Finite Group Databasebeta

← #18804

Subgroups — #18804

Order 256.

484 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1652C212yesmaximal, complemented
7#1646C212yesmaximal, complemented
8#1644C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1486C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×#137—21no—
13#556C2211yescharacteristic
14#555C2211yescharacteristic
15#554C2211yescharacteristic
16#553—21no—
17#552—21no—
18#552—21no—
19#551—22no—
20#547C2211yescharacteristic
21#546—21no—
22#533C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#529—22no—
25#529—21no—
26#525—21no—
27C4×#41—21no—
28C4×#41C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117—21no—
31#498C2212yescomplemented
32#487C2212yescomplemented
33#483—22no—
34#475—22no—
35#474—21no—
36#474—21no—
37#474C2212yes—
38#474C2212yes—
39#462—21no—
40#462C2212yescomplemented
41#461—21no—
42#461C2212yescomplemented
43#431C2211yescharacteristic
44C2×#107—21no—
45#423C2211yescharacteristic
46C2×C4⋊C8C2211yescharacteristic
47#419—21no—
48C2×#102—21no—
49#333C2212yescomplemented
50#332C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yes—
53D4⋊C8C2212yes—
54D4⋊C8C2212yescomplemented
55C2×#41—21no—
56C2×#41D411yescharacteristic
57C22×Q8—21no—
58C22×D4D411yescharacteristic
59C23×C4—21no—
60#137—44no—
61C2×Q16—22no—
62C2×SD16—22no—
63C2×M16—21no—
64C22×C8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8C2312yes—
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71#126—22no—
72#126—22no—
73#126—22no—
74#125—22no—
75#125—21no—
76#125—21no—
77#124—22no—
78#124—21no—
79#123—22no—
80#123—22no—
81#123—22no—
82#123—22no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—22no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8D412yes—
89#121—22no—
90#121—21no—
91#121—21no—
92#121—21no—
93#120—22no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8D411yescharacteristic
97C4×Q8D412yes—
98C4×Q8C2311yescharacteristic
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4D411yescharacteristic
108C4×D4D411yescharacteristic
109C4×D4D411yescharacteristic
110C4×D4C2311yescharacteristic
111C4×D4C2311yescharacteristic
112C4×D4C2311yescharacteristic
113#117—22no—
114#117—22no—
115#117—22no—
116#117—21no—
117#117—21no—
118#117—21no—
119#117—21no—
120#117—21no—
121#117D411yescharacteristic
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4D411yescharacteristic
127C2×C22⋊C4—22no—
128C2×C22⋊C4—21no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#107—22no—
134#106—24no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8C2312yes—
139C4⋊C8C2312yes—
140#103—24no—
141#103—22no—
142#103—22no—
143#102—22no—
144#102—22no—
145C22⋊C8—21no—
146C4×C8C2312yes—
147C24—21no—
148#41—22no—
149#41—22no—
150#41—42no—
151#41—42no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8C2×D411yescharacteristic
158C2×Q8C2×D412yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184Q16—44no—
185SD16—44no—
186M16—42no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—42no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D412yes—
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—22no—
220C22⋊C4—21no—
221C22⋊C4—21no—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—21no—
226C42—22no—
227C42—22no—
228C42—21no—
229C42—22no—
230C42—21no—
231C42—21no—
232C42C2×D411yescharacteristic
233C42C2×D411yescharacteristic
234C42C2×D411yescharacteristic
235C42C2411yescharacteristic
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—81no—
242C23#12011yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—42no—
246Q8—41no—
247Q8—41no—
248Q8—42no—
249Q8—42no—
250Q8—82no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—21no—
255D4—21no—
256D4—42no—
257D4—44no—
258D4—42no—
259D4—82no—
260D4#13712yescomplemented
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#12011yescharacteristic
287C2×C4#12011yescharacteristic
288C2×C4#12011yescharacteristic
289C2×C4#13512yes—
290C2×C4AGL(1,ℤ/8)12yes—
291C2×C4C22×D411yescharacteristic
292C2×C4C22×D411yescharacteristic
293C2×C4C22×D411yescharacteristic
294C8—42no—
295C8—42no—
296C8—42no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—84no—
309C22#44912yes—
310C22C2×#12011yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—42no—
319C4—41no—
320C4—42no—
321C4—82no—
322C4#44912yes—
323C4C2×#13511yescharacteristic
324C4C2×AGL(1,ℤ/8)11yescharacteristic
325C4C2×#13711yescharacteristic
326C4#57511yescharacteristic
327C2—22no—
328C2—42no—
329C2—81no—
330C2—81no—
331C2C2×#44911yescharacteristic
332C2#300711yescharacteristic
333C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3475C2
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. #3229C2semidirect
3. C2×#474C2semidirect
4. #3039C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1646C2semidirect
8. #1644C2semidirect
9. #1550C2semidirect
10. #1486C2semidirect
11. #1477C2semidirect
31. #498C22semidirect
32. #487C22semidirect
40. #462C22semidirect
42. #461C22semidirect
49. #333C22semidirect
50. #332C22semidirect
51. Q8⋊C8C22semidirect
54. D4⋊C8C22semidirect
260. D4#137semidirect

Maximal subgroups

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

subgroupindex
#34752
#32292
C2×#4742
#30392
#16542
#16522
#16462
#16442
#15502
#14862
#14772