Finite Group Databasebeta

← #480997

Subgroups — #480997

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37868C211yesmaximal, characteristic
2#17428C212yesmaximal
3#11571C212yesmaximal, complemented
4#11556C212yesmaximal, complemented
5#11552C212yesmaximal, complemented
6#11536C211yesmaximal, characteristic, complemented
7#11515C211yesmaximal, characteristic, complemented
8#11513C211yesmaximal, characteristic, complemented
9#11482C211yesmaximal, characteristic, complemented
10#11449C211yesmaximal, characteristic, complemented
11#11447C211yesmaximal, characteristic, complemented
12C2×#556C2211yescharacteristic
13C2×#552C411yescharacteristic
14#2984—22no—
15#2857C411yescharacteristic
16#2722—22no—
17#2717—22no—
18#2660C411yescharacteristic
19#2621C411yescharacteristic
20#2567C411yescharacteristic
21#2565C2211yescharacteristic
22#2502C411yescharacteristic
23#2499C411yescharacteristic
24#2499C2211yescharacteristic
25#2499C2211yescharacteristic
26#2478—22no—
27#2462—22no—
28#2327C2211yescharacteristic
29#2317C2211yescharacteristic
30#2290C411yescharacteristic
31#2285C2211yescharacteristic
32#1473—22no—
33#1473—22no—
34#1469—22no—
35#1467C2212yes—
36#1464C2212yes—
37#1464C2212yes—
38#1461C2212yes—
39#1458C2212yes—
40#1458C2212yes—
41#1325C2212yes—
42#1325C2212yes—
43#1324C2211yescharacteristic
44#1324C2211yescharacteristic
45#1324C2212yes—
46#1324C2212yes—
47#1324C2211yescharacteristic
48#1324C2211yescharacteristic
49#1319C2212yes—
50#1319C2212yes—
51#556—42no—
52#556—42no—
53#552—42no—
54#552—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#126—21no—
58C2×#126D412yes—
59C2×#126D412yes—
60C2×#125—21no—
61C2×#125C2×C411yescharacteristic
62C2×#124D412yes—
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D412yes—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79#446—42no—
80#446—42no—
81#446—42no—
82C8×Q8—42no—
83#432—22no—
84#432—22no—
85#432—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89#398—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#397—22no—
94#397—22no—
95#397—22no—
96#397—22no—
97#395—22no—
98#395—22no—
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2311yescharacteristic
105#389—22no—
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#387—22no—
115#387—22no—
116#386—21no—
117#385—22no—
118#385—21no—
119#385—22no—
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#384C2311yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2311yescharacteristic
128#382C2311yescharacteristic
129#380—22no—
130#380—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C411yescharacteristic
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—
146C22×Q8C22⋊C412yes—
147C22×Q8C22×C411yescharacteristic
148C22×Q8C22×C411yescharacteristic
149C22×D4C22⋊C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C4⋊Q8—22no—
155C4⋊Q8—41no—
156C4⋊Q8—42no—
157#126—42no—
158#126—42no—
159#126—42no—
160#126—42no—
161#126—44no—
162#125—42no—
163#125—42no—
164#125—42no—
165#124—42no—
166#124—42no—
167#123—44no—
168C22⋊Q8—42no—
169C22⋊Q8—41no—
170C22⋊Q8—41no—
171C22⋊Q8—42no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×D4—22no—
181C4×D4—22no—
182#117—22no—
183#117—41no—
184#117—41no—
185#117—42no—
186#117—41no—
187#117—42no—
188#117—42no—
189#117—41no—
190#117—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4C22⋊C411yescharacteristic
224C2×C22⋊C4C22⋊C411yescharacteristic
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—41no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252#97—42no—
253#97—42no—
254#97—42no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8—42no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—21no—
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—22no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—82no—
286C2×D4—22no—
287C2×D4—81no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4—82no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D412yes—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4#9912yes—
351C4⋊C4#9912yes—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—22no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—81no—
365C42—21no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—42no—
378C42—41no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42#13112yes—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23C22×C22⋊C411yescharacteristic
394Q8—44no—
395Q8—81no—
396Q8—81no—
397Q8—82no—
398Q8—81no—
399Q8—81no—
400D4—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4C2×#9912yes—
449C2×C4C2×#9911yescharacteristic
450C2×C4C2×#9911yescharacteristic
451C2×C4C2×#9912yes—
452C2×C4C2×#13111yescharacteristic
453C2×C4#56811yescharacteristic
454C8—82no—
455C8—82no—
456C8—82no—
457C8—82no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22#288011yescharacteristic
471C22C22×#9911yescharacteristic
472C22#288911yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—41no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C2—21no—
493C2—21no—
494C2—81no—
495C2—81no—
496C2#1741611yescharacteristic
497C2#3336011yescharacteristic
498C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ 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. #11556C2semidirect
5. #11552C2semidirect
6. #11536C2semidirect
7. #11515C2semidirect
8. #11513C2semidirect
9. #11482C2semidirect
10. #11449C2semidirect
11. #11447C2semidirect

Maximal subgroups

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

subgroupindex
#378682
#174282
#115712
#115562
#115522
#115362
#115152
#115132
#114822
#114492
#114472