Finite Group Databasebeta

← #16857274

Subgroups — #16857274

Order 3968.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758492C211yesmaximal, characteristic, complemented
2C22×#92482C211yesmaximal, characteristic, complemented
3Q8×C22⋊D31C211yesmaximal, characteristic, complemented
4#14521705C211yesmaximal, characteristic
5#14521440C211yesmaximal, characteristic, complemented
6#14521306C211yesmaximal, characteristic, complemented
7#14521276C211yesmaximal, characteristic, complemented
8#14521269C211yesmaximal, characteristic, complemented
9#14521258C211yesmaximal, characteristic, complemented
10#14521208C211yesmaximal, characteristic, complemented
11#14521207C211yesmaximal, characteristic, complemented
12#14521206C211yesmaximal, characteristic, complemented
13#14521153C211yesmaximal, characteristic, complemented
14#14521117C211yesmaximal, characteristic, complemented
15C31×#448C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic
18C2×#92497C2211yescharacteristic
19C22×Dic62C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic
22C31×#121C2211yescharacteristic
23#11758523—21no—
24#11758523C2211yescharacteristic
25Q8×Dic31C2211yescharacteristic
26#11758521—21no—
27#11758521C2211yescharacteristic
28C2×#92482—24no—
29C2×#92482C2211yescharacteristic
30C2×#92482C2211yescharacteristic
31C2×#92482C2211yescharacteristic
32C2×#92482C2211yescharacteristic
33#11758516—21no—
34#11758513C2211yescharacteristic
35D4×Dic31—21no—
36C2×#92480C2211yescharacteristic
37C4×C22⋊D31—21no—
38C4×C22⋊D31C2211yescharacteristic
39#11758501—21no—
40#11758501C2211yescharacteristic
41C22×C31⋊C8C2211yescharacteristic
42#11758492—24no—
43C2×Dic124C2211yescharacteristic
44C2×Dic124C2211yescharacteristic
45C2×#92469C2211yescharacteristic
46#11758474C2211yescharacteristic
47#11758464C2211yescharacteristic
48#11758462—21no—
49#11758455—21no—
50C31×#103C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×C22⋊C8C2211yescharacteristic
53#11758420C2211yescharacteristic
54(C2×D31)⋊C8C2211yescharacteristic
55#11758404C2211yescharacteristic
56#11758404C2211yescharacteristic
57#11758403C2211yescharacteristic
58Dic31⋊C8C2211yescharacteristic
59#11758398C2211yescharacteristic
60#11758396C2211yescharacteristic
61C2×Q8×C31—21no—
62C2×Q8×C31—22no—
63C2×Q8×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×Q8×C31C2311yescharacteristic
66C2×D4×C31—21no—
67C2×D4×C31C2311yescharacteristic
68C22×C4×C31—21no—
69C22×C4×C31C2311yescharacteristic
70C2×C22⋊D31—21no—
71C2×C22⋊D31D411yescharacteristic
72C22×Dic31—21no—
73C22×Dic31—21no—
74Q8×D31—22no—
75Q8×D31—41no—
76#92497—22no—
77#92497—41no—
78C2×C4×D31—21no—
79C2×C4×D31C2311yescharacteristic
80C2×Dic62—22no—
81C2×Dic62—21no—
82C2×Dic62—21no—
83C2×Dic62—41no—
84C2×Dic62C2311yescharacteristic
85C2×Dic62C2311yescharacteristic
86C2×Dic62C2311yescharacteristic
87SD16×C31—22no—
88SD16×C31—41no—
89C2×C8×C31C2311yescharacteristic
90C2×C8×C31C2311yescharacteristic
91C4⋊C4×C31C2311yescharacteristic
92C22⋊C4×C31—21no—
93C22⋊Dic31—21no—
94C22⋊Dic31—21no—
95C22⋊Dic31—21no—
96C22⋊Dic31D411yescharacteristic
97#92482—22no—
98#92482—22no—
99#92482—22no—
100#92482D414yescomplemented
101#92480—22no—
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—21no—
104(C2×D31)⋊C4D411yescharacteristic
105C4⋊Dic31—21no—
106C4⋊Dic31—21no—
107C4⋊Dic31C2311yescharacteristic
108Dic31⋊C4—21no—
109Dic31⋊C4—21no—
110Dic31⋊C4—21no—
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4D411yescharacteristic
114C4×Dic31—21no—
115C4×Dic31—21no—
116C4×Dic31C2311yescharacteristic
117C2×C31⋊C8—22no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120Dic124—22no—
121Dic124—41no—
122#92469—22no—
123C23×C31—21no—
124C23×C31C2×D411yescharacteristic
125C22×D31C2×D411yescharacteristic
126Q8×C31—21no—
127Q8×C31—21no—
128Q8×C31—21no—
129Q8×C31—42no—
130Q8×C31C2×D412yes—
131D4×C31—21no—
132D4×C31—21no—
133D4×C31—42no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—42no—
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C22⋊D31—22no—
143C22⋊D31—42no—
144C2×Dic31—21no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—42no—
149C2×Dic31—42no—
150C2×Dic31C2×D411yescharacteristic
151C2×Dic31C2×D411yescharacteristic
152C2×Dic31C2×D411yescharacteristic
153C4×D31—21no—
154C4×D31—21no—
155C4×D31—42no—
156Dic62—21no—
157Dic62—21no—
158Dic62—21no—
159Dic62—21no—
160Dic62—21no—
161Dic62—42no—
162Dic62—42no—
163Dic62C2×D412yes—
164C8×C31—21no—
165C8×C31—21no—
166C31⋊C8—21no—
167C31⋊C8C2×D412yes—
168#3270—311nomaximal, Sylow
169C22×C31—21no—
170C22×C31—21no—
171C22×C31—41no—
172C22×C31—41no—
173C22×C31—41no—
174C22×C31#13712yescomplemented
175C22×C31C22×D411yescharacteristic
176C2×D31—21no—
177C2×D31—21no—
178C2×D31—21no—
179C4×C31—21no—
180C4×C31—22no—
181C4×C31—41no—
182C4×C31—41no—
183C4×C31C22×D411yescharacteristic
184C4×C31H(5,𝔽2)11yescharacteristic
185Dic31—22no—
186Dic31—21no—
187Dic31—21no—
188Dic31—21no—
189Dic31—41no—
190Dic31—41no—
191Dic31—41no—
192C2×#137—311no—
193C22×Q16—311no—
194D4×Q8—311no—
195#548—311no—
196#497—311no—
197#470—311no—
198#465—311no—
199#464—311no—
200#462—311no—
201#452—311no—
202#451—311no—
203#451—311no—
204#448D3111yescharacteristic, complemented
205#441—311no—
206#435—311no—
207C2×C31—22no—
208C2×C31—81no—
209C2×C31D4211yescharacteristic
210C2×C31C2×#13711yescharacteristic
211C2×C31#57811yescharacteristic
212D31—41no—
213C2×#41—311no—
214C22×Q8—311no—
215C22×Q8—311no—
216C22×Q8C2×D3111yescharacteristic
217#137—624no—
218C2×Q16—311no—
219C2×Q16—311no—
220C2×Q16—311no—
221C2×Q16—311no—
222C2×Q16—311no—
223C2×Q16—311no—
224C2×Q16—624no—
225C2×SD16—311no—
226C2×SD16C2×D3111yescharacteristic
227C2×SD16C2×D3111yescharacteristic
228C2×M16—311no—
229C22×C8—311no—
230C4⋊Q8—311no—
231C4⋊Q8—621no—
232#124—311no—
233#124—621no—
234C22⋊Q8—311no—
235C22⋊Q8—311no—
236C22⋊Q8—311no—
237C22⋊Q8—621no—
238C22⋊Q8—621no—
239C22⋊Q8—621no—
240#121—621no—
241#121C2×D3111yescharacteristic
242C4×Q8—311no—
243C4×Q8—311no—
244C4×D4—311no—
245C4×D4—621no—
246C4×D4—621no—
247#106—311no—
248C4⋊C8—311no—
249#103—311no—
250#103—311no—
251#103—311no—
252#103C2×D3111yescharacteristic
253#103C2×D3111yescharacteristic
254#102—311no—
255C22⋊C8—311no—
256C22⋊C8C2×D3111yescharacteristic
257#97—311no—
258C31#327011yescharacteristic, complemented, Sylow
259#41—622no—
260#41—1241no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—311no—
265C2×Q8—311no—
266C2×Q8—311no—
267C2×Q8—621no—
268C2×Q8—622no—
269C2×Q8—621no—
270C2×Q8—622no—
271C2×Q8—1241no—
272C2×Q8—1241no—
273C2×Q8C22×D3111yescharacteristic
274C2×Q8C22×D3111yescharacteristic
275C2×D4—21no—
276C2×D4—311no—
277C2×D4—621no—
278C2×D4C22×D3111yescharacteristic
279C22×C4—21no—
280C22×C4—311no—
281C22×C4—621no—
282C22×C4—621no—
283C22×C4—621no—
284C22×C4C22×D3111yescharacteristic
285Q16—314no—
286Q16—622no—
287Q16—622no—
288Q16—622no—
289Q16—622no—
290Q16—1241no—
291SD16—22no—
292SD16—41no—
293SD16—622no—
294M16—622no—
295C2×C8—311no—
296C2×C8—311no—
297C2×C8—622no—
298C2×C8C22×D3111yescharacteristic
299C2×C8C22×D3111yescharacteristic
300C4⋊C4—311no—
301C4⋊C4—311no—
302C4⋊C4—621no—
303C4⋊C4—621no—
304C4⋊C4—621no—
305C4⋊C4—621no—
306C4⋊C4—621no—
307C4⋊C4—621no—
308C4⋊C4—621no—
309C4⋊C4C22×D3111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—311no—
312C22⋊C4—311no—
313C22⋊C4—621no—
314C22⋊C4—621no—
315C22⋊C4—621no—
316C22⋊C4—621no—
317C22⋊C4—621no—
318C42—311no—
319C42—621no—
320C42—621no—
321C23—21no—
322C23—311no—
323C23D4×D3111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—312no—
329Q8—621no—
330Q8—621no—
331Q8—621no—
332Q8—621no—
333Q8—621no—
334Q8—1242no—
335Q8—1242no—
336Q8D4×D3112yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—622no—
341D4—1242no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—311no—
349C2×C4—311no—
350C2×C4—311no—
351C2×C4—621no—
352C2×C4—621no—
353C2×C4—621no—
354C2×C4—621no—
355C2×C4—621no—
356C2×C4—621no—
357C2×C4—1242no—
358C2×C4—1242no—
359C2×C4—1242no—
360C2×C4D4×D3111yescharacteristic
361C2×C4C23×D3111yescharacteristic
362C8—21no—
363C8—21no—
364C8—312no—
365C8—621no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—621no—
372C22—621no—
373C22—621no—
374C22#1175849212yescomplemented
375C22C2×D4×D3111yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—621no—
381C4—621no—
382C4—622no—
383C4—621no—
384C4—1241no—
385C4—1241no—
386C4—1241no—
387C4C2×D4×D3111yescharacteristic
388C4#1175856111yescharacteristic
389C2—22no—
390C2—81no—
391C2—1241no—
392C2#1452168111yescharacteristic
393C2C2×#1175849211yescharacteristic
394C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#448C2
socleC22×C31C22×D4
prime radical#14521705C2
hypercenter#448D31
hypocenterC31#3270
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#11758492C2semidirect
2. C22×#92482C2semidirect
3. Q8×C22⋊D31C2semidirect
5. #14521440C2semidirect
6. #14521306C2semidirect
7. #14521276C2semidirect
8. #14521269C2semidirect
9. #14521258C2semidirect
10. #14521208C2semidirect
11. #14521207C2semidirect
12. #14521206C2semidirect
13. #14521153C2semidirect
14. #14521117C2semidirect
15. C31×#448C2semidirect
100. #92482D4semidirect
174. C22×C31#137semidirect
204. #448D31semidirect
258. C31#3270semidirect
374. C22#11758492semidirect

Maximal subgroups

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

subgroupindex
C2×#117584922
C22×#924822
Q8×C22⋊D312
#145217052
#145214402
#145213062
#145212762
#145212692
#145212582
#145212082
#145212072
#145212062
#145211532
#145211172
C31×#4482
#327031