Finite Group Databasebeta

← #10396

Subgroups — #10396

Order 256.

316 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 63 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#2618C211yesmaximal, characteristic
2#1874C211yesmaximal, characteristic
3#1853C211yesmaximal, characteristic, complemented
4C2×#327C211yesmaximal, characteristic
5C2×#324C211yesmaximal, characteristic, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1294C211yesmaximal, characteristic, complemented
8C2×#123—21no—
9C2×#121C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C2×#106—21no—
13C2×#102—21no—
14C2×C22⋊C8C2211yescharacteristic
15C2×C22⋊C8C2211yescharacteristic
16#398—21no—
17#396—21no—
18#396—21no—
19#394—21no—
20#386—21no—
21#380C2211yescharacteristic
22C2×#95—21no—
23C2×#95—21no—
24C2×#95C2211yescharacteristic
25#340—21no—
26#340—21no—
27#327—22no—
28#324—22no—
29C22×D4—21no—
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C23×C4—21no—
33C23×C4D411yescharacteristic
34C23×C4D411yescharacteristic
35C23×C4C2311yescharacteristic
36C22×C8—21no—
37C22×C8—21no—
38#123—42no—
39#123—42no—
40#121—21no—
41#121—21no—
42#121—21no—
43#121—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64#106—42no—
65#102—42no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8D412yes—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—42no—
76#95—42no—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4SD1612yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4#4111yescharacteristic
124C22×C4#4111yescharacteristic
125C22×C4#4111yescharacteristic
126C2×C8—22no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C22⋊C4—42no—
147C22⋊C4—42no—
148C22⋊C4—41no—
149C22⋊C4—41no—
150C22⋊C4—41no—
151C22⋊C4—41no—
152C22⋊C4—81no—
153C22⋊C4—81no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23#12011yescharacteristic
166C23#12111yescharacteristic
167C23#12311yescharacteristic
168C23C2×SD1611yescharacteristic
169C23#13511yescharacteristic
170C23AGL(1,ℤ/8)11yescharacteristic
171C23AGL(1,ℤ/8)11yescharacteristic
172D4—42no—
173D4—81no—
174D4—81no—
175D4—81no—
176D4—81no—
177D4—81no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4#12111yescharacteristic
222C2×C4#12111yescharacteristic
223C2×C4#12311yescharacteristic
224C2×C4#12411yescharacteristic
225C8—42no—
226C8—81no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22#39411yescharacteristic
247C22#44911yescharacteristic
248C22#45011yescharacteristic
249C22#45411yescharacteristic
250C22#45611yescharacteristic
251C22B(4,𝔽2)11yescharacteristic
252C22#45811yescharacteristic
253C22#46511yescharacteristic
254C22#46911yescharacteristic
255C22#48111yescharacteristic
256C22#48211yescharacteristic
257C4—21no—
258C4—21no—
259C4—21no—
260C4—21no—
261C4—42no—
262C4—82no—
263C4—82no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4—81no—
268C2—21no—
269C2—21no—
270C2—21no—
271C2—21no—
272C2—161no—
273C2#159711yescharacteristic
274C2#160411yescharacteristic
275C2#160911yescharacteristic
276C2#161611yescharacteristic
277C2#203411yescharacteristic
278C2#203811yescharacteristic
279C2#204411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1853C2semidirect
5. C2×#324C2semidirect
6. #1441C2semidirect
7. #1294C2semidirect

Maximal subgroups

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

subgroupindex
#26182
#18742
#18532
C2×#3272
C2×#3242
#14412
#12942