Finite Group Databasebeta

← #27142974

Subgroups — #27142974

Order 12416.

318 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15690799C211yesmaximal, characteristic
2C2×#15690734C211yesmaximal, characteristic, complemented
3C2×#15690732C211yesmaximal, characteristic
4C2×#15690730C211yesmaximal, characteristic
5#20260634C211yesmaximal, characteristic, complemented
6#20260489C211yesmaximal, characteristic, complemented
7C2×#103×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690799—22no—
11#15690799—22no—
12C2×C4⋊Dic97—21no—
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095488—21no—
15C2×#13095488—21no—
16C2×C4×Dic97C2211yescharacteristic
17#15690734—22no—
18#15690732—22no—
19#15690730—22no—
20#15690717—21no—
21C22×Q8×C97C2211yescharacteristic
22C22×C8×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24#103×C97—22no—
25C23×D97—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388D412yes—
32C22×Dic97—21no—
33C22×Dic97D411yescharacteristic
34C22×Dic97D411yescharacteristic
35C2×#11715895—21no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C4⋊Dic97—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97—42no—
41#13095488—24no—
42#13095488—24no—
43C4×Dic97—21no—
44C4×Dic97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97D412yes—
51C22×C4×C97—21no—
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54C2×C8×C97—21no—
55C2×C8×C97—21no—
56C2×C8×C97—21no—
57C4⋊C4×C97—21no—
58C4⋊C4×C97D412yes—
59C23×C97C2×D411yescharacteristic
60C22×D97—22no—
61C22×D97—41no—
62C22×D97—41no—
63C22×D97—41no—
64C22×D97—41no—
65C22×D97—41no—
66C22×D97—41no—
67Q8×C97—22no—
68Q8×C97—22no—
69Q8×C97—41no—
70Q8×C97—41no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C2×Dic97—21no—
84C2×Dic97—21no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97—41no—
92D388—22no—
93D388—22no—
94D388—41no—
95D388—41no—
96C8×C97—41no—
97#11715895—41no—
98C22×C97#12011yescharacteristic
99C22×C97#12111yescharacteristic
100C22×C97#12411yescharacteristic
101C22×C97#13511yescharacteristic
102C22×C97#13511yescharacteristic
103C22×C97AGL(1,ℤ/8)11yescharacteristic
104C22×C97#13711yescharacteristic
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—81no—
109C2×D97—81no—
110C2×D97—81no—
111C2×D97—81no—
112C4×C97—42no—
113C4×C97—42no—
114C4×C97#12111yescharacteristic
115C4×C97#12111yescharacteristic
116C4×C97#12311yescharacteristic
117C4×C97#12311yescharacteristic
118Dic97—41no—
119Dic97—41no—
120Dic97—81no—
121C2×C97#39411yescharacteristic
122C2×C97#44911yescharacteristic
123C2×C97#45211yescharacteristic
124C2×C97#46311yescharacteristic
125C2×C97#46411yescharacteristic
126C2×C97#48811yescharacteristic
127C2×C97#48911yescharacteristic
128D97—82no—
129#2042—971nomaximal, Sylow
130C97#204211yescharacteristic, complemented, Sylow
131C2×#124—971no—
132C2×#103—971no—
133C2×#103D9711yescharacteristic, complemented
134C2×#102—971no—
135C2×#102—971no—
136#389—971no—
137#336—971no—
138C22×Q8C2×D9711yescharacteristic
139C22×D4—971no—
140C22×C8—971no—
141C22×C8C2×D9711yescharacteristic
142#124—1942no—
143#124—1942no—
144C2×C4⋊C4—971no—
145C2×C4⋊C4—1941no—
146C2×C4⋊C4C2×D9711yescharacteristic
147C2×C22⋊C4—1941no—
148C2×C22⋊C4—1941no—
149C2×C42—971no—
150#103—22no—
151#103—1942no—
152#102—1942no—
153#102—1942no—
154#95—1941no—
155C24—1941no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8D38812yes—
162C2×D4—972no—
163C2×D4—1941no—
164C2×D4—1941no—
165C2×D4—1941no—
166C2×D4—1941no—
167C2×D4—1941no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—971no—
171C22×C4—971no—
172C22×C4—1941no—
173C22×C4C22×D9711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—1941no—
178C2×C8—1941no—
179C2×C8—1941no—
180C4⋊C4—21no—
181C4⋊C4—3882no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4C22⋊D9712yes—
185C22⋊C4—1944no—
186C22⋊C4—1944no—
187C42—1941no—
188C42—1941no—
189C23—3881no—
190C23—3881no—
191C23—3881no—
192C23—3881no—
193C23—3881no—
194C23—3881no—
195C23—1942no—
196C23D4×D9711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—3881no—
202D4—3881no—
203D4—1942no—
204D4—1942no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—3881no—
212C2×C4—3881no—
213C2×C4—3881no—
214C2×C4—1941no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4—1941no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4D4×D9711yescharacteristic
221C2×C4C2×C22⋊D9711yescharacteristic
222C2×C4C2×D38811yescharacteristic
223C2×C4#1309551011yescharacteristic
224C2×C4#1309551011yescharacteristic
225C2×C4#1309551111yescharacteristic
226C8—41no—
227C8—3881no—
228C22—7761no—
229C22—7761no—
230C22—7761no—
231C22—7761no—
232C22—3882no—
233C22—3882no—
234C22—3882no—
235C22#1569078011yescharacteristic
236C22#1569078211yescharacteristic
237C22#1569079611yescharacteristic
238C22#1569083111yescharacteristic
239C22#1569083311yescharacteristic
240C22#1569083711yescharacteristic
241C22#1569084211yescharacteristic
242C4—42no—
243C4—42no—
244C4—7761no—
245C4—3881no—
246C4—3881no—
247C4#1569078111yescharacteristic
248C4#1569078411yescharacteristic
249C4#1569079211yescharacteristic
250C4#1569079411yescharacteristic
251C2—7762no—
252C2#2026065111yescharacteristic
253C2#2026088311yescharacteristic
254C2#2026090011yescharacteristic
255C2#2026095211yescharacteristic
256C2#2026095911yescharacteristic
257C2#2026109111yescharacteristic
258C2#2026109611yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
hypercenterC2×#103D97
hypocenterC97#2042
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#15690734C2semidirect
5. #20260634C2semidirect
6. #20260489C2semidirect
7. C2×#103×C97C2semidirect
130. C97#2042semidirect
133. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
C2×#156907342
C2×#156907322
C2×#156907302
#202606342
#202604892
C2×#103×C972
#204297