Finite Group Databasebeta

← #29817

Subgroups — #29817

Order 256.

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

normal subgroups only

This group has 433 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×#497C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2621C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic
5#2056C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#2000C212yesmaximal
9#1958C211yesmaximal, characteristic, complemented
10#1956C211yesmaximal, characteristic, complemented
11#1836C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2212yes—
18C2×#121C2211yescharacteristic
19C2×#117C2211yescharacteristic
20#497—24no—
21#433—24no—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28C2×#97C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#398—22no—
31#397—22no—
32#396—22no—
33#394—22no—
34#391C2212yes—
35#391C2211yescharacteristic
36#390C2211yescharacteristic
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2312yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C2×Q16—24no—
47C2×Q16—42no—
48C2×Q16—42no—
49C2×SD16—24no—
50C2×SD16—42no—
51C2×SD16—42no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56#124—22no—
57#124—22no—
58C22⋊Q8—42no—
59C22⋊Q8—42no—
60C22⋊Q8—42no—
61#121—42no—
62#117—22no—
63#117—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4D412yes—
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80C4⋊C8—24no—
81#103—42no—
82#103—42no—
83#103—42no—
84#102—42no—
85C22⋊C8—24no—
86#97D414yescomplemented
87C4×C8—22no—
88#95—22no—
89#95—22no—
90#95—24no—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—82no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D412yes—
121C22×C4C2411yescharacteristic
122Q16—48no—
123SD16—48no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8C2×D414yes—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—82no—
139C22⋊C4—24no—
140C22⋊C4—42no—
141C22⋊C4—44no—
142C22⋊C4—42no—
143C22⋊C4—44no—
144C22⋊C4—82no—
145C42—21no—
146C42—21no—
147C42C2×D412yes—
148C42#4112yes—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—22no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—42no—
158C23—42no—
159C23C22×D411yescharacteristic
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165Q8—42no—
166D4—42no—
167D4—42no—
168D4—84no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4#12114yes—
193C2×C4C22×D411yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4H(5,𝔽2)12yes—
196C2×C4H(5,𝔽2)12yes—
197C8—24no—
198C8—42no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—42no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22C2×#12111yescharacteristic
210C22#53511yescharacteristic
211C22#53511yescharacteristic
212C22#57712yes—
213C22#57812yes—
214C4—24no—
215C4—41no—
216C4—42no—
217C4—81no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4—82no—
222C4#53512yes—
223C4#54012yes—
224C2—81no—
225C2—161no—
226C2#261911yescharacteristic
227C2#306312yes—
228C2#322211yescharacteristic
229C2#322311yescharacteristic
230C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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×#497C2semidirect
2. C2×#433C2semidirect
3. #2621C2semidirect
5. #2056C2semidirect
6. #2055C2semidirect
7. #2001C2semidirect
9. #1958C2semidirect
10. #1956C2semidirect
11. #1836C2semidirect
86. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4332
#26212
#26202
#20562
#20552
#20012
#20002
#19582
#19562
#18362