42no—
138#417—42no—
139#417—42no—
140#416—24no—
141#416—42no—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103C2311yescharacteristic
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2311yescharacteristic
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2311yescharacteristic
156C2×#102C2311yescharacteristic
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160#405—42no—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#400—24no—
164#400—24no—
165#400—24no—
166#399—24no—
167#398—24no—
168#398—24no—
169#398—24no—
170#397—24no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#391C2314yes—
176#390C2312yes—
177#390C2312yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#384D412yes—
199#384C2312yes—
200#384C2312yes—
201#382—21no—
202#382—21no—
203#382D412yes—
204#382C2312yes—
205#382C2312yes—
206#382C2314yes—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380D414yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218#376—21no—
219C2×#95—22no—
220C2×#95—22no—
221C43—21no—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8#4114yes—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8#4114yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2411yescharacteristic
273C22×C8C2411yescharacteristic
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276#124—24no—
277#124—42no—
278#123—48no—
279C22⋊Q8—48no—
280C4×Q8—28no—
281C4×Q8—44no—
282C4×Q8—48no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—48no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—24no—
344#107—24no—
345#107—42no—
346#107—42no—
347#106—44no—
348#106—42no—
349#106—42no—
350C4⋊C8—44no—
351#103—28no—
352#103—24no—
353#103—24no—
354#103—22no—
355#103—22no—
356#103—44no—
357#103—42no—
358#103—42no—
359#103—42no—
360#103—42no—
361#103—42no—
362#102—22no—
363#102—22no—
364#102—42no—
365#102—42no—
366#102—42no—
367#97—42no—
368C4×C8—42no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D414yes—
384#95C2×D414yes—
385C24—21no—
386C24C22×D411yescharacteristic
387#41—44no—
388#41—44no—
389#41—44no—
390#41—44no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8C2×#4114yes—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455M16—44no—
456M16—44no—
457C2×C8—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C2×C8C2×#4112yes—
473C2×C8C2×#4112yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—48no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—82no—
524C42—82no—
525C23—22no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23#53612yes—
533C23#54612yes—
534C23#55012yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—24no—
538Q8—42no—
539Q8—48no—
540Q8—42no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—84no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—162no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4#53612yes—
591C2×C4#53914yes—
592C2×C4#53914yes—
593C2×C4#54612yes—
594C2×C4#55012yes—
595C2×C4#55214yes—
596C2×C4#56214yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C8—22no—
604C8—22no—
605C8—42no—
606C8—42no—
607C8—42no—
608C22—24no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—82no—
613C22—81no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22#269414yes—
618C22C2×#53611yescharacteristic
619C22C2×#54611yescharacteristic
620C22C2×#55011yescharacteristic
621C22#358412yes—
622C22#358612yes—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4#269414yes—
637C4C2×#53911yescharacteristic
638C4C2×#53911yescharacteristic
639C4C2×#55211yescharacteristic
640C4C2×#56211yescharacteristic
641C2—24no—
642C2—161no—
643C2C2×#269411yescharacteristic
644C2#6087612yes—
645C2#6120412yes—
646C2#6141311yescharacteristic
647C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35339C2semidirect
3. #31596C2semidirect
4. #31591C2semidirect
5. #31580C2semidirect
6. #31575C2semidirect
8. C2×#2097C2semidirect
9. #23661C2semidirect
14. #21158C2semidirect
15. #21061C2semidirect
17. #21026C2semidirect
18. #20880C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356732
#353392
#315962
#315912
#315802
#315752
#237372
C2×#20972
#236612
#236312
#236102
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