Finite Group Databasebeta

← #27141944

Subgroups — #27141944

Order 12416.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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#20261209C211yesmaximal, characteristic
2#20260775C211yesmaximal, characteristic, complemented
3C2×#15690726C211yesmaximal, characteristic
4#20260636C211yesmaximal, characteristic
5#20260496C211yesmaximal, characteristic, complemented
6#20260494C211yesmaximal, characteristic, complemented
7#428×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C4×D388—21no—
11C4×D388—21no—
12#15690772—21no—
13#15690772—21no—
14#15690769C2211yescharacteristic
15C2×#13095488—22no—
16C22⋊C4×D97—22no—
17C2×C4×Dic97C2211yescharacteristic
18#15690743—21no—
19#15690741—21no—
20#15690726C414yescomplemented
21C2×M16×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24#107×C97—21no—
25#106×C97—21no—
26C23×D97D412yes—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388C2×C412yes—
32C2×D388Q812yes—
33C22×Dic97—21no—
34C2×C4×D97—22no—
35C2×C4×D97—22no—
36#13095495—21no—
37#13095495C2×C412yescomplemented
38C2×#11715895—21no—
39C4⋊Dic97D411yescharacteristic
40C4⋊Dic97D411yescharacteristic
41C22⋊Dic97—21no—
42#13095488—22no—
43#13095488—22no—
44#13095488—44no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C22×C4×C97—21no—
49C22×C4×C97C2311yescharacteristic
50M16×C97—21no—
51M16×C97C2×C412yescomplemented
52C2×C8×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97—21no—
55C4⋊C4×C97D411yescharacteristic
56C4⋊C4×C97D411yescharacteristic
57C22⋊C4×C97—21no—
58C42×C97—21no—
59C23×C97C2×D411yescharacteristic
60C22×D97—22no—
61C22×D97—22no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97C4⋊C414yes—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—42no—
68C2×C4×C97—41no—
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×Q811yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×Dic97—22no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80D388—22no—
81D388—22no—
82D388—22no—
83D388—22no—
84C4×D97—82no—
85C4×D97—82no—
86C8×C97—22no—
87#11715895—22no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97C2×C4⋊C411yescharacteristic
91C22×C97#12011yescharacteristic
92C22×C97#12311yescharacteristic
93C2×D97—24no—
94C2×D97—42no—
95C2×D97—42no—
96C2×D97—44no—
97C2×D97—82no—
98C4×C97—41no—
99C4×C97—41no—
100C4×C97—81no—
101C4×C97C4×D411yescharacteristic
102C4×C97C4×D411yescharacteristic
103C4×C97C22⋊Q811yescharacteristic
104C4×C97C22⋊Q811yescharacteristic
105Dic97—42no—
106Dic97—41no—
107Dic97—41no—
108C2×C97—21no—
109C2×C97—21no—
110C2×C97#38511yescharacteristic
111C2×C97#45311yescharacteristic
112C2×C97#45511yescharacteristic
113D97—44no—
114#1856—971nomaximal, Sylow
115C97#185611yescharacteristic, complemented, Sylow
116#518—971no—
117#428—971no—
118#428D9711yescharacteristic, complemented
119C2×#100—971no—
120#390—971no—
121#339—971no—
122#339—971no—
123C22×D4—971no—
124C2×M16—971no—
125C2×M16C2×D9711yescharacteristic
126C4×D4—1941no—
127C4×D4—1941no—
128C4×D4—1941no—
129C4×D4—1941no—
130#117—971no—
131#117C2×D9711yescharacteristic
132C2×C4⋊C4C2×D9711yescharacteristic
133C2×C22⋊C4—1942no—
134C2×C22⋊C4—1942no—
135C2×C42—971no—
136#107—21no—
137#107—1941no—
138#106—21no—
139#106—1941no—
140#100—974no—
141C24—972no—
142C2×D4—972no—
143C2×D4—972no—
144C2×D4—1941no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4—1941no—
148C22×C4—21no—
149C22×C4—1942no—
150C22×C4—1942no—
151C22×C4—1941no—
152C22×C4C22×D9711yescharacteristic
153M16—21no—
154M16—972no—
155M16—1941no—
156M16C4×D9712yescomplemented
157C2×C8—21no—
158C2×C8—1941no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—971no—
162C4⋊C4—971no—
163C4⋊C4C22⋊D9711yescharacteristic
164C4⋊C4C22⋊D9711yescharacteristic
165C22⋊C4—21no—
166C22⋊C4—3884no—
167C22⋊C4—1942no—
168C22⋊C4—1942no—
169C22⋊C4—1941no—
170C42—21no—
171C42—1941no—
172C42—1941no—
173C42—1941no—
174C23—974no—
175C23—3882no—
176C23—3882no—
177C23—1942no—
178C23—1942no—
179C23D4×D9711yescharacteristic
180D4—1942no—
181D4—1942no—
182D4—1942no—
183D4—1942no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—7762no—
189C2×C4—7762no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—1942no—
193C2×C4—1941no—
194C2×C4—1941no—
195C2×C4C2×C4×D9711yescharacteristic
196C2×C4D4×D9711yescharacteristic
197C2×C4C2×C22⋊D9711yescharacteristic
198C2×C4Q8×D9711yescharacteristic
199C2×C4#1309551111yescharacteristic
200C2×C4#1309551211yescharacteristic
201C8—22no—
202C8—1942no—
203C22—21no—
204C22—21no—
205C22—7762no—
206C22—3884no—
207C22—3882no—
208C22—3882no—
209C22—1944no—
210C22C4⋊C4×D9711yescharacteristic
211C22#1569077811yescharacteristic
212C22#1569079511yescharacteristic
213C4—41no—
214C4—41no—
215C4—81no—
216C4—3882no—
217C4—3881no—
218C4—3881no—
219C4C4×C22⋊D9711yescharacteristic
220C4#1569077211yescharacteristic
221C4#1569078611yescharacteristic
222C4#1569078811yescharacteristic
223C2—21no—
224C2—21no—
225C2—3884no—
226C2#2026060611yescharacteristic
227C2#2026090111yescharacteristic
228C2#2026091111yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#428×C97C2
socleC22×C97#120
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenter#428D97
hypocenterC97#1856
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #20260775C2semidirect
5. #20260496C2semidirect
6. #20260494C2semidirect
7. #428×C97C2semidirect
20. #15690726C4semidirect
37. #13095495C2×C4semidirect
51. M16×C97C2×C4semidirect
115. C97#1856semidirect
118. #428D97semidirect
156. M16C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202612092
#202607752
C2×#156907262
#202606362
#202604962
#202604942
#428×C972
#185697