Finite Group Databasebeta

← #29926

Subgroups — #29926

Order 256.

364 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 121 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#2857C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic
3#2097C212yesmaximal, complemented
4#2074C212yesmaximal, complemented
5#2036C212yesmaximal
6#2035C212yesmaximal
7#1981C211yesmaximal, characteristic, complemented
8#1981C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2212yes—
24#391C2212yes—
25#389—21no—
26#389—21no—
27#384C2211yescharacteristic
28#383—21no—
29#382C2211yescharacteristic
30#382C2211yescharacteristic
31#381—21no—
32#380—21no—
33#380—21no—
34#380C2211yescharacteristic
35#380C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#376—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C22⋊Q8—42no—
52C22⋊Q8—42no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—42no—
75#106—42no—
76C4⋊C8—22no—
77C4⋊C8—22no—
78#103—42no—
79#103—42no—
80#103—42no—
81#103—42no—
82C22⋊C8—42no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—22no—
92#95—22no—
93C24C2×D411yescharacteristic
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—41no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—42no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C22⋊C4—42no—
148C42—22no—
149C42—22no—
150C42—21no—
151C42—21no—
152C42—42no—
153C42—42no—
154C42#4112yes—
155C42#4112yes—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23#13712yes—
163C23C22×D411yescharacteristic
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—44no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4#12312yes—
204C2×C4#12312yes—
205C2×C4#13512yes—
206C2×C4#13712yes—
207C2×C4C2×#4111yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4H(5,𝔽2)11yescharacteristic
210C2×C4H(5,𝔽2)11yescharacteristic
211C2×C4H(5,𝔽2)11yescharacteristic
212C2×C42−1+411yescharacteristic
213C8—42no—
214C8—42no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22C2×#12311yescharacteristic
223C22#53511yescharacteristic
224C22#53711yescharacteristic
225C22C2×#13511yescharacteristic
226C22C2×#13711yescharacteristic
227C22C2×#13711yescharacteristic
228C22#57511yescharacteristic
229C4—22no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—82no—
234C4—82no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4#48514yes—
240C4#54011yescharacteristic
241C4#54011yescharacteristic
242C4#54211yescharacteristic
243C4#54211yescharacteristic
244C2—42no—
245C2#262411yescharacteristic
246C2C2×#48511yescharacteristic
247C2#309511yescharacteristic
248C2#321211yescharacteristic
249C2#321811yescharacteristic
250C2#324811yescharacteristic
251C2#325211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2097C2semidirect
4. #2074C2semidirect
7. #1981C2semidirect
8. #1981C2semidirect
9. #1948C2semidirect
10. #1947C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22832
#20972
#20742
#20362
#20352
#19812
#19482
#19472
#17972