Finite Group Databasebeta

← #24321

Subgroups — #24321

Order 256.

518 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 137 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#3503C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic, complemented
3C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
4#2563C211yesmaximal, characteristic, complemented
5#2061C212yesmaximal, complemented
6#2027C212yesmaximal, complemented
7#1997C211yesmaximal, characteristic, complemented
8#1986C211yesmaximal, characteristic, complemented
9#1883C212yesmaximal
10#1860C212yesmaximal, complemented
11#1806C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)C2211yescharacteristic
13Q82—21no—
14#557—21no—
15#554—21no—
16#554—22no—
17D4×Q8—22no—
18D4×Q8—21no—
19#544—22no—
20#544C2212yes—
21#533—21no—
22#533C2211yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25#520C2212yes—
26C4×#41C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28#478—22no—
29#476—21no—
30#474—21no—
31#442—21no—
32#440—21no—
33C4×SD16—21no—
34C4×D8—21no—
35#428C2212yes—
36#428C2211yescharacteristic
37#423C2211yescharacteristic
38C2×#104C2212yes—
39#418C2212yes—
40#418C2211yescharacteristic
41#417C2211yescharacteristic
42C4×M16C2211yescharacteristic
43#393—22no—
44#390C2212yes—
45#382C2212yes—
46#380—22no—
47C4×C4⋊C4C2211yescharacteristic
48#343C2212yes—
49#343C2212yes—
50#339C2212yes—
51#339C2212yes—
52#339C2212yes—
53C2×#41C2311yescharacteristic
54C22×Q8—22no—
55C22×Q8D412yes—
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58AGL(1,ℤ/8)—22no—
59AGL(1,ℤ/8)Q814yescomplemented
60C2×SD16—21no—
61C2×D8—21no—
62C2×M16C2311yescharacteristic
63C2×M16C2312yes—
64C2×M16C2311yescharacteristic
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4⋊Q8—24no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8—21no—
74C4⋊Q8D411yescharacteristic
75C4⋊Q8D411yescharacteristic
76#125—24no—
77#125—22no—
78#125—22no—
79#125D412yes—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C22⋊Q8—24no—
83C22⋊Q8—24no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—24no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D411yescharacteristic
93C4×Q8D411yescharacteristic
94C4×D4—22no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—24no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4D411yescharacteristic
103C4×D4D411yescharacteristic
104#117—22no—
105#117—21no—
106#117C2311yescharacteristic
107#117C2312yes—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D412yes—
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2312yes—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—24no—
116C2×C42—21no—
117C2×C42C2312yes—
118C2×C42C2311yescharacteristic
119#107—21no—
120#107—22no—
121#106—21no—
122#106—22no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125#104—24no—
126#103—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—22no—
131#102—22no—
132#97—21no—
133C4×C8—21no—
134#95—21no—
135#95—24no—
136C24—21no—
137#41—21no—
138#41—21no—
139#41C2×Q812yes—
140#41#4112yes—
141C2×Q8—24no—
142C2×Q8—21no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—82no—
147C2×Q8—84no—
148C2×Q8C2×D411yescharacteristic
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4C2×D411yescharacteristic
156C2×D4C2×Q812yescomplemented
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—44no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D414yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2411yescharacteristic
170SD16—24no—
171D8—24no—
172M16—21no—
173M16—22no—
174M16—21no—
175M16—21no—
176M16—22no—
177M16C2×Q812yescomplemented
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—42no—
198C4⋊C4C2×D412yes—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D412yes—
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—24no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—84no—
210C42—24no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—22no—
215C42—22no—
216C42—22no—
217C42C2×D411yescharacteristic
218C42C2×D411yescharacteristic
219C42C2×D411yescharacteristic
220C42C2×D411yescharacteristic
221C23—21no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23C22×D411yescharacteristic
228Q8—21no—
229Q8—44no—
230Q8—42no—
231Q8—42no—
232Q8—82no—
233Q8—84no—
234Q8C22⋊Q812yes—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—21no—
239D4—22no—
240D4—41no—
241D4—41no—
242D4C22⋊Q812yes—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4#12011yescharacteristic
265C2×C4#12011yescharacteristic
266C2×C4#12011yescharacteristic
267C2×C4#12011yescharacteristic
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4C22×Q811yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C42−1+411yescharacteristic
273C2×C42−1+411yescharacteristic
274C8—22no—
275C8—41no—
276C8—42no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—44no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—82no—
285C22—81no—
286C22—81no—
287C22C2×#12011yescharacteristic
288C22#54811yescharacteristic
289C22D4×Q811yescharacteristic
290C4—22no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—41no—
295C4—42no—
296C4—41no—
297C4—44no—
298C4—42no—
299C4—82no—
300C4—81no—
301C4#45412yes—
302C4C2×C22⋊Q811yescharacteristic
303C4#53611yescharacteristic
304C4#54811yescharacteristic
305C4D4×Q811yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—82no—
310C2#240911yescharacteristic
311C2C2×#45411yescharacteristic
312C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3503C2semidirect
2. #3077C2semidirect
4. #2563C2semidirect
5. #2061C2semidirect
6. #2027C2semidirect
7. #1997C2semidirect
8. #1986C2semidirect
10. #1860C2semidirect
11. #1806C2semidirect
59. AGL(1,ℤ/8)Q8semidirect
156. C2×D4C2×Q8semidirect
177. M16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35032
#30772
C4×AGL(1,ℤ/8)2
#25632
#20612
#20272
#19972
#19862
#18832
#18602
#18062