Finite Group Databasebeta

← #24320

Subgroups — #24320

Order 256.

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

normal subgroups only

This group has 546 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#2542C211yesmaximal, characteristic, complemented
5#2060C212yesmaximal, complemented
6#2027C212yesmaximal, complemented
7#1997C211yesmaximal, characteristic, complemented
8#1986C211yesmaximal, characteristic, complemented
9#1883C212yesmaximal
10#1859C212yesmaximal, 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
25C2×C22⋊Q8—22no—
26#520C2212yes—
27C4×#41C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#478—22no—
30#476—21no—
31#474—21no—
32#442—21no—
33#440—21no—
34C4×SD16—21no—
35C4×D8—21no—
36#428C2212yes—
37#428C2211yescharacteristic
38#423C2211yescharacteristic
39C2×#104C2212yes—
40#418C2212yes—
41#418C2211yescharacteristic
42#417C2211yescharacteristic
43C4×M16C2211yescharacteristic
44#399—22no—
45#390C2212yes—
46#384C2212yes—
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—
87C22⋊Q8—48no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8D411yescharacteristic
94C4×Q8D411yescharacteristic
95C4×D4—22no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—24no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4D411yescharacteristic
104C4×D4D411yescharacteristic
105#117—22no—
106#117—21no—
107#117C2311yescharacteristic
108#117C2312yes—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4D412yes—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2312yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—24no—
118C2×C42—21no—
119C2×C42C2312yes—
120C2×C42C2311yescharacteristic
121#107—21no—
122#107—22no—
123#106—21no—
124#106—22no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127#104—24no—
128#103—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—22no—
133#102—22no—
134#97—21no—
135C4×C8—21no—
136#95—21no—
137C24—21no—
138#41—21no—
139#41—21no—
140#41C2×Q812yes—
141#41#4112yes—
142C2×Q8—24no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—44no—
148C2×Q8—84no—
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×Q812yescomplemented
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—44no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D414yes—
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—24no—
172D8—24no—
173M16—21no—
174M16—22no—
175M16—21no—
176M16—21no—
177M16—22no—
178M16C2×Q812yescomplemented
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—22no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—48no—
200C4⋊C4C2×D412yes—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D412yes—
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—24no—
205C22⋊C4—22no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—48no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C42—24no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42C2×D411yescharacteristic
220C42C2×D411yescharacteristic
221C42C2×D411yescharacteristic
222C42C2×D411yescharacteristic
223C23—21no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23C22×D411yescharacteristic
230Q8—21no—
231Q8—42no—
232Q8—44no—
233Q8—42no—
234Q8—82no—
235Q8—84no—
236Q8C22⋊Q812yescomplemented
237D4—22no—
238D4—22no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—41no—
243D4—41no—
244D4C22⋊Q812yescomplemented
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—82no—
266C2×C4#12011yescharacteristic
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4#12011yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×Q811yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C42−1+411yescharacteristic
275C2×C42−1+411yescharacteristic
276C8—22no—
277C8—41no—
278C8—42no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—44no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22C2×#12011yescharacteristic
290C22#54811yescharacteristic
291C22D4×Q811yescharacteristic
292C4—22no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—44no—
297C4—41no—
298C4—44no—
299C4—42no—
300C4—41no—
301C4—42no—
302C4—81no—
303C4#45312yes—
304C4C2×C22⋊Q811yescharacteristic
305C4#53611yescharacteristic
306C4#54811yescharacteristic
307C4D4×Q811yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—82no—
312C2#240911yescharacteristic
313C2C2×#45311yescharacteristic
314C2#302011yescharacteristic
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. #2542C2semidirect
5. #2060C2semidirect
6. #2027C2semidirect
7. #1997C2semidirect
8. #1986C2semidirect
10. #1859C2semidirect
11. #1806C2semidirect
59. AGL(1,ℤ/8)Q8semidirect
157. C2×D4C2×Q8semidirect
178. M16C2×Q8semidirect
236. Q8C22⋊Q8semidirect
244. D4C22⋊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
#25422
#20602
#20272
#19972
#19862
#18832
#18592
#18062