Finite Group Databasebeta

← #24772

Subgroups — #24772

Order 256.

420 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 129 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×#473C212yesmaximal
2#3039C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4C4×#137C211yesmaximal, characteristic
5#2906C211yesmaximal, characteristic
6#2064C212yesmaximal, complemented
7#2047C212yesmaximal, complemented
8#1966C211yesmaximal, characteristic, complemented
9#1950C211yesmaximal, characteristic, complemented
10#1908C212yesmaximal
11#1801C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×#137C2211yescharacteristic
14C22×Q16C2212yes—
15C2×#131C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C4×#41C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19#473—28no—
20#462—21no—
21#462—21no—
22#462—21no—
23#462—21no—
24#462—22no—
25#461—21no—
26#461—21no—
27#441—21no—
28#440—21no—
29C4×Q16—21no—
30C4×SD16—21no—
31C2×#108C2212yes—
32#428C2211yescharacteristic
33#424—22no—
34#424C2212yes—
35#423—21no—
36#423C2211yescharacteristic
37#423C2211yescharacteristic
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2211yescharacteristic
40#419C2212yes—
41#419C2211yescharacteristic
42#417C2211yescharacteristic
43C2×#103C2212yes—
44C2×#101C2212yes—
45C2×#101C2212yes—
46C2×#101C2212yes—
47C4×M16C2211yescharacteristic
48#343C2212yes—
49#341C2212yes—
50C2×#41C2311yescharacteristic
51C22×Q8C2312yes—
52C22×Q8C2311yescharacteristic
53#137—22no—
54#137—24no—
55#137—24no—
56#137D414yes—
57C2×Q16—21no—
58C2×Q16—24no—
59C2×Q16—22no—
60C2×Q16—22no—
61C2×Q16—22no—
62C2×Q16—42no—
63C2×SD16—21no—
64C2×SD16—22no—
65#131—24no—
66#131—42no—
67C2×M16—22no—
68C2×M16C2312yes—
69C2×M16C2312yes—
70C2×M16C2311yescharacteristic
71C2×M16C2311yescharacteristic
72C22×C8—22no—
73C22×C8C2312yes—
74C4⋊Q8—21no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4×Q8—21no—
79C4×Q8—21no—
80C4×Q8—21no—
81C4×Q8—21no—
82C4×Q8D411yescharacteristic
83C4×Q8D411yescharacteristic
84C4×D4—21no—
85C4×D4—21no—
86C4×D4D411yescharacteristic
87C4×D4D411yescharacteristic
88#117—21no—
89#117C2311yescharacteristic
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2311yescharacteristic
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97#108—24no—
98#107—21no—
99#106—21no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8—21no—
105C4⋊C8—21no—
106C4⋊C8D411yescharacteristic
107C4⋊C8D411yescharacteristic
108C4⋊C8D412yes—
109#103—21no—
110#103—22no—
111#103—22no—
112#103—21no—
113#103—21no—
114#103—22no—
115#103—22no—
116#102—21no—
117#101—24no—
118#101—24no—
119#101—42no—
120#97—21no—
121C4×C8—21no—
122#41—21no—
123#41—21no—
124#41C2×D412yes—
125#41#4112yes—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—21no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×D412yes—
138C2×Q8C2×D411yescharacteristic
139C2×D4—21no—
140C2×D4C2×D411yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4#4111yescharacteristic
148C22×C4#4111yescharacteristic
149C22×C4C2411yescharacteristic
150Q16—24no—
151Q16—44no—
152Q16—44no—
153Q16—44no—
154SD16—24no—
155SD16—44no—
156M16—21no—
157M16—22no—
158M16—22no—
159M16—22no—
160M16—21no—
161M16—22no—
162M16—21no—
163M16—44no—
164M16C2×D412yes—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—44no—
172C2×C8C2×D412yes—
173C2×C8C2×D412yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4C2×D411yescharacteristic
185C4⋊C4C2×D411yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—41no—
193C42—41no—
194C42C2×D411yescharacteristic
195C42C2×D411yescharacteristic
196C42C2×D411yescharacteristic
197C42C2×D411yescharacteristic
198C23—21no—
199C23C2×#4111yescharacteristic
200Q8—21no—
201Q8—22no—
202Q8—22no—
203Q8—41no—
204Q8—41no—
205Q8—42no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209Q8#12112yes—
210D4—22no—
211D4—22no—
212D4—21no—
213D4#12112yes—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4#12111yescharacteristic
232C2×C4#12111yescharacteristic
233C2×C4#12111yescharacteristic
234C2×C4#12111yescharacteristic
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C2×#4111yescharacteristic
239C2×C4H(5,𝔽2)11yescharacteristic
240C2×C42−1+411yescharacteristic
241C8—22no—
242C8—22no—
243C8—22no—
244C8—42no—
245C8—41no—
246C8—42no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22C2×#12111yescharacteristic
253C22#54711yescharacteristic
254C22#55311yescharacteristic
255C4—22no—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—41no—
261C4—41no—
262C4—82no—
263C4—81no—
264C4#47312yescomplemented
265C4C2×#12111yescharacteristic
266C4#53711yescharacteristic
267C4D4211yescharacteristic
268C4#54811yescharacteristic
269C2—21no—
270C2—21no—
271C2—42no—
272C2#241611yescharacteristic
273C2C2×#47311yescharacteristic
274C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#462C2semidirect
6. #2064C2semidirect
7. #2047C2semidirect
8. #1966C2semidirect
9. #1950C2semidirect
11. #1801C2semidirect
264. C4#473semidirect

Maximal subgroups

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

subgroupindex
C2×#4732
#30392
C2×#4622
C4×#1372
#29062
#20642
#20472
#19662
#19502
#19082
#18012