Finite Group Databasebeta

← #27142653

Subgroups — #27142653

Order 12416.

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

normal subgroups only

This group has 336 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
1#20261281C211yesmaximal, characteristic
2C2×#15690738C211yesmaximal, characteristic, complemented
3C2×#15690737C211yesmaximal, characteristic, complemented
4C2×#15690736C211yesmaximal, characteristic, complemented
5M16×Dic97C211yesmaximal, characteristic, complemented
6C4×#13095490C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690808C411yescharacteristic, complemented
11#15690806C411yescharacteristic, complemented
12#15690799C411yescharacteristic, complemented
13#15690797C411yescharacteristic, complemented
14#15690782—21no—
15#15690782—21no—
16C4×C22⋊D97—21no—
17C4×C22⋊D97—21no—
18C2×#13095490—21no—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2211yescharacteristic
21#15690738—22no—
22#15690737—22no—
23#15690736—22no—
24#15690720—21no—
25C8×Dic97—21no—
26#15690717—21no—
27C2×#41×C97C2211yescharacteristic
28C2×M16×C97C2211yescharacteristic
29C2×C42×C97C2211yescharacteristic
30#104×C97—22no—
31#13095511—21no—
32#13095511—21no—
33#13095511—21no—
34#13095511—21no—
35C2×Dic194C2×C411yescharacteristic
36C2×D388C2×C411yescharacteristic
37C2×C22⋊D97—21no—
38C2×C22⋊D97—21no—
39C2×C22⋊D97—21no—
40C22×Dic97—21no—
41C22×Dic97D411yescharacteristic
42C22×Dic97D411yescharacteristic
43C2×C4×D97—21no—
44#13095495—21no—
45#13095495D412yescomplemented
46C2×#11715895—21no—
47#13095490—24no—
48#13095490—21no—
49#13095490—21no—
50C22⋊Dic97—21no—
51C22⋊Dic97—21no—
52#13095488—21no—
53#13095488—21no—
54C4×Dic97—21no—
55C4×Dic97—21no—
56C4×Dic97C2×C411yescharacteristic
57C4×Dic97C2×C411yescharacteristic
58C4×Dic97D411yescharacteristic
59C4×Dic97D411yescharacteristic
60#41×C97—21no—
61#41×C97—21no—
62#41×C97—21no—
63#41×C97—21no—
64C2×Q8×C97C2×C411yescharacteristic
65C2×D4×C97—21no—
66C2×D4×C97C2×C411yescharacteristic
67C22×C4×C97—21no—
68C22×C4×C97—21no—
69C22×C4×C97C2311yescharacteristic
70M16×C97—21no—
71M16×C97D412yescomplemented
72C2×C8×C97—21no—
73C42×C97—21no—
74C42×C97—21no—
75C23×C97—21no—
76C23×C97C2×D411yescharacteristic
77C22×D97—21no—
78Q8×C97—21no—
79Q8×C97—21no—
80D4×C97—21no—
81D4×C97—21no—
82D4×C97—41no—
83D4×C97—41no—
84C2×C4×C97—22no—
85C2×C4×C97—21no—
86C2×C4×C97—41no—
87C2×C4×C97—41no—
88C2×C4×C97—41no—
89C2×C4×C97—41no—
90C2×C4×C97C22×C411yescharacteristic
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97#4111yescharacteristic
95C2×C4×C97#4111yescharacteristic
96C22⋊D97—42no—
97C22⋊D97—41no—
98C22⋊D97—41no—
99C22⋊D97—42no—
100C2×Dic97—22no—
101C2×Dic97—22no—
102C2×Dic97—21no—
103C2×Dic97—41no—
104C2×Dic97—41no—
105C2×Dic97—41no—
106C2×Dic97C22⋊C411yescharacteristic
107C2×Dic97C22⋊C411yescharacteristic
108C2×Dic97C22⋊C411yescharacteristic
109C2×Dic97C22⋊C411yescharacteristic
110D388—21no—
111D388—21no—
112C4×D97—41no—
113C4×D97—41no—
114Dic194—21no—
115Dic194—21no—
116C8×C97—22no—
117#11715895—22no—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97—41no—
121C22×C97—41no—
122C22×C97—41no—
123C22×C97C2×C22⋊C411yescharacteristic
124C22×C97#12411yescharacteristic
125C22×C97#12711yescharacteristic
126C2×D97—41no—
127C2×D97—41no—
128C2×D97—41no—
129C4×C97—42no—
130C4×C97—41no—
131C4×C97C4×D411yescharacteristic
132C4×C97C4×D411yescharacteristic
133C4×C97#12111yescharacteristic
134C4×C97#12111yescharacteristic
135Dic97—21no—
136Dic97—21no—
137Dic97—21no—
138Dic97—42no—
139Dic97—41no—
140Dic97#10412yescomplemented
141C2×C97—21no—
142C2×C97—21no—
143C2×C97—81no—
144C2×C97#39011yescharacteristic
145C2×C97C2×#10411yescharacteristic
146C2×C97#42111yescharacteristic
147D97—81no—
148#1980—971nomaximal, Sylow
149C97#198011yescharacteristic, complemented, Sylow
150#532—971no—
151C2×#104—971no—
152C2×#104—971no—
153C2×#104—971no—
154C2×#104D9711yescharacteristic, complemented
155C4×M16—971no—
156C4×C4⋊C4—971no—
157C2×#41—971no—
158C2×#41C2×D9711yescharacteristic
159C2×M16—971no—
160C2×M16C2×D9711yescharacteristic
161C4⋊Q8—971no—
162#127—971no—
163#124—971no—
164#124—971no—
165#121—1941no—
166#121—1941no—
167C4×D4—1941no—
168C4×D4—1941no—
169C2×C4⋊C4—1941no—
170C2×C42—971no—
171C2×C42—971no—
172C2×C42C2×D9711yescharacteristic
173#104—22no—
174#104—1942no—
175#104—1942no—
176#104—1942no—
177#97—1941no—
178C4×C8—1941no—
179#95—1941no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—1941no—
185#41—1941no—
186#41—1941no—
187#41—1941no—
188C2×Q8—971no—
189C2×Q8C4×D9711yescharacteristic
190C2×D4—21no—
191C2×D4—971no—
192C2×D4—1941no—
193C2×D4—1941no—
194C2×D4—1941no—
195C2×D4C4×D9711yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—1941no—
201C22×C4—1941no—
202C22×C4C22×D9711yescharacteristic
203M16—21no—
204M16—972no—
205M16—1941no—
206M16C22⋊D9712yescomplemented
207C2×C8—21no—
208C2×C8—1941no—
209C4⋊C4—1941no—
210C4⋊C4—1941no—
211C4⋊C4—1944no—
212C22⋊C4—1941no—
213C22⋊C4—1941no—
214C22⋊C4—1941no—
215C22⋊C4—1941no—
216C42—21no—
217C42—21no—
218C42—971no—
219C42—971no—
220C42—971no—
221C42—971no—
222C42—1941no—
223C42—1941no—
224C23—21no—
225C23—1941no—
226C23D4×D9711yescharacteristic
227Q8—21no—
228Q8—21no—
229Q8—1941no—
230Q8—1941no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—3882no—
236D4—3881no—
237D4—3881no—
238D4—3882no—
239D4—1941no—
240D4—1941no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—971no—
248C2×C4—971no—
249C2×C4—971no—
250C2×C4—971no—
251C2×C4—3881no—
252C2×C4—3881no—
253C2×C4—3881no—
254C2×C4—3881no—
255C2×C4—3881no—
256C2×C4—1942no—
257C2×C4—1942no—
258C2×C4—1941no—
259C2×C4C2×C4×D9711yescharacteristic
260C2×C4D4×D9711yescharacteristic
261C2×C4D4×D9711yescharacteristic
262C2×C4C2×C22⋊D9711yescharacteristic
263C2×C4#1309551011yescharacteristic
264C2×C4#1309551111yescharacteristic
265C8—22no—
266C8—1942no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—3881no—
273C22—3881no—
274C22—3881no—
275C22C22⋊C4×D9711yescharacteristic
276C22#1569079611yescharacteristic
277C22#1569080611yescharacteristic
278C4—42no—
279C4—41no—
280C4—972no—
281C4—3882no—
282C4—3881no—
283C4—1941no—
284C4—1941no—
285C4—1941no—
286C4#1569077011yescharacteristic
287C4C4×C22⋊D9711yescharacteristic
288C4#1569078211yescharacteristic
289C4#1569078311yescharacteristic
290C2—21no—
291C2—21no—
292C2—81no—
293C2—7761no—
294C2#2026063511yescharacteristic
295C2#104×D9711yescharacteristic
296C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261281C2
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1980
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#15690738C2semidirect
3. C2×#15690737C2semidirect
4. C2×#15690736C2semidirect
5. M16×Dic97C2semidirect
6. C4×#13095490C2semidirect
7. C2×#104×C97C2semidirect
10. #15690808C4semidirect
11. #15690806C4semidirect
12. #15690799C4semidirect
13. #15690797C4semidirect
45. #13095495D4semidirect
71. M16×C97D4semidirect
140. Dic97#104semidirect
149. C97#1980semidirect
154. C2×#104D97semidirect
206. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612812
C2×#156907382
C2×#156907372
C2×#156907362
M16×Dic972
C4×#130954902
C2×#104×C972
#198097