Finite Group Databasebeta

← #29847

Subgroups — #29847

Order 256.

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

normal subgroups only

This group has 420 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×#486C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2628C211yesmaximal, characteristic, complemented
4#2619C211yesmaximal, characteristic
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2034C212yesmaximal, complemented
8#2033C212yesmaximal, complemented
9#1993C211yesmaximal, characteristic, complemented
10#1961C211yesmaximal, characteristic, complemented
11#1846C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#121C2212yes—
15C2×#117C2211yescharacteristic
16#486—24no—
17#433—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×#97C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#400—21no—
31#399—21no—
32#395—21no—
33#394—21no—
34#394—21no—
35#393—21no—
36#392—21no—
37#392—21no—
38#390C2212yes—
39#389C2212yes—
40#382C2211yescharacteristic
41#382C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×D4—22no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4C2312yes—
50C23×C4C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57#127—22no—
58#127—22no—
59#121—42no—
60#121—42no—
61#117—22no—
62#117—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D411yescharacteristic
75C2×C22⋊C4D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82#102—24no—
83#102—24no—
84#102—42no—
85#102—42no—
86C22⋊C8—22no—
87C22⋊C8—22no—
88#97—22no—
89C4×C8D414yescomplemented
90#95—22no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—22no—
96C24—22no—
97C24C2×D411yescharacteristic
98C2×Q8—42no—
99C2×Q8—82no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—44no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—82no—
110C2×D4—82no—
111C2×D4—82no—
112C2×D4—82no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8C2×D412yes—
139C2×C8C2×D412yes—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C22⋊C4—44no—
153C22⋊C4—82no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C42—21no—
157C42—21no—
158C42C2×D412yes—
159C42#4112yes—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—22no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23C22×D411yescharacteristic
173Q8—84no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—42no—
178D4—84no—
179D4—84no—
180D4—88no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4#12112yes—
204C2×C4#12112yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C4H(5,𝔽2)11yescharacteristic
209C2×C42−1+411yescharacteristic
210C2×C42−1+411yescharacteristic
211C8—22no—
212C8—22no—
213C8—41no—
214C8—41no—
215C22—41no—
216C22—42no—
217C22—42no—
218C22—41no—
219C22—41no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22C2×#12111yescharacteristic
228C22#53511yescharacteristic
229C22#53611yescharacteristic
230C22#57612yes—
231C22#57712yes—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—82no—
238C4—82no—
239C4—82no—
240C4#53711yescharacteristic
241C4#53711yescharacteristic
242C4#54211yescharacteristic
243C4#54211yescharacteristic
244C2—81no—
245C2—162no—
246C2#262011yescharacteristic
247C2#305711yescharacteristic
248C2#305811yescharacteristic
249C2#322412yes—
250C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2619C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#486C2semidirect
2. C2×#433C2semidirect
3. #2628C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1993C2semidirect
10. #1961C2semidirect
11. #1846C2semidirect
89. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
C2×#4332
#26282
#26192
#20672
#20652
#20342
#20332
#19932
#19612
#18462