Finite Group Databasebeta

← #497652

Subgroups — #497652

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39163C211yesmaximal, characteristic
2#22513C212yesmaximal, complemented
3#13225C212yesmaximal
4#13225C212yesmaximal
5#13224C212yesmaximal, complemented
6#12800C211yesmaximal, characteristic, complemented
7#12799C211yesmaximal, characteristic
8#12798C211yesmaximal, characteristic, complemented
9#12628C211yesmaximal, characteristic
10#12627C211yesmaximal, characteristic, complemented
11C4×#354C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic, complemented
13C2×#554C2211yescharacteristic
14#2861C411yescharacteristic, complemented
15#2728—22no—
16#2702—22no—
17#2642C411yescharacteristic, complemented
18#2640C411yescharacteristic, complemented
19#2568C411yescharacteristic, complemented
20#2566C411yescharacteristic, complemented
21#2565C411yescharacteristic, complemented
22#2547C2211yescharacteristic
23#2544C2211yescharacteristic
24#2544C2211yescharacteristic
25#2481—22no—
26#2477—22no—
27#2345—22no—
28#2328C2211yescharacteristic
29#2326C2211yescharacteristic
30C4×#124C2211yescharacteristic
31#2295C411yescharacteristic, complemented
32C2×#354C2212yes—
33C2×#354C2212yes—
34C2×#354C2212yes—
35C2×#354C2212yes—
36C2×#353C2212yes—
37C2×#353C2212yes—
38#1996—22no—
39#1996—22no—
40#1996—22no—
41#1799C2212yes—
42#1799C2212yes—
43#1799C2212yes—
44C4×#99C2212yes—
45C4×#99C2212yes—
46C4×#99C2212yes—
47#1393C2211yescharacteristic
48#1393C2211yescharacteristic
49#1393C2211yescharacteristic
50#1393C2211yescharacteristic
51#556—42no—
52#556—42no—
53#554—42no—
54#554—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2311yescharacteristic
58C2×#126—21no—
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#125C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124C2311yescharacteristic
65C2×#124C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8D412yes—
69C2×C22⋊Q8D412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C22×C4⋊C4D412yes—
79C2×#99C2312yes—
80C2×#99C2312yes—
81C2×#99C2312yes—
82C2×#99C2312yes—
83#400—22no—
84#400—22no—
85#400—22no—
86#398—22no—
87#397—22no—
88#397—22no—
89#396—22no—
90#394—22no—
91#394—22no—
92#393—22no—
93#391—22no—
94#391—21no—
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2311yescharacteristic
105#389—21no—
106#389—42no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—21no—
113#387—21no—
114#387—42no—
115#387—42no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#385—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#380—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137#376C2×C411yescharacteristic
138#376C2×C411yescharacteristic
139C43C2×C411yescharacteristic
140#354—24no—
141#354—24no—
142#354—42no—
143#354—42no—
144#353—24no—
145#353—42no—
146#342—22no—
147#342—22no—
148#342—22no—
149#342—22no—
150#342—22no—
151#342—22no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22⋊C412yes—
154C22×Q8C22×C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C22×D4—22no—
157C22×D4C22×C411yescharacteristic
158C22×D4C2411yescharacteristic
159C23×C4—21no—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165#127—22no—
166#127—21no—
167#127—21no—
168#126—44no—
169#126—42no—
170#126—42no—
171#125—42no—
172#125—41no—
173#125—41no—
174#124—22no—
175#124—22no—
176#124—21no—
177#124—21no—
178#124—21no—
179#124—21no—
180#124—41no—
181#124—41no—
182#124—42no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—44no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—42no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×D4—42no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—42no—
199C4×D4—42no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261#99—22no—
262#99—22no—
263#99—22no—
264#99—22no—
265#99—22no—
266#99—22no—
267#99—22no—
268#99#4114yes—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95—42no—
290C24—21no—
291C24C22×D412yes—
292C2×Q8—22no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—81no—
303C2×Q8—82no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×#4112yes—
315C2×D4H(5,𝔽2)11yescharacteristic
316C2×D42−1+411yescharacteristic
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C23×C411yescharacteristic
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—84no—
373C4⋊C4—162no—
374C4⋊C4—162no—
375C4⋊C4#9912yes—
376C4⋊C4#9912yes—
377C22⋊C4—24no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
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—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—22no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—41no—
413C42—42no—
414C42—41no—
415C42—41no—
416C42—81no—
417C42—81no—
418C42—84no—
419C42—162no—
420C42—162no—
421C23—22no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23C22×C22⋊C411yescharacteristic
431C23#54812yes—
432C23#55012yes—
433Q8—82no—
434Q8—81no—
435Q8—81no—
436Q8—81no—
437Q8—81no—
438Q8—162no—
439D4—42no—
440D4—41no—
441D4—41no—
442D4—81no—
443D4—81no—
444D4—162no—
445D4—162no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
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×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4C2×#9911yescharacteristic
503C2×C4C2×#9911yescharacteristic
504C2×C4C2×#9912yes—
505C2×C4C2×#9912yes—
506C2×C4C4×#4111yescharacteristic
507C2×C4#52011yescharacteristic
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22#234111yescharacteristic
523C22C22×#9911yescharacteristic
524C22#288411yescharacteristic
525C4—21no—
526C4—21no—
527C4—22no—
528C4—21no—
529C4—21no—
530C4—22no—
531C4—42no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—42no—
536C4—41no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—84no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4—162no—
546C4—162no—
547C4—162no—
548C2—21no—
549C2—21no—
550C2—42no—
551C2—42no—
552C2—161no—
553C2#2253411yescharacteristic
554C2#3335811yescharacteristic
555C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22513C2semidirect
5. #13224C2semidirect
6. #12800C2semidirect
8. #12798C2semidirect
10. #12627C2semidirect
11. C4×#354C2semidirect
12. C2×#556C4semidirect
14. #2861C4semidirect
17. #2642C4semidirect
18. #2640C4semidirect
19. #2568C4semidirect
20. #2566C4semidirect
21. #2565C4semidirect
31. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#391632
#225132
#132252
#132242
#128002
#127992
#127982
#126282
#126272
C4×#3542