Finite Group Databasebeta

← #19561

Subgroups — #19561

Order 256.

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

normal subgroups only

This group has 306 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#3253C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3#3226C211yesmaximal, characteristic, complemented
4#2979C211yesmaximal, characteristic, complemented
5#1697C212yesmaximal, complemented
6#1695C212yesmaximal, complemented
7#1677C212yesmaximal, complemented
8#1675C212yesmaximal, complemented
9#1574C211yesmaximal, characteristic, complemented
10#1548C211yesmaximal, characteristic, complemented
11#1543C211yesmaximal, characteristic, complemented
12#532C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15#500C2212yes—
16#498—21no—
17#486—21no—
18#486C2212yes—
19#481—22no—
20#480—22no—
21#476—21no—
22#476C2212yes—
23#476C2212yes—
24#476C2212yes—
25#474—21no—
26#474—21no—
27#469—22no—
28#459—21no—
29#459C2212yes—
30#436—21no—
31#436C2212yes—
32C8×D4—21no—
33C8×D4C2212yes—
34#431C2211yescharacteristic
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38#408—22no—
39#335C2212yes—
40#334C2212yes—
41#333C2211yescharacteristic
42#333C2211yescharacteristic
43#333C2212yes—
44#333C2212yes—
45#331C2211yescharacteristic
46#331C2211yescharacteristic
47C2×#41—21no—
48C2×#41—21no—
49C22×Q8—21no—
50C2×D8—22no—
51C2×M16—21no—
52C2×M16—22no—
53C22×C8—22no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8D411yescharacteristic
58C4⋊Q8C2312yes—
59C4⋊Q8C2311yescharacteristic
60#127D411yescharacteristic
61#127C2311yescharacteristic
62#124—21no—
63#124—21no—
64#121—21no—
65#121—21no—
66#121—21no—
67#121D412yes—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4—21no—
71C4×D4C2312yes—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C42C2311yescharacteristic
75#107—22no—
76#107—22no—
77#106—22no—
78#106—22no—
79#106—22no—
80C4⋊C8—21no—
81C4⋊C8D411yescharacteristic
82C4⋊C8D411yescharacteristic
83C4⋊C8C2311yescharacteristic
84C4⋊C8C2311yescharacteristic
85C4⋊C8C2311yescharacteristic
86C4⋊C8C2311yescharacteristic
87C4⋊C8C2312yes—
88#102—22no—
89#102—22no—
90#102—22no—
91#102—22no—
92#102—22no—
93#102—22no—
94C22⋊C8—21no—
95C22⋊C8—21no—
96C22⋊C8—22no—
97C22⋊C8—22no—
98C22⋊C8D412yes—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8C2312yes—
102#41—81no—
103#41—81no—
104#41—82no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—41no—
109C2×Q8—82no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—41no—
117C2×D4C2×D412yes—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4#4111yescharacteristic
124D8—82no—
125M16—82no—
126M16—81no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—82no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4C2×D412yes—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—41no—
146C22⋊C4—41no—
147C42C2×D411yescharacteristic
148C42C2×D411yescharacteristic
149C42#4111yescharacteristic
150C42C2411yescharacteristic
151C23—22no—
152C23—41no—
153C23#12111yescharacteristic
154Q8—81no—
155Q8—82no—
156Q8—81no—
157Q8—82no—
158D4—22no—
159D4—22no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—42no—
165D4—81no—
166D4—81no—
167D4—82no—
168D4—82no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4#12111yescharacteristic
182C2×C4#12111yescharacteristic
183C2×C4#12111yescharacteristic
184C2×C4C22×D411yescharacteristic
185C2×C4C22×D411yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C8—22no—
188C8—22no—
189C8—41no—
190C8—41no—
191C8—41no—
192C8—42no—
193C8—41no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22C2×#12111yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—42no—
208C4—81no—
209C4—82no—
210C4#47212yes—
211C4#57711yescharacteristic
212C4#57711yescharacteristic
213C4#57711yescharacteristic
214C4#57811yescharacteristic
215C2—41no—
216C2—82no—
217C2—161no—
218C2#304211yescharacteristic
219C2#306311yescharacteristic
220C2C2×#47211yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3253C2
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. #3227C2semidirect
3. #3226C2semidirect
4. #2979C2semidirect
5. #1697C2semidirect
6. #1695C2semidirect
7. #1677C2semidirect
8. #1675C2semidirect
9. #1574C2semidirect
10. #1548C2semidirect
11. #1543C2semidirect

Maximal subgroups

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

subgroupindex
#32532
#32272
#32262
#29792
#16972
#16952
#16772
#16752
#15742
#15482
#15432