Finite Group Databasebeta

← #32974

Subgroups — #32974

Order 256.

593 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C4×2−1+4C211yesmaximal, characteristic
2#3080C212yesmaximal, complemented
3#2565C211yesmaximal, characteristic, complemented
4#1860C212yesmaximal, complemented
5#1859C212yesmaximal, complemented
6#1857C211yesmaximal, characteristic, complemented
7#1857C211yesmaximal, characteristic, complemented
8#1806C212yesmaximal, complemented
9#1801C211yesmaximal, characteristic, complemented
10#1793C211yesmaximal, characteristic
11#1793C211yesmaximal, characteristic
12C2×2−1+4C2211yescharacteristic
13C2×#125C2211yescharacteristic
14#520—22no—
15#520—22no—
16#520C2212yes—
17#519—22no—
18#519C2211yescharacteristic
19C4×#41—22no—
20C4×#41—22no—
21C4×#41C2212yes—
22C2×C4×Q8—22no—
23C2×C4×Q8C2211yescharacteristic
24#479—22no—
25#479—22no—
26#478—22no—
27#478—22no—
28#428C2212yes—
29#428C2212yes—
30#423C2212yes—
31C2×#104C2212yes—
32C2×#104C2212yes—
33#417C2212yes—
34#417C2212yes—
35C2×#101C2211yescharacteristic
36C2×#101C2211yescharacteristic
37#398—22no—
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#387—22no—
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44#343C2212yes—
45#339C2212yes—
46#339C2212yes—
472−1+4—24no—
482−1+4—22no—
492−1+4Q814yes—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41C2312yes—
53C22×Q8—22no—
54C22×Q8C2311yescharacteristic
55C2×M16C2312yes—
56C2×M16C2312yes—
57#125—22no—
58#125—22no—
59C4×Q8—22no—
60C4×Q8—24no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8D412yes—
68C4×Q8D412yes—
69C4×D4—22no—
70C4×D4—24no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—24no—
75C4×D4—22no—
76C4×D4D412yes—
77C4×D4D412yes—
78#117—24no—
79#117—24no—
80#117—22no—
81#117—22no—
82#117—22no—
83#117C2312yes—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4D412yes—
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C42—24no—
95C2×C42—22no—
96C2×C42D412yes—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#107—22no—
102#106—22no—
103#106—22no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106#104—24no—
107#104—24no—
108#103—22no—
109#103—22no—
110#102—22no—
111#102—22no—
112#101—22no—
113#101—22no—
114#95—21no—
115#95—22no—
116#95—22no—
117#95—22no—
118#41—28no—
119#41—28no—
120#41—28no—
121#41—28no—
122#41—22no—
123#41—22no—
124#41C2×Q814yes—
125#41#4114yes—
126C2×Q8—22no—
127C2×Q8—28no—
128C2×Q8—28no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8C2×D412yes—
132C2×Q8C2×Q812yes—
133C2×Q8#4112yes—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4C2×D412yes—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C2×D412yes—
147C22×C4C2×D412yes—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151M16—22no—
152M16—22no—
153M16—22no—
154M16—22no—
155C2×C8—22no—
156C2×C8—22no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C42—24no—
178C42—24no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42C2×D412yes—
187C42C2×D412yes—
188C23—22no—
189C23C22×D411yescharacteristic
190Q8—28no—
191Q8—24no—
192Q8—22no—
193Q8—24no—
194Q8C22⋊Q814yes—
195D4—28no—
196D4—24no—
197D4—24no—
198D4—22no—
199D4C22⋊Q814yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—44no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—84no—
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4#12012yes—
228C2×C4C22⋊Q814yes—
229C2×C4C22×D412yes—
230C2×C4C22×Q811yescharacteristic
231C2×C4C2×#4112yes—
232C2×C4C2×#4111yescharacteristic
233C8—42no—
234C8—42no—
235C22—21no—
236C22—21no—
237C22—24no—
238C22—42no—
239C22—42no—
240C22C2×#12011yescharacteristic
241C22C2×C22⋊Q811yescharacteristic
242C22#52511yescharacteristic
243C4—24no—
244C4—24no—
245C4—22no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4C2×C22⋊Q812yes—
256C4#52512yes—
257C2—21no—
258C2—21no—
259C2—44no—
260C2#284811yescharacteristic
261C2#301811yescharacteristic
262C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×2−1+4C22
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. #3080C2semidirect
3. #2565C2semidirect
4. #1860C2semidirect
5. #1859C2semidirect
6. #1857C2semidirect
7. #1857C2semidirect
8. #1806C2semidirect
9. #1801C2semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+42
#30802
#25652
#18602
#18592
#18572
#18062
#18012
#17932