Finite Group Databasebeta

← #480913

Subgroups — #480913

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38092C211yesmaximal, characteristic
2#17427C212yesmaximal, complemented
3#11564C212yesmaximal, complemented
4#11556C212yesmaximal, complemented
5#11551C212yesmaximal, complemented
6#11533C211yesmaximal, characteristic, complemented
7#11508C211yesmaximal, characteristic, complemented
8#11507C211yesmaximal, characteristic, complemented
9#11479C211yesmaximal, characteristic, complemented
10#11432C211yesmaximal, characteristic, complemented
11#11431C211yesmaximal, characteristic, complemented
12C2×#553C2211yescharacteristic
13C2×#547C411yescharacteristic
14#2974—22no—
15#2850C411yescharacteristic
16#2693—22no—
17#2688—22no—
18#2642C411yescharacteristic
19#2620C411yescharacteristic
20#2566C411yescharacteristic
21#2547C2211yescharacteristic
22#2503C2211yescharacteristic
23#2499C2211yescharacteristic
24#2481—22no—
25#2457—22no—
26#2431C411yescharacteristic
27#2431C411yescharacteristic
28#2328C2211yescharacteristic
29#2311C2211yescharacteristic
30#2297C2211yescharacteristic
31#2290C411yescharacteristic
32#1473—22no—
33#1471—22no—
34#1468—22no—
35#1465C2212yes—
36#1464C2212yes—
37#1463C2212yes—
38#1459C2212yes—
39#1458C2212yes—
40C2×#324C2212yes—
41#1324C2212yes—
42#1324C2212yes—
43#1319C2212yes—
44#1319C2212yes—
45#1319C2212yes—
46#1319C2212yes—
47#1318C2211yescharacteristic
48#1318C2211yescharacteristic
49#1318C2211yescharacteristic
50#1318C2211yescharacteristic
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53#553—42no—
54#553—42no—
55#547—42no—
56#547—42no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#126D412yes—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×#123—21no—
65C2×#123D412yes—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×#121—21no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#121D412yes—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D412yes—
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2311yescharacteristic
80C22×C42C2×C411yescharacteristic
81#436—42no—
82#435—42no—
83#435—42no—
84C8×D4—42no—
85#432—22no—
86#431—22no—
87#407—22no—
88#407—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91#400—22no—
92#400—22no—
93#400—22no—
94#397—22no—
95#397—22no—
96#396—22no—
97#396—22no—
98#394—22no—
99#392—22no—
100#392—22no—
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2311yescharacteristic
106#391C2311yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#388—22no—
112#388—21no—
113#387—21no—
114#387—22no—
115#386—22no—
116#386—21no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#385—22no—
124#385—21no—
125#384C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2311yescharacteristic
128#381—22no—
129#380—22no—
130#380—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4—22no—
133C4×C22⋊C4C2×C411yescharacteristic
134C4×C22⋊C4C2×C411yescharacteristic
135#376C2×C411yescharacteristic
136#324—22no—
137#324—22no—
138#324C2312yes—
139#324C2312yes—
140#324C2312yes—
141#324C2312yes—
142#323—22no—
143#323—22no—
144#323—22no—
145#323—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—42no—
151C2×#41—41no—
152C2×#41—41no—
153C2×#41—81no—
154C2×#41—81no—
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22⋊C412yes—
162C22×D4C22⋊C412yes—
163C22×D4C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C2×M16—42no—
174C2×M16—42no—
175C22×C8—42no—
176C22×C8—42no—
177C4⋊Q8—42no—
178#127—22no—
179#127—41no—
180#126—42no—
181#126—42no—
182#125—42no—
183#124—81no—
184#124—81no—
185#123—42no—
186#123—42no—
187#123—44no—
188#123—42no—
189#123—42no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—42no—
193#121—41no—
194#121—41no—
195#121—42no—
196#121—42no—
197#121—42no—
198#121—44no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—42no—
206C4×D4—42no—
207#117—41no—
208#117—41no—
209#117—81no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C411yescharacteristic
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C42—21no—
247C2×C42—81no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D412yes—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264#97—42no—
265C4×C8—42no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279C24—21no—
280C24C2×C22⋊C412yes—
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C411yescharacteristic
283#41—82no—
284#41—82no—
285#41—84no—
286#41—84no—
287#41—82no—
288#41—82no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—82no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4#13112yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—82no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D412yes—
363M16—162no—
364M16—162no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—162no—
370C2×C8—162no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—81no—
387C4⋊C4—81no—
388C4⋊C4—82no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—81no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—81no—
406C22⋊C4—81no—
407C22⋊C4#9912yes—
408C22⋊C4#9912yes—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—21no—
413C42—21no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—81no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—21no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23C2×#9912yes—
438C23C22×C22⋊C411yescharacteristic
439Q8—81no—
440Q8—81no—
441Q8—81no—
442Q8—81no—
443D4—42no—
444D4—42no—
445D4—44no—
446D4—81no—
447D4—81no—
448D4—82no—
449D4—82no—
450D4—84no—
451D4—84no—
452D4—162no—
453D4—164no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4C2×#9911yescharacteristic
506C2×C4C2×#9911yescharacteristic
507C2×C4C2×#9912yes—
508C2×C4C2×#13111yescharacteristic
509C2×C4#56811yescharacteristic
510C8—82no—
511C8—82no—
512C8—82no—
513C8—82no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—44no—
520C22—41no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—42no—
526C22—41no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22#288011yescharacteristic
537C22C22×#9911yescharacteristic
538C22#288811yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—42no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C2—21no—
557C2—21no—
558C2—42no—
559C2—81no—
560C2—82no—
561C2—82no—
562C2#1741111yescharacteristic
563C2#3336111yescharacteristic
564C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#38092C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#553 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17427C2semidirect
3. #11564C2semidirect
4. #11556C2semidirect
5. #11551C2semidirect
6. #11533C2semidirect
7. #11508C2semidirect
8. #11507C2semidirect
9. #11479C2semidirect
10. #11432C2semidirect
11. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#380922
#174272
#115642
#115562
#115512
#115332
#115082
#115072
#114792
#114322
#114312