Finite Group Databasebeta

← #25668

Subgroups — #25668

Order 256.

492 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 123 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#3512C211yesmaximal, characteristic
2#3142C211yesmaximal, characteristic, complemented
3#3080C211yesmaximal, characteristic, complemented
4#2859C211yesmaximal, characteristic, complemented
5#2040C212yesmaximal, complemented
6#2039C212yesmaximal, complemented
7#1998C211yesmaximal, characteristic, complemented
8#1962C211yesmaximal, characteristic, complemented
9#1860C212yesmaximal, complemented
10#1859C212yesmaximal, complemented
11C4×#104C211yesmaximal, characteristic, complemented
12#553—22no—
13#553—24no—
14#552—22no—
15#541—21no—
16#540—21no—
17#537C2212yes—
18#532C2211yescharacteristic
19#529—22no—
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22#520C2212yes—
23C4×#41C2211yescharacteristic
24#491—21no—
25#490—21no—
26#488—22no—
27#479—22no—
28#478—22no—
29#428C2212yes—
30#423C2211yescharacteristic
31C2×#104C2212yes—
32C2×#104C2212yes—
33C2×#104C2211yescharacteristic
34C2×#104C2211yescharacteristic
35#417C2212yes—
36#417C2212yes—
37#417C2212yes—
38C4×M16C2211yescharacteristic
39#389—24no—
40#384C2212yes—
41#382C2212yes—
42C43C2211yescharacteristic
43#343C2212yes—
44#343C2212yes—
45#339C2211yescharacteristic
46#339C2211yescharacteristic
47C2×#41C2311yescharacteristic
48C2×#41C2312yes—
49C2×M16C2311yescharacteristic
50C2×M16C2311yescharacteristic
51C2×M16C2312yes—
52C4⋊Q8D411yescharacteristic
53#127D411yescharacteristic
54#126—24no—
55#126—24no—
56#125—24no—
57#125—21no—
58#125—21no—
59#125—22no—
60#125—22no—
61#125—21no—
62#125—21no—
63#124D412yes—
64#123—24no—
65#123—24no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#121—24no—
69#121—22no—
70#121—22no—
71#121—22no—
72#121—24no—
73C4×Q8—22no—
74C4×Q8D412yes—
75C4×D4—24no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—24no—
79C4×D4—22no—
80C4×D4D412yes—
81#117—24no—
82#117—22no—
83#117—21no—
84#117C2311yescharacteristic
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4D414yes—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C42—21no—
92C2×C42—22no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#106—22no—
98C4⋊C8—22no—
99#104—24no—
100#104—24no—
101#104—22no—
102#104—22no—
103#103—22no—
104#103—22no—
105#103—22no—
106#102—22no—
107#102—22no—
108#102—22no—
109#97—21no—
110C4×C8—21no—
111#95—24no—
112#41—22no—
113#41—22no—
114#41—21no—
115#41—21no—
116#41—22no—
117#41—22no—
118#41—21no—
119#41—21no—
120C2×Q8C2×D412yes—
121C2×Q8C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—24no—
125C2×D4—22no—
126C2×D4C2×D412yes—
127C2×D4C2×D411yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4C2×D414yes—
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138M16—22no—
139M16—22no—
140M16—21no—
141M16—21no—
142M16—21no—
143M16—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—48no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4C2×D412yes—
161C22⋊C4—24no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—21no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C42—24no—
169C42—22no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—44no—
174C42C2×D411yescharacteristic
175C42C2×D411yescharacteristic
176C42C2×D412yes—
177C42#4112yes—
178C42#4112yes—
179C23—22no—
180C23—21no—
181C23C22×D411yescharacteristic
182Q8—21no—
183Q8—21no—
184Q8—22no—
185Q8—22no—
186D4—21no—
187D4—21no—
188D4—22no—
189D4—22no—
190D4—42no—
191D4—42no—
192D4—41no—
193D4—41no—
194D4—82no—
195D4—84no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—84no—
214C2×C4#12012yes—
215C2×C4#12012yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4C2×#4111yescharacteristic
220C2×C42−1+411yescharacteristic
221C2×C42−1+411yescharacteristic
222C8—42no—
223C8—41no—
224C8—41no—
225C22—22no—
226C22—42no—
227C22—41no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22C2×#12011yescharacteristic
232C22#54811yescharacteristic
233C22#54811yescharacteristic
234C4—24no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—41no—
240C4—44no—
241C4—42no—
242C4—82no—
243C4#53211yescharacteristic
244C4#53611yescharacteristic
245C4#54711yescharacteristic
246C4#54711yescharacteristic
247C2—21no—
248C2—21no—
249C2—82no—
250C2—81no—
251C2#243211yescharacteristic
252C2#301812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3512C2
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
2. #3142C2semidirect
3. #3080C2semidirect
4. #2859C2semidirect
5. #2040C2semidirect
6. #2039C2semidirect
7. #1998C2semidirect
8. #1962C2semidirect
9. #1860C2semidirect
10. #1859C2semidirect
11. C4×#104C2semidirect

Maximal subgroups

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

subgroupindex
#35122
#31422
#30802
#28592
#20402
#20392
#19982
#19622
#18602
#18592
C4×#1042