Finite Group Databasebeta

← #27142088

Subgroups — #27142088

Order 12416.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#13095499C211yesmaximal, characteristic
2C2×#15690775C211yesmaximal, characteristic, complemented
3C2×#15690735C211yesmaximal, characteristic, complemented
4#20260638C211yesmaximal, characteristic
5#20260497C211yesmaximal, characteristic, complemented
6#20260489C211yesmaximal, characteristic, complemented
7C2×#102×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095499—22no—
10C2×#13095499—22no—
11C2×#13095499C414yescomplemented
12C22×#11715895C2211yescharacteristic
13#15690775—22no—
14#15690775—22no—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—21no—
17C2×C22⋊Dic97—21no—
18C2×C22⋊Dic97—21no—
19C2×C4×Dic97—21no—
20C2×C4×Dic97C2211yescharacteristic
21#15690735—22no—
22C22×D4×C97C2211yescharacteristic
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25#102×C97—22no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C412yes—
30C2×Dic194D412yes—
31C2×Dic194D412yes—
32C22×Dic97—21no—
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C22×Dic97D411yescharacteristic
36#13095499—28no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C2×#11715895C2×C412yes—
40C4⋊Dic97—21no—
41C4⋊Dic97—21no—
42#13095490—24no—
43C22⋊Dic97—42no—
44C22⋊Dic97—42no—
45C4×Dic97—24no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C24×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97—21no—
53C2×D4×C97—21no—
54C2×D4×C97C2×C412yes—
55C22×C4×C97—21no—
56C22×C4×C97C2311yescharacteristic
57C2×C8×C97—21no—
58C2×C8×C97—21no—
59C2×C8×C97—21no—
60C4⋊C4×C97—21no—
61C4⋊C4×C97D412yes—
62C23×C97—22no—
63C23×C97—41no—
64C23×C97—41no—
65C23×C97—41no—
66C23×C97—41no—
67C23×C97—41no—
68C23×C97—41no—
69C23×C97C2×D411yescharacteristic
70D4×C97—22no—
71D4×C97—22no—
72D4×C97—41no—
73D4×C97—41no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97C22×C411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C2×Dic97—22no—
84C2×Dic97—21no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97—21no—
91C2×Dic97—41no—
92C2×Dic97—41no—
93C2×Dic97—41no—
94C2×Dic97SD1612yescomplemented
95Dic194—22no—
96Dic194—22no—
97Dic194—22no—
98Dic194C22⋊C414yescomplemented
99C8×C97—41no—
100#11715895—22no—
101C22×C97—42no—
102C22×C97—42no—
103C22×C97—42no—
104C22×C97—81no—
105C22×C97—81no—
106C22×C97—81no—
107C22×C97—81no—
108C22×C97C4×D411yescharacteristic
109C22×C97#12011yescharacteristic
110C22×C97#12111yescharacteristic
111C22×C97C2×SD1611yescharacteristic
112C22×C97#13511yescharacteristic
113C22×C97AGL(1,ℤ/8)11yescharacteristic
114C22×C97#13711yescharacteristic
115C4×C97—42no—
116C4×C97C2×C22⋊C411yescharacteristic
117C4×C97C4×D411yescharacteristic
118C4×C97#12111yescharacteristic
119C4×C97#12311yescharacteristic
120Dic97—24no—
121Dic97—22no—
122Dic97—41no—
123Dic97—81no—
124C2×C97—82no—
125C2×C97#38611yescharacteristic
126C2×C97C4×SD1611yescharacteristic
127C2×C97#44011yescharacteristic
128C2×C97#44811yescharacteristic
129C2×C97#44911yescharacteristic
130C2×C97#46011yescharacteristic
131C2×C97#46411yescharacteristic
132#1880—971nomaximal, Sylow
133C97#188011yescharacteristic, complemented, Sylow
134C22×SD16—971no—
135C2×C4×Q8—971no—
136C2×#103—971no—
137C2×#102D9711yescharacteristic
138#390—971no—
139#340—971no—
140#336—971no—
141C22×Q8—971no—
142C22×D4C2×D9711yescharacteristic
143C2×SD16—974no—
144C2×SD16—1942no—
145C2×SD16—1942no—
146C22×C8—971no—
147C22×C8C2×D9711yescharacteristic
148C4×Q8—1942no—
149C4×Q8—1942no—
150C2×C4⋊C4—971no—
151C2×C4⋊C4—1941no—
152C2×C4⋊C4C2×D9711yescharacteristic
153C2×C22⋊C4—1941no—
154C2×C22⋊C4—1941no—
155C2×C42—971no—
156C2×C42—1941no—
157#103—1942no—
158#102—22no—
159C24—21no—
160C2×Q8—972no—
161C2×Q8—972no—
162C2×Q8—972no—
163C2×Q8—1941no—
164C2×Q8—1941no—
165C2×Q8—1941no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C4×D9712yes—
172C22×C4—21no—
173C22×C4—971no—
174C22×C4—971no—
175C22×C4—1941no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178SD16—1948no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—972no—
183C2×C8—1941no—
184C2×C8—1941no—
185C4⋊C4—21no—
186C4⋊C4—1944no—
187C4⋊C4—1941no—
188C4⋊C4—1941no—
189C4⋊C4C22⋊D9712yes—
190C22⋊C4—3882no—
191C22⋊C4—3882no—
192C42—1944no—
193C42—1941no—
194C42—1941no—
195C23—22no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23D4×D9711yescharacteristic
203Q8—974no—
204Q8—1942no—
205Q8—1942no—
206Q8—1942no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—972no—
215C2×C4—3881no—
216C2×C4—3881no—
217C2×C4—3881no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4—1941no—
221C2×C4—1942no—
222C2×C4—1941no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4—1941no—
226C2×C4C2×C4×D9711yescharacteristic
227C2×C4D4×D9711yescharacteristic
228C2×C4D4×D9711yescharacteristic
229C2×C4C2×C22⋊D9711yescharacteristic
230C2×C4#1309551011yescharacteristic
231C2×C4#1309551111yescharacteristic
232C8—41no—
233C8—1942no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22#1569077011yescharacteristic
242C22#1569077811yescharacteristic
243C22#1569078311yescharacteristic
244C22SD16×D9711yescharacteristic
245C22#1569082911yescharacteristic
246C22#1569083411yescharacteristic
247C22#1569083911yescharacteristic
248C4—42no—
249C4—7761no—
250C4—3881no—
251C4—1944no—
252C4—1942no—
253C4C22⋊C4×D9711yescharacteristic
254C4C4×C22⋊D9711yescharacteristic
255C4#1569078111yescharacteristic
256C4#1569079211yescharacteristic
257C2—82no—
258C2#2026061111yescharacteristic
259C2#2026082611yescharacteristic
260C2#2026083211yescharacteristic
261C2#2026087211yescharacteristic
262C2#2026087711yescharacteristic
263C2#2026093411yescharacteristic
264C2#2026095611yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
prime radicalC22×D4×C97C22
cosocleC22×C4×C97C23
hypercenterC2×#102D97
hypocenterC97#1880
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#15690775C2semidirect
3. C2×#15690735C2semidirect
5. #20260497C2semidirect
6. #20260489C2semidirect
11. C2×#13095499C4semidirect
94. C2×Dic97SD16semidirect
98. Dic194C22⋊C4semidirect
133. C97#1880semidirect

Maximal subgroups

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

subgroupindex
C22×#130954992
C2×#156907752
C2×#156907352
#202606382
#202604972
#202604892
C2×#102×C972
#188097