Finite Group Databasebeta

← #16848392

Subgroups — #16848392

Order 3968.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758473C211yesmaximal, characteristic
2C2×C4×Dic62C211yesmaximal, characteristic, complemented
3C2×#11758469C211yesmaximal, characteristic
4#14520953C211yesmaximal, characteristic, complemented
5#14520948C211yesmaximal, characteristic, complemented
6#14520946C211yesmaximal, characteristic, complemented
7#14520946C211yesmaximal, characteristic, complemented
8#14520941C211yesmaximal, characteristic
9#14520941C211yesmaximal, characteristic
10#14520940C211yesmaximal, characteristic
11#14520938C211yesmaximal, characteristic
12#14520930C211yesmaximal, characteristic, complemented
13#14520908C211yesmaximal, characteristic
14#14520900C211yesmaximal, characteristic, complemented
15C31×#387C211yesmaximal, characteristic, complemented
16C22×C4×D31C2211yescharacteristic
17C22×Dic62C2211yescharacteristic
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C4⋊C4×C31C2211yescharacteristic
20C2×C42×C31C2211yescharacteristic
21C2×(C2×D31)⋊C4C2211yescharacteristic
22C2×(C2×D31)⋊C4C2211yescharacteristic
23C2×(C2×D31)⋊C4C2211yescharacteristic
24C2×(C2×D31)⋊C4C2211yescharacteristic
25C2×(C2×D31)⋊C4C2211yescharacteristic
26C2×(C2×D31)⋊C4C2211yescharacteristic
27C2×C4⋊Dic31C2211yescharacteristic
28C2×C4⋊Dic31C2211yescharacteristic
29C2×C4⋊Dic31C2211yescharacteristic
30C2×Dic31⋊C4C2211yescharacteristic
31C2×Dic31⋊C4C2211yescharacteristic
32C2×Dic31⋊C4C2211yescharacteristic
33C2×Dic31⋊C4C2211yescharacteristic
34C2×Dic31⋊C4C2211yescharacteristic
35C2×Dic31⋊C4C2211yescharacteristic
36C2×C4×Dic31C2211yescharacteristic
37C2×C4×Dic31C2211yescharacteristic
38C2×C4×Dic31C2211yescharacteristic
39C2×C4×Dic31C2211yescharacteristic
40C2×C4×Dic31C2211yescharacteristic
41#11758473C418yescomplemented
42#11758469—24no—
43C4×Dic62—24no—
44C31×#95C2211yescharacteristic
45C31×#95C2211yescharacteristic
46C31×#95C2211yescharacteristic
47C31×#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C31⋊#95C2211yescharacteristic
53C31⋊#95C2211yescharacteristic
54C23×D31C2311yescharacteristic
55C22×C4×C31C2311yescharacteristic
56C22×C4×C31C2311yescharacteristic
57C22×C4×C31C2311yescharacteristic
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2311yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×Dic31C2311yescharacteristic
63C22×Dic31C2311yescharacteristic
64C22×Dic31C2311yescharacteristic
65C22×Dic31C2311yescharacteristic
66C22×Dic31C2311yescharacteristic
67C22×Dic31C2311yescharacteristic
68C22×Dic31C2311yescharacteristic
69C2×C4×D31—22no—
70C2×C4×D31—22no—
71C2×C4×D31C2×C414yes—
72C2×Dic62—22no—
73C2×Dic62—22no—
74C2×Dic62C2×C414yes—
75C4⋊C4×C31—22no—
76C4⋊C4×C31C2×C414yes—
77C42×C31—22no—
78(C2×D31)⋊C4—22no—
79(C2×D31)⋊C4—22no—
80(C2×D31)⋊C4—41no—
81(C2×D31)⋊C4—41no—
82(C2×D31)⋊C4C2×C414yes—
83(C2×D31)⋊C4C2×C414yes—
84C4⋊Dic31—22no—
85C4⋊Dic31—22no—
86C4⋊Dic31—41no—
87Dic31⋊C4—22no—
88Dic31⋊C4—22no—
89Dic31⋊C4—22no—
90Dic31⋊C4—41no—
91Dic31⋊C4C2×C414yes—
92Dic31⋊C4C2×C414yes—
93C4×Dic31—22no—
94C4×Dic31—22no—
95C4×Dic31—41no—
96C4×Dic31—41no—
97C4×Dic31D414yes—
98C23×C31C2411yescharacteristic
99C22×D31—21no—
100C22×D31—21no—
101C22×D31—21no—
102C22×D31—21no—
103C22×D31—21no—
104C22×D31—21no—
105C22×D31C22×C412yes—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31C22×C412yes—
124C2×C4×C31C22×C412yes—
125C2×C4×C31C22×C412yes—
126C2×C4×C31C2×D412yes—
127C2×Dic31—21no—
128C2×Dic31—21no—
129C2×Dic31—21no—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—21no—
138C2×Dic31—21no—
139C2×Dic31—21no—
140C2×Dic31—21no—
141C2×Dic31—21no—
142C2×Dic31C22×C412yes—
143C2×Dic31C22×C412yes—
144C2×Dic31C22×C412yes—
145C2×Dic31C2×D412yes—
146C2×Dic31C2×D412yes—
147C2×Dic31#4112yes—
148C4×D31—28no—
149Dic62—28no—
150#2323—311nomaximal, Sylow
151C22×C31C23×C411yescharacteristic
152C22×C31C22×D411yescharacteristic
153C22×C31C2×#4111yescharacteristic
154C22×C31H(5,𝔽2)11yescharacteristic
155C22×C312−1+411yescharacteristic
156C22×C312−1+411yescharacteristic
157C22×C312−1+411yescharacteristic
158C2×D31—22no—
159C2×D31—22no—
160C2×D31—22no—
161C2×D31—41no—
162C2×D31—41no—
163C2×D31—41no—
164C2×D31—41no—
165C4×C31—22no—
166C4×C31—22no—
167C4×C31—22no—
168C4×C31—22no—
169C4×C31—41no—
170C4×C31—41no—
171C4×C31—41no—
172Dic31—22no—
173Dic31—22no—
174Dic31—22no—
175Dic31—41no—
176Dic31—41no—
177Dic31—41no—
178Dic31C4×D414yes—
179C2×C22⋊Q8—311no—
180C2×C4×Q8—311no—
181C2×#117—311no—
182#391—311no—
183#389—311no—
184#389—311no—
185#389—311no—
186#388—311no—
187#388—311no—
188#388—311no—
189#388—311no—
190#387—311no—
191#387D3111yescharacteristic, complemented
192#380—311no—
193C4×C4⋊C4—311no—
194C2×C31C2×C4×D411yescharacteristic
195C2×C31#51911yescharacteristic
196C2×C31#52011yescharacteristic
197C2×C31#53611yescharacteristic
198C2×C31#53711yescharacteristic
199C2×C31#54111yescharacteristic
200C2×C31#54211yescharacteristic
201D31—42no—
202C22×Q8—311no—
203C23×C4—311no—
204C22⋊Q8—318no—
205C4×Q8—624no—
206#117—624no—
207C2×C4⋊C4—311no—
208C2×C4⋊C4—311no—
209C2×C4⋊C4—311no—
210C2×C4⋊C4—311no—
211C2×C4⋊C4—311no—
212C2×C4⋊C4—311no—
213C2×C4⋊C4—311no—
214C2×C4⋊C4—311no—
215C2×C4⋊C4—311no—
216C2×C4⋊C4C2×D3111yescharacteristic
217C2×C4⋊C4C2×D3111yescharacteristic
218C2×C22⋊C4—311no—
219C2×C22⋊C4—311no—
220C2×C22⋊C4—311no—
221C2×C22⋊C4—311no—
222C2×C22⋊C4—311no—
223C2×C22⋊C4—311no—
224C2×C42—311no—
225C2×C42—311no—
226C2×C42—311no—
227C2×C42—311no—
228C2×C42—311no—
229C2×C42C2×D3111yescharacteristic
230#95—311no—
231#95—311no—
232#95—311no—
233#95—311no—
234#95—311no—
235#95—311no—
236#95C2×D3111yescharacteristic
237#95C2×D3111yescharacteristic
238#95C2×D3111yescharacteristic
239#95C2×D3111yescharacteristic
240C31#232311yescharacteristic, complemented, Sylow
241C24—311no—
242C2×Q8—314no—
243C2×Q8—622no—
244C2×Q8—622no—
245C22×C4—311no—
246C22×C4—311no—
247C22×C4—311no—
248C22×C4—311no—
249C22×C4—314no—
250C22×C4—311no—
251C22×C4—311no—
252C22×C4—311no—
253C22×C4—622no—
254C22×C4—622no—
255C22×C4C22×D3111yescharacteristic
256C22×C4C22×D3111yescharacteristic
257C22×C4C22×D3111yescharacteristic
258C22×C4C22×D3111yescharacteristic
259C22×C4C22×D3111yescharacteristic
260C22×C4C22×D3111yescharacteristic
261C22×C4C22×D3111yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—314no—
264C4⋊C4—314no—
265C4⋊C4—622no—
266C4⋊C4—622no—
267C4⋊C4—622no—
268C4⋊C4—622no—
269C4⋊C4—622no—
270C4⋊C4—1241no—
271C4⋊C4—1241no—
272C4⋊C4C4×D3114yes—
273C22⋊C4—314no—
274C22⋊C4—314no—
275C22⋊C4—622no—
276C22⋊C4—622no—
277C22⋊C4—1241no—
278C22⋊C4—1241no—
279C42—22no—
280C42—314no—
281C42—622no—
282C42—622no—
283C42—1241no—
284C42—1241no—
285C23—312no—
286C23—621no—
287C23—621no—
288C23—621no—
289C23—621no—
290C23—621no—
291C23—621no—
292C23C23×D3111yescharacteristic
293Q8—628no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—312no—
312C2×C4—312no—
313C2×C4—312no—
314C2×C4—312no—
315C2×C4—312no—
316C2×C4—312no—
317C2×C4—628no—
318C2×C4—621no—
319C2×C4—621no—
320C2×C4—621no—
321C2×C4—621no—
322C2×C4—621no—
323C2×C4—621no—
324C2×C4—621no—
325C2×C4—621no—
326C2×C4—621no—
327C2×C4—621no—
328C2×C4—621no—
329C2×C4—621no—
330C2×C4—621no—
331C2×C4—621no—
332C2×C4—621no—
333C2×C4C2×C4×D3112yes—
334C2×C4C2×C4×D3112yes—
335C2×C4C2×C4×D3112yes—
336C2×C4D4×D3112yes—
337C22—622no—
338C22—622no—
339C22—622no—
340C22—1241no—
341C22—1241no—
342C22—1241no—
343C22—1241no—
344C22C22×C4×D3111yescharacteristic
345C22C2×D4×D3111yescharacteristic
346C22#1175856411yescharacteristic
347C22#41×D3111yescharacteristic
348C22#1175856611yescharacteristic
349C22#1175856711yescharacteristic
350C22#1175856711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—314no—
359C4—622no—
360C4—622no—
361C4—622no—
362C4—1241no—
363C4—1241no—
364C4—1241no—
365C2—1242no—
366C2C4×D4×D3111yescharacteristic
367C2#1452152511yescharacteristic
368C2#1452152911yescharacteristic
369C2#1452162111yescharacteristic
370C2#1452162911yescharacteristic
371C2#1452165811yescharacteristic
372C2#1452166511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#387C2
socle, cosocleC23×C31C24
prime radicalC23×D31C23
hypercenter#387D31
hypocenterC31#2323
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic62C2semidirect
4. #14520953C2semidirect
5. #14520948C2semidirect
6. #14520946C2semidirect
7. #14520946C2semidirect
12. #14520930C2semidirect
14. #14520900C2semidirect
15. C31×#387C2semidirect
41. #11758473C4semidirect
191. #387D31semidirect
240. C31#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#117584732
C2×C4×Dic622
C2×#117584692
#145209532
#145209482
#145209462
#145209412
#145209402
#145209382
#145209302
#145209082
#145209002
C31×#3872
#232331