Finite Group Databasebeta

← #24255

Subgroups — #24255

Order 256.

696 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×D4×Q8C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#451C212yesmaximal, complemented
4C2×#441C211yesmaximal, characteristic, complemented
5#2536C211yesmaximal, characteristic
6#2035C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic
10#1805C211yesmaximal, characteristic, complemented
11#1787C212yesmaximal, complemented
12C23×Q8C2212yes—
13C22×Q16C2212yes—
14C22×Q16C2211yescharacteristic
15D4×Q8—24no—
16D4×Q8—24no—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—22no—
20C2×C22⋊Q8C2212yes—
21C2×C22⋊Q8C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26#462—24no—
27#451—28no—
28#441—24no—
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#103C2212yes—
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×#97C2211yescharacteristic
40#398—21no—
41#394—21no—
42#388—22no—
43#382C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46C22×Q8—24no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—24no—
50C22×Q8—24no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8C2311yescharacteristic
54C22×Q8C2312yes—
55C22×Q8C2311yescharacteristic
56C22×D4D411yescharacteristic
57C23×C4—22no—
58C23×C4C2312yes—
59C2×Q16—22no—
60C2×Q16—22no—
61C2×Q16—24no—
62C2×Q16—24no—
63C2×Q16—42no—
64C2×Q16D414yescomplemented
65C22×C8C2312yes—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C4⋊Q8—28no—
69C4⋊Q8—22no—
70C4⋊Q8D414yes—
71C22⋊Q8—216no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—42no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8D414yes—
80C4×D4—28no—
81C4×D4—22no—
82C4×D4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4D412yes—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100#106—22no—
101C4⋊C8—22no—
102#103—22no—
103#103—24no—
104#103—24no—
105#103—24no—
106#103—22no—
107#103—42no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110#97—22no—
111#95—22no—
112#95—21no—
113#95—21no—
114C24C2×D412yes—
115C2×Q8—28no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—48no—
132C2×Q8—48no—
133C2×Q8—48no—
134C2×Q8—48no—
135C2×Q8C2×D414yes—
136C2×Q8C2×D412yes—
137C2×Q8C2×D412yes—
138C2×Q8C2×D412yes—
139C2×Q8#4112yes—
140C2×D4—24no—
141C2×D4—22no—
142C22×C4—21no—
143C22×C4—28no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161Q16—28no—
162Q16—48no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8C2×D412yes—
172C4⋊C4—28no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4C2×D412yes—
185C22⋊C4—28no—
186C22⋊C4—24no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C42—24no—
190C42—42no—
191C42C2×D412yes—
192C42C2×D412yes—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23#13714yes—
200C23C22×D411yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—22no—
204Q8—22no—
205Q8—22no—
206Q8—24no—
207Q8—24no—
208Q8—416no—
209Q8—42no—
210Q8—41no—
211Q8—42no—
212Q8—41no—
213Q8#12114yes—
214D4—28no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—416no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4#12012yes—
242C2×C4#12012yes—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)11yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C8—22no—
250C8—41no—
251C8—42no—
252C22—24no—
253C22—24no—
254C22—24no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22C2×#12011yescharacteristic
260C22D4211yescharacteristic
261C22#54611yescharacteristic
262C22C2×#13712yes—
263C22#57812yes—
264C4—24no—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—44no—
269C4—42no—
270C4—42no—
271C4—81no—
272C4—82no—
273C4C2×#12111yescharacteristic
274C4#53511yescharacteristic
275C4D4211yescharacteristic
276C4#54611yescharacteristic
277C2—44no—
278C2#240811yescharacteristic
279C2#300411yescharacteristic
280C2#301111yescharacteristic
281C2#327012yes—
282C2#331212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
3. C2×#451C2semidirect
4. C2×#441C2semidirect
6. #2035C2semidirect
7. #2001C2semidirect
8. #1960C2semidirect
10. #1805C2semidirect
11. #1787C2semidirect
64. C2×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4622
C2×#4512
C2×#4412
#25362
#20352
#20012
#19602
#19552
#18052
#17872