Finite Group Databasebeta

← #205968

Subgroups — #205968

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3205C211yesmaximal, characteristic
2#21218C211yesmaximal, characteristic, complemented
3C2×#1965C211yesmaximal, characteristic, complemented
4#17813C211yesmaximal, characteristic, complemented
5#17704C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic, complemented
7#17319C211yesmaximal, characteristic, complemented
8#9811C214yesmaximal, complemented
9#9804C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3205—22no—
13#3205—22no—
14#3205—22no—
15#3205—22no—
16C2×#493C2212yes—
17C2×#486C2212yes—
18C2×#459C414yes—
19C2×#447C414yes—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#413—22no—
25C22×#100—22no—
26C2×C4×M16C2211yescharacteristic
27#1967—22no—
28#1966C2212yescomplemented
29#1965—22no—
30#1965C2214yes—
31#1965C2212yescomplemented
32#1570—22no—
33#1570—22no—
34#1566C2214yescomplemented
35#1547—22no—
36#1546C2212yescomplemented
37#1545—22no—
38#1545C2214yes—
39#1545C2212yescomplemented
40#1540—22no—
41C2×#331—22no—
42#1452—22no—
43#1451—22no—
44#1451C2214yescomplemented
45C22×#41—21no—
46C23×D4—21no—
47C23×D4—21no—
48C23×D4C2×C412yes—
49C22×D8C2×C418yes—
50C22×M16—21no—
51C22×M16—21no—
52C22×M16D412yes—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61C2×C4⋊Q8C2311yescharacteristic
62C2×#127—21no—
63C2×#127—21no—
64C2×#127—21no—
65C2×#127—21no—
66C2×#127—21no—
67C2×#127—21no—
68C2×#127—21no—
69C2×#127—21no—
70C2×#127—21no—
71C2×#127—21no—
72C2×#127C2×C412yes—
73C2×#127C2312yes—
74C2×#127C2311yescharacteristic
75C2×#124—21no—
76C2×#121—21no—
77C2×#121C2×C412yes—
78C2×C4×D4—21no—
79C2×C4×D4C2×C412yes—
80C22×C42C2311yescharacteristic
81#493—24no—
82#493—24no—
83#486—24no—
84#486—24no—
85#459—28no—
86#459—28no—
87#447—28no—
88#447—28no—
89#431—21no—
90#431—21no—
91#431D412yes—
92#431D412yes—
93#431D412yes—
94#431D412yes—
95#431D412yes—
96#431C2312yes—
97#423—21no—
98#423—21no—
99#423—21no—
100#423—21no—
101#423—21no—
102#423—21no—
103#423C2312yes—
104#423C2312yes—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8D412yes—
111C2×#102C2×C418yes—
112#413—48no—
113C2×#101—24no—
114C2×#100—28no—
115C2×#100—24no—
116C2×#100—48no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8C2×C412yes—
119C4×M16—21no—
120C4×M16—21no—
121C4×M16D412yes—
122C4×M16C2312yes—
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2312yes—
127#334—28no—
128#332—24no—
129#331—28no—
130#331—24no—
131#322—28no—
132C25—22no—
133C2×#41—41no—
134C2×#41—41no—
135C2×#41—41no—
136C2×#41—41no—
137C2×#41—44no—
138C2×#41—44no—
139C22×Q8—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—84no—
163C22×D4—84no—
164C22×D4C22×C412yes—
165C22×D4C22×C414yes—
166C23×C4—22no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C2×D8—216no—
171C2×D8—88no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—41no—
178C2×M16—41no—
179C2×M16—44no—
180C2×M16—44no—
181C2×M16C2×D414yes—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—41no—
185C22×C8—41no—
186C22×C8C22×C414yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191#127—22no—
192#127—22no—
193#127—22no—
194#127—22no—
195#127—22no—
196#127—22no—
197#127—22no—
198#127—21no—
199#127—21no—
200#127—21no—
201#127—22no—
202#127—21no—
203#127—41no—
204#127—41no—
205#127—41no—
206#127—41no—
207#127—41no—
208#127—41no—
209#127—41no—
210#127—41no—
211#124—41no—
212#124—41no—
213#124—41no—
214#124—41no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—41no—
220#121—41no—
221#121—41no—
222#121—41no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C22×C412yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C22⋊C414yes—
257C4⋊C8C22⋊C414yes—
258C4⋊C8C2×D414yes—
259#102—216no—
260#101—48no—
261#100—416no—
262#100—48no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8C22⋊C414yes—
266C22⋊C8C22⋊C414yes—
267#97—22no—
268#97—22no—
269C4×C8—22no—
270C4×C8—22no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8C2×D414yes—
274C4×C8C2×D414yes—
275C24—22no—
276C24—24no—
277C24—42no—
278C24—42no—
279C24—48no—
280C24—84no—
281C24—84no—
282C24C4×D411yescharacteristic
283#41—84no—
284#41—84no—
285#41—82no—
286#41—82no—
287#41—82no—
288#41—82no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—48no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—48no—
321C2×D4—84no—
322C2×D4—84no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—82no—
326C2×D4—84no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—82no—
338C2×D4—82no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—48no—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12411yescharacteristic
357C22×C4#12411yescharacteristic
358C22×C4#12711yescharacteristic
359C22×C4#12711yescharacteristic
360C22×C4AGL(1,ℤ/8)14yes—
361C22×C4AGL(1,ℤ/8)14yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365D8—816no—
366M16—24no—
367M16—24no—
368M16—44no—
369M16—44no—
370M16—44no—
371M16—82no—
372M16—82no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—82no—
380C2×C8—82no—
381C2×C8—82no—
382C2×C8C2×C22⋊C418yes—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42C2×C22⋊C411yescharacteristic
392C42C2×C22⋊C411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42#12111yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42#12411yescharacteristic
400C42#12411yescharacteristic
401C42#12711yescharacteristic
402C42#12711yescharacteristic
403C42C22×D411yescharacteristic
404C42C22×D411yescharacteristic
405C42C22×D411yescharacteristic
406C42C2×#4111yescharacteristic
407C23—41no—
408C23—41no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—48no—
413C23—84no—
414C23—84no—
415C23—84no—
416C23—84no—
417C23—84no—
418C23—82no—
419C23—84no—
420C23—82no—
421C23—84no—
422C23—84no—
423C23—1616no—
424C23#39011yescharacteristic
425C23#39011yescharacteristic
426C23#44214yes—
427C23C2×C4×D411yescharacteristic
428Q8—81no—
429Q8—81no—
430Q8—82no—
431D4—44no—
432D4—48no—
433D4—82no—
434D4—84no—
435D4—82no—
436D4—84no—
437D4—84no—
438D4—84no—
439D4—84no—
440D4—84no—
441D4—84no—
442D4—84no—
443D4—82no—
444D4—82no—
445D4—81no—
446D4—81no—
447D4—84no—
448D4—84no—
449D4—84no—
450D4—82no—
451D4—82no—
452D4—84no—
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—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4#39011yescharacteristic
470C2×C4#39011yescharacteristic
471C2×C4#39011yescharacteristic
472C2×C4#39011yescharacteristic
473C2×C4#39011yescharacteristic
474C2×C4#39011yescharacteristic
475C2×C4#44214yes—
476C2×C4#46914yes—
477C2×C4#46914yes—
478C2×C4#49612yes—
479C2×C4#49612yes—
480C2×C4#49612yes—
481C2×C4#49612yes—
482C2×C4C22×C22⋊C411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12411yescharacteristic
487C2×C4C2×#12711yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)12yes—
489C2×C4C2×AGL(1,ℤ/8)12yes—
490C8—28no—
491C8—44no—
492C8—44no—
493C8—82no—
494C8—82no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—84no—
503C22—82no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—84no—
508C22—164no—
509C22—168no—
510C22—168no—
511C22—168no—
512C22#197212yes—
513C22#197212yes—
514C22C2×#39011yescharacteristic
515C22C2×#44212yes—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—82no—
523C4#197212yes—
524C4#197212yes—
525C4#288812yes—
526C4C2×#46912yes—
527C4#306812yes—
528C4C2×#49611yescharacteristic
529C4C2×#49611yescharacteristic
530C2—22no—
531C2—22no—
532C2—22no—
533C2—162no—
534C2—164no—
535C2#2121911yescharacteristic
536C2C2×#197211yescharacteristic
537C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21218C2semidirect
3. C2×#1965C2semidirect
4. #17813C2semidirect
5. #17704C2semidirect
6. C2×#1545C2semidirect
7. #17319C2semidirect
8. #9811C2semidirect
9. #9804C2semidirect
28. #1966C22semidirect
31. #1965C22semidirect
34. #1566C22semidirect
36. #1546C22semidirect
39. #1545C22semidirect
44. #1451C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32052
#212182
C2×#19652
#178132
#177042
C2×#15452
#173192
#98112
#98042