Finite Group Databasebeta

← #17521

Subgroups — #17521

Order 256.

505 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3481C211yesmaximal, characteristic
2#2990C211yesmaximal, characteristic, complemented
3#2922C211yesmaximal, characteristic, complemented
4#1496C211yesmaximal, characteristic, complemented
5#1495C211yesmaximal, characteristic, complemented
6#1494C211yesmaximal, characteristic, complemented
7#1490C211yesmaximal, characteristic, complemented
8#1488C211yesmaximal, characteristic, complemented
9#1486C211yesmaximal, characteristic, complemented
10#1483C211yesmaximal, characteristic, complemented
11#1482C211yesmaximal, characteristic, complemented
12#1480C211yesmaximal, characteristic, complemented
13#1478C211yesmaximal, characteristic, complemented
14#1477C211yesmaximal, characteristic, complemented
15C2×D4⋊C8C211yesmaximal, characteristic, complemented
16#559C411yescharacteristic
17#554C411yescharacteristic
18#554C411yescharacteristic
19#553—21no—
20#553C411yescharacteristic
21#553C411yescharacteristic
22#551—21no—
23#551—21no—
24#546—21no—
25#546C411yescharacteristic
26#546C411yescharacteristic
27D42C411yescharacteristic
28#532C2211yescharacteristic
29#529—21no—
30#529C2211yescharacteristic
31#529C2211yescharacteristic
32#525—21no—
33#525—21no—
34C4×#41C2211yescharacteristic
35C4×#41C2211yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39#446—21no—
40#435—21no—
41#435—21no—
42C8×D4—21no—
43#433—21no—
44#433—21no—
45#432C2211yescharacteristic, complemented
46#432C2211yescharacteristic, complemented
47#431C2211yescharacteristic, complemented
48#431C2211yescharacteristic, complemented
49#431C2211yescharacteristic, complemented
50#431C2211yescharacteristic, complemented
51#431C2211yescharacteristic, complemented
52#424—21no—
53#423C2211yescharacteristic, complemented
54C2×C4⋊C8C2211yescharacteristic, complemented
55C2×C4⋊C8C2211yescharacteristic, complemented
56#408—21no—
57C4×M16C2211yescharacteristic, complemented
58C2×C4×C8C2211yescharacteristic, complemented
59Q8⋊C8C2211yescharacteristic, complemented
60Q8⋊C8C2211yescharacteristic, complemented
61Q8⋊C8C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic, complemented
68D4⋊C8C2211yescharacteristic, complemented
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71D4⋊C8C2211yescharacteristic, complemented
72D4⋊C8C2211yescharacteristic, complemented
73D4⋊C8C2211yescharacteristic, complemented
74D4⋊C8C2211yescharacteristic, complemented
75C2×#41—21no—
76C2×#41C2×C411yescharacteristic
77C2×#41D411yescharacteristic
78C22×D4—21no—
79C22×D4C2×C411yescharacteristic
80C22×D4C2×C411yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C4⋊Q8C2×C411yescharacteristic
89#127C2×C411yescharacteristic
90#126—21no—
91#126—21no—
92#126—21no—
93#126—21no—
94#125—21no—
95#125C2×C411yescharacteristic
96#125C2×C411yescharacteristic
97#124—21no—
98#124—21no—
99#124C2×C411yescharacteristic
100#124C2×C411yescharacteristic
101#123—21no—
102#123—21no—
103#123—21no—
104#123—21no—
105#123—21no—
106#123—21no—
107#123—21no—
108#123—21no—
109C22⋊Q8—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8C2×C411yescharacteristic
115C22⋊Q8C2×C411yescharacteristic
116#121—21no—
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121C2×C411yescharacteristic
123#121C2×C411yescharacteristic
124#121C2×C411yescharacteristic
125#121C2×C411yescharacteristic
126#120—21no—
127#120—21no—
128#120—21no—
129#120—21no—
130C4×Q8C2×C411yescharacteristic
131C4×Q8C2×C411yescharacteristic
132C4×Q8D411yescharacteristic
133C4×Q8C2311yescharacteristic
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2×C411yescharacteristic
148C4×D4C2×C411yescharacteristic
149C4×D4C2×C411yescharacteristic
150C4×D4C2×C411yescharacteristic
151C4×D4C2×C411yescharacteristic
152C4×D4C2×C411yescharacteristic
153C4×D4C2×C411yescharacteristic
154C4×D4C2×C411yescharacteristic
155C4×D4D411yescharacteristic
156C4×D4D411yescharacteristic
157C4×D4D411yescharacteristic
158C4×D4C2311yescharacteristic
159C4×D4C2311yescharacteristic
160C4×D4C2311yescharacteristic
161#117—21no—
162#117—21no—
163#117—21no—
164#117C2×C411yescharacteristic
165#117D411yescharacteristic
166#117D411yescharacteristic
167#117D411yescharacteristic
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×C411yescharacteristic
170C2×C4⋊C4C2×C411yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C42—21no—
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8C2311yescharacteristic
187C4⋊C8C2311yescharacteristic
188C4⋊C8C2311yescharacteristic
189C4⋊C8C2311yescharacteristic
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197#97—21no—
198#97—21no—
199#97—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8C2311yescharacteristic
203C4×C8C2311yescharacteristic
204C4×C8C2311yescharacteristic
205C4×C8C2311yescharacteristic
206C24—21no—
207C24—21no—
208#41—22no—
209#41—42no—
210#41—41no—
211#41—81no—
212#41—81no—
213C2×Q8—21no—
214C2×Q8C22⋊C411yescharacteristic
215C2×Q8C22×C411yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—44no—
230C2×D4C22⋊C411yescharacteristic
231C2×D4C22⋊C411yescharacteristic
232C2×D4C22⋊C411yescharacteristic
233C2×D4C22×C411yescharacteristic
234C2×D4C22×C411yescharacteristic
235C2×D4C22×C411yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C22×C411yescharacteristic
252C22×C4C22×C411yescharacteristic
253C22×C4C22×C411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D411yescharacteristic
260M16—42no—
261M16—81no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—42no—
271C2×C8—81no—
272C2×C8—81no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4C22⋊C411yescharacteristic
285C4⋊C4C22⋊C411yescharacteristic
286C4⋊C4C22⋊C411yescharacteristic
287C4⋊C4C22⋊C411yescharacteristic
288C4⋊C4C22⋊C411yescharacteristic
289C4⋊C4C22⋊C411yescharacteristic
290C4⋊C4C22×C411yescharacteristic
291C4⋊C4C22×C411yescharacteristic
292C4⋊C4C22×C411yescharacteristic
293C4⋊C4C22×C411yescharacteristic
294C22⋊C4—21no—
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—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4C22⋊C411yescharacteristic
314C22⋊C4C22⋊C411yescharacteristic
315C22⋊C4C22⋊C411yescharacteristic
316C22⋊C4C22⋊C411yescharacteristic
317C22⋊C4C22⋊C411yescharacteristic
318C22⋊C4C22⋊C411yescharacteristic
319C42—21no—
320C42—21no—
321C42—21no—
322C42C22×C411yescharacteristic
323C42C22×C411yescharacteristic
324C42C22×C411yescharacteristic
325C42C2×D411yescharacteristic
326C42C2×D411yescharacteristic
327C42C2×D411yescharacteristic
328C42C2×D411yescharacteristic
329C42C2×D411yescharacteristic
330C42C2×D411yescharacteristic
331C42C2411yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—81no—
340C23—81no—
341C23C2×C22⋊C411yescharacteristic
342C23C2×C22⋊C411yescharacteristic
343C23C2×C22⋊C411yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—41no—
347Q8—81no—
348D4—22no—
349D4—22no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4#13112yes—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4C2×C22⋊C411yescharacteristic
401C2×C4C2×C22⋊C411yescharacteristic
402C2×C4C2×C22⋊C411yescharacteristic
403C2×C4C2×C22⋊C411yescharacteristic
404C2×C4C2×C22⋊C411yescharacteristic
405C2×C4C2×C22⋊C411yescharacteristic
406C2×C4C2×C22⋊C411yescharacteristic
407C2×C4C2×C22⋊C411yescharacteristic
408C2×C4C2×C22⋊C411yescharacteristic
409C2×C4C23×C411yescharacteristic
410C2×C4C22×D411yescharacteristic
411C2×C4C22×D411yescharacteristic
412C8—41no—
413C8—41no—
414C8—41no—
415C8—41no—
416C8—41no—
417C8—41no—
418C8—41no—
419C8—41no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—84no—
438C22—84no—
439C22#42112yes—
440C22C22×C22⋊C411yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4C2×#13111yescharacteristic
457C4#56811yescharacteristic
458C4#57611yescharacteristic
459C4#57711yescharacteristic
460C2—22no—
461C2—41no—
462C2—41no—
463C2—42no—
464C2—81no—
465C2—81no—
466C2#288011yescharacteristic
467C2#289711yescharacteristic
468C2C2×#42111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3481C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2990C2semidirect
3. #2922C2semidirect
4. #1496C2semidirect
5. #1495C2semidirect
6. #1494C2semidirect
7. #1490C2semidirect
8. #1488C2semidirect
9. #1486C2semidirect
10. #1483C2semidirect
11. #1482C2semidirect
12. #1480C2semidirect
13. #1478C2semidirect
14. #1477C2semidirect
15. C2×D4⋊C8C2semidirect
45. #432C22semidirect
46. #432C22semidirect
47. #431C22semidirect
48. #431C22semidirect
49. #431C22semidirect
50. #431C22semidirect
51. #431C22semidirect
53. #423C22semidirect
54. C2×C4⋊C8C22semidirect
55. C2×C4⋊C8C22semidirect
57. C4×M16C22semidirect
58. C2×C4×C8C22semidirect
59. Q8⋊C8C22semidirect
60. Q8⋊C8C22semidirect
61. Q8⋊C8C22semidirect
62. Q8⋊C8C22semidirect
67. D4⋊C8C22semidirect
68. D4⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect
71. D4⋊C8C22semidirect
72. D4⋊C8C22semidirect
73. D4⋊C8C22semidirect
74. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34812
#29902
#29222
#14962
#14952
#14942
#14902
#14882
#14862
#14832
#14822
#14802
#14782
#14772
C2×D4⋊C82