Finite Group Databasebeta

← #27140185

Subgroups — #27140185

Order 12416.

276 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 75 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#20261282C211yesmaximal, characteristic
2#20260757C211yesmaximal, characteristic, complemented
3C4×C8×D97C211yesmaximal, characteristic, complemented
4#20260464C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095511—21no—
9C2×#13095494—21no—
10C2×C8×D97—21no—
11#15690809C411yescharacteristic
12#15690806C411yescharacteristic
13#15690796—21no—
14#15690785C411yescharacteristic
15#15690783—21no—
16#15690781C411yescharacteristic
17C4×D388C2211yescharacteristic
18C4×C22⋊D97—21no—
19C42×D97C2211yescharacteristic
20C4⋊#11715895C2211yescharacteristic
21#15690739—21no—
22C4×#11715895C2211yescharacteristic
23C8×Dic97—21no—
24C4×D4×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27#13095511—42no—
28#13095511—42no—
29C2×Dic194—21no—
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39#13095494—42no—
40C2×#11715895—21no—
41C2×#11715895—21no—
42C8×D97—24no—
43C4⋊Dic97C2×C411yescharacteristic
44#13095490—21no—
45C22⋊Dic97—21no—
46#13095488—21no—
47#13095488—21no—
48C4×Dic97C2×C411yescharacteristic
49C4×Dic97D411yescharacteristic
50C4×Dic97D411yescharacteristic
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C2×C8×C97—21no—
54C2×C8×C97—21no—
55C4⋊C4×C97C2×C411yescharacteristic
56C22⋊C4×C97—21no—
57C42×C97C2311yescharacteristic
58C23×C97—21no—
59C22×D97—21no—
60C22×D97C22⋊C411yescharacteristic
61D4×C97—21no—
62D4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C22⋊D97—42no—
70C22⋊D97—42no—
71C22⋊D97—42no—
72C2×Dic97—21no—
73C2×Dic97C22⋊C411yescharacteristic
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C22⋊C411yescharacteristic
76D388—21no—
77D388—21no—
78D388—42no—
79C4×D97—21no—
80C4×D97—21no—
81C4×D97—22no—
82C4×D97—41no—
83C4×D97—41no—
84C4×D97D812yes—
85C4×D97SD1612yes—
86Dic194—42no—
87C8×C97—22no—
88C8×C97—41no—
89#11715895—22no—
90#11715895—41no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97C2×C22⋊C411yescharacteristic
95C2×D97—21no—
96C2×D97—21no—
97C2×D97—41no—
98C2×D97—41no—
99C2×D97—41no—
100C2×D97#10212yes—
101C4×C97—21no—
102C4×C97—41no—
103C4×C97#13111yescharacteristic
104C4×C97#13111yescharacteristic
105C4×C97C2×D811yescharacteristic
106C4×C97C2×SD1611yescharacteristic
107Dic97—21no—
108Dic97—41no—
109Dic97#10212yes—
110Dic97#10412yes—
111C2×C97—81no—
112C2×C97#40811yescharacteristic
113C2×C97C2×#10211yescharacteristic
114C2×C97C2×#10411yescharacteristic
115D97—22no—
116D97—81no—
117#1492—971nomaximal, Sylow
118C97#149211yescharacteristic, complemented, Sylow
119#532—971no—
120#423—971no—
121C2×C4×C8—971no—
122D4⋊C8—971no—
123D4⋊C8—971no—
124D4⋊C8—971no—
125D4⋊C8D9711yescharacteristic, complemented
126C2×#41—1941no—
127C2×M16—1941no—
128C22×C8—1941no—
129C4⋊Q8—971no—
130#127—971no—
131#124—1941no—
132#121—971no—
133#121—971no—
134#121—1941no—
135C4×D4—971no—
136C4×D4—1941no—
137C4×D4C2×D9711yescharacteristic
138C2×C42—971no—
139C4⋊C8—971no—
140C4⋊C8—1941no—
141C4⋊C8C2×D9711yescharacteristic
142C4×C8—971no—
143C4×C8—1941no—
144C4×C8C2×D9711yescharacteristic
145#41—3882no—
146#41—3882no—
147C2×Q8—1941no—
148C2×D4—971no—
149C2×D4—1941no—
150C2×D4—1941no—
151C2×D4—1941no—
152C2×D4—1941no—
153C2×D4C4×D9711yescharacteristic
154C22×C4—21no—
155C22×C4—971no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—1941no—
159M16—3882no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—1944no—
163C2×C8—1941no—
164C2×C8—1941no—
165C4⋊C4—971no—
166C4⋊C4—1941no—
167C4⋊C4C4×D9711yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—1941no—
170C22⋊C4—1941no—
171C22⋊C4—1941no—
172C42—971no—
173C42—971no—
174C42—971no—
175C42C22×D9711yescharacteristic
176C23—21no—
177C23—971no—
178C23—1941no—
179Q8—3882no—
180D4—21no—
181D4—21no—
182D4—3882no—
183D4—3882no—
184D4—3882no—
185D4—3882no—
186D4—1941no—
187D4—1941no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—971no—
192C2×C4—971no—
193C2×C4—972no—
194C2×C4—972no—
195C2×C4—971no—
196C2×C4—3881no—
197C2×C4—3881no—
198C2×C4—1942no—
199C2×C4—1941no—
200C2×C4—1941no—
201C2×C4—1941no—
202C2×C4C2×C4×D9711yescharacteristic
203C2×C4D4×D9711yescharacteristic
204C2×C4D4×D9711yescharacteristic
205C8—22no—
206C8—41no—
207C8—3881no—
208C8—1942no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—972no—
213C22—3881no—
214C22—3881no—
215C22—3881no—
216C22—1941no—
217C22—1941no—
218C22C22⋊C4×D9711yescharacteristic
219C4—21no—
220C4—41no—
221C4—972no—
222C4—972no—
223C4—3881no—
224C4—1941no—
225C4#1569081611yescharacteristic
226C4#1569081811yescharacteristic
227C4D8×D9711yescharacteristic
228C4SD16×D9711yescharacteristic
229C2—81no—
230C2—7761no—
231C2—1942no—
232C2#2026070011yescharacteristic
233C2#102×D9711yescharacteristic
234C2#104×D9711yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC42×C97C23
hypercenterD4⋊C8D97
hypocenterC97#1492
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260757C2semidirect
3. C4×C8×D97C2semidirect
4. #20260464C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. D4⋊C8×C97C2semidirect
118. C97#1492semidirect
125. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612822
#202607572
C4×C8×D972
#202604642
D4⋊#117158952
#202604622
D4⋊C8×C972
#149297