Finite Group Databasebeta

← #16846012

Subgroups — #16846012

Order 3968.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D31C211yesmaximal, characteristic, complemented
2#14521561C211yesmaximal, characteristic
3C2×#11758393C211yesmaximal, characteristic, complemented
4C2×#11758413C211yesmaximal, characteristic, complemented
5C4×(C2×D31)⋊C4C211yesmaximal, characteristic, complemented
6#14520808C211yesmaximal, characteristic, complemented
7C31×#339C211yesmaximal, characteristic, complemented
8C2×#92495C2211yescharacteristic
9C22×C4×D31C2211yescharacteristic
10C2×M16×C31C2211yescharacteristic
11C31×#117C2211yescharacteristic
12C2×C42×C31C2211yescharacteristic
13C2×(C2×D31)⋊C4—21no—
14C2×C4×Dic31C2211yescharacteristic
15C2×#92474C2211yescharacteristic
16M16×D31—22no—
17M16×D31—22no—
18C2×#92469—21no—
19C2×C8×D31—21no—
20#11758474C411yescharacteristic, complemented
21#11758474C411yescharacteristic, complemented
22#11758472C411yescharacteristic, complemented
23#11758472C411yescharacteristic, complemented
24#11758463—21no—
25#11758459—21no—
26C4×D124—21no—
27C4×D124—21no—
28C31⋊#95—21no—
29#11758413—22no—
30#11758393—22no—
31C23×D31D411yescharacteristic
32C22×C4×C31—21no—
33C22×C4×C31C2311yescharacteristic
34C2×C22⋊D31—21no—
35C22×Dic31—21no—
36C22×Dic31D411yescharacteristic
37#92495—21no—
38#92495—21no—
39#92495—21no—
40#92495—21no—
41C2×D124—21no—
42C2×D124—21no—
43C2×D124C2×C411yescharacteristic
44C2×C4×D31—21no—
45C2×C4×D31—21no—
46C2×C4×D31—21no—
47C2×C4×D31—21no—
48C2×C4×D31—21no—
49C2×C4×D31C2×C411yescharacteristic
50C2×C4×D31C2×C411yescharacteristic
51C2×C4×D31D411yescharacteristic
52C2×C4×D31D411yescharacteristic
53C2×Dic62C2×C411yescharacteristic
54M16×C31—21no—
55M16×C31D412yescomplemented
56C2×C8×C31—21no—
57C4⋊C4×C31C2×C411yescharacteristic
58C4⋊C4×C31C2×C411yescharacteristic
59C22⋊C4×C31—21no—
60C42×C31—21no—
61C42×C31—21no—
62C42×C31—21no—
63(C2×D31)⋊C4—24no—
64(C2×D31)⋊C4—21no—
65(C2×D31)⋊C4—21no—
66(C2×D31)⋊C4—21no—
67(C2×D31)⋊C4—21no—
68C4⋊Dic31—21no—
69C4⋊Dic31—21no—
70C4×Dic31—21no—
71C4×Dic31—21no—
72#92474—21no—
73#92474D412yescomplemented
74C2×C31⋊C8—21no—
75#92469—24no—
76C8×D31—24no—
77C23×C31C2×D411yescharacteristic
78C22×D31—21no—
79C22×D31—21no—
80C22×D31—22no—
81C22×D31—21no—
82C22×D31—41no—
83C22×D31C22⋊C411yescharacteristic
84C22×D31C22⋊C411yescharacteristic
85C2×C4×C31—22no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—41no—
89C2×C4×C31—41no—
90C2×C4×C31C22×C411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31#4111yescharacteristic
95C2×C4×C31#4111yescharacteristic
96C22⋊D31—41no—
97C22⋊D31—41no—
98C2×Dic31—21no—
99C2×Dic31—21no—
100C2×Dic31—22no—
101C2×Dic31—21no—
102C2×Dic31—41no—
103C2×Dic31—41no—
104C2×Dic31C22⋊C411yescharacteristic
105C2×Dic31C22⋊C411yescharacteristic
106D124—21no—
107D124—21no—
108D124—42no—
109D124—42no—
110C4×D31—21no—
111C4×D31—21no—
112C4×D31—21no—
113C4×D31—21no—
114C4×D31—21no—
115C4×D31—21no—
116C4×D31—21no—
117C4×D31—21no—
118C4×D31—41no—
119C4×D31—41no—
120Dic62—21no—
121Dic62—21no—
122C8×C31—22no—
123C31⋊C8—22no—
124#1871—311nomaximal, Sylow
125C22×C31—21no—
126C22×C31—21no—
127C22×C31C2×C22⋊C411yescharacteristic
128C22×C31#12111yescharacteristic
129C22×C31#12111yescharacteristic
130C2×D31—21no—
131C2×D31—21no—
132C2×D31—21no—
133C2×D31—21no—
134C2×D31—21no—
135C2×D31—42no—
136C2×D31—42no—
137C2×D31—41no—
138C2×D31—41no—
139C2×D31—41no—
140C2×D31#10412yescomplemented
141C4×C31—42no—
142C4×C31—41no—
143C4×C31—41no—
144C4×C31C4×D411yescharacteristic
145C4×C31C4×D411yescharacteristic
146C4×C31#12011yescharacteristic
147C4×C31#12311yescharacteristic
148Dic31—21no—
149Dic31—21no—
150Dic31—42no—
151Dic31—41no—
152C22×M16—311no—
153#525—311no—
154C2×#104—311no—
155C2×#104—311no—
156C4×C22⋊C4—311no—
157#339—311no—
158#339D3111yescharacteristic, complemented
159C2×C31—21no—
160C2×C31—21no—
161C2×C31#38611yescharacteristic
162C2×C31C2×#10411yescharacteristic
163C2×C31#42111yescharacteristic
164D31—22no—
165D31—41no—
166D31—81no—
167C2×#41—311no—
168C23×C4—311no—
169C2×M16—311no—
170C2×M16—621no—
171C2×M16—622no—
172C2×M16—622no—
173C2×M16C2×D3111yescharacteristic
174C22×C8—621no—
175#123—621no—
176C22⋊Q8—311no—
177C22⋊Q8—311no—
178#121—311no—
179#121—311no—
180#120—621no—
181C4×D4—621no—
182C4×D4—621no—
183#117C2×D3111yescharacteristic
184C2×C22⋊C4—621no—
185C2×C42—311no—
186C2×C42C2×D3111yescharacteristic
187#104—622no—
188#104—622no—
189#95—621no—
190C31#187111yescharacteristic, complemented, Sylow
191C24—311no—
192#41—621no—
193#41—621no—
194#41—621no—
195#41—621no—
196C2×Q8—311no—
197C2×D4—311no—
198C2×D4—621no—
199C2×D4—621no—
200C2×D4—621no—
201C22×C4—21no—
202C22×C4—311no—
203C22×C4—311no—
204C22×C4—311no—
205C22×C4—311no—
206C22×C4—311no—
207C22×C4—621no—
208C22×C4—621no—
209C22×C4—621no—
210C22×C4—621no—
211C22×C4—621no—
212C22×C4—621no—
213C22×C4C22×D3111yescharacteristic
214M16—21no—
215M16—312no—
216M16—624no—
217M16—621no—
218M16D12412yescomplemented
219C2×C8—21no—
220C2×C8—624no—
221C2×C8—621no—
222C4⋊C4—621no—
223C4⋊C4—621no—
224C4⋊C4C4×D3111yescharacteristic
225C4⋊C4C4×D3111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—621no—
228C22⋊C4—621no—
229C22⋊C4—621no—
230C22⋊C4—621no—
231C22⋊C4—624no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—621no—
236C42—621no—
237C23—311no—
238C23—311no—
239C23—622no—
240C23—621no—
241C23—621no—
242C23—621no—
243C23—1241no—
244C23D4×D3111yescharacteristic
245Q8—621no—
246Q8—621no—
247D4—621no—
248D4—621no—
249D4—1242no—
250D4—1242no—
251D4—1241no—
252D4—1241no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—311no—
259C2×C4—311no—
260C2×C4—621no—
261C2×C4—621no—
262C2×C4—621no—
263C2×C4—621no—
264C2×C4—621no—
265C2×C4—621no—
266C2×C4—621no—
267C2×C4—621no—
268C2×C4—621no—
269C2×C4—621no—
270C2×C4—622no—
271C2×C4—621no—
272C2×C4—1241no—
273C2×C4—1241no—
274C2×C4—1241no—
275C2×C4—1241no—
276C2×C4C2×C4×D3111yescharacteristic
277C2×C4C2×D12411yescharacteristic
278C2×C4#9249511yescharacteristic
279C2×C4D4×D3111yescharacteristic
280C2×C4D4×D3111yescharacteristic
281C2×C4#9249911yescharacteristic
282C8—22no—
283C8—622no—
284C22—21no—
285C22—21no—
286C22—312no—
287C22—621no—
288C22—621no—
289C22—621no—
290C22—621no—
291C22—621no—
292C22—1242no—
293C22—1242no—
294C22—1241no—
295C22—1241no—
296C22—1241no—
297C22C22⋊C4×D3111yescharacteristic
298C22#1175846111yescharacteristic
299C22#1175847211yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—621no—
304C4—621no—
305C4—1242no—
306C4—1241no—
307C4C4×D12411yescharacteristic
308C4#1175845911yescharacteristic
309C4#1175847011yescharacteristic
310C4#1175847111yescharacteristic
311C2—21no—
312C2—21no—
313C2—622no—
314C2—1241no—
315C2—2481no—
316C2#1452092711yescharacteristic
317C2#104×D3111yescharacteristic
318C2#1452106411yescharacteristic
subgroupG/H
centerC2×C4D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#339C2
socleC22×C31C2×C22⋊C4
prime radical#14521561C2
cosocleC22×C4×C31C23
hypercenter#339D31
hypocenterC31#1871
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×M16×D31C2semidirect
3. C2×#11758393C2semidirect
4. C2×#11758413C2semidirect
5. C4×(C2×D31)⋊C4C2semidirect
6. #14520808C2semidirect
7. C31×#339C2semidirect
20. #11758474C4semidirect
21. #11758474C4semidirect
22. #11758472C4semidirect
23. #11758472C4semidirect
55. M16×C31D4semidirect
73. #92474D4semidirect
140. C2×D31#104semidirect
158. #339D31semidirect
190. C31#1871semidirect
218. M16D124semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D312
#145215612
C2×#117583932
C2×#117584132
C4×(C2×D31)⋊C42
#145208082
C31×#3392
#187131