Finite Group Databasebeta

← #213794

Subgroups — #213794

Order 512.

1143 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#9106C212yesmaximal, complemented
5#9105C212yesmaximal, complemented
6#8964C212yesmaximal, complemented
7#8963C212yesmaximal, complemented
8#8731C212yesmaximal, complemented
9#8322C212yesmaximal, complemented
10C2×#535C411yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#518C2211yescharacteristic
13#2983C2212yescomplemented
14#2975C2212yescomplemented
15#2363—22no—
16#2361—22no—
17#2358—22no—
18#2358—22no—
19#2328C411yescharacteristic
20#2328C2211yescharacteristic
21#2326C411yescharacteristic
22#2326C2211yescharacteristic
23#2317C411yescharacteristic
24C4×#127C411yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C411yescharacteristic
27C4×#124C2211yescharacteristic
28#2297C411yescharacteristic
29C2×#390C411yescharacteristic
30C4×#117C2211yescharacteristic
31#1478C2214yes—
32#1478C2214yes—
33#1466—22no—
34#1465—22no—
35#1464—22no—
36#1463—22no—
37#1461C2212yescomplemented
38#1460C2212yescomplemented
39#1458C2212yes—
40#1458C2212yes—
41#1276C2214yescomplemented
42#1274C2214yescomplemented
43C22×#41C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45#535—24no—
46#535—24no—
47C2×#127C2×C411yescharacteristic
48C2×#127C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#121—22no—
52C2×#120—22no—
53#520—44no—
54#518—44no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117D412yes—
72C2×#117C2311yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D412yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42C2×C411yescharacteristic
79#446—44no—
80#436—44no—
81#432—24no—
82#431C2314yes—
83#431C2314yes—
84#407—24no—
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#390—22no—
88#390—22no—
89#390—22no—
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#390C2×C411yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C411yescharacteristic
95#390C2×C411yescharacteristic
96#389—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120#376C2×C411yescharacteristic
121#376C2311yescharacteristic
122C43C2×C411yescharacteristic
123C43C2311yescharacteristic
124D4⋊C8—44no—
125D4⋊C8—44no—
126#324—22no—
127#324—22no—
128#324—24no—
129#323—22no—
130#323—22no—
131#323—24no—
132C25—21no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—42no—
136C2×#41—42no—
137C22×Q8C22×C411yescharacteristic
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4C22⋊C412yes—
145C22×D4M1614yescomplemented
146C22×D4C22×C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—44no—
160#127—22no—
161#127—22no—
162#127—22no—
163#127—22no—
164#124—22no—
165#124—22no—
166#124—22no—
167#124—22no—
168#121—216no—
169#120—216no—
170C4×Q8—24no—
171C4×Q8—42no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—44no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—44no—
182C4×D4—84no—
183C4×D4—84no—
184#117—22no—
185#117—24no—
186#117—44no—
187#117—42no—
188#117M1614yescomplemented
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—84no—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C22⋊C4C2×D411yescharacteristic
223C2×C22⋊C4C2×D412yes—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42C22⋊C412yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—24no—
240C4⋊C8—24no—
241C4⋊C8—44no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—44no—
245#97—44no—
246C4×C8—24no—
247C4×C8—24no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—21no—
256C24—21no—
257C24—21no—
258C24—24no—
259C24—42no—
260C24—42no—
261C24—82no—
262C24—82no—
263C24C2×C22⋊C412yes—
264C24C2×C22⋊C411yescharacteristic
265#41—48no—
266#41—48no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×D4—24no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—48no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—48no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4C2×M1612yes—
296C2×D4C2×M1612yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×M1612yes—
332C22×C4C2×M1612yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—164no—
337C2×C8—44no—
338C2×C8—44no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C22⋊C4—28no—
352C22⋊C4—28no—
353C22⋊C4—24no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—24no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—22no—
370C42—24no—
371C42—21no—
372C42—21no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—81no—
381C42—81no—
382C42C2×M1612yes—
383C42C2×M1612yes—
384C42H(5,𝔽2)14yes—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—48no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—88no—
406C23—164no—
407C23—162no—
408C23—162no—
409C23#40712yes—
410C23C22×C22⋊C411yescharacteristic
411Q8—42no—
412Q8—42no—
413D4—24no—
414D4—24no—
415D4—48no—
416D4—48no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—88no—
422D4—88no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—48no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—162no—
460C2×C4—162no—
461C2×C4—162no—
462C2×C4—162no—
463C2×C4—162no—
464C2×C4—162no—
465C2×C4#40712yes—
466C2×C4#51812yes—
467C2×C4#53412yes—
468C2×C4#53412yes—
469C2×C4C22×M1611yescharacteristic
470C2×C4C22×M1611yescharacteristic
471C2×C4#56812yes—
472C8—84no—
473C8—84no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—41no—
483C22—41no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—168no—
497C22#230811yescharacteristic
498C22C2×#40711yescharacteristic
499C22#287911yescharacteristic
500C4—22no—
501C4—22no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—41no—
509C4—41no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4#297312yes—
515C4#297512yes—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—82no—
520C2—82no—
521C2—82no—
522C2#2174211yescharacteristic
523C2#2178211yescharacteristic
524C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #17370C2semidirect
4. #9106C2semidirect
5. #9105C2semidirect
6. #8964C2semidirect
7. #8963C2semidirect
8. #8731C2semidirect
9. #8322C2semidirect
13. #2983C22semidirect
14. #2975C22semidirect
37. #1461C22semidirect
38. #1460C22semidirect
41. #1276C22semidirect
42. #1274C22semidirect
145. C22×D4M16semidirect
188. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342752
#173722
#173702
#91062
#91052
#89642
#89632
#87312
#83222