Finite Group Databasebeta

← #481230

Subgroups — #481230

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60323C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#33485C211yesmaximal, characteristic, complemented
4#11636C212yesmaximal, complemented
5#11633C211yesmaximal, characteristic, complemented
6#11627C212yesmaximal, complemented
7#11626C211yesmaximal, characteristic, complemented
8#11620C212yesmaximal, complemented
9#11617C211yesmaximal, characteristic, complemented
10#11587C212yesmaximal, complemented
11#11586C211yesmaximal, characteristic, complemented
12#3481C411yescharacteristic
13#2994C412yes—
14#2993—21no—
15#2991C411yescharacteristic
16#2990C411yescharacteristic
17#2979C411yescharacteristic
18#2976—22no—
19#2974C412yes—
20#2972—21no—
21#2969—21no—
22#2966C2212yes—
23#2965C2211yescharacteristic
24C8×#41C2211yescharacteristic
25D4×M16—21no—
26#2933—22no—
27#2931C2211yescharacteristic
28C2×#436C2212yes—
29#2183—21no—
30Q8×C16—21no—
31#2169—21no—
32#2169—22no—
33#2169—21no—
34#2168—22no—
35#2163C2212yescomplemented
36#2163C2211yescharacteristic, complemented
37#2163C2212yescomplemented
38#2162C2211yescharacteristic, complemented
39#2150C2212yescomplemented
40C2×C4⋊C16C2211yescharacteristic, complemented
41C4×M32C2211yescharacteristic, complemented
42C4×M32C2212yescomplemented
43#1338C2211yescharacteristic, complemented
44#1338C2211yescharacteristic, complemented
45#1338C2212yescomplemented
46#1330C2211yescharacteristic
47#1330C2212yes—
48#1330C2211yescharacteristic
49#1330C2212yescomplemented
50#1330C2212yescomplemented
51#1330C2212yescomplemented
52#1330C2212yescomplemented
53C2×#131—21no—
54C22×M16—22no—
55#559C811yescharacteristic
56#554C811yescharacteristic
57#554C811yescharacteristic
58#553—21no—
59#553C812yes—
60#551—22no—
61#546—21no—
62#546C812yes—
63D42C811yescharacteristic
64#532C2×C411yescharacteristic
65#529—21no—
66#529C2×C412yes—
67#525—22no—
68C4×#41C2×C411yescharacteristic
69C4×#41C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×M32—22no—
73C2×M32—21no—
74C2×M32—22no—
75C22×C16—21no—
76#446—21no—
77#446C2×C411yescharacteristic
78#446C2311yescharacteristic
79C8×Q8C2×C412yes—
80C8×Q8D411yescharacteristic
81C8×Q8D411yescharacteristic
82#436—21no—
83#436—22no—
84#436—22no—
85#436—22no—
86#436—21no—
87#436C2×C412yes—
88#436C2×C412yes—
89#436C2×C411yescharacteristic
90#436C2311yescharacteristic
91#436C2312yes—
92#435—21no—
93#435—22no—
94#435—21no—
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99C8×D4—22no—
100C8×D4—21no—
101C8×D4—21no—
102C8×D4—22no—
103C8×D4—21no—
104C8×D4C2×C412yes—
105C8×D4C2×C412yes—
106C8×D4D412yes—
107C8×D4D411yescharacteristic
108C8×D4D411yescharacteristic
109C8×D4D412yes—
110#433—22no—
111#433—22no—
112#432C2×C412yes—
113#431—21no—
114#431—21no—
115#431C2×C411yescharacteristic
116#431C2×C412yes—
117#431C2×C411yescharacteristic
118#424—22no—
119#423—21no—
120#423C2×C411yescharacteristic
121C2×C4⋊C8C2×C412yes—
122#408—21no—
123#408—22no—
124#407—22no—
125C2×C22⋊C8—22no—
126C2×C22⋊C8—22no—
127#405—22no—
128C4×M16—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2312yes—
131C4⋊C16—21no—
132C4⋊C16—22no—
133C4⋊C16—21no—
134C4⋊C16—22no—
135C4⋊C16C2312yes—
136C4⋊C16C2311yescharacteristic
137C4⋊C16C2311yescharacteristic
138C22⋊C16—21no—
139C22⋊C16—22no—
140C22⋊C16—22no—
141C22⋊C16—21no—
142#346—22no—
143#346C2311yescharacteristic
144#346C2312yes—
145#346C2311yescharacteristic
146C4×C16—21no—
147C4×C16—22no—
148C4×C16—21no—
149C2×#41—21no—
150C2×#41C2×C811yescharacteristic
151C2×#41M1611yescharacteristic
152C22×D4—21no—
153C22×D4C2×C812yes—
154C23×C4—22no—
155#131—81no—
156#131—81no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—82no—
164C2×M16—82no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×D412yes—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D412yes—
171C22×C8C2×D411yescharacteristic
172C4⋊Q8C2×C811yescharacteristic
173#127C2×C811yescharacteristic
174#126—22no—
175#126—22no—
176#125—21no—
177#125C2×C812yes—
178#124—21no—
179#124—21no—
180#124C2×C812yes—
181#123—22no—
182#123—22no—
183#123—22no—
184#123—22no—
185C22⋊Q8—21no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8C2×C812yes—
189#121—22no—
190#121—21no—
191#121—21no—
192#121—22no—
193#121C2×C812yes—
194#121C2×C812yes—
195#120—22no—
196#120—22no—
197C4×Q8C22⋊C411yescharacteristic
198C4×Q8C2×C811yescharacteristic
199C4×Q8C2×C811yescharacteristic
200C4×Q8C22×C411yescharacteristic
201C4×D4—21no—
202C4×D4—22no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—22no—
207C4×D4—21no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4C22⋊C411yescharacteristic
211C4×D4C22⋊C412yes—
212C4×D4C2×C812yes—
213C4×D4C2×C812yes—
214C4×D4C2×C811yescharacteristic
215C4×D4C2×C811yescharacteristic
216C4×D4C2×C812yes—
217C4×D4C22×C411yescharacteristic
218C4×D4C22×C412yes—
219#117—22no—
220#117—21no—
221#117C2×C811yescharacteristic
222#117M1612yes—
223#117M1611yescharacteristic
224C2×C4⋊C4—21no—
225C2×C4⋊C4C2×C812yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C42—21no—
229C2×C42C22×C412yes—
230C2×C42C22×C411yescharacteristic
231M32—44no—
232M32—42no—
233M32—82no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—22no—
238C2×C16—21no—
239C2×C16—22no—
240C2×C16—81no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8C22⋊C411yescharacteristic
247C4⋊C8C22⋊C412yes—
248C4⋊C8C22⋊C411yescharacteristic
249C4⋊C8C22⋊C412yes—
250C4⋊C8C22×C411yescharacteristic
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C411yescharacteristic
253C22⋊C8—22no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8C22⋊C412yes—
263C22⋊C8C22⋊C412yes—
264C22⋊C8C22⋊C412yes—
265#97—21no—
266#97—21no—
267#97—22no—
268C4×C8—22no—
269C4×C8—21no—
270C4×C8C22×C411yescharacteristic
271C4×C8C22×C412yes—
272C4×C8C2×D411yescharacteristic
273C4×C8C2×D411yescharacteristic
274C4×C8C2×D412yes—
275C4×C8C2×D412yes—
276C4×C8C2411yescharacteristic
277C24—22no—
278#41—42no—
279#41—81no—
280#41—81no—
281#41—81no—
282C2×Q8—21no—
283C2×Q8C22⋊C811yescharacteristic
284C2×Q8C22×C811yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—44no—
292C2×D4—82no—
293C2×D4C22⋊C811yescharacteristic
294C2×D4C22⋊C812yes—
295C2×D4C22×C811yescharacteristic
296C2×D4C22×C812yes—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C22×C812yes—
309C22×C4C22×C811yescharacteristic
310C22×C4C2×M1611yescharacteristic
311C22×C4C2×M1612yes—
312M16—41no—
313M16—42no—
314M16—41no—
315M16—42no—
316M16—81no—
317M16—82no—
318M16—82no—
319M16—81no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—82no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8C2×C22⋊C412yes—
333C2×C8C2×C22⋊C411yescharacteristic
334C2×C8C2×C22⋊C411yescharacteristic
335C2×C8C2×C22⋊C412yes—
336C2×C8C22×D411yescharacteristic
337C2×C8C22×D411yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4C22⋊C811yescharacteristic
346C4⋊C4C22⋊C812yes—
347C4⋊C4C22⋊C811yescharacteristic
348C4⋊C4C22⋊C812yes—
349C4⋊C4C22×C811yescharacteristic
350C4⋊C4C22×C812yes—
351C4⋊C4C22×C811yescharacteristic
352C22⋊C4—22no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—21no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4C22⋊C812yes—
364C22⋊C4C22⋊C812yes—
365C22⋊C4C22⋊C812yes—
366C42—21no—
367C42—21no—
368C42—21no—
369C42C2×C22⋊C411yescharacteristic
370C42C2×C22⋊C412yes—
371C42C22×C812yes—
372C42C22×C811yescharacteristic
373C42C2×M1611yescharacteristic
374C42C2×M1612yes—
375C42C23×C411yescharacteristic
376C16—22no—
377C16—41no—
378C16—42no—
379C16—42no—
380C16—41no—
381C16—41no—
382C23—22no—
383C23—21no—
384C23—44no—
385C23—82no—
386C23—82no—
387C23C2×C22⋊C812yes—
388C23C2×C22⋊C811yescharacteristic
389Q8—22no—
390Q8—41no—
391Q8—81no—
392D4—22no—
393D4—42no—
394D4—41no—
395D4—42no—
396D4—44no—
397D4—42no—
398D4—82no—
399D4—81no—
400D4—81no—
401D4—82no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4C2×C22⋊C811yescharacteristic
424C2×C4C2×C22⋊C812yes—
425C2×C4C2×C22⋊C812yes—
426C2×C4C2×C22⋊C812yes—
427C2×C4C2×C22⋊C811yescharacteristic
428C2×C4C2×C22⋊C811yescharacteristic
429C2×C4C22×C22⋊C411yescharacteristic
430C2×C4C23×C811yescharacteristic
431C2×C4C22×M1611yescharacteristic
432C8—21no—
433C8—22no—
434C8—21no—
435C8—22no—
436C8—21no—
437C8—42no—
438C8—41no—
439C8—41no—
440C8#57611yescharacteristic
441C8#57711yescharacteristic
442C22—22no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—21no—
447C22—22no—
448C22—41no—
449C22—42no—
450C22—41no—
451C22—42no—
452C22—41no—
453C22—42no—
454C22—164no—
455C22C22×C22⋊C811yescharacteristic
456C4—22no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—21no—
462C4—22no—
463C4—41no—
464C4—42no—
465C4—41no—
466C4—41no—
467C4#211712yes—
468C4#289711yescharacteristic
469C4#290111yescharacteristic
470C2—41no—
471C2—42no—
472C2—82no—
473C2—81no—
474C2#1743811yescharacteristic
475C2#3333711yescharacteristic
476C2C2×#211711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C22×C22⋊C8
prime radical#3481C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33503C2semidirect
3. #33485C2semidirect
4. #11636C2semidirect
5. #11633C2semidirect
6. #11627C2semidirect
7. #11626C2semidirect
8. #11620C2semidirect
9. #11617C2semidirect
10. #11587C2semidirect
11. #11586C2semidirect
35. #2163C22semidirect
36. #2163C22semidirect
37. #2163C22semidirect
38. #2162C22semidirect
39. #2150C22semidirect
40. C2×C4⋊C16C22semidirect
41. C4×M32C22semidirect
42. C4×M32C22semidirect
43. #1338C22semidirect
44. #1338C22semidirect
45. #1338C22semidirect
49. #1330C22semidirect
50. #1330C22semidirect
51. #1330C22semidirect
52. #1330C22semidirect

Maximal subgroups

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

subgroupindex
#603232
#335032
#334852
#116362
#116332
#116272
#116262
#116202
#116172
#115872
#115862