Finite Group Databasebeta

← #29839

Subgroups — #29839

Order 256.

401 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#474C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic
3#2640C211yesmaximal, characteristic, complemented
4#2319C211yesmaximal, characteristic, complemented
5#2068C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#2055C212yesmaximal, complemented
8#2054C212yesmaximal
9#1987C211yesmaximal, characteristic, complemented
10#1983C211yesmaximal, characteristic, complemented
11#1843C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×D8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#474—24no—
18#463—24no—
19C2×#107C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29#397—22no—
30#393—21no—
31#393—21no—
32#391C2212yes—
33#390C2212yes—
34#390C2212yes—
35#389C2212yes—
36#388—22no—
37#381—21no—
38#380—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×SD16—22no—
49C2×SD16—42no—
50C2×D8—22no—
51C2×D8—42no—
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57#124—22no—
58#124—22no—
59C4×D4—22no—
60C4×D4D414yes—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—24no—
77C4⋊C8—22no—
78C4⋊C8—22no—
79C4⋊C8—42no—
80C4⋊C8—42no—
81#103—22no—
82#103—42no—
83#102—24no—
84#102—22no—
85#102—42no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D411yescharacteristic
91#95D411yescharacteristic
92#95D412yes—
93C24—21no—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—42no—
100C2×Q8—82no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4C2×D412yes—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—82no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120SD16—44no—
121D8—44no—
122C2×C8—22no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4C2×D412yes—
137C22⋊C4—24no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—82no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C42—22no—
145C42—22no—
146C42—81no—
147C42C2×D412yes—
148C42#4112yes—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23—42no—
157C23—41no—
158C23—41no—
159C23C22×D411yescharacteristic
160Q8—41no—
161Q8—41no—
162Q8—42no—
163Q8—84no—
164D4—22no—
165D4—22no—
166D4—41no—
167D4—41no—
168D4—42no—
169D4—42no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4#12112yes—
191C2×C4#12112yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)12yes—
195C2×C42−1+412yes—
196C8—42no—
197C8—41no—
198C8—41no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—81no—
205C22—82no—
206C22—82no—
207C22—81no—
208C22—81no—
209C22C2×#12111yescharacteristic
210C22#53712yes—
211C22#57612yes—
212C22#57711yescharacteristic
213C22#57811yescharacteristic
214C4—22no—
215C4—22no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—81no—
220C4—82no—
221C4—82no—
222C4#53511yescharacteristic
223C4#53611yescharacteristic
224C4#54212yes—
225C2—82no—
226C2—161no—
227C2#262011yescharacteristic
228C2#304111yescharacteristic
229C2#304611yescharacteristic
230C2#325712yes—
231C2#326212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#474C2semidirect
3. #2640C2semidirect
4. #2319C2semidirect
5. #2068C2semidirect
6. #2067C2semidirect
7. #2055C2semidirect
9. #1987C2semidirect
10. #1983C2semidirect
11. #1843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4632
#26402
#23192
#20682
#20672
#20552
#20542
#19872
#19832
#18432