Finite Group Databasebeta

← #28290

Subgroups — #28290

Order 256.

429 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 135 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×#476C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3#2331C211yesmaximal, characteristic, complemented
4#2328C211yesmaximal, characteristic
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#1982C212yesmaximal
8#1949C212yesmaximal, complemented
9#1843C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#476—24no—
17#474—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28#391—22no—
29#390C2211yescharacteristic
30#390C2211yescharacteristic
31#389C2212yes—
32#387—22no—
33#386—22no—
34#386—22no—
35#382C2211yescharacteristic
36#382C2211yescharacteristic
37C4×C4⋊C4C2212yes—
38#376C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41C22×Q8—21no—
42C22×D4—22no—
43C22×D4C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50C4⋊Q8D414yescomplemented
51#127—22no—
52#127Q814yescomplemented
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C42—22no—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—24no—
72#106—24no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75C4⋊C8—24no—
76C4⋊C8—42no—
77#102—24no—
78#102—24no—
79#102—42no—
80#102—42no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95—21no—
85#95—22no—
86#95D412yes—
87#95D412yes—
88C24—21no—
89C24—21no—
90C2×Q8—24no—
91C2×Q8—82no—
92C2×D4—28no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—82no—
104C2×D4—82no—
105C2×D4C2×Q812yes—
106C2×D4C2×Q812yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—82no—
114C22×C4—82no—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—82no—
135C4⋊C4C2×D414yes—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—21no—
141C42—21no—
142C42—82no—
143C42C2×D412yes—
144C42C2×Q812yes—
145C42#4112yes—
146C42#4112yes—
147C23—22no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23C22×D411yescharacteristic
158Q8—84no—
159D4—22no—
160D4—22no—
161D4—22no—
162D4—22no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—88no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—84no—
186C2×C4—84no—
187C2×C4#12112yes—
188C2×C4#12112yes—
189C2×C4C22⋊Q814yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C22×Q811yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)11yescharacteristic
195C2×C42−1+411yescharacteristic
196C8—42no—
197C8—42no—
198C22—44no—
199C22—44no—
200C22—42no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22C2×#12111yescharacteristic
209C22C2×C22⋊Q811yescharacteristic
210C22#53711yescharacteristic
211C22#57612yes—
212C22#57712yes—
213C4—24no—
214C4—22no—
215C4—22no—
216C4—42no—
217C4—44no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4D4×Q811yescharacteristic
222C4#55011yescharacteristic
223C4#55411yescharacteristic
224C4#55611yescharacteristic
225C2—82no—
226C2—82no—
227C2#253811yescharacteristic
228C2#304111yescharacteristic
229C2#304311yescharacteristic
230C2#308112yes—
231C2#325712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#476C2semidirect
2. C2×#474C2semidirect
3. #2331C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
8. #1949C2semidirect
9. #1843C2semidirect
10. #1804C2semidirect
11. #1804C2semidirect
50. C4⋊Q8D4semidirect
52. #127Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×#4742
#23312
#23282
#20672
#20652
#19822
#19492
#18432
#18042