Finite Group Databasebeta

← #19565

Subgroups — #19565

Order 256.

292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 107 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#3254C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic
4#2986C211yesmaximal, characteristic, complemented
5#1704C212yesmaximal, complemented
6#1686C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1666C212yesmaximal, complemented
9#1574C211yesmaximal, characteristic, complemented
10#1549C211yesmaximal, characteristic, complemented
11#1543C211yesmaximal, characteristic
12#533C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15#498—21no—
16#492C2212yes—
17#487—21no—
18#487C2212yes—
19#483—22no—
20#477C2212yes—
21#475C2212yes—
22#475C2212yes—
23#474—21no—
24#468—22no—
25#461—21no—
26#461—21no—
27#461—21no—
28#460C2212yes—
29#448—22no—
30#446—21no—
31#446C2212yes—
32C8×Q8—21no—
33C8×Q8C2212yes—
34#433—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#335C2212yes—
40#334C2212yes—
41#333C2211yescharacteristic
42#333C2211yescharacteristic
43#332C2212yes—
44#332C2212yes—
45#331C2211yescharacteristic
46#331C2211yescharacteristic
47C2×#41—21no—
48C22×Q8—21no—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×M16—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8D411yescharacteristic
57C4⋊Q8D411yescharacteristic
58C4⋊Q8C2312yes—
59C4⋊Q8C2311yescharacteristic
60#127C2311yescharacteristic
61#125—21no—
62#124—21no—
63C22⋊Q8—21no—
64C22⋊Q8D412yes—
65#121—21no—
66#121—21no—
67C4×Q8—21no—
68C4×Q8C2312yes—
69C4×D4—21no—
70C4×D4—21no—
71#117—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C42C2311yescharacteristic
75#107—22no—
76#106—22no—
77#106—22no—
78C4⋊C8—22no—
79C4⋊C8—22no—
80C4⋊C8—21no—
81C4⋊C8D411yescharacteristic
82C4⋊C8D411yescharacteristic
83C4⋊C8C2311yescharacteristic
84C4⋊C8C2311yescharacteristic
85C4⋊C8C2311yescharacteristic
86C4⋊C8C2311yescharacteristic
87C4⋊C8C2312yes—
88#103—22no—
89#103—22no—
90#103—22no—
91#103—22no—
92#103—22no—
93#102—22no—
94C22⋊C8—21no—
95C22⋊C8—21no—
96C22⋊C8D412yes—
97#97—22no—
98C4×C8—22no—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8C2312yes—
102#41—82no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—21no—
106C2×Q8—41no—
107C2×Q8—81no—
108C2×Q8—81no—
109C2×Q8C2×D412yes—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4#4111yescharacteristic
119SD16—82no—
120SD16—82no—
121SD16—82no—
122M16—81no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8C2×D412yes—
129C2×C8C2×D412yes—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4C2×D412yes—
142C22⋊C4—22no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C42—22no—
146C42C2×D411yescharacteristic
147C42C2×D411yescharacteristic
148C42#4111yescharacteristic
149C42C2411yescharacteristic
150C23—41no—
151C23#12111yescharacteristic
152Q8—22no—
153Q8—22no—
154Q8—42no—
155Q8—42no—
156Q8—42no—
157Q8—81no—
158Q8—82no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—81no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4#12111yescharacteristic
177C2×C4#12111yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4C22×D411yescharacteristic
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C8—22no—
183C8—22no—
184C8—41no—
185C8—41no—
186C8—41no—
187C8—42no—
188C8—41no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22C2×#12111yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—81no—
203C4#47112yes—
204C4#57711yescharacteristic
205C4#57711yescharacteristic
206C4#57711yescharacteristic
207C4#57811yescharacteristic
208C2—41no—
209C2—161no—
210C2#304311yescharacteristic
211C2#306311yescharacteristic
212C2C2×#47111yescharacteristic
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
2. #3229C2semidirect
4. #2986C2semidirect
5. #1704C2semidirect
6. #1686C2semidirect
7. #1684C2semidirect
8. #1666C2semidirect
9. #1574C2semidirect
10. #1549C2semidirect

Maximal subgroups

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

subgroupindex
#32542
#32292
#32062
#29862
#17042
#16862
#16842
#16662
#15742
#15492
#15432