Finite Group Databasebeta

← #29197

Subgroups — #29197

Order 256.

379 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 127 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#2859C211yesmaximal, characteristic, complemented
2C4×#124C211yesmaximal, characteristic
3#2068C212yesmaximal, complemented
4#2065C212yesmaximal
5#1959C211yesmaximal, characteristic
6#1959C211yesmaximal, characteristic
7#1865C212yesmaximal, complemented
8#1862C212yesmaximal
9#1847C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1767C211yesmaximal, characteristic, complemented
12C2×#125—21no—
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#97C2211yescharacteristic
24C2×#97C2211yescharacteristic
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#389—21no—
28#389—21no—
29#389C2212yes—
30#389C2212yes—
31#388—21no—
32#388—21no—
33#384C2211yescharacteristic
34#384C2211yescharacteristic
35#382—21no—
36#382C2211yescharacteristic
37C4×C22⋊C4—21no—
38C4×C22⋊C4—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48#125—44no—
49#124—22no—
50#124Q814yes—
51C4×Q8—42no—
52C4×D4—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—42no—
78#106—42no—
79C4⋊C8—24no—
80#103—24no—
81#103—42no—
82#102—24no—
83#102—42no—
84#97—22no—
85#97—22no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92C24—21no—
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8C2×Q812yes—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C2×Q812yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8#4114yes—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—41no—
145C42—41no—
146C42—42no—
147C42C2×Q812yes—
148C42#4112yes—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23C22×D411yescharacteristic
157Q8—24no—
158Q8—42no—
159D4—24no—
160D4—42no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—84no—
191C2×C4C22⋊Q812yes—
192C2×C4C22⋊Q812yes—
193C2×C4AGL(1,ℤ/8)12yes—
194C2×C4#13712yes—
195C2×C4C22×Q811yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4112yes—
198C2×C4H(5,𝔽2)12yes—
199C8—24no—
200C8—42no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22C2×C22⋊Q811yescharacteristic
209C22#53211yescharacteristic
210C22#53511yescharacteristic
211C22C2×AGL(1,ℤ/8)11yescharacteristic
212C22C2×#13711yescharacteristic
213C22#57511yescharacteristic
214C22#57511yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—41no—
221C4—41no—
222C4—82no—
223C4—82no—
224C4—81no—
225C4—81no—
226C4#55312yes—
227C4#55412yes—
228C2—82no—
229C2#256611yescharacteristic
230C2#307911yescharacteristic
231C2#308011yescharacteristic
232C2#317111yescharacteristic
233C2#318111yescharacteristic
234C2#321311yescharacteristic
235C2#321611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2859C2semidirect
3. #2068C2semidirect
7. #1865C2semidirect
9. #1847C2semidirect
10. #1836C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
#28592
C4×#1242
#20682
#20652
#19592
#18652
#18622
#18472
#18362
#17672