Finite Group Databasebeta

← #16848872

Subgroups — #16848872

Order 3968.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758523C211yesmaximal, characteristic
2#14520956C211yesmaximal, characteristic, complemented
3#14520946C211yesmaximal, characteristic, complemented
4#14520944C211yesmaximal, characteristic, complemented
5#14520941C211yesmaximal, characteristic
6#14520940C211yesmaximal, characteristic
7#14520939C211yesmaximal, characteristic
8#14520938C211yesmaximal, characteristic
9#14520934C211yesmaximal, characteristic, complemented
10#14520933C211yesmaximal, characteristic, complemented
11#14520932C211yesmaximal, characteristic, complemented
12#14520931C211yesmaximal, characteristic, complemented
13#14520911C211yesmaximal, characteristic
14#14520908C211yesmaximal, characteristic
15C31×#391C211yesmaximal, characteristic, complemented
16C22×Q8×C31C2211yescharacteristic
17C22×C4×D31C2211yescharacteristic
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C42×C31C2211yescharacteristic
20#11758523C418yescomplemented
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×Dic31⋊C4C2211yescharacteristic
30C2×Dic31⋊C4C2211yescharacteristic
31C2×Dic31⋊C4C2211yescharacteristic
32C2×Dic31⋊C4C2211yescharacteristic
33C2×Dic31⋊C4C2211yescharacteristic
34C2×Dic31⋊C4C2211yescharacteristic
35C2×C4×Dic31C2211yescharacteristic
36C2×C4×Dic31C2211yescharacteristic
37C2×C4×Dic31C2211yescharacteristic
38C31×#95C2211yescharacteristic
39C31×#95C2211yescharacteristic
40C31×#95C2211yescharacteristic
41C31×#95C2211yescharacteristic
42C31⋊#95C2211yescharacteristic
43C31⋊#95C2211yescharacteristic
44C31⋊#95C2211yescharacteristic
45C31⋊#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C23×D31C2311yescharacteristic
53C2×Q8×C31—41no—
54C2×Q8×C31—41no—
55C2×Q8×C31C2×C414yes—
56C22×C4×C31C2311yescharacteristic
57C22×C4×C31C2311yescharacteristic
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2311yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×Dic31C2311yescharacteristic
64C22×Dic31C2311yescharacteristic
65C22×Dic31C2311yescharacteristic
66C22×Dic31C2311yescharacteristic
67C22×Dic31C2311yescharacteristic
68C22×Dic31C2311yescharacteristic
69C22×Dic31C2311yescharacteristic
70C2×C4×D31—41no—
71C2×C4×D31—41no—
72C2×C4×D31C2×C414yes—
73C4⋊C4×C31—41no—
74C42×C31—41no—
75(C2×D31)⋊C4—41no—
76(C2×D31)⋊C4—41no—
77(C2×D31)⋊C4—41no—
78(C2×D31)⋊C4—41no—
79(C2×D31)⋊C4C2×C414yes—
80(C2×D31)⋊C4C2×C414yes—
81C4⋊Dic31—41no—
82C4⋊Dic31C2×C414yes—
83Dic31⋊C4—41no—
84Dic31⋊C4—41no—
85Dic31⋊C4—41no—
86Dic31⋊C4—41no—
87Dic31⋊C4C2×C414yes—
88Dic31⋊C4C2×C414yes—
89C4×Dic31—41no—
90C4×Dic31—41no—
91C4×Dic31—41no—
92C23×C31C2411yescharacteristic
93C22×D31—21no—
94C22×D31—21no—
95C22×D31—21no—
96C22×D31—21no—
97C22×D31—21no—
98C22×D31—21no—
99C22×D31C22×C412yes—
100Q8×C31—44no—
101C2×C4×C31—21no—
102C2×C4×C31—21no—
103C2×C4×C31—21no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
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×C31C22×C412yes—
120C2×C4×C31C22×C412yes—
121C2×C4×C31C22×C412yes—
122C2×Dic31—21no—
123C2×Dic31—21no—
124C2×Dic31—21no—
125C2×Dic31—21no—
126C2×Dic31—21no—
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×Dic31C22×C412yes—
141C2×Dic31C22×C412yes—
142C2×Dic31C22×C412yes—
143C4×D31—44no—
144#2378—311nomaximal, Sylow
145C22×C31C23×C411yescharacteristic
146C22×C31H(5,𝔽2)11yescharacteristic
147C22×C31H(5,𝔽2)11yescharacteristic
148C22×C31H(5,𝔽2)11yescharacteristic
149C22×C312−1+411yescharacteristic
150C22×C312−1+411yescharacteristic
151C22×C312−1+411yescharacteristic
152C2×D31—22no—
153C2×D31—22no—
154C2×D31—22no—
155C2×D31—41no—
156C2×D31—41no—
157C2×D31—41no—
158C2×D31—41no—
159C4×C31—22no—
160C4×C31—22no—
161C4×C31—22no—
162C4×C31—41no—
163C4×C31—41no—
164C4×C31—41no—
165C4×C31—41no—
166Dic31—22no—
167Dic31—22no—
168Dic31—22no—
169Dic31—41no—
170Dic31—41no—
171Dic31—41no—
172Dic31—41no—
173C2×C22⋊Q8—311no—
174#391—311no—
175#391D3111yescharacteristic, complemented
176#389—311no—
177#389—311no—
178#388—311no—
179#388—311no—
180#388—311no—
181#388—311no—
182#387—311no—
183#387—311no—
184#387—311no—
185#387—311no—
186#381—311no—
187#380—311no—
188C2×C31#51811yescharacteristic
189C2×C31#51911yescharacteristic
190C2×C31#52011yescharacteristic
191C2×C31#56211yescharacteristic
192C2×C31#56211yescharacteristic
193C2×C31#56311yescharacteristic
194C2×C31#56311yescharacteristic
195D31—42no—
196C22×Q8C2×D3111yescharacteristic
197C23×C4—311no—
198C22⋊Q8—318no—
199C2×C4⋊C4—311no—
200C2×C4⋊C4—311no—
201C2×C4⋊C4—311no—
202C2×C4⋊C4—311no—
203C2×C4⋊C4—311no—
204C2×C4⋊C4—311no—
205C2×C4⋊C4—311no—
206C2×C4⋊C4—311no—
207C2×C4⋊C4C2×D3111yescharacteristic
208C2×C22⋊C4—311no—
209C2×C22⋊C4—311no—
210C2×C22⋊C4—311no—
211C2×C22⋊C4—311no—
212C2×C22⋊C4—311no—
213C2×C22⋊C4—311no—
214C2×C42—311no—
215C2×C42—311no—
216C2×C42—311no—
217C2×C42C2×D3111yescharacteristic
218#95—311no—
219#95—311no—
220#95—311no—
221#95—311no—
222#95—311no—
223#95—311no—
224#95—311no—
225#95—311no—
226#95—311no—
227#95—311no—
228#95C2×D3111yescharacteristic
229#95C2×D3111yescharacteristic
230#95C2×D3111yescharacteristic
231#95C2×D3111yescharacteristic
232C31#237811yescharacteristic, complemented, Sylow
233C24—311no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8C4×D3114yes—
237C22×C4—311no—
238C22×C4—311no—
239C22×C4—311no—
240C22×C4—314no—
241C22×C4—311no—
242C22×C4—311no—
243C22×C4—311no—
244C22×C4—311no—
245C22×C4—1241no—
246C22×C4—1241no—
247C22×C4C22×D3111yescharacteristic
248C22×C4C22×D3111yescharacteristic
249C22×C4C22×D3111yescharacteristic
250C22×C4C22×D3111yescharacteristic
251C22×C4C22×D3111yescharacteristic
252C22×C4C22×D3111yescharacteristic
253C22×C4C22×D3111yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—314no—
256C4⋊C4—314no—
257C4⋊C4—314no—
258C4⋊C4—1241no—
259C4⋊C4—1241no—
260C4⋊C4—1241no—
261C4⋊C4—1241no—
262C4⋊C4—1241no—
263C22⋊C4—314no—
264C22⋊C4—314no—
265C22⋊C4—1241no—
266C22⋊C4—1241no—
267C22⋊C4—1241no—
268C22⋊C4—1241no—
269C42—41no—
270C42—1241no—
271C42—1241no—
272C42—1241no—
273C23—312no—
274C23—621no—
275C23—621no—
276C23—621no—
277C23—621no—
278C23—621no—
279C23—621no—
280C23C23×D3111yescharacteristic
281Q8—44no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—312no—
301C2×C4—312no—
302C2×C4—312no—
303C2×C4—621no—
304C2×C4—621no—
305C2×C4—621no—
306C2×C4—621no—
307C2×C4—621no—
308C2×C4—621no—
309C2×C4—621no—
310C2×C4—621no—
311C2×C4—621no—
312C2×C4—621no—
313C2×C4—621no—
314C2×C4—621no—
315C2×C4—621no—
316C2×C4—621no—
317C2×C4—621no—
318C2×C4—621no—
319C2×C4—621no—
320C2×C4—621no—
321C2×C4—1244no—
322C2×C4C2×C4×D3112yes—
323C2×C4C2×C4×D3112yes—
324C2×C4C2×C4×D3112yes—
325C22—622no—
326C22—622no—
327C22—622no—
328C22—1241no—
329C22—1241no—
330C22—1241no—
331C22—1241no—
332C22C22×C4×D3111yescharacteristic
333C22#1175856111yescharacteristic
334C22#1175856111yescharacteristic
335C22#1175856411yescharacteristic
336C22#1175856411yescharacteristic
337C22#1175856611yescharacteristic
338C22#1175856711yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—622no—
347C4—622no—
348C4—622no—
349C4—1241no—
350C4—1241no—
351C4—1241no—
352C4—1241no—
353C2—1242no—
354C2#1452152211yescharacteristic
355C2#1452152511yescharacteristic
356C2#1452153111yescharacteristic
357C2#1452179311yescharacteristic
358C2#1452179511yescharacteristic
359C2#1452179811yescharacteristic
360C2#1452180011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#391C2
socle, cosocleC23×C31C24
prime radicalC23×D31C23
hypercenter#391D31
hypocenterC31#2378
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #14520956C2semidirect
3. #14520946C2semidirect
4. #14520944C2semidirect
9. #14520934C2semidirect
10. #14520933C2semidirect
11. #14520932C2semidirect
12. #14520931C2semidirect
15. C31×#391C2semidirect
20. #11758523C4semidirect
175. #391D31semidirect
232. C31#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#117585232
#145209562
#145209462
#145209442
#145209412
#145209402
#145209392
#145209382
#145209342
#145209332
#145209322
#145209312
#145209112
#145209082
C31×#3912
#237831