Finite Group Databasebeta

← #123×#87604 #27158201

Subgroups — #123×#87604 #27158201

Order 12416.

688 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#87604C211yesmaximal, characteristic
2C22×C4×#87604C211yesmaximal, characteristic, complemented
3#20261685C212yesmaximal, complemented
4#20261684C212yesmaximal, complemented
5#20261683C211yesmaximal, characteristic
6#20261682C211yesmaximal, characteristic, complemented
7#20261678C211yesmaximal, characteristic, complemented
8C4⋊C4×#87604C212yesmaximal, complemented
9C22⋊C4×#87604C211yesmaximal, characteristic, complemented
10C22⋊C4×#87604C212yesmaximal, complemented
11#123×D97C211yesmaximal, characteristic
12C23×#87604C2211yescharacteristic
13C23×#87604C2211yescharacteristic
14C23×#87604C2211yescharacteristic
15D4×#87604—24no—
16C2×C4⋊#87604C2212yes—
17C2×C4⋊#87604C2211yescharacteristic
18C2×C22⋊#87604C2211yescharacteristic
19C2×C22⋊#87604C2212yes—
20C2×C22⋊#87604C2211yescharacteristic
21C2×C22⋊#87604C2211yescharacteristic
22C2×C4×#87604—24no—
23C2×C4×#87604C2212yes—
24C2×C4×#87604C2212yes—
25C2×C4×#87604C2211yescharacteristic
26C2×C4×#87604C2212yes—
27(C2×#87604)⋊C4C2212yes—
28(C2×#87604)⋊C4C2212yes—
29(C2×#87604)⋊C4C2212yes—
30(C2×#87604)⋊C4C2212yes—
31(C2×#87604)⋊C4C2212yes—
32C2×D4×D97C2211yescharacteristic
33C22×C4×D97C2211yescharacteristic
34#15690795C412yescomplemented
35#15690794C411yescharacteristic, complemented
36#15690793C411yescharacteristic, complemented
37#15690792C412yescomplemented
38#15690791C411yescharacteristic, complemented
39C4⋊C4×D97C2212yes—
40C22⋊C4×D97C2211yescharacteristic
41C22⋊C4×D97C2212yes—
42#123×C97C411yescharacteristic, complemented
43C22×#87604—22no—
44C22×#87604—24no—
45C22×#87604—24no—
46C22×#87604—22no—
47C22×#87604—24no—
48C22×#87604D414yes—
49C22×#87604C2312yes—
50C22×#87604C2311yescharacteristic
51C22×#87604C2311yescharacteristic
52C22×#87604C2311yescharacteristic
53C22×#87604C2312yes—
54C22×#87604C2311yescharacteristic
55C4⋊#87604—22no—
56C4⋊#87604—24no—
57C22⋊#87604—22no—
58C22⋊#87604—22no—
59C22⋊#87604—22no—
60C22⋊#87604—24no—
61C4×#87604—24no—
62C4×#87604—24no—
63C4×#87604—22no—
64C4×#87604—24no—
65C23×D97C2311yescharacteristic
66C23×D97C2311yescharacteristic
67C2×D388C2×C411yescharacteristic
68C2×C22⋊D97C2×C411yescharacteristic
69C2×C22⋊D97C2×C411yescharacteristic
70D4×D97—22no—
71D4×D97—22no—
72C22×Dic97C2×C411yescharacteristic, complemented
73C2×C4×D97—22no—
74C2×C4×D97—22no—
75C2×C4×D97C2×C412yescomplemented
76C2×C4×D97C2312yes—
77C2×C4×D97C2312yes—
78C2×C4×D97C2311yescharacteristic
79C4⋊Dic97C2×C412yescomplemented
80#13095490C2×C412yescomplemented
81#13095490C2×C412yescomplemented
82C22⋊Dic97C2×C412yescomplemented
83C22⋊Dic97C2×C411yescharacteristic, complemented
84#13095488C2×C412yescomplemented
85#13095488C2×C412yescomplemented
86#13095488C2×C412yescomplemented
87C2×D4×C97C2×C411yescharacteristic
88C22×C4×C97C2×C411yescharacteristic, complemented
89C4⋊C4×C97C2×C412yescomplemented
90C22⋊C4×C97C2×C412yescomplemented
91C22⋊C4×C97C2×C411yescharacteristic, complemented
92C23×C97C22×C411yescharacteristic
93C23×C97C22×C411yescharacteristic
94C22×D97—22no—
95C22×D97—22no—
96C22×D97—21no—
97C22×D97—21no—
98C22×D97—22no—
99C22×D97—22no—
100C22×D97C22×C411yescharacteristic
101C22×D97C22×C411yescharacteristic
102C22×D97C2×D412yes—
103C22×D97#4112yes—
104C22×D97C2411yescharacteristic
105C2×#87604—28no—
106C2×#87604—22no—
107C2×#87604—22no—
108C2×#87604—22no—
109C2×#87604—28no—
110C2×#87604—21no—
111C2×#87604—21no—
112C2×#87604—22no—
113C2×#87604—22no—
114C2×#87604—22no—
115C2×#87604—22no—
116C2×#87604—22no—
117C2×#87604—44no—
118C2×#87604C2×D412yes—
119C2×#87604C2×D412yes—
120C2×#87604#4114yes—
121C2×#87604#4114yes—
122D4×C97—22no—
123C2×C4×C97—22no—
124C2×C4×C97C22×C412yes—
125C2×C4×C97C22×C412yes—
126C2×C4×C97C22×C411yescharacteristic
127C22⋊D97—22no—
128C22⋊D97—22no—
129C2×Dic97—22no—
130C2×Dic97C22×C412yes—
131C2×Dic97C22×C412yes—
132C2×Dic97C22×C411yescharacteristic
133D388—22no—
134C4×D97—24no—
135C4×D97—22no—
136C4×D97—21no—
137C4×D97—22no—
138C4×D97—21no—
139C4×D97—22no—
140C4×D97—22no—
141C22×C97—22no—
142C22×C97—21no—
143C22×C97—22no—
144C22×C97C4×D412yes—
145C22×C97C23×C411yescharacteristic
146C2×D97—22no—
147C2×D97—22no—
148C2×D97—22no—
149C2×D97—24no—
150C2×D97—21no—
151C2×D97—22no—
152C2×D97—41no—
153C2×D97—41no—
154C2×D97—42no—
155C2×D97C4×D412yes—
156C2×D97C22×D411yescharacteristic
157C2×D97C2×#4112yes—
158C2×D97C2×#4112yes—
159C2×D97C2×#4111yescharacteristic
160#87604—22no—
161#87604—24no—
162#87604—41no—
163#87604—41no—
164#87604—42no—
165#87604#12314yesdirect product factor
166C4×C97—22no—
167C4×C97—21no—
168C4×C97—22no—
169Dic97—22no—
170Dic97—21no—
171Dic97—22no—
172C2×C97—22no—
173C2×C97—41no—
174C2×C97C2×C4×D411yescharacteristic
175C2×C97C4×#4112yes—
176D97—22no—
177D97—41no—
178D97C2×#12311yescharacteristic
179D97#52511yescharacteristic
180D97#52912yes—
181C4×#123—971nomaximal, Sylow
182C97C4×#12311yescharacteristic, complemented, Sylow
183C2×#123—971no—
184C2×C4×D4—971no—
185C22×C42—971no—
186#388—972no—
187#387—972no—
188#386—971no—
189#385—971no—
190#381—971no—
191C4×C4⋊C4—972no—
192C4×C22⋊C4—972no—
193C4×C22⋊C4—971no—
194C22×D4—971no—
195C23×C4—971no—
196C23×C4—971no—
197C23×C4—971no—
198C23×C4—971no—
199#123—971no—
200#123—971no—
201#123—972no—
202#123—972no—
203#123—971no—
204#123#8760411yescharacteristic, direct product factor
205C4×D4—1944no—
206C2×C4⋊C4—972no—
207C2×C4⋊C4—971no—
208C2×C4⋊C4—972no—
209C2×C22⋊C4—971no—
210C2×C22⋊C4—972no—
211C2×C22⋊C4—971no—
212C2×C22⋊C4—971no—
213C2×C22⋊C4—971no—
214C2×C22⋊C4—972no—
215C2×C42—971no—
216C2×C42—972no—
217C2×C42—972no—
218C2×C42—972no—
219C2×C42—1944no—
220#95—972no—
221#95—972no—
222#95—972no—
223#95—972no—
224#95—972no—
225C24—971no—
226C24—971no—
227C2×D4—971no—
228C2×D4—971no—
229C2×D4—971no—
230C2×D4—1942no—
231C2×D4—1942no—
232C2×D4C2×#8760411yescharacteristic
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—972no—
236C22×C4—974no—
237C22×C4—971no—
238C22×C4—972no—
239C22×C4—972no—
240C22×C4—972no—
241C22×C4—971no—
242C22×C4—972no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—1942no—
246C22×C4—1942no—
247C22×C4—1942no—
248C22×C4—1944no—
249C22×C4—1944no—
250C22×C4—1944no—
251C22×C4—1942no—
252C22×C4C2×#8760411yescharacteristic, complemented
253C4⋊C4—972no—
254C4⋊C4—972no—
255C4⋊C4—972no—
256C4⋊C4—1944no—
257C4⋊C4—1942no—
258C4⋊C4C2×#8760412yescomplemented
259C22⋊C4—971no—
260C22⋊C4—972no—
261C22⋊C4—972no—
262C22⋊C4—972no—
263C22⋊C4—972no—
264C22⋊C4—1944no—
265C22⋊C4—1942no—
266C22⋊C4—1942no—
267C22⋊C4—1942no—
268C22⋊C4C2×#8760411yescharacteristic, complemented
269C22⋊C4C2×#8760412yescomplemented
270C42—1944no—
271C42—1944no—
272C42—1944no—
273C42—1942no—
274C23—971no—
275C23—972no—
276C23—972no—
277C23—971no—
278C23—971no—
279C23—1941no—
280C23—1942no—
281C23—1942no—
282C23—1941no—
283C23—1942no—
284C23—1942no—
285C23C22×#8760411yescharacteristic
286C23C22×#8760411yescharacteristic
287D4—22no—
288D4—1942no—
289D4—1942no—
290D4—1942no—
291C2×C4—22no—
292C2×C4—972no—
293C2×C4—974no—
294C2×C4—972no—
295C2×C4—974no—
296C2×C4—972no—
297C2×C4—972no—
298C2×C4—971no—
299C2×C4—3884no—
300C2×C4—1942no—
301C2×C4—1941no—
302C2×C4—1942no—
303C2×C4—1942no—
304C2×C4—1942no—
305C2×C4—1944no—
306C2×C4—1941no—
307C2×C4—1948no—
308C2×C4—1941no—
309C2×C4—1942no—
310C2×C4—1941no—
311C2×C4—1942no—
312C2×C4—1942no—
313C2×C4—1942no—
314C2×C4—1942no—
315C2×C4—1942no—
316C2×C4—1942no—
317C2×C4—1942no—
318C2×C4—1948no—
319C2×C4—1942no—
320C2×C4C22×#8760412yes—
321C2×C4C22×#8760412yes—
322C2×C4C22×#8760411yescharacteristic
323C22—22no—
324C22—21no—
325C22—22no—
326C22—971no—
327C22—971no—
328C22—972no—
329C22—972no—
330C22—972no—
331C22—3881no—
332C22—3881no—
333C22—3882no—
334C22—1942no—
335C22—1942no—
336C22—1944no—
337C22—1941no—
338C22—1942no—
339C22—1942no—
340C22D4×#8760412yes—
341C22C23×#8760411yescharacteristic
342C4—22no—
343C4—21no—
344C4—22no—
345C4—974no—
346C4—3881no—
347C4—3881no—
348C4—3882no—
349C4—1942no—
350C4—1944no—
351C4—1942no—
352C4—1941no—
353C4—1942no—
354C2—22no—
355C2—41no—
356C2—971no—
357C2—971no—
358C2—972no—
359C2—3881no—
360C2—1942no—
361C2C2×D4×#8760411yescharacteristic
362C2#41×#8760412yes—
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#123×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenter#123#87604
hypocenterC97C4×#123
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×#87604C2semidirect
3. #20261685C2semidirect
4. #20261684C2semidirect
6. #20261682C2semidirect
7. #20261678C2semidirect
8. C4⋊C4×#87604C2semidirect
9. C22⋊C4×#87604C2semidirect
10. C22⋊C4×#87604C2semidirect
34. #15690795C4semidirect
35. #15690794C4semidirect
36. #15690793C4semidirect
37. #15690792C4semidirect
38. #15690791C4semidirect
42. #123×C97C4semidirect
72. C22×Dic97C2×C4semidirect
75. C2×C4×D97C2×C4semidirect
79. C4⋊Dic97C2×C4semidirect
80. #13095490C2×C4semidirect
81. #13095490C2×C4semidirect
82. C22⋊Dic97C2×C4semidirect
83. C22⋊Dic97C2×C4semidirect
84. #13095488C2×C4semidirect
85. #13095488C2×C4semidirect
86. #13095488C2×C4semidirect
88. C22×C4×C97C2×C4semidirect
89. C4⋊C4×C97C2×C4semidirect
90. C22⋊C4×C97C2×C4semidirect
91. C22⋊C4×C97C2×C4semidirect
165. #87604#123direct (N × H)
182. C97C4×#123semidirect
204. #123#87604direct (N × H)
252. C22×C4C2×#87604semidirect
258. C4⋊C4C2×#87604semidirect
268. C22⋊C4C2×#87604semidirect
269. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#876042
C22×C4×#876042
#202616852
#202616842
#202616832
#202616822
#202616782
C4⋊C4×#876042
C22⋊C4×#876042
#123×D972
C4×#12397