Finite Group Databasebeta

← #30109

Subgroups — #30109

Order 256.

397 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 119 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
2C4×C22⋊Q8C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2036C212yesmaximal
7#2016C211yesmaximal, characteristic, complemented
8#2001C212yesmaximal
9#1928C212yesmaximal, complemented
10#1878C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#107C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2212yes—
24#391C2212yes—
25#391C2211yescharacteristic
26#389—21no—
27#389—21no—
28#389C2211yescharacteristic
29#387—21no—
30#385—21no—
31#384C2211yescharacteristic
32#380—21no—
33#380—21no—
34#380C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C2×Q16—24no—
49C2×Q16—42no—
50C2×Q16—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—42no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—42no—
57C22⋊Q8—42no—
58C4×Q8—42no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—42no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#107—42no—
82#103—24no—
83#103—42no—
84#103—42no—
85#103—42no—
86C22⋊C8—42no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94C24C2×D411yescharacteristic
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8#4112yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4Q1614yescomplemented
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2411yescharacteristic
127Q16—48no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C4⋊C4—24no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4#4112yes—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C22⋊C4—44no—
151C42—22no—
152C42—22no—
153C42—41no—
154C42—41no—
155C42—42no—
156C42—42no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23C2×Q1612yes—
164C23C22×D411yescharacteristic
165Q8—24no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169Q8—42no—
170Q8—42no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4C2×Q1612yes—
206C2×C4C2×Q1612yes—
207C2×C4#13712yes—
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)12yes—
211C2×C4H(5,𝔽2)12yes—
212C8—42no—
213C8—42no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22#52511yescharacteristic
222C22#53411yescharacteristic
223C22#53511yescharacteristic
224C22C22×Q1611yescharacteristic
225C22C2×#13711yescharacteristic
226C22#57511yescharacteristic
227C22#57511yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—42no—
232C4—41no—
233C4—42no—
234C4—41no—
235C4—82no—
236C4—81no—
237C4—82no—
238C4—81no—
239C4#53812yes—
240C4#54012yes—
241C2—42no—
242C2#263511yescharacteristic
243C2#310411yescharacteristic
244C2#311611yescharacteristic
245C2#318911yescharacteristic
246C2#319611yescharacteristic
247C2#320911yescharacteristic
248C2#321911yescharacteristic
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. #2100C2semidirect
4. #2095C2semidirect
5. #2043C2semidirect
7. #2016C2semidirect
9. #1928C2semidirect
10. #1878C2semidirect
120. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#28572
C4×C22⋊Q82
#21002
#20952
#20432
#20362
#20162
#20012
#19282
#18782
#18172