Finite Group Databasebeta

← #27170188

Subgroups — #27170188

Order 12928.

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

normal subgroups only

This group has 336 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×D404C211yesmaximal, characteristic, complemented
2#20265554C211yesmaximal, characteristic, complemented
3#20265373C211yesmaximal, characteristic, complemented
4#20265372C211yesmaximal, characteristic, complemented
5#20264920C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C22×D404—21no—
9C2×D4×D101—21no—
10C2×D808—22no—
11C2×D808—21no—
12C2×#13108059—21no—
13#15692252C2211yescharacteristic
14#15692252C2211yescharacteristic, complemented
15#15692230—21no—
16#15692229—21no—
17#15692225—22no—
18C4×D404C2211yescharacteristic
19C4⋊#11717579C2211yescharacteristic, complemented
20#15692177—21no—
21#15692175—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic, complemented
24C4×C8×C101C2211yescharacteristic, complemented
25C23×D101—22no—
26C2×D404—22no—
27C2×D404—21no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—21no—
31C2×D404—21no—
32C2×D404—42no—
33C2×D404—42no—
34C2×D404D411yescharacteristic
35C2×D404D412yes—
36C2×C22⋊D101—22no—
37D4×D101—42no—
38D4×D101—42no—
39C2×C4×D101—21no—
40D808—44no—
41D808—42no—
42#13108059—42no—
43C2×#11717579—21no—
44C4⋊Dic101D411yescharacteristic
45#13108051—22no—
46#13108051—21no—
47C2×D4×C101D411yescharacteristic
48C22×C4×C101—21no—
49C2×C8×C101—21no—
50C2×C8×C101—21no—
51C4⋊C4×C101D411yescharacteristic
52C22⋊C4×C101—21no—
53C42×C101C2311yescharacteristic
54C23×C101—21no—
55C22×D101—21no—
56C22×D101—22no—
57C22×D101—44no—
58C22×D101—44no—
59C22×D101—41no—
60C22×D101—82no—
61D4×C101—21no—
62D4×C101D812yescomplemented
63C2×C4×C101—21no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C22⋊D101—44no—
70C2×Dic101—21no—
71D404—21no—
72D404—42no—
73D404—42no—
74D404—44no—
75D404—41no—
76D404—41no—
77D404—41no—
78D404—41no—
79D404—41no—
80D404—41no—
81D404—41no—
82D404—41no—
83D404D812yescomplemented
84C4×D101—41no—
85C4×D101—41no—
86C8×C101—22no—
87C8×C101—41no—
88#11717579—41no—
89C22×C101—22no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101#12011yescharacteristic
93C2×D101—22no—
94C2×D101—41no—
95C2×D101—41no—
96C2×D101—42no—
97C2×D101—42no—
98C2×D101—42no—
99C2×D101—88no—
100C2×D101—81no—
101C2×D101—81no—
102C2×D101—81no—
103C4×C101—21no—
104C4×C101—41no—
105C4×C101C2×D811yescharacteristic
106C4×C101C2×D811yescharacteristic
107C4×C101AGL(1,ℤ/8)11yescharacteristic
108C4×C101AGL(1,ℤ/8)11yescharacteristic
109Dic101—41no—
110C2×C101—42no—
111C2×C101#44711yescharacteristic
112C2×C101#44711yescharacteristic
113C2×C101#45311yescharacteristic
114D101—42no—
115D101—82no—
116D101—161no—
117#1619—1011nomaximal, Sylow
118C101#161911yescharacteristic, complemented, Sylow
119D42—1011no—
120#493—1011no—
121#459—1011no—
122#459—1011no—
123#331—1011no—
124D4⋊C8—1011no—
125D4⋊C8D10111yescharacteristic, complemented
126C22×D4—2021no—
127C22×D4—2021no—
128C2×D8—2021no—
129C2×D8—2022no—
130C2×D8—2021no—
131#127—1011no—
132#127—1011no—
133#121—2021no—
134#121—2021no—
135#120—2022no—
136C4×D4—1011no—
137C4×D4C2×D10111yescharacteristic
138C4⋊C8—1011no—
139C4⋊C8C2×D10111yescharacteristic, complemented
140#102—2021no—
141#102—2021no—
142C4×C8C2×D10111yescharacteristic, complemented
143C24—2022no—
144C2×D4—1011no—
145C2×D4—1012no—
146C2×D4—4042no—
147C2×D4—4042no—
148C2×D4—4042no—
149C2×D4—4042no—
150C2×D4—2022no—
151C2×D4—2022no—
152C2×D4—2021no—
153C2×D4—2021no—
154C2×D4—2021no—
155C2×D4—2021no—
156C2×D4—2021no—
157C2×D4D40411yescharacteristic
158C22×C4—21no—
159C22×C4—2021no—
160D8—4044no—
161D8—4042no—
162D8—4042no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—2021no—
166C4⋊C4—1011no—
167C4⋊C4D40411yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—2022no—
170C22⋊C4—2021no—
171C42C22×D10111yescharacteristic
172C23—21no—
173C23—8082no—
174C23—4044no—
175C23—4044no—
176C23—4041no—
177C23—2022no—
178C23—2021no—
179D4—21no—
180D4—1012no—
181D4—4042no—
182D4—4042no—
183D4—4044no—
184D4—4044no—
185D4—4041no—
186D4—4041no—
187D4—4041no—
188D4—4041no—
189D4—4041no—
190D4—4041no—
191D4—4041no—
192D4—4041no—
193D4—2021no—
194D4D80812yescomplemented
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—4041no—
199C2×C4—4041no—
200C2×C4—2021no—
201C2×C4D4×D10111yescharacteristic
202C2×C4D4×D10111yescharacteristic
203C2×C4C2×D40411yescharacteristic
204C8—22no—
205C8—41no—
206C8—4041no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—8088no—
211C22—8081no—
212C22—8081no—
213C22—8081no—
214C22—4042no—
215C22—4042no—
216C22—4042no—
217C22—4041no—
218C22—4041no—
219C22—2022no—
220C22#1569222511yescharacteristic
221C4—21no—
222C4—41no—
223C4—4041no—
224C4D8×D10111yescharacteristic
225C4C2×D80811yescharacteristic
226C4#1569228211yescharacteristic
227C4#1569228311yescharacteristic
228C2—42no—
229C2—8082no—
230C2—16161no—
231C2—4042no—
232C2#2026531211yescharacteristic
233C2#2026531311yescharacteristic
234C2#2026534611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
hypercenterD4⋊C8D101
hypocenterC101#1619
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. D4×D404C2semidirect
2. #20265554C2semidirect
3. #20265373C2semidirect
4. #20265372C2semidirect
5. #20264920C2semidirect
6. #20264906C2semidirect
7. D4⋊C8×C101C2semidirect
14. #15692252C22semidirect
19. C4⋊#11717579C22semidirect
23. C4⋊C8×C101C22semidirect
24. C4×C8×C101C22semidirect
62. D4×C101D8semidirect
83. D404D8semidirect
118. C101#1619semidirect
125. D4⋊C8D101semidirect
139. C4⋊C8C2×D101semidirect
142. C4×C8C2×D101semidirect
194. D4D808semidirect

Maximal subgroups

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

subgroupindex
D4×D4042
#202655542
#202653732
#202653722
#202649202
#202649062
D4⋊C8×C1012
#1619101