Finite Group Databasebeta

← #480994

Subgroups — #480994

Order 512.

781 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37865C211yesmaximal, characteristic
2#17426C212yesmaximal
3#11571C212yesmaximal, complemented
4#11555C212yesmaximal, complemented
5#11551C212yesmaximal, complemented
6#11536C211yesmaximal, characteristic, complemented
7#11514C211yesmaximal, characteristic, complemented
8#11511C211yesmaximal, characteristic, complemented
9#11482C211yesmaximal, characteristic, complemented
10#11448C211yesmaximal, characteristic, complemented
11C2×#1324C211yesmaximal, characteristic, complemented
12C2×#554C2211yescharacteristic
13C2×#552C411yescharacteristic
14#2974—22no—
15#2857C411yescharacteristic
16#2719—22no—
17#2698—22no—
18#2660C411yescharacteristic
19#2621C411yescharacteristic
20#2568C411yescharacteristic
21#2565C2211yescharacteristic
22#2503C411yescharacteristic
23#2503C2211yescharacteristic
24#2502C2211yescharacteristic
25#2499C411yescharacteristic
26#2438—22no—
27#2419—22no—
28#2327C2211yescharacteristic
29#2317C2211yescharacteristic
30#2290C411yescharacteristic
31C2×#382C2211yescharacteristic
32#1472—22no—
33#1472—22no—
34#1470—22no—
35#1467C2212yes—
36#1463C2212yes—
37#1463C2212yes—
38#1461C2212yes—
39C2×#324C2212yes—
40C2×#324C2212yes—
41#1325C2212yes—
42#1325C2212yes—
43#1324C2211yescharacteristic
44#1324C2211yescharacteristic
45#1324C2212yes—
46#1324C2212yes—
47#1324C2211yescharacteristic
48#1324C2211yescharacteristic
49#1319C2212yes—
50#1319C2212yes—
51#554—42no—
52#554—42no—
53#552—42no—
54#552—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#126—21no—
58C2×#125—21no—
59C2×#125C2×C411yescharacteristic
60C2×#125D412yes—
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8D412yes—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D412yes—
72C2×#117—21no—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C42C2×C411yescharacteristic
79#436—42no—
80#435—42no—
81#435—42no—
82C8×D4—42no—
83#432—22no—
84#431—22no—
85#407—22no—
86#407—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89#400—22no—
90#400—22no—
91#400—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#397—22no—
97#397—22no—
98#395—22no—
99#391—22no—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2311yescharacteristic
106#389—22no—
107#389—22no—
108#388—21no—
109#387—22no—
110#387—21no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—21no—
115#385—22no—
116#385—22no—
117#385—21no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#384C2311yescharacteristic
121#382—22no—
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#382C2311yescharacteristic
127#382C2311yescharacteristic
128#381—22no—
129#381—22no—
130#380—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4C2×C411yescharacteristic
133C4×C22⋊C4C2×C411yescharacteristic
134#376C2×C411yescharacteristic
135C2×#95—22no—
136#324—22no—
137#324—22no—
138#324—22no—
139#324—22no—
140#324—22no—
141#324—22no—
142#324C2312yes—
143#324C2312yes—
144#324C2312yes—
145#324C2312yes—
146C22×Q8C22⋊C412yes—
147C22×Q8C22×C411yescharacteristic
148C22×Q8C22×C411yescharacteristic
149C22×D4C22⋊C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C2×M16—42no—
157C2×M16—42no—
158C22×C8—42no—
159C22×C8—42no—
160C4⋊Q8—22no—
161C4⋊Q8—42no—
162C4⋊Q8—41no—
163#126—44no—
164#125—42no—
165#125—42no—
166#125—42no—
167#125—44no—
168#123—42no—
169#123—42no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188#117—22no—
189#117—41no—
190#117—42no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—41no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D412yes—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—42no—
255C4×C8—42no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—82no—
285C2×D4—42no—
286C2×D4#9912yes—
287C2×D4#9912yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—81no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D412yes—
326M16—162no—
327M16—162no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—162no—
333C2×C8—162no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—81no—
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—81no—
374C42—21no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—42no—
383C42—41no—
384C42—42no—
385C42—42no—
386C42#13112yes—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C2×#9912yes—
396C23C2×#9911yescharacteristic
397C23C2×#9911yescharacteristic
398C23C22×C22⋊C411yescharacteristic
399Q8—81no—
400Q8—81no—
401Q8—81no—
402Q8—81no—
403Q8—162no—
404D4—24no—
405D4—42no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4C2×#9912yes—
461C2×C4C2×#13111yescharacteristic
462C2×C4#56811yescharacteristic
463C8—82no—
464C8—82no—
465C8—82no—
466C8—82no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—22no—
475C22—21no—
476C22—22no—
477C22—42no—
478C22—42no—
479C22—81no—
480C22—81no—
481C22#213312yes—
482C22#288011yescharacteristic
483C22C22×#9911yescharacteristic
484C22#288911yescharacteristic
485C4—21no—
486C4—21no—
487C4—22no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—41no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C2—21no—
504C2—21no—
505C2—22no—
506C2—41no—
507C2—42no—
508C2#1741611yescharacteristic
509C2#3336111yescharacteristic
510C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#554 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11571C2semidirect
4. #11555C2semidirect
5. #11551C2semidirect
6. #11536C2semidirect
7. #11514C2semidirect
8. #11511C2semidirect
9. #11482C2semidirect
10. #11448C2semidirect
11. C2×#1324C2semidirect

Maximal subgroups

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

subgroupindex
#378652
#174262
#115712
#115552
#115512
#115362
#115142
#115112
#114822
#114482
C2×#13242