Finite Group Databasebeta

← #25844

Subgroups — #25844

Order 256.

612 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 155 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×#501C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic, complemented
5#2058C212yesmaximal, complemented
6#2057C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal
10#1862C212yesmaximal
11#1762C211yesmaximal, characteristic
12C23×Q8—21no—
13D4×Q8—24no—
14D4×Q8—24no—
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8—22no—
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2212yes—
22C2×C4×D4C2211yescharacteristic
23#501—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
33C2×#97C2211yescharacteristic
34#398—21no—
35#395—21no—
36#389—22no—
37#389C2212yes—
38#389C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—28no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—48no—
49C22×Q8D411yescharacteristic
50C22×D4C2311yescharacteristic
51C23×C4—22no—
52C23×C4—21no—
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C22×C8C2312yes—
56C4⋊Q8—24no—
57C4⋊Q8—24no—
58C4⋊Q8—22no—
59C4⋊Q8D414yes—
60#125—22no—
61#125—22no—
62C22⋊Q8—28no—
63C22⋊Q8—216no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—24no—
67C4×D4—24no—
68C4×D4—22no—
69C4×D4D414yes—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D412yes—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C42—21no—
86C2×C42C2312yes—
87C2×C42C2311yescharacteristic
88#107—22no—
89#107—24no—
90#106—22no—
91#106—24no—
92C4⋊C8—22no—
93#102—22no—
94#102—22no—
95#102Q818yescomplemented
96#97—22no—
97#95—21no—
98#95—22no—
99#95—22no—
100C24—21no—
101C2×Q8—28no—
102C2×Q8—22no—
103C2×Q8—24no—
104C2×Q8—24no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—432no—
108C2×Q8—44no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4C2×D412yes—
112C2×D4C2×Q814yes—
113C22×C4—28no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2411yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8C2×Q814yescomplemented
135C4⋊C4—28no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—82no—
150C4⋊C4C2×D414yes—
151C4⋊C4C2×D412yes—
152C4⋊C4C2×Q814yescomplemented
153C22⋊C4—28no—
154C22⋊C4—24no—
155C42—22no—
156C42—22no—
157C42—81no—
158C42C2×D412yes—
159C42C2×D412yes—
160C23—22no—
161C23—24no—
162C23—42no—
163C23—41no—
164C23—41no—
165C23C22×D411yescharacteristic
166Q8—28no—
167Q8—48no—
168Q8—416no—
169D4—22no—
170D4—24no—
171D4C4⋊Q814yes—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—48no—
184C2×C4—416no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4#12012yes—
196C2×C4#12012yes—
197C2×C4AGL(1,ℤ/8)14yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C22×Q812yes—
201C2×C42−1+412yes—
202C8—24no—
203C8—41no—
204C8—41no—
205C22—24no—
206C22—44no—
207C22—44no—
208C22—42no—
209C22—42no—
210C22C2×#12011yescharacteristic
211C22D4×Q812yes—
212C22C2×AGL(1,ℤ/8)12yes—
213C22#57711yescharacteristic
214C22#57811yescharacteristic
215C4—22no—
216C4—24no—
217C4—22no—
218C4—44no—
219C4—44no—
220C4—42no—
221C4—42no—
222C4—81no—
223C4—81no—
224C4C2×C4⋊Q811yescharacteristic
225C4#53611yescharacteristic
226C4D4×Q812yes—
227C2—44no—
228C2#243411yescharacteristic
229C2#300311yescharacteristic
230C2#301311yescharacteristic
231C2#338412yes—
232C2#338612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#501C2semidirect
4. #2584C2semidirect
5. #2058C2semidirect
6. #2057C2semidirect
7. #1987C2semidirect
8. #1985C2semidirect
95. #102Q8semidirect
134. C2×C8C2×Q8semidirect
152. C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#5012
C2×#4782
#25842
#20582
#20572
#19872
#19852
#18642
#18622
#17622