Finite Group Databasebeta

← #205088

Subgroups — #205088

Order 512.

1191 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#21359C211yesmaximal, characteristic, complemented
3#21252C211yesmaximal, characteristic, complemented
4C2×#1980C211yesmaximal, characteristic, complemented
5#17574C211yesmaximal, characteristic, complemented
6C2×#1502C211yesmaximal, characteristic, complemented
7#17319C211yesmaximal, characteristic, complemented
8#9721C218yesmaximal, complemented
9C2×#532C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3245—22no—
12#3226C412yescomplemented
13#3226C412yescomplemented
14#3205C412yescomplemented
15#3205C412yescomplemented
16#3186—22no—
17#3169C2214yes—
18#3100—22no—
19#3100—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#421—22no—
23C22×#104—22no—
24C2×C4×M16C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1980—22no—
28#1980—22no—
29#1980C2214yescomplemented
30#1980C2214yescomplemented
31#1948—22no—
32#1853—22no—
33C4×#106C2214yescomplemented
34#1519—22no—
35#1517—22no—
36#1502—22no—
37#1502C2214yescomplemented
38#1502C2214yescomplemented
39C2×#327—22no—
40#1452—22no—
41#1451—22no—
42#1451C2214yescomplemented
43C22×#41—21no—
44C22×#41—21no—
45C22×M16D412yes—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532C2×C412yes—
59#532C2×C412yes—
60#532C2312yes—
61#532C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#124—21no—
67C2×#124—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#500C2×C414yescomplemented
81#493C2×C414yescomplemented
82#486C2×C414yescomplemented
83#486C2×C414yescomplemented
84#480—24no—
85#480—24no—
86#474—24no—
87#474—24no—
88#466—28no—
89#459—24no—
90#459—24no—
91#447—24no—
92#447—24no—
93C4×D8—28no—
94#431—21no—
95#431—21no—
96#431C2×C412yes—
97#431C2×C412yes—
98#431D412yes—
99#431D412yes—
100#431D412yes—
101#431C2312yes—
102C2×#106—24no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D412yes—
105#421—48no—
106C2×#104—28no—
107C2×#104—28no—
108C2×#104—48no—
109C2×C22⋊C8D412yes—
110C2×C22⋊C8D412yes—
111C4×M16—21no—
112C4×M16—21no—
113C4×M16D412yes—
114C4×M16C2312yes—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2312yes—
119#384—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2312yes—
129C2×#95—21no—
130#340—24no—
131#327—28no—
132#327—28no—
133#322—28no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—41no—
139C2×#41—41no—
140C2×#41—41no—
141C2×#41—41no—
142C2×#41—44no—
143C2×#41—44no—
144C2×#41—44no—
145C2×#41—44no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—41no—
155C22×D4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×D8—28no—
163C2×M16—22no—
164C2×M16—24no—
165C2×M16—24no—
166C2×M16C2×D414yes—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C2×D414yes—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C22×C412yes—
177C4⋊Q8C22×C412yes—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127—21no—
184#127C22×C412yes—
185#127C22×C412yes—
186#124—22no—
187#124—22no—
188#124—22no—
189#124—22no—
190#124—41no—
191#124—41no—
192#124—41no—
193#124—41no—
194#121—22no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—41no—
203#121—41no—
204#121—41no—
205#121—41no—
206#121—41no—
207#121—41no—
208#121—41no—
209#121—41no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226#117—44no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—41no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42C2411yescharacteristic
258#107—28no—
259#106—48no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8C22⋊C414yes—
263C4⋊C8C22⋊C414yes—
264#104—416no—
265#104—416no—
266#102—28no—
267#102—28no—
268C22⋊C8—22no—
269C22⋊C8C22⋊C414yes—
270C22⋊C8C22⋊C414yes—
271C22⋊C8C2×D414yes—
272#97—22no—
273#97—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8C22×C414yes—
279C4×C8C2×D414yes—
280#95—22no—
281#95—22no—
282#95—41no—
283#95—41no—
284C24—41no—
285C24—41no—
286C24#12711yescharacteristic
287#41—44no—
288#41—44no—
289#41—44no—
290#41—44no—
291#41—48no—
292#41—48no—
293#41—84no—
294#41—84no—
295#41—82no—
296#41—82no—
297#41—82no—
298#41—82no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—82no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
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×C4#12711yescharacteristic
361C22×C4AGL(1,ℤ/8)14yes—
362C22×C4AGL(1,ℤ/8)14yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366D8—88no—
367M16—24no—
368M16—24no—
369M16—44no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—82no—
375C2×C8C2×C22⋊C418yes—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—82no—
399C42—44no—
400C42—44no—
401C42—82no—
402C42C2×C22⋊C411yescharacteristic
403C42C2×C22⋊C411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42C4×D411yescharacteristic
407C42C4×D411yescharacteristic
408C42#12111yescharacteristic
409C42#12111yescharacteristic
410C42#12111yescharacteristic
411C42#12111yescharacteristic
412C42#12411yescharacteristic
413C42#12411yescharacteristic
414C42C23×C411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C23—41no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23#39011yescharacteristic
429C23#39011yescharacteristic
430C23#49612yes—
431C23#49612yes—
432C23C2×#12711yescharacteristic
433Q8—44no—
434Q8—44no—
435Q8—82no—
436Q8—82no—
437D4—44no—
438D4—44no—
439D4—44no—
440D4—44no—
441D4—82no—
442D4—82no—
443D4—82no—
444D4—82no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—82no—
449D4—82no—
450D4—82no—
451D4—84no—
452D4—84no—
453D4—82no—
454D4—82no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
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—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—88no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#39011yescharacteristic
483C2×C4#39011yescharacteristic
484C2×C4#39011yescharacteristic
485C2×C4#39011yescharacteristic
486C2×C4#39011yescharacteristic
487C2×C4#39011yescharacteristic
488C2×C4#44214yes—
489C2×C4#44214yes—
490C2×C4#46914yes—
491C2×C4#46914yes—
492C2×C4#49612yes—
493C2×C4#49612yes—
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12411yescharacteristic
500C2×C4C2×AGL(1,ℤ/8)12yes—
501C2×C4C2×AGL(1,ℤ/8)12yes—
502C8—28no—
503C8—44no—
504C8—44no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—164no—
520C22—164no—
521C22#197212yes—
522C22#197212yes—
523C22C2×#39011yescharacteristic
524C22C2×#49611yescharacteristic
525C22C2×#49611yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—162no—
537C4#197212yes—
538C4#197212yes—
539C4#290112yes—
540C4C2×#44212yes—
541C4C2×#46912yes—
542C2—21no—
543C2—21no—
544C2—22no—
545C2—22no—
546C2—162no—
547C2—162no—
548C2C2×#197211yescharacteristic
549C2#2124411yescharacteristic
550C2#2125311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60940C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21359C2semidirect
3. #21252C2semidirect
4. C2×#1980C2semidirect
5. #17574C2semidirect
6. C2×#1502C2semidirect
7. #17319C2semidirect
8. #9721C2semidirect
12. #3226C4semidirect
13. #3226C4semidirect
14. #3205C4semidirect
15. #3205C4semidirect
29. #1980C22semidirect
30. #1980C22semidirect
33. C4×#106C22semidirect
37. #1502C22semidirect
38. #1502C22semidirect
42. #1451C22semidirect
80. #500C2×C4semidirect
81. #493C2×C4semidirect
82. #486C2×C4semidirect
83. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609402
#213592
#212522
C2×#19802
#175742
C2×#15022
#173192
#97212