Finite Group Databasebeta

← #498483

Subgroups — #498483

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33436C211yesmaximal, characteristic, complemented
2#33436C211yesmaximal, characteristic, complemented
3#21548C211yesmaximal, characteristic
4#13125C211yesmaximal, characteristic, complemented
5#13124C211yesmaximal, characteristic, complemented
6#13106C211yesmaximal, characteristic, complemented
7#13106C211yesmaximal, characteristic, complemented
8#12980C211yesmaximal, characteristic, complemented
9#12980C211yesmaximal, characteristic, complemented
10#12978C211yesmaximal, characteristic, complemented
11#12978C211yesmaximal, characteristic, complemented
12#12615C211yesmaximal, characteristic, complemented
13#12614C211yesmaximal, characteristic, complemented
14#12600C211yesmaximal, characteristic, complemented
15#12549C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18C22×M32C2211yescharacteristic
19C22×M32C2211yescharacteristic
20#3054C411yescharacteristic
21C2×#443C2211yescharacteristic
22#2929C2211yescharacteristic
23#2929C2211yescharacteristic
24C2×#405C2211yescharacteristic
25#2169—21no—
26#2169—21no—
27#2168—21no—
28#2168—21no—
29#2168—21no—
30#2168—21no—
31D4×C16—21no—
32D4×C16—21no—
33#2151C2211yescharacteristic
34#2151C2211yescharacteristic
35C2×#350C2211yescharacteristic
36C2×#350C2211yescharacteristic
37C2×#350C2211yescharacteristic
38C2×#350C2211yescharacteristic
39#2113C2211yescharacteristic
40#2113C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×C22⋊C16C2211yescharacteristic
43#2108C2211yescharacteristic
44#2108C2211yescharacteristic
45C2×#346C2211yescharacteristic
46C2×C4×C16C2211yescharacteristic
47#1975C411yescharacteristic
48#1946C411yescharacteristic
49#1938C411yescharacteristic
50#1938C411yescharacteristic
51#1897C411yescharacteristic
52#1897C411yescharacteristic
53#1811C411yescharacteristic
54#1772C2211yescharacteristic
55#1764C2211yescharacteristic
56#1764C2211yescharacteristic
57#1379C2211yescharacteristic
58#1379C2211yescharacteristic
59#1379C2211yescharacteristic
60#1379C2211yescharacteristic
61#1376C2211yescharacteristic
62#1376C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68C2×#131C2311yescharacteristic
69C2×#131C2311yescharacteristic
70C22×M16C2×C411yescharacteristic
71C23×C8C2311yescharacteristic
72#525C2×C411yescharacteristic
73#525C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75#504—22no—
76#504—22no—
77#504—41no—
78#504—41no—
79C2×M32—21no—
80C2×M32—21no—
81C2×M32—21no—
82C2×M32—21no—
83C2×M32—22no—
84C2×M32—22no—
85C2×M32—41no—
86C2×M32—41no—
87C2×M32C2311yescharacteristic
88C2×M32C2311yescharacteristic
89C2×M32C2311yescharacteristic
90C2×M32C2311yescharacteristic
91C22×C16—21no—
92C22×C16—21no—
93C22×C16—21no—
94C22×C16—21no—
95C22×C16C2311yescharacteristic
96C22×C16C2311yescharacteristic
97C22×C16C2311yescharacteristic
98C22×C16C2311yescharacteristic
99#470C811yescharacteristic
100#470C811yescharacteristic
101#469C811yescharacteristic
102#469C811yescharacteristic
103#468C811yescharacteristic
104#468C811yescharacteristic
105#468C811yescharacteristic
106#468C811yescharacteristic
107#443—41no—
108#443—41no—
109#435—21no—
110#435—21no—
111#435—21no—
112#435—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C2×#108C2×C411yescharacteristic
118C2×#108C2×C411yescharacteristic
119#427C2×C411yescharacteristic
120#424C2×C411yescharacteristic
121#424C2×C411yescharacteristic
122C2×#104C2×C411yescharacteristic
123C2×#104C2×C411yescharacteristic
124C2×#104C2×C411yescharacteristic
125C2×#104C2×C411yescharacteristic
126#416C2×C411yescharacteristic
127#416C2×C411yescharacteristic
128#408C2×C411yescharacteristic
129#408C2×C411yescharacteristic
130#407C2×C411yescharacteristic
131#407C2×C411yescharacteristic
132#405—41no—
133#405—41no—
134#405C2311yescharacteristic
135#405C2311yescharacteristic
136C2×#97C2×C411yescharacteristic
137C2×#97C2×C411yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C4⋊C16—21no—
141C4⋊C16—21no—
142C4⋊C16—21no—
143C4⋊C16—21no—
144#350—22no—
145#350—22no—
146#350—41no—
147#350—41no—
148C22⋊C16—21no—
149C22⋊C16—21no—
150C22⋊C16—21no—
151C22⋊C16—21no—
152C22⋊C16—21no—
153C22⋊C16—21no—
154C22⋊C16D411yescharacteristic
155C22⋊C16D411yescharacteristic
156C22⋊C16D411yescharacteristic
157C22⋊C16D411yescharacteristic
158#346—21no—
159#346—21no—
160#346—41no—
161C4×C16—21no—
162C4×C16—21no—
163C4×C16—41no—
164#341C2×C411yescharacteristic
165#341C2×C411yescharacteristic
166#339C2×C411yescharacteristic
167#339C2×C411yescharacteristic
168#339C2×C411yescharacteristic
169#339C2×C411yescharacteristic
170C2×#41C22×C411yescharacteristic
171C2×#41C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173#135—41no—
174#135—41no—
175C2×Q16C2×C811yescharacteristic
176C2×SD16C2×C811yescharacteristic
177C2×SD16C2×C811yescharacteristic
178C2×D8C2×C811yescharacteristic
179#131—21no—
180#131—21no—
181#131—21no—
182#131—21no—
183#131—21no—
184#131—21no—
185#131—21no—
186#131—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—41no—
190C2×M16—41no—
191C2×M16C2×C811yescharacteristic
192C2×M16C2×C811yescharacteristic
193C2×M16C2×C811yescharacteristic
194C2×M16C2×C811yescharacteristic
195C2×M16C22×C411yescharacteristic
196C2×M16C22×C411yescharacteristic
197C2×M16C22×C411yescharacteristic
198C2×M16C22×C411yescharacteristic
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8#4111yescharacteristic
211C22×C8#4111yescharacteristic
212C22×C8C2411yescharacteristic
213#123—21no—
214#123—21no—
215C22⋊Q8C2×C811yescharacteristic
216C22⋊Q8C2×C811yescharacteristic
217C22⋊Q8C2×C811yescharacteristic
218C22⋊Q8C2×C811yescharacteristic
219#121C2×C811yescharacteristic
220#121C2×C811yescharacteristic
221#121C2×C811yescharacteristic
222#121C2×C811yescharacteristic
223#120—21no—
224#120—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229#117—22no—
230#117—41no—
231#117C22×C411yescharacteristic
232#117C22×C411yescharacteristic
233C2×C4⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237M32—21no—
238M32—21no—
239M32—21no—
240M32—21no—
241M32—21no—
242M32—21no—
243M32—42no—
244M32—42no—
245M32—42no—
246M32—42no—
247M32#4112yes—
248M32#4112yes—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—21no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—21no—
263C2×C16—42no—
264C2×C16—42no—
265C2×C16—42no—
266C2×C16—42no—
267C2×C16C2×D411yescharacteristic
268C2×C16C2×D411yescharacteristic
269C2×C16C2×D411yescharacteristic
270C2×C16C2×D411yescharacteristic
271#108—41no—
272#108—41no—
273#107C2×C811yescharacteristic
274#107C2×C811yescharacteristic
275#106C2×C811yescharacteristic
276#106C2×C811yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281#104—22no—
282#104—22no—
283#104—41no—
284#104—41no—
285#103C2×C811yescharacteristic
286#103C2×C811yescharacteristic
287#103C2×C811yescharacteristic
288#103C2×C811yescharacteristic
289#102C2×C811yescharacteristic
290#102C2×C811yescharacteristic
291#102C2×C811yescharacteristic
292#102C2×C811yescharacteristic
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301#97—21no—
302#97—21no—
303#97—41no—
304#97—41no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C24#13111yescharacteristic
312#41—21no—
313#41—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—21no—
319#41—21no—
320C2×Q8C22×C811yescharacteristic
321C2×Q8C22×C811yescharacteristic
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4C22×C811yescharacteristic
329C2×D4C22×C811yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C22×C811yescharacteristic
341C22×C4C22×C811yescharacteristic
342C22×C4#13111yescharacteristic
343C22×C4C23×C411yescharacteristic
344Q16—41no—
345SD16—41no—
346SD16—41no—
347D8—41no—
348M16—21no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8C4×D411yescharacteristic
387C2×C8C4×D411yescharacteristic
388C2×C8C22×C811yescharacteristic
389C2×C8C22×C811yescharacteristic
390C2×C8C22×C811yescharacteristic
391C2×C8C22×C811yescharacteristic
392C2×C8C22×D411yescharacteristic
393C2×C8C2×#4111yescharacteristic
394C2×C8C2×#4111yescharacteristic
395C2×C8C2×#4111yescharacteristic
396C2×C8H(5,𝔽2)11yescharacteristic
397C2×C8H(5,𝔽2)11yescharacteristic
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4C22×C811yescharacteristic
404C4⋊C4C22×C811yescharacteristic
405C4⋊C4C22×C811yescharacteristic
406C4⋊C4C22×C811yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—42no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42#13112yes—
424C16—21no—
425C16—21no—
426C16—21no—
427C16—21no—
428C16—22no—
429C16—22no—
430C16—41no—
431C16—41no—
432C16—41no—
433C16—41no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—41no—
439C23—41no—
440C23C8×D411yescharacteristic
441C23C8×D411yescharacteristic
442C23C2×#13111yescharacteristic
443Q8—21no—
444Q8—21no—
445Q8—21no—
446Q8—21no—
447D4—21no—
448D4—21no—
449D4—21no—
450D4—21no—
451D4—42no—
452D4—42no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—81no—
458D4—81no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4C8×D411yescharacteristic
486C2×C4C8×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C4×#4111yescharacteristic
489C2×C4#51811yescharacteristic
490C2×C4C23×C811yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4#56811yescharacteristic
493C8—21no—
494C8—21no—
495C8—21no—
496C8—21no—
497C8—21no—
498C8—21no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8#54611yescharacteristic
504C8#54611yescharacteristic
505C8#55311yescharacteristic
506C8#55311yescharacteristic
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—81no—
522C22—81no—
523C22C2×C8×D411yescharacteristic
524C22#296811yescharacteristic
525C22#297711yescharacteristic
526C4—21no—
527C4—21no—
528C4—42no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—81no—
536C4#236411yescharacteristic
537C4#293511yescharacteristic
538C4C8×#4111yescharacteristic
539C4#297211yescharacteristic
540C2—21no—
541C2—21no—
542C2—41no—
543C2—41no—
544C2—81no—
545C2—81no—
546C2#2311611yescharacteristic
547C2#3344111yescharacteristic
548C2#3344211yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#3054C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C22⋊C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33436C2semidirect
2. #33436C2semidirect
4. #13125C2semidirect
5. #13124C2semidirect
6. #13106C2semidirect
7. #13106C2semidirect
8. #12980C2semidirect
9. #12980C2semidirect
10. #12978C2semidirect
11. #12978C2semidirect
12. #12615C2semidirect
13. #12614C2semidirect
14. #12600C2semidirect
15. #12549C2semidirect

Maximal subgroups

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

subgroupindex
#334362
#215482
#131252
#131242
#131062
#129802
#129782
#126152
#126142
#126002
#125492