Finite Group Databasebeta

← #19176

Subgroups — #19176

Order 256.

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

normal subgroups only

This group has 309 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#3141C211yesmaximal, characteristic
2#3103C211yesmaximal, characteristic
3C2×#477C211yesmaximal, characteristic, complemented
4Q8×M16C211yesmaximal, characteristic, complemented
5#1704C212yesmaximal, complemented
6#1698C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1678C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic, complemented
10#1548C211yesmaximal, characteristic, complemented
11#1547C211yesmaximal, characteristic
12#532C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#489—22no—
16#487C2212yes—
17#486C2212yes—
18#481—21no—
19#477—21no—
20#477—21no—
21#477C2212yes—
22#477C2212yes—
23#475C2211yescharacteristic
24#475C2211yescharacteristic
25#460C2211yescharacteristic
26#460C2211yescharacteristic
27#448—21no—
28#446—22no—
29#446C2212yes—
30C8×Q8C2212yes—
31C4×SD16—21no—
32C4×SD16—21no—
33#431C2211yescharacteristic
34C2×#107—21no—
35#423—21no—
36#423C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38#417—21no—
39#417—21no—
40C2×#103—21no—
41C4×M16—21no—
42C4×M16C2211yescharacteristic
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#333C2212yes—
47#332C2212yes—
48#331C2212yes—
49C2×#41—21no—
50C22×Q8—21no—
51C22×Q8D411yescharacteristic
52C2×SD16—21no—
53C2×SD16—21no—
54C2×M16—21no—
55C2×M16D411yescharacteristic
56C2×M16D411yescharacteristic
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8C2312yes—
62C4⋊Q8C2311yescharacteristic
63#127C2311yescharacteristic
64#124—21no—
65#121—21no—
66#121—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8D411yescharacteristic
70C4×Q8D411yescharacteristic
71C4×Q8C2311yescharacteristic
72C4×Q8C2311yescharacteristic
73C4×D4—21no—
74C4×D4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D411yescharacteristic
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81#107—22no—
82#107—22no—
83#106—21no—
84#106—21no—
85C4⋊C8—22no—
86C4⋊C8—22no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8C2312yes—
90C4⋊C8C2312yes—
91C4⋊C8C2311yescharacteristic
92C4⋊C8C2311yescharacteristic
93#103—22no—
94#103—22no—
95#103—21no—
96#103—21no—
97#103—22no—
98#103—22no—
99#102—22no—
100#102—22no—
101#102—21no—
102#102—21no—
103C22⋊C8—21no—
104#97—22no—
105#97D412yes—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8C2312yes—
109#41—42no—
110#41—42no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—82no—
118C2×Q8C2×D411yescharacteristic
119C2×Q8C2×D411yescharacteristic
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—41no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4#4111yescharacteristic
131SD16—42no—
132SD16—42no—
133M16—22no—
134M16—22no—
135M16—42no—
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—22no—
140C2×C8—42no—
141C2×C8C2×D412yes—
142C2×C8C2×D412yes—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4C2×D411yescharacteristic
155C4⋊C4C2×D411yescharacteristic
156C22⋊C4—41no—
157C22⋊C4—41no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42#4111yescharacteristic
165C42C2411yescharacteristic
166C23—41no—
167C23#12111yescharacteristic
168Q8—21no—
169Q8—21no—
170Q8—21no—
171Q8—42no—
172Q8—41no—
173Q8—41no—
174Q8—82no—
175Q8—82no—
176Q8#13712yes—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—81no—
182D4—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4#12111yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4#12111yescharacteristic
199C2×C4#13512yes—
200C2×C4#13712yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C8—22no—
205C8—22no—
206C8—42no—
207C8—41no—
208C8—42no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#46412yes—
216C22C2×#12111yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—81no—
225C4—82no—
226C4#46412yes—
227C4C2×#13511yescharacteristic
228C4C2×#13711yescharacteristic
229C4C2×#13711yescharacteristic
230C4#57511yescharacteristic
231C2—22no—
232C2—161no—
233C2C2×#46411yescharacteristic
234C2#305611yescharacteristic
235C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#532C22
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
3. C2×#477C2semidirect
4. Q8×M16C2semidirect
5. #1704C2semidirect
6. #1698C2semidirect
7. #1684C2semidirect
8. #1678C2semidirect
9. #1565C2semidirect
10. #1548C2semidirect

Maximal subgroups

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

subgroupindex
#31412
#31032
C2×#4772
Q8×M162
#17042
#16982
#16842
#16782
#15652
#15482
#15472