Finite Group Databasebeta

← #19501

Subgroups — #19501

Order 256.

322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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#3227C211yesmaximal, characteristic, complemented
2#3106C211yesmaximal, characteristic
3#3039C211yesmaximal, characteristic, complemented
4#2991C211yesmaximal, characteristic, complemented
5#1701C212yesmaximal, complemented
6#1691C212yesmaximal, complemented
7#1682C212yesmaximal, complemented
8#1672C212yesmaximal, complemented
9#1574C211yesmaximal, characteristic, complemented
10#1560C211yesmaximal, characteristic, complemented
11#1549C211yesmaximal, characteristic, complemented
12C2×#137—21no—
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C4×#41C2211yescharacteristic
16#498C2212yes—
17#486C2212yes—
18#482—21no—
19#481—22no—
20#477C2211yescharacteristic
21#476—21no—
22#476—21no—
23#476C2211yescharacteristic
24#462—21no—
25#462C2212yes—
26#461—21no—
27#461C2212yes—
28#460C2211yescharacteristic
29#459C2211yescharacteristic
30#449—21no—
31C8×Q8C2212yes—
32C4×SD16—21no—
33C4×D8—21no—
34#436—22no—
35#436C2212yes—
36#431—21no—
37#431C2211yescharacteristic
38#431C2211yescharacteristic
39#423—21no—
40#423C2211yescharacteristic
41#423C2211yescharacteristic
42#419—21no—
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#331C2212yes—
49C2×#41—21no—
50C2×#41D411yescharacteristic
51C22×Q8—21no—
52#137—44no—
53C2×Q16—22no—
54C2×SD16—21no—
55C2×SD16—22no—
56C2×D8—21no—
57C2×M16—21no—
58C2×M16—21no—
59C2×M16—22no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8C2312yes—
64C4⋊Q8C2311yescharacteristic
65#127C2311yescharacteristic
66#124—21no—
67#121—21no—
68#121—21no—
69C4×Q8D411yescharacteristic
70C4×Q8C2311yescharacteristic
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4D411yescharacteristic
76C4×D4C2311yescharacteristic
77#117—21no—
78#117D411yescharacteristic
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C42—21no—
82C2×C42C2311yescharacteristic
83#107—21no—
84#106—22no—
85#106—22no—
86#106—21no—
87C4⋊C8—22no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8D411yescharacteristic
91C4⋊C8D411yescharacteristic
92C4⋊C8C2312yes—
93C4⋊C8C2312yes—
94C4⋊C8C2311yescharacteristic
95C4⋊C8C2311yescharacteristic
96#103—21no—
97#103—22no—
98#103—22no—
99#102—21no—
100#102—22no—
101#102—22no—
102#102—21no—
103#102—21no—
104C22⋊C8—21no—
105C22⋊C8—22no—
106C22⋊C8D412yes—
107C4×C8—22no—
108C4×C8—21no—
109C4×C8C2312yes—
110#41—22no—
111#41—41no—
112#41—82no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—41no—
118C2×Q8C2×D411yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—41no—
124C2×D4C2×D411yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4#4111yescharacteristic
133Q16—44no—
134SD16—44no—
135SD16—42no—
136D8—42no—
137M16—42no—
138M16—82no—
139M16—81no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C2×D412yes—
145C2×C8C2×D412yes—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4C2×D411yescharacteristic
155C4⋊C4C2×D411yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42#4111yescharacteristic
167C42C2411yescharacteristic
168C23—21no—
169C23—41no—
170C23#12111yescharacteristic
171Q8—21no—
172Q8—42no—
173Q8—42no—
174Q8—81no—
175Q8—82no—
176Q8AGL(1,ℤ/8)12yes—
177D4—21no—
178D4—21no—
179D4—42no—
180D4—41no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—81no—
185D4—82no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#12111yescharacteristic
203C2×C4#12111yescharacteristic
204C2×C4#12111yescharacteristic
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4111yescharacteristic
208C8—22no—
209C8—22no—
210C8—42no—
211C8—41no—
212C8—41no—
213C8—42no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22C2×#12111yescharacteristic
224C4—21no—
225C4—21no—
226C4—22no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—82no—
231C4—81no—
232C4#47112yes—
233C4C2×AGL(1,ℤ/8)11yescharacteristic
234C4#57511yescharacteristic
235C4#57711yescharacteristic
236C4#57811yescharacteristic
237C2—41no—
238C2—81no—
239C2—161no—
240C2#304611yescharacteristic
241C2#305511yescharacteristic
242C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3106C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3227C2semidirect
3. #3039C2semidirect
4. #2991C2semidirect
5. #1701C2semidirect
6. #1691C2semidirect
7. #1682C2semidirect
8. #1672C2semidirect
9. #1574C2semidirect
10. #1560C2semidirect
11. #1549C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#31062
#30392
#29912
#17012
#16912
#16822
#16722
#15742
#15602
#15492