Finite Group Databasebeta

← #16857600

Subgroups — #16857600

Order 3968.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521739C211yesmaximal, characteristic, complemented
2#14521708C211yesmaximal, characteristic
3#14521404C211yesmaximal, characteristic, complemented
4#14521391C211yesmaximal, characteristic, complemented
5#14521363C211yesmaximal, characteristic, complemented
6#14521357C211yesmaximal, characteristic, complemented
7#14521290C211yesmaximal, characteristic, complemented
8#14521278C211yesmaximal, characteristic, complemented
9#14521272C211yesmaximal, characteristic, complemented
10#14521213C211yesmaximal, characteristic, complemented
11#14521207C211yesmaximal, characteristic, complemented
12C4×Dic124C211yesmaximal, characteristic, complemented
13#14521046C211yesmaximal, characteristic, complemented
14C2×#11758404C211yesmaximal, characteristic, complemented
15C8×D4×C31C211yesmaximal, characteristic, complemented
16C2×#92495C2211yescharacteristic
17C22×Dic62C2211yescharacteristic
18C22×C8×C31C2211yescharacteristic
19C22×C8×C31C2211yescharacteristic
20C4×D4×C31C2211yescharacteristic
21#11758516—21no—
22#11758512—21no—
23#11758507C2211yescharacteristic
24#11758503C2211yescharacteristic
25C2×C4⋊Dic31C2211yescharacteristic
26#11758501C2211yescharacteristic
27#11758501C2211yescharacteristic
28#11758501C2211yescharacteristic
29C2×Dic124C2211yescharacteristic
30C2×Dic124C2211yescharacteristic
31C2×#92470C2211yescharacteristic
32#11758473—21no—
33#11758466—21no—
34#11758462—21no—
35(C22⋊D31)⋊C4—21no—
36#11758456—21no—
37#11758454—21no—
38#11758452C2211yescharacteristic
39#11758447C2211yescharacteristic
40C4×Dic62C2211yescharacteristic
41C4×Dic62C2211yescharacteristic
42C31×C4⋊C8C2211yescharacteristic
43C31×C22⋊C8C2211yescharacteristic
44C31×C22⋊C8C2211yescharacteristic
45C4×C8×C31C2211yescharacteristic
46#11758409C2211yescharacteristic
47#11758409C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758406C2211yescharacteristic
50#11758406C2211yescharacteristic
51#11758405C2211yescharacteristic
52#11758404C2211yescharacteristic
53#11758404C2211yescharacteristic
54#11758404C2211yescharacteristic
55#11758404C2211yescharacteristic
56#11758404C2211yescharacteristic
57#11758404C2211yescharacteristic
58#11758404C2211yescharacteristic
59C2×D4×C31D411yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C2×C22⋊D31—21no—
63C2×C22⋊D31—21no—
64C22×Dic31—21no—
65C22×Dic31—21no—
66#92495—22no—
67#92495—41no—
68C2×D124C2311yescharacteristic
69C2×C4×D31—21no—
70C2×Dic62—22no—
71C2×Dic62—21no—
72C2×Dic62—41no—
73C2×Dic62C2311yescharacteristic
74C2×Dic62C2311yescharacteristic
75C2×Dic62C2311yescharacteristic
76C2×C8×C31—41no—
77C2×C8×C31—41no—
78C2×C8×C31C2311yescharacteristic
79C2×C8×C31C2311yescharacteristic
80C2×C8×C31C2311yescharacteristic
81C2×C8×C31C2311yescharacteristic
82C4⋊C4×C31D411yescharacteristic
83C22⋊C4×C31D411yescharacteristic
84C22⋊C4×C31D411yescharacteristic
85C42×C31C2311yescharacteristic
86C22⋊Dic31—21no—
87C22⋊Dic31—21no—
88C22⋊Dic31—21no—
89C22⋊Dic31—21no—
90(C2×D31)⋊C4—21no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93C4⋊Dic31—21no—
94C4⋊Dic31C2311yescharacteristic
95C4⋊Dic31C2311yescharacteristic
96C4⋊Dic31C2311yescharacteristic
97C4⋊Dic31C2311yescharacteristic
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100Dic31⋊C4—21no—
101Dic31⋊C4—21no—
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31—21no—
108Dic124—22no—
109Dic124—41no—
110#92470—41no—
111C23×C31C2×D411yescharacteristic
112C23×C31C2×D411yescharacteristic
113C22×D31—21no—
114D4×C31—22no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31C2×D411yescharacteristic
120C2×C4×C31C2×D411yescharacteristic
121C2×C4×C31C2×D411yescharacteristic
122C2×C4×C31C2×D411yescharacteristic
123C2×C4×C31C2411yescharacteristic
124C22⋊D31—42no—
125C22⋊D31—42no—
126C2×Dic31—21no—
127C2×Dic31—21no—
128C2×Dic31—21no—
129C2×Dic31—21no—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—42no—
134C2×Dic31—42no—
135D124—21no—
136D124—21no—
137C4×D31—42no—
138Dic62—21no—
139Dic62—21no—
140Dic62—21no—
141Dic62—21no—
142Dic62—21no—
143Dic62—42no—
144Dic62#4112yes—
145C8×C31—21no—
146C8×C31—21no—
147C8×C31—21no—
148C8×C31—21no—
149#3300—311nomaximal, Sylow
150C22×C31—21no—
151C22×C31—21no—
152C22×C31—21no—
153C22×C31—21no—
154C22×C31—21no—
155C22×C31#13512yes—
156C22×C31C22×D411yescharacteristic
157C2×D31—41no—
158C2×D31—41no—
159C2×D31—41no—
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—21no—
164C4×C31C2×#4111yescharacteristic
165C4×C31H(5,𝔽2)11yescharacteristic
166Dic31—22no—
167Dic31—41no—
168Dic31—41no—
169Dic31—41no—
170Dic31—41no—
171Dic31—41no—
172Dic31—41no—
173#552—311no—
174#548—311no—
175#488—311no—
176#485—311no—
177#480—311no—
178#479—311no—
179#467—311no—
180#465—311no—
181#464—311no—
182#452—311no—
183#451—311no—
184C4×Q16—311no—
185C8×D4D3111yescharacteristic, complemented
186#416—311no—
187C2×#103—311no—
188C2×C31—22no—
189C2×C31—41no—
190C2×C31#54611yescharacteristic
191C2×C31C2×#13511yescharacteristic
192C2×C31#57811yescharacteristic
193D31—81no—
194C2×#41—311no—
195C22×Q8—311no—
196C2×Q16—311no—
197C2×Q16—311no—
198C2×SD16—311no—
199C22×C8C2×D3111yescharacteristic
200C22×C8C2×D3111yescharacteristic
201#126—621no—
202#125—311no—
203#125—621no—
204#124—311no—
205#124—621no—
206#123—621no—
207C22⋊Q8—311no—
208C22⋊Q8—311no—
209C22⋊Q8—311no—
210C22⋊Q8—621no—
211#121—311no—
212#121—621no—
213C4×Q8—311no—
214C4×Q8—311no—
215C4×D4—621no—
216C4×D4C2×D3111yescharacteristic
217#117—311no—
218#117—621no—
219C2×C4⋊C4—311no—
220#107—311no—
221#107—311no—
222#106—311no—
223C4⋊C8C2×D3111yescharacteristic
224#103—311no—
225#103—311no—
226#103—311no—
227#103—311no—
228#103—311no—
229#103—311no—
230#103—311no—
231#102—311no—
232#102—311no—
233#102—311no—
234C22⋊C8C2×D3111yescharacteristic
235C22⋊C8C2×D3111yescharacteristic
236C4×C8C2×D3111yescharacteristic
237C31#330011yescharacteristic, complemented, Sylow
238#41—622no—
239#41—1241no—
240C2×Q8—311no—
241C2×Q8—311no—
242C2×Q8—311no—
243C2×Q8—622no—
244C2×Q8—621no—
245C2×Q8—1241no—
246C2×D4—311no—
247C2×D4—621no—
248C2×D4—621no—
249C2×D4D12411yescharacteristic
250C22×C4—621no—
251C22×C4—621no—
252C22×C4—621no—
253C22×C4C22×D3111yescharacteristic
254C22×C4C22×D3111yescharacteristic
255Q16—622no—
256Q16—1241no—
257SD16—1241no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D3111yescharacteristic
261C2×C8C22×D3111yescharacteristic
262C2×C8C22×D3111yescharacteristic
263C2×C8C22×D3111yescharacteristic
264C4⋊C4—311no—
265C4⋊C4—311no—
266C4⋊C4—311no—
267C4⋊C4—311no—
268C4⋊C4—621no—
269C4⋊C4—621no—
270C4⋊C4—621no—
271C4⋊C4—621no—
272C4⋊C4—621no—
273C4⋊C4—621no—
274C4⋊C4—621no—
275C4⋊C4—621no—
276C4⋊C4D12411yescharacteristic
277C22⋊C4—621no—
278C22⋊C4—621no—
279C22⋊C4—621no—
280C22⋊C4—621no—
281C22⋊C4—621no—
282C22⋊C4—621no—
283C22⋊C4—621no—
284C22⋊C4D12411yescharacteristic
285C22⋊C4D12411yescharacteristic
286C42—621no—
287C42—621no—
288C42—621no—
289C42C22×D3111yescharacteristic
290C23—621no—
291C23C2×D12411yescharacteristic
292C23C2×D12411yescharacteristic
293Q8—312no—
294Q8—621no—
295Q8—621no—
296Q8—621no—
297Q8—621no—
298Q8—621no—
299Q8—1242no—
300D4—22no—
301D4—621no—
302D4—621no—
303D4—1242no—
304D4—1242no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—621no—
310C2×C4—621no—
311C2×C4—621no—
312C2×C4—621no—
313C2×C4—621no—
314C2×C4—621no—
315C2×C4—621no—
316C2×C4—1242no—
317C2×C4—1242no—
318C2×C4—1242no—
319C2×C4C2×D12411yescharacteristic
320C2×C4C2×D12411yescharacteristic
321C2×C4C2×D12411yescharacteristic
322C2×C4C2×D12411yescharacteristic
323C2×C4C23×D3111yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—1241no—
334C22—1241no—
335C22—1241no—
336C22#1175848112yes—
337C22C22×D12411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—622no—
343C4—1241no—
344C4—1241no—
345C4—1241no—
346C4—1241no—
347C4—1241no—
348C4—1241no—
349C4#1175856111yescharacteristic
350C4#41×D3111yescharacteristic
351C2—22no—
352C2—41no—
353C2—2481no—
354C2#1452169411yescharacteristic
355C2C2×#1175848111yescharacteristic
356C2#1452186611yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC8×D4×C31C2
socleC22×C31C22×D4
prime radical#14521708C2
hypercenterC8×D4D31
hypocenterC31#3300
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #14521739C2semidirect
3. #14521404C2semidirect
4. #14521391C2semidirect
5. #14521363C2semidirect
6. #14521357C2semidirect
7. #14521290C2semidirect
8. #14521278C2semidirect
9. #14521272C2semidirect
10. #14521213C2semidirect
11. #14521207C2semidirect
12. C4×Dic124C2semidirect
13. #14521046C2semidirect
14. C2×#11758404C2semidirect
15. C8×D4×C31C2semidirect
185. C8×D4D31semidirect
237. C31#3300semidirect

Maximal subgroups

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

subgroupindex
#145217392
#145217082
#145214042
#145213912
#145213632
#145213572
#145212902
#145212782
#145212722
#145212132
#145212072
C4×Dic1242
#145210462
C2×#117584042
C8×D4×C312
#330031