Finite Group Databasebeta

← #27140682

Subgroups — #27140682

Order 12416.

320 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 53 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#20260885C211yesmaximal, characteristic, complemented
2#20260868C211yesmaximal, characteristic, complemented
3#20260644C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic, complemented
6#20260459C211yesmaximal, characteristic, complemented
7#538×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×C22⋊D97—21no—
10C2×#13095498—21no—
11C2×#13095496—21no—
12C2×#13095488—21no—
13#15690730—21no—
14#15690730—21no—
15C22⋊#11715895C2211yescharacteristic
16C22⋊#11715895C2211yescharacteristic
17#15690717C2211yescharacteristic
18#126×C97—21no—
19#124×C97—21no—
20#123×C97—21no—
21C22⋊Q8×C97C2211yescharacteristic
22#121×C97—21no—
23#121×C97C2211yescharacteristic
24#120×C97—21no—
25C4×D4×C97—21no—
26C2×C22⋊C4×C97C2211yescharacteristic
27C23×D97—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—42no—
31C2×C22⋊D97—42no—
32C2×C22⋊D97—42no—
33C2×C22⋊D97—42no—
34C22×Dic97—21no—
35#13095498—42no—
36#13095496—42no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39#13095488—42no—
40C24×C97D411yescharacteristic
41C2×Q8×C97D411yescharacteristic, complemented
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97—21no—
45C2×D4×C97D411yescharacteristic, complemented
46C22×C4×C97—21no—
47C22×C4×C97D411yescharacteristic
48C22×C4×C97C2311yescharacteristic
49C4⋊C4×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97D411yescharacteristic, complemented
52C4⋊C4×C97D411yescharacteristic, complemented
53C22⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97—21no—
56C22⋊C4×C97—21no—
57C22⋊C4×C97—21no—
58C22⋊C4×C97—21no—
59C22⋊C4×C97—21no—
60C42×C97—21no—
61C23×C97—21no—
62C23×C97—21no—
63C23×C97—21no—
64C23×C97—21no—
65C23×C97—41no—
66C23×C97—41no—
67C23×C97C2×D411yescharacteristic
68C22×D97—22no—
69C22×D97—41no—
70C22×D97—41no—
71C22×D97—41no—
72C22×D97—41no—
73C22×D97—41no—
74C22×D97—41no—
75Q8×C97—41no—
76D4×C97—41no—
77D4×C97—41no—
78D4×C97—41no—
79D4×C97—81no—
80D4×C97—81no—
81C2×C4×C97—21no—
82C2×C4×C97—21no—
83C2×C4×C97—21no—
84C2×C4×C97—21no—
85C2×C4×C97—41no—
86C2×C4×C97—41no—
87C2×C4×C97—81no—
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C22⋊D97—48no—
91C2×Dic97—22no—
92C2×Dic97—41no—
93C2×Dic97—41no—
94D388—22no—
95D388—22no—
96D388—41no—
97D388—41no—
98#11715895—41no—
99#11715895—41no—
100C22×C97—21no—
101C22×C97—21no—
102C22×C97—42no—
103C22×C97—41no—
104C22×C97—41no—
105C22×C97—41no—
106C22×C97—41no—
107C22×C97—41no—
108C22×C97—41no—
109C22×C97—81no—
110C22×C97#12011yescharacteristic
111C22×C97#13511yescharacteristic
112C22×C97AGL(1,ℤ/8)11yescharacteristic
113C2×D97—42no—
114C2×D97—42no—
115C2×D97—42no—
116C2×D97—81no—
117C2×D97—81no—
118C2×D97—81no—
119C2×D97—81no—
120C4×C97—21no—
121C4×C97—21no—
122C4×C97—41no—
123C4×C97—41no—
124C4×C97—41no—
125C4×C97—41no—
126Dic97—42no—
127C2×C97—21no—
128C2×C97—21no—
129C2×C97—81no—
130C2×C97—81no—
131C2×C97#44911yescharacteristic
132C2×C97#45311yescharacteristic
133C2×C97B(4,𝔽2)11yescharacteristic
134D97—82no—
135#1600—971nomaximal, Sylow
136C97#160011yescharacteristic, complemented, Sylow
137#538D9711yescharacteristic, complemented
138#450—971no—
139#447—971no—
140#392—971no—
141#328—971no—
142#327—971no—
143#323—971no—
144C22×D4—971no—
145C22×D4—1941no—
146C2×SD16—1941no—
147C2×D8—1941no—
148#126—21no—
149#124—21no—
150#123—21no—
151C22⋊Q8C2×D9711yescharacteristic
152#121—21no—
153#121C2×D9711yescharacteristic
154#120—21no—
155C4×D4—21no—
156C2×C22⋊C4—1941no—
157C2×C22⋊C4C2×D9711yescharacteristic
158#102—1941no—
159#102—1941no—
160C22⋊C8—971no—
161C22⋊C8—971no—
162#95—971no—
163C24—1941no—
164C24C22⋊D9711yescharacteristic
165C2×Q8C22⋊D9711yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—3882no—
170C2×D4—3882no—
171C2×D4—3882no—
172C2×D4—3882no—
173C2×D4—1941no—
174C2×D4—1941no—
175C2×D4C22⋊D9711yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—1941no—
178C22×C4C22⋊D9711yescharacteristic
179C22×C4C22×D9711yescharacteristic
180SD16—3882no—
181D8—3882no—
182C2×C8—1941no—
183C2×C8—1941no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C22⋊D9711yescharacteristic, complemented
187C4⋊C4C22⋊D9711yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—3882no—
196C42—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—3881no—
204C23—3881no—
205C23—3881no—
206C23—3881no—
207C23—3881no—
208C23—3881no—
209C23—1942no—
210C23C2×C22⋊D9711yescharacteristic
211Q8—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—3881no—
218D4—3881no—
219D4—3888no—
220D4—1942no—
221D4—1942no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—3881no—
230C2×C4—3881no—
231C2×C4—1942no—
232C2×C4C2×C22⋊D9711yescharacteristic
233C2×C4C2×C22⋊D9711yescharacteristic
234C8—3881no—
235C8—3881no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—7761no—
247C22—7761no—
248C22—7761no—
249C22—7761no—
250C22—3882no—
251C22—3882no—
252C22—3882no—
253C22#1569077911yescharacteristic
254C22#1569083011yescharacteristic
255C22#1569083511yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—3882no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—81no—
267C2—7762no—
268C2#2026088011yescharacteristic
269C2#2026090311yescharacteristic
270C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#538×C97C2
socleC22×C97#120
hypercenter#538D97
hypocenterC97#1600
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #20260885C2semidirect
2. #20260868C2semidirect
3. #20260644C2semidirect
4. #20260472C2semidirect
5. #20260469C2semidirect
6. #20260459C2semidirect
7. #538×C97C2semidirect
41. C2×Q8×C97D4semidirect
45. C2×D4×C97D4semidirect
51. C4⋊C4×C97D4semidirect
52. C4⋊C4×C97D4semidirect
136. C97#1600semidirect
137. #538D97semidirect
165. C2×Q8C22⋊D97semidirect
175. C2×D4C22⋊D97semidirect
186. C4⋊C4C22⋊D97semidirect
187. C4⋊C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202608852
#202608682
#202606442
#202604722
#202604692
#202604592
#538×C972
#160097