Finite Group Databasebeta

← #497653

Subgroups — #497653

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39161C211yesmaximal, characteristic
2#22513C212yesmaximal, complemented
3#13225C212yesmaximal, complemented
4#13225C212yesmaximal, complemented
5#13223C212yesmaximal
6#12800C211yesmaximal, characteristic, complemented
7#12799C211yesmaximal, characteristic, complemented
8#12797C211yesmaximal, characteristic, complemented
9#12628C211yesmaximal, characteristic, complemented
10#12627C211yesmaximal, characteristic, complemented
11C4×#353C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic, complemented
13C2×#554C2211yescharacteristic
14#2861C411yescharacteristic, complemented
15#2730—22no—
16#2681—22no—
17#2641C411yescharacteristic, complemented
18#2640C411yescharacteristic, complemented
19#2566C411yescharacteristic, complemented
20#2565C411yescharacteristic, complemented
21#2544C2211yescharacteristic
22#2544C2211yescharacteristic
23#2542C411yescharacteristic, complemented
24#2541C2211yescharacteristic
25#2463—22no—
26#2443—22no—
27#2345—22no—
28#2328C2211yescharacteristic
29#2326C2211yescharacteristic
30C4×#127C2211yescharacteristic
31#2295C411yescharacteristic, complemented
32C2×#354C2212yes—
33C2×#354C2212yes—
34C2×#353C2212yes—
35C2×#353C2212yes—
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#559—42no—
52#559—42no—
53#554—42no—
54#554—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#127C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#125C2×C411yescharacteristic
60C2×#125C2×C411yescharacteristic
61C2×#124—21no—
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124C2311yescharacteristic
65C2×#123—21no—
66C2×#123—21no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8D412yes—
69C2×C22⋊Q8D412yes—
70C2×C22⋊Q8D412yes—
71C2×#121—22no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C22×C4⋊C4D412yes—
82C2×#99C2312yes—
83C2×#99C2312yes—
84C2×#99C2312yes—
85C2×#99C2312yes—
86#399—22no—
87#398—22no—
88#397—22no—
89#396—22no—
90#394—22no—
91#394—22no—
92#394—22no—
93#394—22no—
94#393—22no—
95#392—22no—
96#391—21no—
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#390—22no—
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2311yescharacteristic
108#389—21no—
109#389—42no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—21no—
115#387—21no—
116#387—42no—
117#387—42no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#385—22no—
123#385—22no—
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#380—22no—
128#380—22no—
129#380—22no—
130C4×C4⋊C4—42no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
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—42no—
142#353—24no—
143#353—24no—
144#353—42no—
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×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C22×D4C2411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C4⋊Q8—42no—
168#127—22no—
169#127—22no—
170#127—21no—
171#127—21no—
172#127—21no—
173#127—21no—
174#127—42no—
175#125—41no—
176#125—41no—
177#125—42no—
178#124—22no—
179#124—21no—
180#124—21no—
181#124—41no—
182#124—41no—
183#124—42no—
184#124—44no—
185#123—44no—
186#123—42no—
187#123—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—84no—
195#121—84no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—42no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—84no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D412yes—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268#99—22no—
269#99—22no—
270#99—22no—
271#99—22no—
272#99—22no—
273#99—22no—
274#99—22no—
275#99#4114yes—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293C24—21no—
294C24—21no—
295C24C22×D412yes—
296C2×Q8—22no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—81no—
303C2×Q8—82no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4C2×#4112yes—
327C2×D4H(5,𝔽2)11yescharacteristic
328C2×D42−1+411yescharacteristic
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C23×C411yescharacteristic
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—162no—
386C4⋊C4—162no—
387C4⋊C4#9912yes—
388C4⋊C4#9912yes—
389C22⋊C4—24no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C42—21no—
417C42—21no—
418C42—22no—
419C42—22no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—41no—
427C42—42no—
428C42—42no—
429C42—81no—
430C42—84no—
431C42—162no—
432C42—162no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C22×C22⋊C411yescharacteristic
448C23#54812yes—
449C23#55012yes—
450Q8—82no—
451Q8—81no—
452Q8—81no—
453Q8—162no—
454D4—42no—
455D4—41no—
456D4—41no—
457D4—81no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—84no—
462D4—84no—
463D4—84no—
464D4—162no—
465D4—162no—
466D4—162no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—161no—
520C2×C4—161no—
521C2×C4—162no—
522C2×C4C2×#9912yes—
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×#9912yes—
526C2×C4C4×#4111yescharacteristic
527C2×C4#52011yescharacteristic
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#234111yescharacteristic
548C22C22×#9911yescharacteristic
549C22#288411yescharacteristic
550C4—21no—
551C4—21no—
552C4—22no—
553C4—21no—
554C4—21no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—84no—
567C4—81no—
568C4—81no—
569C4—162no—
570C4—162no—
571C4—162no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—161no—
577C2—161no—
578C2#2253411yescharacteristic
579C2#3335811yescharacteristic
580C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radical#2541C22
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
3. #13225C2semidirect
4. #13225C2semidirect
6. #12800C2semidirect
7. #12799C2semidirect
8. #12797C2semidirect
9. #12628C2semidirect
10. #12627C2semidirect
11. C4×#353C2semidirect
12. C2×#559C4semidirect
14. #2861C4semidirect
17. #2641C4semidirect
18. #2640C4semidirect
19. #2566C4semidirect
20. #2565C4semidirect
23. #2542C4semidirect
31. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#391612
#225132
#132252
#132232
#128002
#127992
#127972
#126282
#126272
C4×#3532