Finite Group Databasebeta

← #29741

Subgroups — #29741

Order 256.

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

normal subgroups only

This group has 512 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
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#408C211yesmaximal, characteristic, complemented
3#2616C211yesmaximal, characteristic, complemented
4#2615C211yesmaximal, characteristic
5#2077C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#2034C212yesmaximal, complemented
8#2000C212yesmaximal, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1894C211yesmaximal, characteristic, complemented
11#1807C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×SD16C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2212yes—
18C2×#121C2211yescharacteristic
19C2×#121C2212yes—
20#465—24no—
21C2×#106C2211yescharacteristic
22C2×#106C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27#408—24no—
28C2×C22⋊C8C2212yes—
29C2×C22⋊C8C2212yes—
30#395—21no—
31#394—22no—
32#394—21no—
33#393—21no—
34#392—22no—
35#392—21no—
36#380C2211yescharacteristic
37#380C2211yescharacteristic
38#380C2212yes—
39#380C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—42no—
46C2×#41—42no—
47C22×Q8D411yescharacteristic
48C22×Q8C2311yescharacteristic
49C22×D4—22no—
50C22×D4—21no—
51C22×D4D412yes—
52C22×D4D411yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C2×SD16—22no—
57C2×SD16—42no—
58C2×M16—22no—
59C2×M16—42no—
60C22×C8—42no—
61C22×C8D414yescomplemented
62C22×C8C2312yes—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—42no—
68C22⋊Q8—42no—
69#121—22no—
70#121—22no—
71#121—42no—
72#121—42no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87#106—22no—
88#106—42no—
89#103—22no—
90#103—42no—
91#102—22no—
92#102—42no—
93C22⋊C8—24no—
94C22⋊C8—24no—
95#95—22no—
96#95—21no—
97#95—22no—
98#95—21no—
99C24—21no—
100C24C2×D412yes—
101C24C2×D411yescharacteristic
102#41—44no—
103#41—44no—
104#41—48no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—42no—
111C2×D4—24no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—82no—
123C2×D4—82no—
124C2×D4—82no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4#4112yes—
142C22×C4C2411yescharacteristic
143SD16—44no—
144M16—44no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—44no—
151C2×C8C2×D412yes—
152C2×C8C2×D412yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—81no—
168C22⋊C4—82no—
169C22⋊C4—81no—
170C22⋊C4—82no—
171C23—22no—
172C23—21no—
173C23—21no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23—42no—
186C23—41no—
187C23#12112yes—
188C23C22×D411yescharacteristic
189Q8—44no—
190Q8—41no—
191Q8—41no—
192Q8—42no—
193D4—44no—
194D4—48no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—84no—
199D4—88no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4#12112yes—
225C2×C4C22×D411yescharacteristic
226C2×C4C2×#4111yescharacteristic
227C2×C4H(5,𝔽2)12yes—
228C2×C4H(5,𝔽2)12yes—
229C8—22no—
230C8—22no—
231C8—42no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—81no—
245C22—82no—
246C22—82no—
247C22—81no—
248C22—81no—
249C22C2×#12111yescharacteristic
250C22#53412yes—
251C22#57711yescharacteristic
252C22#57712yes—
253C22#57711yescharacteristic
254C4—22no—
255C4—42no—
256C4—81no—
257C4—82no—
258C4—82no—
259C4—81no—
260C4—81no—
261C4#53411yescharacteristic
262C4#53511yescharacteristic
263C4#53812yes—
264C2—42no—
265C2—82no—
266C2—161no—
267C2#261711yescharacteristic
268C2#305711yescharacteristic
269C2#306011yescharacteristic
270C2#320012yes—
271C2#320412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#465C2semidirect
2. C2×#408C2semidirect
3. #2616C2semidirect
5. #2077C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2000C2semidirect
9. #1895C2semidirect
10. #1894C2semidirect
11. #1807C2semidirect
61. C22×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4082
#26162
#26152
#20772
#20352
#20342
#20002
#18952
#18942
#18072