Finite Group Databasebeta

← #13095

Subgroups — #13095

Order 256.

347 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 63 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×#413C211yesmaximal, characteristic
2#2432C211yesmaximal, characteristic
3C2×#356C212yesmaximal, complemented
4#1785C211yesmaximal, characteristic
5#1394C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C22×M16—21no—
8C2×C4⋊Q8C2211yescharacteristic
9C2×#124C411yescharacteristic
10C2×C22⋊Q8C411yescharacteristic
11#413—24no—
12C2×#101C2212yes—
13C2×#101C2212yes—
14C2×#100—21no—
15C2×#100—21no—
16#407—21no—
17#394—21no—
18#394—21no—
19#391C2211yescharacteristic
20#390C411yescharacteristic
21#385—21no—
22#385—21no—
23C4×C22⋊C4C411yescharacteristic
24#356—24no—
25C2×#41—22no—
26C2×#41—22no—
27C2×#41—41no—
28C2×#41—41no—
29C2×#41—42no—
30C22×Q8C2×C411yescharacteristic
31C22×Q8C2311yescharacteristic
32C22×D4C2×C411yescharacteristic
33C22×D4D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4C2×C411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C2×M16—24no—
39C2×M16—22no—
40C2×M16—22no—
41C2×M16—42no—
42C2×M16—41no—
43C22×C8—41no—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46#124—42no—
47C22⋊Q8—41no—
48C22⋊Q8—41no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2×C411yescharacteristic
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C42C2×C411yescharacteristic
60C2×C42C2×C411yescharacteristic
61#101—22no—
62#101—22no—
63#101—22no—
64#101D414yes—
65#100—24no—
66#100—42no—
67C22⋊C8—41no—
68C22⋊C8—41no—
69#95—21no—
70#95—21no—
71#95—21no—
72C24C22⋊C411yescharacteristic
73C24C22⋊C411yescharacteristic
74C24C22⋊C411yescharacteristic
75#41—44no—
76#41—44no—
77#41—44no—
78#41—44no—
79C2×Q8—22no—
80C2×Q8—41no—
81C2×Q8—41no—
82C2×Q8—41no—
83C2×Q8—41no—
84C2×Q8C2×D412yes—
85C2×Q8#4112yes—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—21no—
90C2×D4—22no—
91C2×D4—21no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4C22⋊C411yescharacteristic
114C22×C4C22×C411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117M16—24no—
118M16—44no—
119M16—42no—
120M16—42no—
121M16—42no—
122C2×C8—44no—
123C2×C8—42no—
124C2×C8—42no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C4⋊C4—41no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C42—22no—
138C42—41no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23#9912yes—
154C23C2×C22⋊C411yescharacteristic
155Q8—42no—
156Q8—42no—
157Q8—82no—
158D4—44no—
159D4—44no—
160D4—42no—
161D4—42no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4#9911yescharacteristic
191C2×C4#9911yescharacteristic
192C2×C4C4×D412yes—
193C2×C4#12011yescharacteristic
194C2×C4#12112yes—
195C2×C4#12311yescharacteristic
196C8—44no—
197C8—82no—
198C22—21no—
199C22—21no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#38611yescharacteristic
216C22C2×#9911yescharacteristic
217C22C2×#9911yescharacteristic
218C4—21no—
219C4—22no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—22no—
224C4—42no—
225C4—81no—
226C4—81no—
227C4—81no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—81no—
232C2—81no—
233C2#189811yescharacteristic
234C2#213211yescharacteristic
235C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#356C2semidirect
5. #1394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4132
#24322
C2×#3562
#17852
#13942