Finite Group Databasebeta

← #16854589

Subgroups — #16854589

Order 3968.

484 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×#11758493C211yesmaximal, characteristic
2#14521531C211yesmaximal, characteristic, complemented
3#14521528C211yesmaximal, characteristic, complemented
4#14521159C211yesmaximal, characteristic, complemented
5#14521158C211yesmaximal, characteristic, complemented
6#14521154C211yesmaximal, characteristic, complemented
7#14521153C211yesmaximal, characteristic, complemented
8C4×#92470C211yesmaximal, characteristic, complemented
9SD16×Dic31C211yesmaximal, characteristic, complemented
10#14521136C211yesmaximal, characteristic, complemented
11#106×D31C211yesmaximal, characteristic, complemented
12#14521049C211yesmaximal, characteristic, complemented
13#14521047C211yesmaximal, characteristic, complemented
14#14521003C211yesmaximal, characteristic, complemented
15C4×SD16×C31C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×#92497C2211yescharacteristic
18C2×SD16×C31C2211yescharacteristic
19C4×Q8×C31C2211yescharacteristic
20C4×D4×C31C2211yescharacteristic
21Q8×Dic31C2211yescharacteristic
22C2×#92482C2211yescharacteristic
23D4×Dic31C2211yescharacteristic
24C2×#92479C2211yescharacteristic
25C4×C22⋊D31—21no—
26C2×Dic31⋊C4—21no—
27#11758493C418yescomplemented
28C2×#92470C2211yescharacteristic
29C2×#92469C2211yescharacteristic
30C2×C8×D31C2211yescharacteristic
31#11758470—21no—
32#11758470C2211yescharacteristic
33C4⋊C4×D31—21no—
34C4⋊C4×D31C2211yescharacteristic
35C4⋊C4×D31C2211yescharacteristic
36#11758464C2211yescharacteristic
37(C22⋊D31)⋊C4—21no—
38#11758454—21no—
39C4×D124—21no—
40C4×D124C2211yescharacteristic
41#11758449—21no—
42#11758449C2211yescharacteristic
43C4×Dic62C2211yescharacteristic
44C31×#106C2211yescharacteristic
45C31×#103C2211yescharacteristic
46C31×#102C2211yescharacteristic
47C4×C8×C31C2211yescharacteristic
48#11758423C2211yescharacteristic
49#11758420C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758405C2211yescharacteristic
52#11758404C2211yescharacteristic
53#11758403C2211yescharacteristic
54C8×Dic31C2211yescharacteristic
55#11758398C2211yescharacteristic
56#11758397C2211yescharacteristic
57#11758396C2211yescharacteristic
58#11758396C2211yescharacteristic
59#11758391C2211yescharacteristic
60C2×Q8×C31C2311yescharacteristic
61C2×D4×C31C2311yescharacteristic
62C22×C4×C31—21no—
63C2×C22⋊D31—21no—
64C22×Dic31—21no—
65C22×Dic31—21no—
66#92499—22no—
67#92499C2×C414yescomplemented
68#92497—22no—
69#92497C2×C414yescomplemented
70C2×D124—21no—
71C2×D124C2311yescharacteristic
72C2×C4×D31—21no—
73C2×C4×D31—21no—
74C2×C4×D31—21no—
75C2×C4×D31—21no—
76C2×C4×D31—21no—
77C2×C4×D31C2311yescharacteristic
78C2×Dic62C2311yescharacteristic
79SD16×C31C2×C414yescomplemented
80C2×C8×C31C2311yescharacteristic
81C2×C8×C31C2311yescharacteristic
82C4⋊C4×C31—21no—
83C4⋊C4×C31C2311yescharacteristic
84C4⋊C4×C31C2311yescharacteristic
85C22⋊C4×C31—21no—
86C42×C31—21no—
87C42×C31C2311yescharacteristic
88C22⋊Dic31—21no—
89#92482C2×C414yescomplemented
90#92479C2×C414yescomplemented
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4D411yescharacteristic
94(C2×D31)⋊C4D411yescharacteristic
95C4⋊Dic31—21no—
96C4⋊Dic31C2311yescharacteristic
97C4⋊Dic31C2311yescharacteristic
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100Dic31⋊C4—21no—
101Dic31⋊C4—21no—
102Dic31⋊C4D411yescharacteristic
103Dic31⋊C4D411yescharacteristic
104C4×Dic31—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110#92470C2×C414yescomplemented
111#92469—22no—
112C8×D31C2×C414yescomplemented
113C23×C31—21no—
114C22×D31—21no—
115C22×D31C2×D411yescharacteristic
116Q8×C31—21no—
117Q8×C31C22×C412yes—
118D4×C31—21no—
119D4×C31C22×C412yes—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—81no—
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2411yescharacteristic
127C22⋊D31—24no—
128C2×Dic31—21no—
129C2×Dic31—24no—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—81no—
133C2×Dic31C2×D411yescharacteristic
134C2×Dic31C2×D411yescharacteristic
135C2×Dic31C2×D411yescharacteristic
136D124—24no—
137D124—21no—
138D124C22×C412yes—
139C4×D31—24no—
140C4×D31—81no—
141C4×D31—81no—
142C4×D31—81no—
143C4×D31—81no—
144C4×D31C22×C412yes—
145C4×D31#4112yes—
146Dic62—21no—
147Dic62C22×C412yes—
148C8×C31—21no—
149C8×C31C22×C412yes—
150C31⋊C8—21no—
151C31⋊C8C22×C412yes—
152#2952—311nomaximal, Sylow
153C22×C31—22no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31C22×D411yescharacteristic
157C2×D31—21no—
158C2×D31—21no—
159C2×D31—22no—
160C2×D31—41no—
161C2×D31—41no—
162C2×D31C4×D412yes—
163C4×C31—22no—
164C4×C31—21no—
165C4×C31—21no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31C23×C411yescharacteristic
169C4×C31C2×#4111yescharacteristic
170Dic31—22no—
171Dic31—21no—
172Dic31—21no—
173Dic31—41no—
174Dic31—41no—
175Dic31C4×D412yes—
176C2×#135—311no—
177#520—311no—
178#520—311no—
179#442—311no—
180#442—311no—
181#441—311no—
182#441—311no—
183C4×SD16—311no—
184C4×SD16—311no—
185C4×SD16—311no—
186C4×SD16D3111yescharacteristic, complemented
187C2×#106—311no—
188#417—311no—
189#417—311no—
190#405—311no—
191C2×C31—42no—
192C2×C31C2×C4×D411yescharacteristic
193C2×C31#57711yescharacteristic
194C2×C31#57711yescharacteristic
195D31—22no—
196D31—42no—
197C2×#41—311no—
198C2×#41—311no—
199#135—318no—
200C2×Q16—311no—
201C2×SD16—311no—
202C2×SD16C2×D3111yescharacteristic
203C2×D8—311no—
204C2×M16—311no—
205C22×C8—311no—
206C4×Q8—311no—
207C4×Q8—311no—
208C4×Q8—311no—
209C4×Q8C2×D3111yescharacteristic
210C4×D4—311no—
211C4×D4—311no—
212C4×D4—311no—
213C4×D4—621no—
214C4×D4—621no—
215C4×D4—621no—
216C4×D4—621no—
217C4×D4C2×D3111yescharacteristic
218#117—311no—
219#117—621no—
220#117—621no—
221C2×C4⋊C4—311no—
222C2×C4⋊C4—311no—
223C2×C4⋊C4—621no—
224C2×C4⋊C4—621no—
225#106—311no—
226#106—311no—
227#106—311no—
228#106C2×D3111yescharacteristic
229#103—311no—
230#103—311no—
231#103—311no—
232#103C2×D3111yescharacteristic
233#102—311no—
234#102—311no—
235#102—311no—
236#102C2×D3111yescharacteristic
237#97—311no—
238#97—311no—
239C4×C8—311no—
240C4×C8C2×D3111yescharacteristic
241C31#295211yescharacteristic, complemented, Sylow
242#41—314no—
243#41—314no—
244#41—622no—
245#41—622no—
246C2×Q8—311no—
247C2×Q8C22×D3111yescharacteristic
248C2×D4—311no—
249C2×D4—621no—
250C2×D4—621no—
251C2×D4C22×D3111yescharacteristic
252C22×C4—21no—
253C22×C4—311no—
254C22×C4—621no—
255C22×C4—621no—
256C22×C4—621no—
257C22×C4—621no—
258C22×C4—621no—
259C22×C4—621no—
260C22×C4—621no—
261Q16—314no—
262SD16—314no—
263SD16C4×D3114yescomplemented
264D8—314no—
265M16—622no—
266C2×C8—311no—
267C2×C8—314no—
268C2×C8—311no—
269C2×C8C22×D3111yescharacteristic
270C2×C8C22×D3111yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—311no—
273C4⋊C4—311no—
274C4⋊C4—311no—
275C4⋊C4—311no—
276C4⋊C4—621no—
277C4⋊C4—621no—
278C4⋊C4—621no—
279C4⋊C4—621no—
280C4⋊C4—621no—
281C4⋊C4C22×D3111yescharacteristic
282C4⋊C4C22×D3111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—311no—
285C22⋊C4—311no—
286C22⋊C4—621no—
287C22⋊C4—621no—
288C22⋊C4—621no—
289C42—21no—
290C42—311no—
291C42—621no—
292C42—621no—
293C42—621no—
294C42C22×D3111yescharacteristic
295C23—21no—
296C23—311no—
297C23—621no—
298Q8—21no—
299Q8—312no—
300Q8—621no—
301Q8C2×C4×D3112yes—
302D4—21no—
303D4—312no—
304D4—624no—
305D4—624no—
306D4—621no—
307D4C2×C4×D3112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—311no—
313C2×C4—312no—
314C2×C4—312no—
315C2×C4—311no—
316C2×C4—311no—
317C2×C4—624no—
318C2×C4—624no—
319C2×C4—621no—
320C2×C4—621no—
321C2×C4—621no—
322C2×C4—2481no—
323C2×C4—2481no—
324C2×C4—2481no—
325C2×C4—2481no—
326C2×C4—2481no—
327C2×C4D4×D3111yescharacteristic
328C2×C4D4×D3111yescharacteristic
329C2×C4C23×D3111yescharacteristic
330C8—21no—
331C8—312no—
332C8—621no—
333C8C2×C4×D3112yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—312no—
338C22—621no—
339C22—621no—
340C22—622no—
341C22—1241no—
342C22—1241no—
343C22C2×D4×D3111yescharacteristic
344C4—21no—
345C4—22no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—312no—
350C4—621no—
351C4—622no—
352C4—621no—
353C4—1241no—
354C4—1241no—
355C4C22×C4×D3111yescharacteristic
356C4#41×D3111yescharacteristic
357C2—42no—
358C2—622no—
359C2—1242no—
360C2C4×D4×D3111yescharacteristic
361C2#1452185511yescharacteristic
362C2#1452185611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutatorC4×C31C23×C4
FrattiniC2×C4C23×D31
FittingC4×SD16×C31C2
socleC22×C31C22×D4
prime radicalC2×#11758493C2
cosocleC2×C4×C31C24
hypercenterC4×SD16D31
hypocenterC31#2952
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #14521531C2semidirect
3. #14521528C2semidirect
4. #14521159C2semidirect
5. #14521158C2semidirect
6. #14521154C2semidirect
7. #14521153C2semidirect
8. C4×#92470C2semidirect
9. SD16×Dic31C2semidirect
10. #14521136C2semidirect
11. #106×D31C2semidirect
12. #14521049C2semidirect
13. #14521047C2semidirect
14. #14521003C2semidirect
15. C4×SD16×C31C2semidirect
27. #11758493C4semidirect
67. #92499C2×C4semidirect
69. #92497C2×C4semidirect
79. SD16×C31C2×C4semidirect
89. #92482C2×C4semidirect
90. #92479C2×C4semidirect
110. #92470C2×C4semidirect
112. C8×D31C2×C4semidirect
186. C4×SD16D31semidirect
241. C31#2952semidirect
263. SD16C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584932
#145215312
#145215282
#145211592
#145211582
#145211542
#145211532
C4×#924702
SD16×Dic312
#145211362
#106×D312
#145210492
#145210472
#145210032
C4×SD16×C312
#295231