Finite Group Databasebeta

← #496948

Subgroups — #496948

Order 512.

543 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33435C211yesmaximal, characteristic, complemented
2#33421C211yesmaximal, characteristic, complemented
3#21517C211yesmaximal, characteristic
4#13196C211yesmaximal, characteristic, complemented
5#13195C211yesmaximal, characteristic, complemented
6#13193C211yesmaximal, characteristic, complemented
7#13163C211yesmaximal, characteristic, complemented
8#13125C211yesmaximal, characteristic, complemented
9#13124C211yesmaximal, characteristic, complemented
10#12980C211yesmaximal, characteristic, complemented
11#12980C211yesmaximal, characteristic, complemented
12#12917C211yesmaximal, characteristic, complemented
13#12917C211yesmaximal, characteristic, complemented
14#12787C211yesmaximal, characteristic, complemented
15#12663C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18#3035C411yescharacteristic
19C2×#443C2211yescharacteristic
20C2×#443C2211yescharacteristic
21#2931C2211yescharacteristic
22C4×#131C2211yescharacteristic
23#2168—21no—
24#2168—21no—
25#2168—21no—
26#2168—21no—
27#2164—21no—
28#2164—21no—
29#2163—21no—
30#2163—21no—
31C2×#364C2211yescharacteristic
32C2×#364C2211yescharacteristic
33#2151C2211yescharacteristic
34#2151C2211yescharacteristic
35#2150C2211yescharacteristic
36C2×C4⋊C16C2211yescharacteristic
37C2×#350C2211yescharacteristic
38C2×#350C2211yescharacteristic
39C2×#350C2211yescharacteristic
40C2×#350C2211yescharacteristic
41C2×#350C2211yescharacteristic
42C2×#350C2211yescharacteristic
43#2113C2211yescharacteristic
44#2113C2211yescharacteristic
45#2113C2211yescharacteristic
46#2113C2211yescharacteristic
47C4×M32C2211yescharacteristic
48C2×#346C2211yescharacteristic
49C2×#346C2211yescharacteristic
50C2×C4×C16C2211yescharacteristic
51#1980C411yescharacteristic
52#1950C411yescharacteristic
53#1929C411yescharacteristic
54#1929C411yescharacteristic
55#1876C411yescharacteristic
56#1876C411yescharacteristic
57#1806C411yescharacteristic
58C8×C4⋊C4C2211yescharacteristic
59#1764C2211yescharacteristic
60#1764C2211yescharacteristic
61#1379C2211yescharacteristic
62#1379C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68C2×#135C2×C411yescharacteristic
69C2×#131C2×C411yescharacteristic
70C2×#131C2311yescharacteristic
71C2×#131C2311yescharacteristic
72C2×#131C2311yescharacteristic
73#532C2×C411yescharacteristic
74C4×#41C2×C411yescharacteristic
75#504—41no—
76#504—41no—
77#504—41no—
78#504—41no—
79C2×M32—21no—
80C2×M32—21no—
81C2×M32C2311yescharacteristic
82C2×M32C2311yescharacteristic
83C2×M32C2311yescharacteristic
84C2×M32C2311yescharacteristic
85C22×C16—21no—
86C22×C16—21no—
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91#464C811yescharacteristic
92#464C811yescharacteristic
93#463C811yescharacteristic
94#463C811yescharacteristic
95#462C811yescharacteristic
96#461C811yescharacteristic
97#460C811yescharacteristic
98#459C811yescharacteristic
99#443—41no—
100#443—41no—
101#443—41no—
102#443—41no—
103#436—21no—
104#436—21no—
105C8×D4—21no—
106C8×D4—21no—
107C2×#108C2×C411yescharacteristic
108C2×#108C2×C411yescharacteristic
109#423C2×C411yescharacteristic
110C2×C4⋊C8C2×C411yescharacteristic
111C2×#104C2×C411yescharacteristic
112C2×#104C2×C411yescharacteristic
113C2×#104C2×C411yescharacteristic
114C2×#104C2×C411yescharacteristic
115C2×#104C2×C411yescharacteristic
116C2×#104C2×C411yescharacteristic
117#416C2×C411yescharacteristic
118#416C2×C411yescharacteristic
119#408C2×C411yescharacteristic
120#408C2×C411yescharacteristic
121#405—21no—
122#405—21no—
123#405C2×C411yescharacteristic
124#405C2311yescharacteristic
125C4×M16—21no—
126C4×M16C2×C411yescharacteristic
127C2×C4×C8—21no—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132#364—41no—
133#364—41no—
134C4⋊C16—21no—
135C4⋊C16—21no—
136C4⋊C16—21no—
137C4⋊C16—21no—
138C4⋊C16—21no—
139C4⋊C16—21no—
140C4⋊C16—21no—
141C4⋊C16—21no—
142#350—41no—
143#350—41no—
144#350—41no—
145#350—41no—
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⋊C16—21no—
155C22⋊C16—21no—
156#346—21no—
157#346—41no—
158#346D411yescharacteristic
159#346D411yescharacteristic
160#346D411yescharacteristic
161#346D411yescharacteristic
162C4×C16—21no—
163C4×C16—41no—
164#341C2×C411yescharacteristic
165#341C2×C411yescharacteristic
166#339C2×C411yescharacteristic
167#339C2×C411yescharacteristic
168#336C2×C411yescharacteristic
169#336C2×C411yescharacteristic
170C2×#41C22×C411yescharacteristic
171C2×#41C22×C411yescharacteristic
172C2×#41C22×C411yescharacteristic
173#135—41no—
174#135—41no—
175#135—41no—
176#135—41no—
177C2×Q16C2×C811yescharacteristic
178C2×Q16C2×C811yescharacteristic
179C2×SD16C2×C811yescharacteristic
180C2×SD16C2×C811yescharacteristic
181C2×SD16C2×C811yescharacteristic
182C2×SD16C2×C811yescharacteristic
183C2×D8C2×C811yescharacteristic
184C2×D8C2×C811yescharacteristic
185#131—21no—
186#131—21no—
187#131—21no—
188#131—21no—
189#131—21no—
190#131—21no—
191#131—21no—
192#131—21no—
193#131—21no—
194#131—21no—
195#131—21no—
196#131—21no—
197#131—41no—
198#131—41no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16C22×C411yescharacteristic
204C2×M16C22×C411yescharacteristic
205C2×M16C22×C411yescharacteristic
206C2×M16C22×C411yescharacteristic
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C2411yescharacteristic
215C4⋊Q8C2×C811yescharacteristic
216#127C2×C811yescharacteristic
217#124C2×C811yescharacteristic
218#124C2×C811yescharacteristic
219#121—21no—
220#121—21no—
221C4×Q8C2×C811yescharacteristic
222C4×Q8C2×C811yescharacteristic
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4C2×C811yescharacteristic
228C4×D4C2×C811yescharacteristic
229#117—21no—
230#117C22×C411yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—21no—
241M32—21no—
242M32—21no—
243M32—21no—
244M32—42no—
245M32—42no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
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—42no—
261C2×C16—42no—
262C2×C16C2×D411yescharacteristic
263C2×C16C2×D411yescharacteristic
264C2×C16C2×D411yescharacteristic
265C2×C16C2×D411yescharacteristic
266#108—41no—
267#108—41no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8C2×C811yescharacteristic
271C4⋊C8C2×C811yescharacteristic
272C4⋊C8C2×C811yescharacteristic
273C4⋊C8C2×C811yescharacteristic
274#104—41no—
275#104—41no—
276#104—41no—
277#104—41no—
278#104—41no—
279#104—41no—
280#103C2×C811yescharacteristic
281#103C2×C811yescharacteristic
282#103C2×C811yescharacteristic
283#103C2×C811yescharacteristic
284#102C2×C811yescharacteristic
285#102C2×C811yescharacteristic
286#102C2×C811yescharacteristic
287#102C2×C811yescharacteristic
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292#97—21no—
293#97—21no—
294#97—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8C2×D411yescharacteristic
303C4×C8C2×D411yescharacteristic
304C4×C8#4111yescharacteristic
305C4×C8#4111yescharacteristic
306#95—21no—
307#41—21no—
308#41—21no—
309#41—21no—
310#41—21no—
311#41—21no—
312#41—21no—
313#41—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—21no—
319C2×Q8C22×C811yescharacteristic
320C2×Q8C22×C811yescharacteristic
321C2×Q8C22×C811yescharacteristic
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4C22×C811yescharacteristic
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#13111yescharacteristic
338C22×C4#13111yescharacteristic
339C22×C4C23×C411yescharacteristic
340Q16—41no—
341Q16—41no—
342SD16—41no—
343SD16—41no—
344SD16—41no—
345SD16—41no—
346D8—41no—
347D8—41no—
348M16—21no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—42no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363C2×C8—21no—
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—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8C4×D411yescharacteristic
381C2×C8C4×D411yescharacteristic
382C2×C8C22×C811yescharacteristic
383C2×C8C22×C811yescharacteristic
384C2×C8C22×C811yescharacteristic
385C2×C8C22×C811yescharacteristic
386C2×C8C22×D411yescharacteristic
387C2×C8C2×#4111yescharacteristic
388C2×C8H(5,𝔽2)11yescharacteristic
389C2×C8H(5,𝔽2)11yescharacteristic
390C2×C8H(5,𝔽2)11yescharacteristic
391C2×C82−1+411yescharacteristic
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—42no—
397C4⋊C4C22×C811yescharacteristic
398C4⋊C4C22×C811yescharacteristic
399C22⋊C4—21no—
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42C4×D411yescharacteristic
414C42C4×D411yescharacteristic
415C42C22×C811yescharacteristic
416C42C22×C811yescharacteristic
417C16—21no—
418C16—21no—
419C16—21no—
420C16—21no—
421C16—41no—
422C16—41no—
423C16—41no—
424C16—41no—
425C16—41no—
426C16—41no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23C2×#13111yescharacteristic
431Q8—21no—
432Q8—21no—
433Q8—21no—
434Q8—21no—
435Q8—21no—
436Q8—21no—
437D4—21no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—21no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—41no—
449D4—81no—
450D4—81no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4C8×D411yescharacteristic
473C2×C4C8×D411yescharacteristic
474C2×C4C8×D411yescharacteristic
475C2×C4C8×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#51811yescharacteristic
478C2×C4#52011yescharacteristic
479C2×C4C23×C811yescharacteristic
480C2×C4#56811yescharacteristic
481C2×C4#56811yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—21no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8#53511yescharacteristic
493C8#53711yescharacteristic
494C8#54011yescharacteristic
495C8#54211yescharacteristic
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22C2×C8×D411yescharacteristic
508C22#298111yescharacteristic
509C22#299311yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—81no—
521C4#232411yescharacteristic
522C4#293011yescharacteristic
523C4#297211yescharacteristic
524C4#298811yescharacteristic
525C2—21no—
526C2—21no—
527C2—81no—
528C2—81no—
529C2—81no—
530C2#2213311yescharacteristic
531C2#3344211yescharacteristic
532C2#3344311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#3035C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×#346 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33435C2semidirect
2. #33421C2semidirect
4. #13196C2semidirect
5. #13195C2semidirect
6. #13193C2semidirect
7. #13163C2semidirect
8. #13125C2semidirect
9. #13124C2semidirect
10. #12980C2semidirect
11. #12980C2semidirect
12. #12917C2semidirect
13. #12917C2semidirect
14. #12787C2semidirect
15. #12663C2semidirect

Maximal subgroups

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

subgroupindex
#334352
#334212
#215172
#131962
#131952
#131932
#131632
#131252
#131242
#129802
#129172
#127872
#126632