Finite Group Databasebeta

← #18628

Subgroups — #18628

Order 256.

499 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic
2#3248C211yesmaximal, characteristic, complemented
3#3131C211yesmaximal, characteristic, complemented
4#3110C211yesmaximal, characteristic, complemented
5#1646C211yesmaximal, characteristic, complemented
6#1646C211yesmaximal, characteristic, complemented
7#1644C211yesmaximal, characteristic, complemented
8#1643C211yesmaximal, characteristic, complemented
9#1642C211yesmaximal, characteristic, complemented
10#1641C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic, complemented
13#1551C211yesmaximal, characteristic, complemented
14#1494C211yesmaximal, characteristic, complemented
15C2×D4⋊C8C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#554C2211yescharacteristic
18#553—21no—
19#553—21no—
20#550C2211yescharacteristic
21#546—22no—
22#546—21no—
23D42—21no—
24D42C2211yescharacteristic
25#540—21no—
26#538—22no—
27#533C2211yescharacteristic, complemented
28#532C2211yescharacteristic
29#529C2211yescharacteristic
30#525—21no—
31C2×#121—21no—
32#518—21no—
33C2×C4×D4C2211yescharacteristic
34#488—22no—
35#487C2211yescharacteristic, complemented
36#487C2211yescharacteristic, complemented
37#486C2211yescharacteristic, complemented
38#486C2211yescharacteristic, complemented
39#484—21no—
40#479—21no—
41#475C2211yescharacteristic, complemented
42#474C2211yescharacteristic
43#474C2211yescharacteristic, complemented
44#474C2211yescharacteristic, complemented
45#467—21no—
46#465—21no—
47#464—21no—
48#462C2211yescharacteristic, complemented
49#461C2211yescharacteristic
50#461C2211yescharacteristic, complemented
51#461C2211yescharacteristic, complemented
52#450—21no—
53#440—21no—
54#440—21no—
55#432C2211yescharacteristic, complemented
56C2×C4⋊C8C2211yescharacteristic, complemented
57#416—22no—
58C2×C4×C8C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62#331C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic, complemented
68D4⋊C8C2211yescharacteristic, complemented
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41D412yes—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4D411yescharacteristic
77C23×C4—21no—
78C2×Q16—21no—
79C2×SD16—21no—
80C2×SD16—21no—
81C2×SD16—21no—
82C22×C8—21no—
83C22×C8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127—21no—
88#127C2311yescharacteristic
89#126—22no—
90#126—21no—
91#125—21no—
92#125—21no—
93#125D411yescharacteristic
94#124—22no—
95#124D411yescharacteristic
96#124D412yes—
97#123—22no—
98#123—22no—
99#123—22no—
100#123—21no—
101C22⋊Q8—21no—
102C22⋊Q8—22no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8D411yescharacteristic
106#121—22no—
107#121—22no—
108#121—22no—
109#121—22no—
110#121—21no—
111#121—22no—
112#121—21no—
113#121—21no—
114#121—21no—
115#121—21no—
116#121—21no—
117#121—21no—
118#121D411yescharacteristic
119#120—22no—
120#120—22no—
121#120—22no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8C2311yescharacteristic
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4D411yescharacteristic
136C4×D4D411yescharacteristic
137C4×D4C2311yescharacteristic
138C4×D4C2311yescharacteristic
139C4×D4C2311yescharacteristic
140#117—21no—
141#117—21no—
142#117—21no—
143#117—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4D411yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C42C2311yescharacteristic
150#107—21no—
151#107—21no—
152#107—21no—
153#106—21no—
154C4⋊C8—21no—
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159#103—22no—
160#103—22no—
161#103—21no—
162#103—21no—
163#103—21no—
164#103—21no—
165#103—21no—
166#102—22no—
167#102—22no—
168#102—21no—
169#102—21no—
170#102—21no—
171C22⋊C8—21no—
172#97—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—22no—
177C24—21no—
178#41—44no—
179#41—81no—
180#41—81no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8C2×D411yescharacteristic
184C2×Q8C2×D412yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—84no—
202C2×D4C2×D411yescharacteristic
203C2×D4C2×D411yescharacteristic
204C2×D4C2×D411yescharacteristic
205C2×D4C2×D412yes—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220Q16—81no—
221SD16—42no—
222SD16—42no—
223SD16—81no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—42no—
231C2×C8—81no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4C2×D411yescharacteristic
248C4⋊C4C2×D411yescharacteristic
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D411yescharacteristic
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—22no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—22no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—41no—
268C42—22no—
269C42—21no—
270C42—41no—
271C42C2×D411yescharacteristic
272C42C2×D411yescharacteristic
273C42C2×D411yescharacteristic
274C42C2411yescharacteristic
275C23—21no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—42no—
280C23—44no—
281C23—42no—
282C23—81no—
283C23—82no—
284C23—82no—
285C23#12011yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—42no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—42no—
300D4—44no—
301D4—42no—
302D4—44no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—82no—
311D4#13512yes—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#12011yescharacteristic
338C2×C4#12011yescharacteristic
339C2×C4#12011yescharacteristic
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C22×D411yescharacteristic
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C8—41no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—164no—
365C22#45512yes—
366C22C2×#12011yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4C2×#13511yescharacteristic
380C4#57511yescharacteristic
381C4#57611yescharacteristic
382C4#57711yescharacteristic
383C2—22no—
384C2—42no—
385C2—81no—
386C2—82no—
387C2—81no—
388C2#300211yescharacteristic
389C2#301211yescharacteristic
390C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3519C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3248C2semidirect
3. #3131C2semidirect
4. #3110C2semidirect
5. #1646C2semidirect
6. #1646C2semidirect
7. #1644C2semidirect
8. #1643C2semidirect
9. #1642C2semidirect
10. #1641C2semidirect
11. #1639C2semidirect
12. #1639C2semidirect
13. #1551C2semidirect
14. #1494C2semidirect
15. C2×D4⋊C8C2semidirect
27. #533C22semidirect
35. #487C22semidirect
36. #487C22semidirect
37. #486C22semidirect
38. #486C22semidirect
41. #475C22semidirect
43. #474C22semidirect
44. #474C22semidirect
48. #462C22semidirect
50. #461C22semidirect
51. #461C22semidirect
55. #432C22semidirect
56. C2×C4⋊C8C22semidirect
58. C2×C4×C8C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #331C22semidirect
62. #331C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
67. D4⋊C8C22semidirect
68. D4⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35192
#32482
#31312
#31102
#16462
#16442
#16432
#16422
#16412
#16392
#15512
#14942
C2×D4⋊C82