Finite Group Databasebeta

← #27170234

Subgroups — #27170234

Order 12928.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic202C211yesmaximal, characteristic, complemented
2#20265523C211yesmaximal, characteristic, complemented
3#20265374C211yesmaximal, characteristic, complemented
4#20264922C211yesmaximal, characteristic, complemented
5#20264907C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#459×C101C211yesmaximal, characteristic
8C22×Dic202—21no—
9C2×#13108062—21no—
10#15692255C2211yescharacteristic
11#15692254—21no—
12#15692234—21no—
13#15692232—22no—
14C4×Dic202C2211yescharacteristic
15D4×Dic101—21no—
16C4⋊#11717579C2211yescharacteristic
17#15692176—22no—
18#15692176—21no—
19C4×#11717579C2211yescharacteristic
20C2×D8×C101—21no—
21#127×C101C2211yescharacteristic
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24#102×C101—21no—
25C2×Dic202—22no—
26C2×Dic202—21no—
27C2×Dic202—42no—
28C2×Dic202—42no—
29C2×Dic202D411yescharacteristic
30C22×Dic101—22no—
31#13108062—42no—
32C2×#11717579—21no—
33C2×#11717579—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36C4⋊Dic101D412yes—
37#13108053—22no—
38#13108053—21no—
39C22⋊Dic101—22no—
40C4×Dic101—21no—
41C2×D4×C101—21no—
42C2×D4×C101—21no—
43C2×D4×C101—21no—
44C2×D4×C101D411yescharacteristic
45C22×C4×C101—21no—
46D8×C101—42no—
47C2×C8×C101—21no—
48C4⋊C4×C101D411yescharacteristic
49C22⋊C4×C101—21no—
50C42×C101C2311yescharacteristic
51C23×C101—21no—
52C23×C101—41no—
53D4×C101—21no—
54D4×C101—42no—
55D4×C101—41no—
56D4×C101—41no—
57D4×C101—41no—
58D4×C101—41no—
59D4×C101SD1612yescomplemented
60C2×C4×C101—21no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×Dic101—21no—
67C2×Dic101—21no—
68C2×Dic101—22no—
69C2×Dic101—44no—
70Dic202—21no—
71Dic202—44no—
72Dic202—41no—
73Dic202—41no—
74Dic202D812yescomplemented
75C8×C101—41no—
76#11717579—41no—
77#11717579—41no—
78C22×C101—22no—
79C22×C101—41no—
80C22×C101—41no—
81C22×C101—81no—
82C22×C101—81no—
83C22×C101—81no—
84C22×C101#12011yescharacteristic
85C4×C101—21no—
86C4×C101—41no—
87C4×C101C2×D811yescharacteristic
88C4×C101C2×SD1611yescharacteristic
89C4×C101AGL(1,ℤ/8)11yescharacteristic
90C4×C101#13711yescharacteristic
91Dic101—22no—
92Dic101—42no—
93Dic101—41no—
94C2×C101—42no—
95C2×C101—161no—
96C2×C101#44711yescharacteristic
97C2×C101#44811yescharacteristic
98C2×C101#45411yescharacteristic
99#1627—1011nomaximal, Sylow
100C101#162711yescharacteristic, complemented, Sylow
101D4×Q8—1011no—
102#486—1011no—
103#460—1011no—
104#459D10111yescharacteristic
105#332—1011no—
106Q8⋊C8—1011no—
107D4⋊C8—1011no—
108C22×Q8—2021no—
109C2×SD16—2021no—
110C2×D8—21no—
111C4⋊Q8—1011no—
112C4⋊Q8—2021no—
113#127C2×D10111yescharacteristic
114C22⋊Q8—2022no—
115C22⋊Q8—2021no—
116C4×Q8—1011no—
117C4×D4—2021no—
118C4×D4C2×D10111yescharacteristic
119C4⋊C8—1011no—
120C4⋊C8C2×D10111yescharacteristic
121#102—21no—
122#102—2021no—
123#102—2022no—
124C4×C8—1011no—
125C2×Q8—1011no—
126C2×Q8—4042no—
127C2×Q8—4042no—
128C2×Q8—2022no—
129C2×Q8—2021no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C22⋊D10111yescharacteristic
134C22×C4—21no—
135C22×C4—2022no—
136SD16—4042no—
137D8—42no—
138C2×C8—21no—
139C2×C8—2021no—
140C2×C8—2021no—
141C4⋊C4—1011no—
142C4⋊C4—1012no—
143C4⋊C4—2022no—
144C4⋊C4—2021no—
145C4⋊C4—2021no—
146C4⋊C4C22⋊D10111yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—2022no—
149C42—2021no—
150C42C22×D10111yescharacteristic
151C23—21no—
152C23—41no—
153Q8—1012no—
154Q8—4044no—
155Q8—4041no—
156Q8—4041no—
157Q8—2021no—
158D4—21no—
159D4—42no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4#1310806212yescomplemented
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—4044no—
169C2×C4—2021no—
170C2×C4—2022no—
171C2×C4—2021no—
172C2×C4D4×D10111yescharacteristic
173C2×C4D4×D10111yescharacteristic
174C2×C4C2×C22⋊D10111yescharacteristic
175C8—41no—
176C8—4041no—
177C8—4041no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22#1569222311yescharacteristic
185C4—21no—
186C4—41no—
187C4—4042no—
188C4—4041no—
189C4—2022no—
190C4D8×D10111yescharacteristic
191C4C2×#1310806211yescharacteristic
192C4#1569227911yescharacteristic
193C4#1569228511yescharacteristic
194C2—42no—
195C2—161no—
196C2#2026530911yescharacteristic
197C2#2026531511yescharacteristic
198C2#2026534811yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radical#459×C101C2
socleC22×C101#120
hypercenter#459D101
hypocenterC101#1627
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. D4×Dic202C2semidirect
2. #20265523C2semidirect
3. #20265374C2semidirect
4. #20264922C2semidirect
5. #20264907C2semidirect
6. D4⋊#11717579C2semidirect
59. D4×C101SD16semidirect
74. Dic202D8semidirect
100. C101#1627semidirect
164. D4#13108062semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2022
#202655232
#202653742
#202649222
#202649072
D4⋊#117175792
#459×C1012
#1627101