Finite Group Databasebeta

← #27169599

Subgroups — #27169599

Order 12928.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265724C211yesmaximal, characteristic
2#20265233C211yesmaximal, characteristic, complemented
3#20265228C211yesmaximal, characteristic, complemented
4#20264906C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108074—21no—
9#15692254C411yescharacteristic
10#15692251C411yescharacteristic
11#15692241—21no—
12#15692230C411yescharacteristic
13#15692228—21no—
14#15692226C411yescharacteristic
15C4×D404C2211yescharacteristic
16C4×C22⋊D101—21no—
17C42×D101C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692184—21no—
20#15692167—21no—
21#15692167—21no—
22#15692165—21no—
23C4×#11717579C2211yescharacteristic
24C4×D4×C101C2211yescharacteristic
25C4⋊C8×C101C2211yescharacteristic
26C4×C8×C101C2211yescharacteristic
27#13108074—42no—
28#13108074—42no—
29C2×Dic202—21no—
30C2×D404—21no—
31C2×D404C2×C411yescharacteristic
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35C2×C4×D101—21no—
36C2×C4×D101C2×C411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×#11717579—21no—
40C2×#11717579—21no—
41C4⋊Dic101C2×C411yescharacteristic
42#13108053—21no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101C2×C411yescharacteristic
47C4×Dic101D411yescharacteristic
48C4×Dic101D411yescharacteristic
49C2×D4×C101C2×C411yescharacteristic
50C22×C4×C101—21no—
51C2×C8×C101—21no—
52C2×C8×C101—21no—
53C4⋊C4×C101C2×C411yescharacteristic
54C22⋊C4×C101—21no—
55C42×C101C2311yescharacteristic
56C23×C101—21no—
57C22×D101—21no—
58C22×D101C22⋊C411yescharacteristic
59D4×C101—21no—
60D4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101C22×C411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C22⋊D101—42no—
68C22⋊D101—42no—
69C22⋊D101—42no—
70C2×Dic101—21no—
71C2×Dic101C22⋊C411yescharacteristic
72C2×Dic101C22⋊C411yescharacteristic
73C2×Dic101C22⋊C411yescharacteristic
74D404—21no—
75D404—21no—
76D404—42no—
77C4×D101—21no—
78C4×D101—21no—
79C4×D101—21no—
80C4×D101—21no—
81C4×D101—41no—
82C4×D101—41no—
83C4×D101—41no—
84Dic202—42no—
85C8×C101—41no—
86C8×C101—41no—
87#11717579—41no—
88#11717579—41no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101C2×C22⋊C411yescharacteristic
93C2×D101—21no—
94C2×D101—21no—
95C2×D101—21no—
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—41no—
99C4×C101—21no—
100C4×C101—41no—
101C4×C101#13111yescharacteristic
102C4×C101#13111yescharacteristic
103C4×C101AGL(1,ℤ/8)11yescharacteristic
104C4×C101AGL(1,ℤ/8)11yescharacteristic
105Dic101—21no—
106Dic101—21no—
107Dic101—21no—
108Dic101—41no—
109C2×C101—81no—
110C2×C101#40811yescharacteristic
111C2×C101#41811yescharacteristic
112C2×C101#42111yescharacteristic
113D101—41no—
114D101—81no—
115#1496—1011nomaximal, Sylow
116C101#149611yescharacteristic, complemented, Sylow
117#532—1011no—
118#432—1011no—
119#431—1011no—
120D4⋊C8—1011no—
121D4⋊C8—1011no—
122D4⋊C8—1011no—
123D4⋊C8D10111yescharacteristic, complemented
124C2×#41—2021no—
125C4⋊Q8—1011no—
126#127—1011no—
127#124—2021no—
128#121—1011no—
129#121—1011no—
130#121—2021no—
131C4×D4—1011no—
132C4×D4—2021no—
133C4×D4C2×D10111yescharacteristic
134C2×C42—1011no—
135C4⋊C8—1011no—
136C4⋊C8—2021no—
137C4⋊C8C2×D10111yescharacteristic
138C22⋊C8—2021no—
139C22⋊C8—2021no—
140#97—2021no—
141C4×C8—1011no—
142C4×C8C2×D10111yescharacteristic
143#41—4042no—
144#41—4042no—
145C2×Q8—2021no—
146C2×D4—1011no—
147C2×D4—2021no—
148C2×D4—2021no—
149C2×D4—2021no—
150C2×D4—2021no—
151C2×D4C4×D10111yescharacteristic
152C22×C4—21no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—1011no—
156C22×C4—2021no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—2021no—
160C2×C8—2021no—
161C4⋊C4—1011no—
162C4⋊C4—2021no—
163C4⋊C4C4×D10111yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C42—1011no—
169C42—1011no—
170C42—1011no—
171C42C22×D10111yescharacteristic
172C23—21no—
173C23—1011no—
174C23—2021no—
175Q8—4042no—
176D4—21no—
177D4—21no—
178D4—4042no—
179D4—4042no—
180D4—4042no—
181D4—4042no—
182D4—2021no—
183D4—2021no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1011no—
188C2×C4—1011no—
189C2×C4—1011no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4—2021no—
198C2×C4C2×C4×D10111yescharacteristic
199C2×C4D4×D10111yescharacteristic
200C2×C4D4×D10111yescharacteristic
201C8—41no—
202C8—41no—
203C8—4041no—
204C8—4041no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—4041no—
209C22—4041no—
210C22—4041no—
211C22—2021no—
212C22—2021no—
213C22—2021no—
214C22C22⋊C4×D10111yescharacteristic
215C4—21no—
216C4—41no—
217C4—4041no—
218C4—2021no—
219C4—2021no—
220C4—2021no—
221C4#1569226111yescharacteristic
222C4#1569226311yescharacteristic
223C4#1569227911yescharacteristic
224C4#1569228211yescharacteristic
225C2—81no—
226C2—8081no—
227C2—4041no—
228C2#2026514211yescharacteristic
229C2#2026518011yescharacteristic
230C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265724C2
cosocleC42×C101C23
hypercenterD4⋊C8D101
hypocenterC101#1496
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20265233C2semidirect
3. #20265228C2semidirect
4. #20264906C2semidirect
5. D4⋊#11717579C2semidirect
6. #20264904C2semidirect
7. D4⋊C8×C101C2semidirect
116. C101#1496semidirect
123. D4⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202657242
#202652332
#202652282
#202649062
D4⋊#117175792
#202649042
D4⋊C8×C1012
#1496101