Finite Group Databasebeta

← #27142667

Subgroups — #27142667

Order 12416.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15690797C211yesmaximal, characteristic
2C2×#15690739C211yesmaximal, characteristic, complemented
3C2×#15690735C211yesmaximal, characteristic, complemented
4C2×#15690733C211yesmaximal, characteristic, complemented
5C2×#15690731C211yesmaximal, characteristic
6C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
7C2×#102×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690797—22no—
11#15690797C414yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C22⋊Dic97—21no—
14C2×C22⋊Dic97—21no—
15C2×C4×Dic97—21no—
16C2×C4×Dic97C2211yescharacteristic
17#15690739—22no—
18#15690735—22no—
19#15690733—22no—
20#15690731—22no—
21#15690717—21no—
22C22×D4×C97C2211yescharacteristic
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25#102×C97—22no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194C2×C412yes—
32C22×Dic97—21no—
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C22×Dic97D411yescharacteristic
36C2×#11715895—21no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97D412yes—
41C22⋊Dic97—24no—
42C22⋊Dic97—24no—
43C4×Dic97—42no—
44C4×Dic97C2×C412yes—
45C4×Dic97D412yes—
46C24×C97—21no—
47C2×D4×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97C2×C412yes—
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55C2×C8×C97—21no—
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C4⋊C4×C97—21no—
59C4⋊C4×C97D412yes—
60C23×C97—22no—
61C23×C97—41no—
62C23×C97—41no—
63C23×C97—41no—
64C23×C97—41no—
65C23×C97—41no—
66C23×C97—41no—
67C23×C97C2×D411yescharacteristic
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—22no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97C22×C411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97#4111yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×Dic97—21no—
82C2×Dic97—21no—
83C2×Dic97—21no—
84C2×Dic97—21no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97C22⋊C412yes—
92C2×Dic97C22⋊C412yes—
93Dic194—22no—
94Dic194—22no—
95Dic194—41no—
96Dic194—41no—
97C8×C97—41no—
98#11715895—41no—
99C22×C97—42no—
100C22×C97—42no—
101C22×C97—42no—
102C22×C97—81no—
103C22×C97—81no—
104C22×C97—81no—
105C22×C97—81no—
106C22×C97C2×C22⋊C411yescharacteristic
107C22×C97#12111yescharacteristic
108C22×C97#12111yescharacteristic
109C22×C97#13511yescharacteristic
110C22×C97#13511yescharacteristic
111C22×C97AGL(1,ℤ/8)11yescharacteristic
112C22×C97#13711yescharacteristic
113C4×C97—42no—
114C4×C97C4×D411yescharacteristic
115C4×C97C4×D411yescharacteristic
116C4×C97#12411yescharacteristic
117C4×C97#12711yescharacteristic
118Dic97—22no—
119Dic97—22no—
120Dic97—42no—
121Dic97—42no—
122C2×C97—82no—
123C2×C97#39011yescharacteristic
124C2×C97#41611yescharacteristic
125C2×C97#41711yescharacteristic
126C2×C97#46311yescharacteristic
127C2×C97#46311yescharacteristic
128C2×C97#46411yescharacteristic
129C2×C97#46411yescharacteristic
130#1983—971nomaximal, Sylow
131C97#198311yescharacteristic, complemented, Sylow
132C2×#124—971no—
133C2×C4⋊C8—971no—
134C2×#103—971no—
135C2×#103—971no—
136C2×#102—971no—
137C2×#102D9711yescharacteristic
138C4×C4⋊C4—971no—
139C22×Q8—971no—
140C22×D4C2×D9711yescharacteristic
141C22×C8—971no—
142C22×C8C2×D9711yescharacteristic
143#124—974no—
144#124—1942no—
145C2×C4⋊C4—971no—
146C2×C4⋊C4C2×D9711yescharacteristic
147C2×C22⋊C4—1941no—
148C2×C22⋊C4—1941no—
149C2×C42—971no—
150C2×C42—1941no—
151C4⋊C8—1942no—
152#103—1942no—
153#103—1942no—
154#102—22no—
155#102—1942no—
156#95—1941no—
157C24—21no—
158C2×Q8—972no—
159C2×Q8—1941no—
160C2×Q8—1941no—
161C2×Q8—1941no—
162C2×Q8—1941no—
163C2×Q8—1941no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4C4×D9712yes—
170C22×C4—21no—
171C22×C4—971no—
172C22×C4—971no—
173C22×C4—1941no—
174C22×C4—1941no—
175C22×C4C22×D9711yescharacteristic
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—1941no—
180C2×C8—1941no—
181C2×C8—1941no—
182C4⋊C4—21no—
183C4⋊C4—972no—
184C4⋊C4—1941no—
185C4⋊C4C22⋊D9712yes—
186C22⋊C4—1944no—
187C22⋊C4—1944no—
188C42—972no—
189C42—972no—
190C42—3882no—
191C23—22no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23D4×D9711yescharacteristic
199Q8—3881no—
200Q8—3881no—
201Q8—1942no—
202Q8—1942no—
203D4—22no—
204D4—22no—
205D4—41no—
206D4—41no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—972no—
211C2×C4—972no—
212C2×C4—3881no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—3881no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4—1941no—
221C2×C4—1941no—
222C2×C4C2×C4×D9711yescharacteristic
223C2×C4D4×D9711yescharacteristic
224C2×C4D4×D9711yescharacteristic
225C2×C4C2×C22⋊D9711yescharacteristic
226C2×C4#1309551011yescharacteristic
227C2×C4#1309551111yescharacteristic
228C8—41no—
229C8—3881no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22C22⋊C4×D9711yescharacteristic
238C22#1569078211yescharacteristic
239C22#1569078311yescharacteristic
240C22#1569082911yescharacteristic
241C22#1569083111yescharacteristic
242C22#1569083411yescharacteristic
243C22#1569083911yescharacteristic
244C4—42no—
245C4—3882no—
246C4—3882no—
247C4—1942no—
248C4—1942no—
249C4#1569077011yescharacteristic
250C4C4×C22⋊D9711yescharacteristic
251C4#1569079611yescharacteristic
252C4#1569080611yescharacteristic
253C2—82no—
254C2#2026063511yescharacteristic
255C2#2026072811yescharacteristic
256C2#2026073311yescharacteristic
257C2#2026094811yescharacteristic
258C2#2026094911yescharacteristic
259C2#2026095511yescharacteristic
260C2#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#1983
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×#15690739C2semidirect
3. C2×#15690735C2semidirect
4. C2×#15690733C2semidirect
6. C4⋊C4×Dic97C2semidirect
11. #15690797C4semidirect
131. C97#1983semidirect

Maximal subgroups

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

subgroupindex
C2×#156907972
C2×#156907392
C2×#156907352
C2×#156907332
C2×#156907312
C4⋊C4×Dic972
C2×#102×C972
#198397