Finite Group Databasebeta

← #497552

Subgroups — #497552

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61234C211yesmaximal, characteristic
2D4×M32C211yesmaximal, characteristic, complemented
3D4×#106C211yesmaximal, characteristic, complemented
4#13287C211yesmaximal, characteristic, complemented
5#13278C211yesmaximal, characteristic, complemented
6#13065C212yesmaximal, complemented
7#13001C212yesmaximal, complemented
8#12870C212yesmaximal, complemented
9#12816C211yesmaximal, characteristic, complemented
10C4×#360C211yesmaximal, characteristic, complemented
11C2×#1386C212yesmaximal, complemented
12#3475—21no—
13C22×M32C2212yes—
14#3342C411yescharacteristic, complemented
15#3341C411yescharacteristic, complemented
16#3340—21no—
17#3339C412yescomplemented
18#3338C411yescharacteristic, complemented
19#3337—22no—
20#3336C412yescomplemented
21#3335—21no—
22#3334C411yescharacteristic, complemented
23#3160C2211yescharacteristic
24#3091—22no—
25#3076—21no—
26#3050C2212yes—
27C4×#135C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#427C2212yes—
30C22×#106C2212yes—
31D4×C4⋊C4—21no—
32#2169—21no—
33#2168—22no—
34D4×C16—21no—
35#2150C2211yescharacteristic
36C2×#360C2211yescharacteristic
37C2×#360C2211yescharacteristic
38C2×#360C2212yes—
39#2112C2212yes—
40C4×M32C2211yescharacteristic
41#1948—21no—
42#1853—22no—
43#1844C2211yescharacteristic
44#1807C2212yes—
45C4×#106C2211yescharacteristic
46#1386C2212yes—
47#1386C2212yes—
48#1386C2212yes—
49#1386C2211yescharacteristic
50#1386C2212yes—
51#1386C2211yescharacteristic
52#1386C2212yes—
53C2×#135C2311yescharacteristic
54C23×C8C2312yes—
55#556—21no—
56#555—21no—
57#554—21no—
58#553—22no—
59#552—22no—
60#552—41no—
61#551—42no—
62#547—21no—
63#546—41no—
64#533—21no—
65#529—42no—
66#529—41no—
67#525—22no—
68C4×#41—21no—
69C4×#41—41no—
70C2×C4×D4—41no—
71C2×C4×D4D411yescharacteristic
72C2×#117—22no—
73C22×C4⋊C4—22no—
74C2×M32—22no—
75C2×M32—24no—
76C2×M32—42no—
77C2×M32C2311yescharacteristic
78C2×M32C2312yes—
79C2×M32C2311yescharacteristic
80C22×C16—22no—
81#500C2×C411yescharacteristic, complemented
82#500C2×C411yescharacteristic, complemented
83#499—21no—
84#499C2×C412yescomplemented
85#498—21no—
86#485—22no—
87#485—22no—
88#484—22no—
89#483—22no—
90#482—22no—
91#482—22no—
92#481—22no—
93#480—22no—
94#479—22no—
95#478—22no—
96#477—21no—
97#476—21no—
98#475—21no—
99#474—21no—
100#467C2×C412yescomplemented
101#467C2×C412yescomplemented
102#466C2×C412yescomplemented
103#466C2×C412yescomplemented
104#465—22no—
105#465—22no—
106C4×Q16C2×C411yescharacteristic, complemented
107C4×Q16C2×C411yescharacteristic, complemented
108C4×SD16—21no—
109C4×SD16—21no—
110C4×D8C2×C411yescharacteristic, complemented
111C4×D8C2×C411yescharacteristic, complemented
112C8×D4C2×C411yescharacteristic
113C8×D4C2×C412yes—
114C8×D4C2×C411yescharacteristic
115C8×D4D411yescharacteristic
116C8×D4D412yes—
117C8×D4D411yescharacteristic
118#427—22no—
119#427—22no—
120#427—21no—
121#427C2×C412yescomplemented
122#427C2×C412yescomplemented
123#427C2312yes—
124#427C2311yescharacteristic
125C2×#107C2×C412yescomplemented
126C2×#107C2×C412yescomplemented
127C2×#106—22no—
128C2×#106—24no—
129C2×#106—22no—
130C2×#106—21no—
131C2×#106—42no—
132C2×#106C2311yescharacteristic
133C2×#106C2312yes—
134C2×#106C2311yescharacteristic
135C2×C4⋊C8D411yescharacteristic
136#416—22no—
137#416—21no—
138C2×C22⋊C8D412yes—
139C2×C4×C8C2311yescharacteristic
140#389—41no—
141#385—42no—
142#384—21no—
143#380—22no—
144C4×C4⋊C4—21no—
145C4⋊C16—21no—
146C4⋊C16—21no—
147#360—22no—
148#360—22no—
149#360—42no—
150C22⋊C16—22no—
151C22⋊C16—22no—
152#346—21no—
153C4×C16—21no—
154#340—22no—
155#340—21no—
156C2×#41—21no—
157C2×#41—41no—
158C22×D4D811yescharacteristic
159C23×C4—42no—
160C23×C4C2×D412yes—
161#135—21no—
162#135—21no—
163#135—21no—
164#135—21no—
165C2×Q16C22×C411yescharacteristic
166C2×SD16—21no—
167C2×D8C22×C411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C22×C412yes—
173C22×C8C22×C412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2411yescharacteristic
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181#126—42no—
182#126—42no—
183#126—42no—
184#125—22no—
185#125—42no—
186#125—41no—
187#124—41no—
188#124—42no—
189#123—42no—
190#123—42no—
191#123—42no—
192#123—42no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—42no—
199#121—22no—
200#121—22no—
201#121—42no—
202#121—41no—
203#120—42no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—84no—
218C4×D4C22⋊C411yescharacteristic
219C4×D4C22⋊C412yes—
220C4×D4C22⋊C411yescharacteristic
221C4×D4SD1611yescharacteristic
222C4×D4SD1611yescharacteristic
223C4×D4SD1612yes—
224#117—22no—
225#117—22no—
226#117—21no—
227#117—22no—
228#117—42no—
229#117—42no—
230#117—41no—
231#117—42no—
232#117—41no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4D811yescharacteristic
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4D812yes—
247C2×C42—42no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42C2×D411yescharacteristic
251M32—21no—
252M32—21no—
253M32—22no—
254M32—22no—
255M32—21no—
256M32—44no—
257M32#4112yescomplemented
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—22no—
261C2×C16—44no—
262#107—22no—
263#107—22no—
264#107—21no—
265#107—21no—
266#107C22×C411yescharacteristic
267#107C22×C411yescharacteristic
268#107C22×C412yes—
269#107C22×C412yes—
270#106—22no—
271#106—22no—
272#106—22no—
273#106—21no—
274#106—22no—
275#106—21no—
276#106—21no—
277#106—22no—
278#106—22no—
279#106—21no—
280#106—44no—
281#106—81no—
282#106#4112yescomplemented
283C4⋊C8C22⋊C411yescharacteristic
284C4⋊C8C22⋊C411yescharacteristic
285C4⋊C8C22⋊C412yes—
286#103—21no—
287#103—21no—
288#103—22no—
289#103—22no—
290#102—21no—
291#102—21no—
292#102—22no—
293#102—22no—
294C22⋊C8C22⋊C412yes—
295C22⋊C8C22⋊C412yes—
296C22⋊C8C22⋊C412yes—
297C22⋊C8C22⋊C412yes—
298C4×C8C22×C411yescharacteristic
299C4×C8C22×C411yescharacteristic
300C4×C8C2×D411yescharacteristic
301C4×C8C2×D411yescharacteristic
302#95—41no—
303#95—42no—
304C24C2×D812yes—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—84no—
310C2×Q8—21no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4#10212yes—
319C2×D4#10211yescharacteristic
320C2×D4#10211yescharacteristic
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—162no—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×D812yes—
339C22×C4C2×D811yescharacteristic
340C22×C4C2×D811yescharacteristic
341C22×C4C2×SD1612yes—
342C22×C4C2×SD1612yes—
343C22×C4C22×D411yescharacteristic
344Q16—21no—
345Q16—21no—
346SD16—41no—
347SD16—41no—
348D8—21no—
349D8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8C2×C22⋊C411yescharacteristic
363C2×C8C2×C22⋊C411yescharacteristic
364C2×C8C2×C22⋊C412yes—
365C2×C8C2×C22⋊C412yes—
366C2×C8C23×C411yescharacteristic
367C2×C8C22×D411yescharacteristic
368C2×C8C2×#4111yescharacteristic
369C2×C8C2×#4111yescharacteristic
370C2×C8H(5,𝔽2)11yescharacteristic
371C2×C82−1+411yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—84no—
393C4⋊C4—82no—
394C4⋊C4—161no—
395C4⋊C4#10211yescharacteristic
396C4⋊C4#10211yescharacteristic
397C4⋊C4#10212yes—
398C22⋊C4—41no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—84no—
409C22⋊C4#10212yes—
410C22⋊C4#10212yes—
411C22⋊C4#10212yes—
412C22⋊C4#10212yes—
413C42—41no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—41no—
418C42—41no—
419C42—82no—
420C42C2×C22⋊C411yescharacteristic
421C42C2×C22⋊C411yescharacteristic
422C42C2×SD1611yescharacteristic
423C42C2×SD1611yescharacteristic
424C16—22no—
425C16—42no—
426C16—41no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—42no—
431C23—41no—
432C23C2×#10212yes—
433C23C2×#10212yes—
434C23C22×D811yescharacteristic
435Q8—41no—
436Q8—41no—
437Q8—82no—
438Q8—161no—
439D4—24no—
440D4—22no—
441D4—41no—
442D4—41no—
443D4—81no—
444D4—84no—
445D4—81no—
446D4—162no—
447D4#36014yescomplemented
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—84no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—161no—
479C2×C4—162no—
480C2×C4—168no—
481C2×C4C2×#10212yes—
482C2×C4C2×#10211yescharacteristic
483C2×C4C2×#10211yescharacteristic
484C2×C4C2×#10211yescharacteristic
485C2×C4C2×#10212yes—
486C2×C4C2×#10211yescharacteristic
487C2×C4C22×C22⋊C411yescharacteristic
488C2×C4#54611yescharacteristic
489C2×C4#54711yescharacteristic
490C2×C4C22×SD1611yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57811yescharacteristic
493C8—22no—
494C8—22no—
495C8—21no—
496C8—21no—
497C8C4×#4111yescharacteristic
498C8#52011yescharacteristic
499C8#54611yescharacteristic
500C8#54711yescharacteristic
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22—44no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22C2×#36014yes—
514C22C22×#10211yescharacteristic
515C22#329411yescharacteristic
516C22#333411yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—22no—
521C4—22no—
522C4—82no—
523C4—81no—
524C4—82no—
525C4—81no—
526C4—82no—
527C4—81no—
528C4—161no—
529C4—162no—
530C4C2×#36012yes—
531C4#233911yescharacteristic
532C4#289811yescharacteristic
533C4#329811yescharacteristic
534C4#333511yescharacteristic
535C2—21no—
536C2—21no—
537C2—24no—
538C2—42no—
539C2—161no—
540C2#2248511yescharacteristic
541C2C22×#36011yescharacteristic
542C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#61234C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×M32C2semidirect
3. D4×#106C2semidirect
4. #13287C2semidirect
5. #13278C2semidirect
6. #13065C2semidirect
7. #13001C2semidirect
8. #12870C2semidirect
9. #12816C2semidirect
10. C4×#360C2semidirect
11. C2×#1386C2semidirect
14. #3342C4semidirect
15. #3341C4semidirect
17. #3339C4semidirect
18. #3338C4semidirect
20. #3336C4semidirect
22. #3334C4semidirect
81. #500C2×C4semidirect
82. #500C2×C4semidirect
84. #499C2×C4semidirect
100. #467C2×C4semidirect
101. #467C2×C4semidirect
102. #466C2×C4semidirect
103. #466C2×C4semidirect
106. C4×Q16C2×C4semidirect
107. C4×Q16C2×C4semidirect
110. C4×D8C2×C4semidirect
111. C4×D8C2×C4semidirect
121. #427C2×C4semidirect
122. #427C2×C4semidirect
125. C2×#107C2×C4semidirect
126. C2×#107C2×C4semidirect
257. M32#41semidirect
282. #106#41semidirect
447. D4#360semidirect

Maximal subgroups

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

subgroupindex
#612342
D4×M322
D4×#1062
#132872
#132782
#130652
#130012
#128702
#128162
C4×#3602
C2×#13862