Finite Group Databasebeta

← #480136

Subgroups — #480136

Order 512.

1167 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#34275C211yesmaximal, characteristic
2#17351C212yesmaximal, complemented
3#11480C212yesmaximal, complemented
4#11479C212yesmaximal, complemented
5#11476C212yesmaximal, complemented
6#11475C212yesmaximal, complemented
7#11448C212yesmaximal, complemented
8C2×#1318C212yesmaximal, complemented
9C2×#535C411yescharacteristic
10C2×#520C2211yescharacteristic
11C2×#518C411yescharacteristic
12#2972—22no—
13#2363—22no—
14#2361—22no—
15#2358—22no—
16#2358—22no—
17#2328C411yescharacteristic
18#2328C2211yescharacteristic
19#2326C411yescharacteristic
20#2326C2211yescharacteristic
21#2317C2211yescharacteristic
22C4×#127C411yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26#2297C411yescharacteristic
27C2×#390C2211yescharacteristic
28C4×#117C411yescharacteristic
29#1461C2214yescomplemented
30#1459—22no—
31#1459C2214yescomplemented
32#1458—22no—
33C2×#324—22no—
34C2×#324C2214yes—
35#1325C2214yescomplemented
36#1324C2214yescomplemented
37#1318C2214yes—
38#1318C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41#535—24no—
42#535C818yescomplemented
43C2×#127C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#121—22no—
48C2×#120—22no—
49#520—44no—
50#518—44no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2311yescharacteristic
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42C2×C411yescharacteristic
75C8×D4—48no—
76#431—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#390—22no—
82#390—22no—
83#390—22no—
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#390C2311yescharacteristic
89#390C2311yescharacteristic
90#389—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#384C2311yescharacteristic
105#382C2×C411yescharacteristic
106#382C2311yescharacteristic
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114#376C2×C411yescharacteristic
115#376C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118#324—24no—
119#324—24no—
120#324C2318yes—
121#323—24no—
122C25—21no—
123C2×#41—42no—
124C2×#41—42no—
125C2×#41C2×C814yes—
126C2×#41M1614yes—
127C22×Q8C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4C22⋊C412yes—
135C22×D4C2×C814yescomplemented
136C22×D4C22×C411yescharacteristic
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C22×C8—44no—
150C22×C8—44no—
151#127—22no—
152#127—22no—
153#127C2×C814yescomplemented
154#127C2×C814yescomplemented
155#124—22no—
156#124—22no—
157#124C2×C814yescomplemented
158#124C2×C814yescomplemented
159#121—216no—
160#120—216no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—28no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—44no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—84no—
174C4×D4—84no—
175#117—22no—
176#117—24no—
177#117—44no—
178#117—42no—
179#117C2×C814yescomplemented
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—84no—
211C2×C22⋊C4C22⋊C411yescharacteristic
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C411yescharacteristic
214C2×C22⋊C4C2×D412yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42C22⋊C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C4⋊C8—44no—
231C22⋊C8—28no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C4×C8—44no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—21no—
244C24—21no—
245C24—24no—
246C24—42no—
247C24—42no—
248C24—82no—
249C24—82no—
250C24C2×C22⋊C412yes—
251C24C2×C22⋊C411yescharacteristic
252#41—48no—
253#41—48no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8C22×C812yes—
257C2×Q8C2×M1612yes—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—28no—
261C2×D4—28no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—24no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—48no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—48no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4C22⋊C818yes—
280C2×D4C22×C812yes—
281C2×D4C22×C812yes—
282C2×D4C22×C812yes—
283C2×D4C2×M1612yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4C22⋊C818yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C22×C812yes—
320C22×C4C2×M1612yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D412yes—
323C2×C8—48no—
324C2×C8—164no—
325C2×C8—164no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—24no—
339C22⋊C4—22no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—24no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4#9914yes—
356C42—21no—
357C42—21no—
358C42—24no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—81no—
369C42—81no—
370C42C22×C812yes—
371C42C22×C812yes—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—48no—
389C23—88no—
390C23—162no—
391C23—162no—
392C23—164no—
393C23C2×C22⋊C814yes—
394C23C2×C22⋊C812yes—
395C23C2×#9912yes—
396C23#41312yes—
397C23C22×C22⋊C411yescharacteristic
398Q8—42no—
399Q8—42no—
400D4—216no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—48no—
406D4—42no—
407D4—42no—
408D4—88no—
409D4—88no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—48no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—162no—
449C2×C4—162no—
450C2×C4—162no—
451C2×C4—162no—
452C2×C4—162no—
453C2×C4C2×C22⋊C814yes—
454C2×C4C2×C22⋊C812yes—
455C2×C4C2×#9912yes—
456C2×C4C2×#9912yes—
457C2×C4C23×C811yescharacteristic
458C2×C4C22×M1611yescharacteristic
459C8—88no—
460C22—21no—
461C22—22no—
462C22—21no—
463C22—22no—
464C22—24no—
465C22—42no—
466C22—42no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—168no—
482C22#145914yes—
483C22C22×C22⋊C811yescharacteristic
484C22C22×#9911yescharacteristic
485C22C2×#41311yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—21no—
490C4—21no—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—81no—
501C4—81no—
502C2—21no—
503C2—21no—
504C2—24no—
505C2—44no—
506C2—82no—
507C2—82no—
508C2C2×#145911yescharacteristic
509C2#3335911yescharacteristic
510C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#535C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17351C2semidirect
3. #11480C2semidirect
4. #11479C2semidirect
5. #11476C2semidirect
6. #11475C2semidirect
7. #11448C2semidirect
8. C2×#1318C2semidirect
29. #1461C22semidirect
31. #1459C22semidirect
35. #1325C22semidirect
36. #1324C22semidirect
42. #535C8semidirect
135. C22×D4C2×C8semidirect
153. #127C2×C8semidirect
154. #127C2×C8semidirect
157. #124C2×C8semidirect
158. #124C2×C8semidirect
179. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
#342752
#173512
#114802
#114792
#114762
#114752
#114482
C2×#13182