Finite Group Databasebeta

← #19175

Subgroups — #19175

Order 256.

298 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 115 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#3139C211yesmaximal, characteristic
2#3104C211yesmaximal, characteristic
3C2×#475C211yesmaximal, characteristic
4Q8×M16C211yesmaximal, characteristic
5#1704C212yesmaximal, complemented
6#1700C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1680C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic
10#1550C211yesmaximal, characteristic
11#1546C211yesmaximal, characteristic
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#491—22no—
16#487C2212yes—
17#487C2212yes—
18#484—21no—
19#477C2211yescharacteristic
20#477C2211yescharacteristic
21#475—21no—
22#475—21no—
23#475C2212yes—
24#475C2212yes—
25#462C2211yescharacteristic
26#462C2211yescharacteristic
27#451—21no—
28#446—22no—
29#446C2212yes—
30C8×Q8C2212yes—
31C4×Q16—21no—
32C4×Q16—21no—
33#431C2211yescharacteristic
34C2×#106—21no—
35#423—21no—
36#423C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38#419—21no—
39#419—21no—
40C2×#103—21no—
41C4×M16—21no—
42C4×M16C2211yescharacteristic
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#332C2212yes—
49C22×Q8—21no—
50C22×Q8D411yescharacteristic
51C2×Q16—21no—
52C2×Q16—21no—
53C2×M16—21no—
54C2×M16D411yescharacteristic
55C2×M16D411yescharacteristic
56C22×C8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8C2312yes—
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#125—21no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8D411yescharacteristic
71C4×Q8D411yescharacteristic
72C4×Q8C2311yescharacteristic
73C4×Q8C2311yescharacteristic
74#117—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D411yescharacteristic
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81#107—21no—
82#107—21no—
83#106—22no—
84#106—22no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8C2312yes—
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2311yescharacteristic
92C4⋊C8C2312yes—
93#103—22no—
94#103—22no—
95#103—21no—
96#103—21no—
97#103—22no—
98#103—22no—
99#103—22no—
100#103—22no—
101#103—21no—
102#103—21no—
103C22⋊C8—21no—
104#97—22no—
105#97D412yes—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8C2312yes—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8C2×D411yescharacteristic
119C2×Q8C2×D411yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4#4111yescharacteristic
126Q16—42no—
127Q16—42no—
128M16—22no—
129M16—22no—
130M16—42no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—42no—
136C2×C8C2×D412yes—
137C2×C8C2×D412yes—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4C2×D411yescharacteristic
154C4⋊C4C2×D411yescharacteristic
155C22⋊C4—41no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—41no—
161C42C2×D411yescharacteristic
162C42C2×D411yescharacteristic
163C42#4111yescharacteristic
164C42C2411yescharacteristic
165C23#12111yescharacteristic
166Q8—21no—
167Q8—21no—
168Q8—21no—
169Q8—42no—
170Q8—42no—
171Q8—41no—
172Q8—42no—
173Q8—41no—
174Q8—81no—
175Q8—82no—
176Q8#13712yes—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—81no—
189C2×C4—82no—
190C2×C4#12111yescharacteristic
191C2×C4#12111yescharacteristic
192C2×C4#12111yescharacteristic
193C2×C4#13512yes—
194C2×C4#13712yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C8—22no—
199C8—22no—
200C8—42no—
201C8—41no—
202C8—42no—
203C8—41no—
204C22—21no—
205C22—21no—
206C22#46412yes—
207C22C2×#12111yescharacteristic
208C4—21no—
209C4—22no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4#46412yes—
219C4C2×#13511yescharacteristic
220C4C2×#13711yescharacteristic
221C4C2×#13711yescharacteristic
222C4#57511yescharacteristic
223C2—22no—
224C2C2×#46411yescharacteristic
225C2#305611yescharacteristic
226C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC23#121
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
5. #1704C2semidirect
6. #1700C2semidirect
7. #1684C2semidirect
8. #1680C2semidirect

Maximal subgroups

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

subgroupindex
#31392
#31042
C2×#4752
Q8×M162
#17042
#17002
#16842
#16802
#15652
#15502
#15462