Finite Group Databasebeta

← #214690

Subgroups — #214690

Order 512.

917 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35624C211yesmaximal, characteristic
2#17369C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#9103C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8961C212yesmaximal, complemented
7#8957C212yesmaximal, complemented
8#8732C212yesmaximal, complemented
9#8327C212yesmaximal, complemented
10C2×#541C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2975C2212yes—
13#2972C2212yes—
14#2566C411yescharacteristic
15#2566C411yescharacteristic
16#2547C411yescharacteristic
17#2547C411yescharacteristic
18#2499C411yescharacteristic
19#2499C411yescharacteristic
20#2480—22no—
21#2480—22no—
22#2460—22no—
23#2460—22no—
24#2435C411yescharacteristic
25#2435C411yescharacteristic
26#2326C2211yescharacteristic
27#2326C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#124C2211yescharacteristic
30#2298C2211yescharacteristic
31#1479C2214yescomplemented
32#1479C2214yescomplemented
33#1465—22no—
34#1462—22no—
35#1460C2212yes—
36#1460C2212yes—
37#1459—22no—
38#1458C2212yes—
39#1458C2212yes—
40C2×#323—22no—
41#1276C2214yescomplemented
42#1274C2214yescomplemented
43C23×D4C2×C411yescharacteristic
44#541—44no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#126—21no—
47C2×#126—21no—
48C2×#125—21no—
49C2×#125—21no—
50C2×#125C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#124C2×C411yescharacteristic
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2×C411yescharacteristic
57#518—44no—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C22⋊C4D412yes—
69C22×C22⋊C4D412yes—
70#436—44no—
71C8×D4—44no—
72#431C2314yes—
73#431C2314yes—
74#407—24no—
75C2×C22⋊C8—24no—
76#397—22no—
77#397—22no—
78#397—22no—
79#397—22no—
80#396—22no—
81#396—22no—
82#394—22no—
83#394—22no—
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—21no—
97#388—21no—
98#387—21no—
99#387—21no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#384C2×C411yescharacteristic
107#384C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#380—22no—
111#380—22no—
112#379—22no—
113#379—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120#376C2311yescharacteristic
121C43C2311yescharacteristic
122Q8⋊C8—44no—
123Q8⋊C8—44no—
124#324—22no—
125#324—22no—
126#323—22no—
127#323—24no—
128#323—22no—
129#323—24no—
130C25C22⋊C412yes—
131C22×Q8C22×C411yescharacteristic
132C22×Q8C22×C411yescharacteristic
133C22×D4—24no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4C22⋊C412yes—
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C2×M16—44no—
145C22×C8—44no—
146C4⋊Q8—42no—
147#126—44no—
148#126—44no—
149#125—42no—
150#125—44no—
151#125—44no—
152#124—24no—
153#124—24no—
154#124—42no—
155#124—42no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—44no—
165C4×D4—44no—
166#117—24no—
167#117—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C2×D411yescharacteristic
204C2×C22⋊C4C2×D411yescharacteristic
205C2×C22⋊C4C2×D412yes—
206C2×C22⋊C4C2×D412yes—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C22⋊C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2411yescharacteristic
218C4⋊C8—24no—
219C4⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8—44no—
223C22⋊C8—44no—
224C4×C8—24no—
225C4×C8—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238C24—24no—
239C24—42no—
240C24—42no—
241C24—82no—
242C24—82no—
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C412yes—
245C24C2×C22⋊C412yes—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—416no—
261C2×D4#13114yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289M16—164no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—164no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—81no—
339C42—81no—
340C42H(5,𝔽2)14yes—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—22no—
348C23—21no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—48no—
354C23—41no—
355C23—41no—
356C23—162no—
357C23—162no—
358C23—164no—
359C23—164no—
360C23#40814yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—44no—
363Q8—44no—
364D4—28no—
365D4—48no—
366D4—44no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—162no—
406C2×C4—162no—
407C2×C4#51812yes—
408C2×C4#53412yes—
409C2×C4#53412yes—
410C2×C4C2×#13112yes—
411C2×C4#56812yes—
412C8—84no—
413C8—84no—
414C22—21no—
415C22—21no—
416C22—24no—
417C22—44no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—42no—
427C22—82no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22—168no—
432C22#230811yescharacteristic
433C22C2×#40811yescharacteristic
434C22#287911yescharacteristic
435C4—22no—
436C4—22no—
437C4—42no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4#297614yes—
451C2—21no—
452C2—21no—
453C2—44no—
454C2—82no—
455C2—81no—
456C2#2174111yescharacteristic
457C2#2178411yescharacteristic
458C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
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
4. #9103C2semidirect
5. #9099C2semidirect
6. #8961C2semidirect
7. #8957C2semidirect
8. #8732C2semidirect
9. #8327C2semidirect
31. #1479C22semidirect
32. #1479C22semidirect
41. #1276C22semidirect
42. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#356242
#173692
#173502
#91032
#90992
#89612
#89572
#87322
#83272