Finite Group Databasebeta

← #27171323

Subgroups — #27171323

Order 12928.

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

normal subgroups only

This group has 360 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
1C2×C4⋊C4×D101C211yesmaximal, characteristic
2C2×#15692186C211yesmaximal, characteristic, complemented
3C2×#15692167C211yesmaximal, characteristic
4#20265034C211yesmaximal, characteristic
5#20264940C211yesmaximal, characteristic, complemented
6#20264939C211yesmaximal, characteristic, complemented
7C2×#106×C101C211yesmaximal, characteristic, complemented
8C22×C4×D101—21no—
9C22×C4×D101C2211yescharacteristic
10C22×#11717579C2211yescharacteristic
11C2×C4⋊Dic101C2211yescharacteristic
12C2×C4⋊Dic101C2211yescharacteristic
13C2×#13108053—21no—
14C4⋊C4×D101—22no—
15C4⋊C4×D101—22no—
16C2×#13108051—21no—
17#15692186—22no—
18#15692167C414yescomplemented
19#15692162—21no—
20C22×C8×C101C2211yescharacteristic
21C2×C4⋊C4×C101C2211yescharacteristic
22C2×C4⋊C4×C101C2211yescharacteristic
23#106×C101—22no—
24C23×D101D411yescharacteristic
25C22×Dic101—21no—
26C22×Dic101—21no—
27C22×Dic101D411yescharacteristic
28C2×C4×D101—21no—
29C2×C4×D101—21no—
30C2×C4×D101—21no—
31C2×C4×D101—21no—
32C2×C4×D101—24no—
33C2×C4×D101—42no—
34C2×C4×D101—42no—
35C2×C4×D101C2×C412yes—
36C2×C4×D101Q812yes—
37C2×#11717579—21no—
38C2×#11717579—21no—
39C2×#11717579C2×C412yescomplemented
40C4⋊Dic101—21no—
41C4⋊Dic101—21no—
42C4⋊Dic101—21no—
43C4⋊Dic101D412yes—
44#13108053—24no—
45#13108051—42no—
46C22×C4×C101—21no—
47C22×C4×C101—21no—
48C22×C4×C101C2311yescharacteristic
49C2×C8×C101—21no—
50C2×C8×C101—21no—
51C2×C8×C101C2×C412yescomplemented
52C4⋊C4×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101—21no—
55C4⋊C4×C101D412yes—
56C23×C101C2×D411yescharacteristic
57C22×D101—21no—
58C22×D101—21no—
59C22×D101—21no—
60C22×D101—21no—
61C22×D101C4⋊C412yes—
62C22×D101SD1612yes—
63C22×D101SD1612yes—
64C2×C4×C101—22no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C22×C411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×Q811yescharacteristic
74C2×C4×C101#4111yescharacteristic
75C2×C4×C101#4111yescharacteristic
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—22no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83C2×Dic101—41no—
84C2×Dic101C4⋊C412yes—
85C4×D101—22no—
86C4×D101—22no—
87C4×D101—22no—
88C4×D101—22no—
89C4×D101—48no—
90C8×C101—22no—
91#11717579—22no—
92C22×C101C2×C4⋊C411yescharacteristic
93C22×C101#12011yescharacteristic
94C22×C101#12311yescharacteristic
95C22×C101C2×SD1611yescharacteristic
96C22×C101C2×SD1611yescharacteristic
97C22×C101AGL(1,ℤ/8)11yescharacteristic
98C22×C101#13711yescharacteristic
99C2×D101—22no—
100C2×D101—22no—
101C2×D101—22no—
102C2×D101—22no—
103C2×D101—22no—
104C2×D101—22no—
105C2×D101#10614yescomplemented
106C4×C101—42no—
107C4×C101—81no—
108C4×C101C4×D411yescharacteristic
109C4×C101C4×D411yescharacteristic
110C4×C101C22⋊Q811yescharacteristic
111C4×C101C22⋊Q811yescharacteristic
112Dic101—22no—
113Dic101—42no—
114Dic101—81no—
115C2×C101#38511yescharacteristic
116C2×C101C2×#10611yescharacteristic
117C2×C101#42811yescharacteristic
118C2×C101#44811yescharacteristic
119C2×C101#45011yescharacteristic
120C2×C101#48111yescharacteristic
121C2×C101#48311yescharacteristic
122D101—24no—
123#1853—1011nomaximal, Sylow
124C101#185311yescharacteristic, complemented, Sylow
125C22×C4⋊C4—1011no—
126C2×#106—1011no—
127C2×#106D10111yescharacteristic, complemented
128C2×C22⋊C8—1011no—
129#380—1011no—
130#340—1011no—
131#340—1011no—
132C23×C4—1011no—
133C23×C4—2021no—
134C22×C8—1011no—
135C22×C8C2×D10111yescharacteristic
136C2×C4⋊C4—1011no—
137C2×C4⋊C4—1011no—
138C2×C4⋊C4—2022no—
139C2×C4⋊C4—2021no—
140C2×C4⋊C4—2022no—
141C2×C4⋊C4C2×D10111yescharacteristic
142C2×C4⋊C4C2×D10111yescharacteristic
143C2×C22⋊C4—2021no—
144#106—22no—
145#106—2022no—
146C22⋊C8—1014no—
147#95—2021no—
148C24—1011no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1011no—
152C22×C4—1012no—
153C22×C4—1012no—
154C22×C4—4042no—
155C22×C4—4042no—
156C22×C4—2024no—
157C22×C4—2021no—
158C22×C4—2021no—
159C22×C4—2021no—
160C22×C4—2021no—
161C22×C4—2021no—
162C22×C4—2021no—
163C22×C4C22×D10111yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—1012no—
167C2×C8—2021no—
168C2×C8—2021no—
169C2×C8C4×D10112yescomplemented
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—1012no—
174C4⋊C4—2024no—
175C4⋊C4—2021no—
176C4⋊C4—2021no—
177C4⋊C4—2021no—
178C4⋊C4C22⋊D10112yes—
179C22⋊C4—4042no—
180C23—1012no—
181C23—1012no—
182C23—1012no—
183C23—2021no—
184C23—2021no—
185C23—2021no—
186C23—2021no—
187C23D4×D10111yescharacteristic
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—1012no—
195C2×C4—4048no—
196C2×C4—4041no—
197C2×C4—4041no—
198C2×C4—4041no—
199C2×C4—4041no—
200C2×C4—4041no—
201C2×C4—2022no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2022no—
205C2×C4—2022no—
206C2×C4—2022no—
207C2×C4—2022no—
208C2×C4C2×C4×D10111yescharacteristic
209C2×C4D4×D10111yescharacteristic
210C2×C4C2×C22⋊D10111yescharacteristic
211C2×C4Q8×D10111yescharacteristic
212C2×C4#1310807411yescharacteristic
213C2×C4#1310807511yescharacteristic
214C8—22no—
215C8—2022no—
216C22—1014no—
217C22—2022no—
218C22—2022no—
219C22—2022no—
220C22—2022no—
221C22—2022no—
222C22—2022no—
223C22C4⋊C4×D10111yescharacteristic
224C22#1569222311yescharacteristic
225C22#1569224011yescharacteristic
226C22SD16×D10111yescharacteristic
227C22SD16×D10111yescharacteristic
228C22#1569228211yescharacteristic
229C22#1569228411yescharacteristic
230C4—42no—
231C4—81no—
232C4—8081no—
233C4—4042no—
234C4—2022no—
235C4C4×C22⋊D10111yescharacteristic
236C4#1569221711yescharacteristic
237C4#1569223111yescharacteristic
238C4#1569223311yescharacteristic
239C2—2024no—
240C2#2026504811yescharacteristic
241C2#106×D10111yescharacteristic
242C2#2026521611yescharacteristic
243C2#2026531411yescharacteristic
244C2#2026532611yescharacteristic
245C2#2026549711yescharacteristic
246C2#2026550911yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#106×C101C2
socleC23×C101C2×D4
prime radicalC23×D101D4
cosocleC22×C4×C101C23
hypercenterC2×#106D101
hypocenterC101#1853
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#15692186C2semidirect
5. #20264940C2semidirect
6. #20264939C2semidirect
7. C2×#106×C101C2semidirect
18. #15692167C4semidirect
39. C2×#11717579C2×C4semidirect
51. C2×C8×C101C2×C4semidirect
105. C2×D101#106semidirect
124. C101#1853semidirect
127. C2×#106D101semidirect
169. C2×C8C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1012
C2×#156921862
C2×#156921672
#202650342
#202649402
#202649392
C2×#106×C1012
#1853101