Finite Group Databasebeta

← #16846751

Subgroups — #16846751

Order 3968.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#11758509C211yesmaximal, characteristic, complemented
2#14521627C211yesmaximal, characteristic, complemented
3C2×#11758425C211yesmaximal, characteristic, complemented
4C2×#11758413C211yesmaximal, characteristic, complemented
5#14520952C211yesmaximal, characteristic, complemented
6#14520817C211yesmaximal, characteristic, complemented
7C2×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C22×D4×C31C2211yescharacteristic
10C2×#92495C2211yescharacteristic
11C31×AGL(1,ℤ/8)—22no—
12C31×AGL(1,ℤ/8)—22no—
13C2×SD16×C31—21no—
14C2×D8×C31—21no—
15C2×M16×C31C2211yescharacteristic
16C2×C22⋊Dic31—22no—
17#11758523—21no—
18#11758516—22no—
19#11758515—21no—
20C2×#92480—21no—
21#11758509—22no—
22#11758509—22no—
23C2×#92479—21no—
24#11758507—21no—
25C2×C4⋊Dic31C2211yescharacteristic
26#11758501—21no—
27C2×C4×Dic31C2211yescharacteristic
28C2×#92474C2211yescharacteristic
29#11758425—22no—
30#11758413—22no—
31C24×C31—21no—
32#41×C31—21no—
33#41×C31—21no—
34#41×C31—21no—
35#41×C31—21no—
36C2×Q8×C31D411yescharacteristic
37C2×D4×C31—21no—
38C2×D4×C31—21no—
39C2×D4×C31—21no—
40C2×D4×C31—21no—
41C2×D4×C31—21no—
42C2×D4×C31—21no—
43C2×D4×C31D411yescharacteristic
44C2×D4×C31D412yescomplemented
45C22×C4×C31—21no—
46C22×C4×C31C2311yescharacteristic
47C2×C22⋊D31—21no—
48C2×C22⋊D31—22no—
49C22×Dic31—21no—
50C22×Dic31D412yes—
51#92495—21no—
52#92495—21no—
53#92495—21no—
54#92495—21no—
55C2×D124D411yescharacteristic
56C2×C4×D31—21no—
57C2×Dic62D411yescharacteristic
58SD16×C31—24no—
59D8×C31—24no—
60M16×C31—21no—
61M16×C31D412yescomplemented
62C2×C8×C31—21no—
63C22⋊Dic31—22no—
64C22⋊Dic31—44no—
65#92480—24no—
66#92479—24no—
67(C2×D31)⋊C4—22no—
68C4⋊Dic31—21no—
69C4⋊Dic31—21no—
70Dic31⋊C4—22no—
71C4×Dic31—21no—
72C4×Dic31—21no—
73#92474—21no—
74#92474D412yescomplemented
75C2×C31⋊C8—21no—
76C23×C31—21no—
77C23×C31—22no—
78C23×C31—42no—
79C23×C31—42no—
80C23×C31—41no—
81C23×C31—41no—
82C23×C31C2×D411yescharacteristic
83C22×D31—21no—
84Q8×C31—21no—
85Q8×C31—21no—
86D4×C31—22no—
87D4×C31—21no—
88D4×C31—22no—
89D4×C31—21no—
90D4×C31—41no—
91D4×C31—41no—
92D4×C31—41no—
93D4×C31—41no—
94C2×C4×C31—21no—
95C2×C4×C31—41no—
96C2×C4×C31—41no—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31#4111yescharacteristic
103C22⋊D31—41no—
104C22⋊D31—41no—
105C22⋊D31—82no—
106C2×Dic31—22no—
107C2×Dic31—21no—
108C2×Dic31—22no—
109C2×Dic31—41no—
110C2×Dic31—41no—
111C2×Dic31—42no—
112D124—21no—
113D124—21no—
114C4×D31—41no—
115C4×D31—41no—
116Dic62—21no—
117Dic62—21no—
118C8×C31—22no—
119C31⋊C8—22no—
120#2006—311nomaximal, Sylow
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—44no—
124C22×C31—41no—
125C22×C31—42no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31—82no—
129C22×C31—81no—
130C22×C31—81no—
131C22×C31#12011yescharacteristic
132C22×C31#12111yescharacteristic
133C22×C31#12711yescharacteristic
134C2×D31—41no—
135C2×D31—41no—
136C2×D31—41no—
137C4×C31—41no—
138C4×C31#12011yescharacteristic
139C4×C31#12011yescharacteristic
140C4×C31#12111yescharacteristic
141C4×C31#12111yescharacteristic
142Dic31—42no—
143Dic31—41no—
144Dic31—42no—
145C2×AGL(1,ℤ/8)—311no—
146C2×AGL(1,ℤ/8)D3111yescharacteristic, complemented
147#537—311no—
148C2×#104—311no—
149C2×#104—311no—
150#390—311no—
151#343—311no—
152C2×C31—21no—
153C2×C31—21no—
154C2×C31—81no—
155C2×C31—82no—
156C2×C31#39211yescharacteristic
157C2×C31#45311yescharacteristic
158C2×C31#45411yescharacteristic
159D31—81no—
160C2×#41—311no—
161C2×#41C2×D3111yescharacteristic
162C22×D4C2×D3111yescharacteristic
163AGL(1,ℤ/8)—22no—
164AGL(1,ℤ/8)—22no—
165AGL(1,ℤ/8)—622no—
166AGL(1,ℤ/8)—622no—
167C2×SD16—21no—
168C2×SD16—621no—
169C2×D8—21no—
170C2×D8—621no—
171C2×M16—311no—
172C2×M16C2×D3111yescharacteristic
173C22⋊Q8—621no—
174C22⋊Q8—621no—
175#121—621no—
176#121—622no—
177#121—621no—
178C2×C4⋊C4—311no—
179C2×C22⋊C4—622no—
180C2×C42—311no—
181#104—622no—
182#104—622no—
183C31#200611yescharacteristic, complemented, Sylow
184C24—21no—
185#41—21no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—621no—
190#41—621no—
191#41—621no—
192#41—621no—
193C2×Q8—311no—
194C2×Q8C22⋊D3111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—311no—
202C2×D4—621no—
203C2×D4—622no—
204C2×D4C22⋊D3111yescharacteristic
205C2×D4C22⋊D3112yescomplemented
206C22×C4—21no—
207C22×C4—312no—
208C22×C4—621no—
209C22×C4—621no—
210C22×C4C22×D3111yescharacteristic
211SD16—24no—
212SD16—624no—
213D8—24no—
214D8—624no—
215M16—21no—
216M16—312no—
217M16—621no—
218M16C22⋊D3112yescomplemented
219C2×C8—21no—
220C2×C8—621no—
221C4⋊C4—622no—
222C4⋊C4—621no—
223C4⋊C4—621no—
224C22⋊C4—622no—
225C22⋊C4—622no—
226C22⋊C4—1244no—
227C42—621no—
228C42—621no—
229C23—21no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—621no—
236C23D4×D3111yescharacteristic
237Q8—21no—
238Q8—21no—
239Q8—621no—
240Q8—621no—
241D4—22no—
242D4—21no—
243D4—21no—
244D4—22no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—621no—
250D4—621no—
251D4—1241no—
252D4—1241no—
253D4—2482no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—622no—
258C2×C4—622no—
259C2×C4—621no—
260C2×C4—1241no—
261C2×C4—1241no—
262C2×C4—1241no—
263C2×C4—1241no—
264C2×C4—1242no—
265C2×C4D4×D3111yescharacteristic
266C2×C4D4×D3111yescharacteristic
267C2×C4#9249711yescharacteristic
268C2×C4C2×C22⋊D3111yescharacteristic
269C2×C4C2×C22⋊D3111yescharacteristic
270C2×C4C2×C22⋊D3111yescharacteristic
271C8—22no—
272C8—622no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22—1241no—
284C22—1241no—
285C22—1241no—
286C22#1175851411yescharacteristic
287C22#1175851611yescharacteristic
288C22#1175851711yescharacteristic
289C4—41no—
290C4—1242no—
291C4—1241no—
292C4—1242no—
293C4#1175851411yescharacteristic
294C4#1175851511yescharacteristic
295C4#1175851611yescharacteristic
296C4#1175853011yescharacteristic
297C2—21no—
298C2—21no—
299C2—82no—
300C2—81no—
301C2—2481no—
302C2#1452095711yescharacteristic
303C2#1452121511yescharacteristic
304C2#1452122011yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
FittingC2×C31×AGL(1,ℤ/8)C2
socleC22×C31#120
hypercenterC2×AGL(1,ℤ/8)D31
hypocenterC31#2006
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11758509C2semidirect
2. #14521627C2semidirect
3. C2×#11758425C2semidirect
4. C2×#11758413C2semidirect
5. #14520952C2semidirect
6. #14520817C2semidirect
7. C2×C31×AGL(1,ℤ/8)C2semidirect
44. C2×D4×C31D4semidirect
61. M16×C31D4semidirect
74. #92474D4semidirect
146. C2×AGL(1,ℤ/8)D31semidirect
183. C31#2006semidirect
205. C2×D4C22⋊D31semidirect
218. M16C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585092
#145216272
C2×#117584252
C2×#117584132
#145209522
#145208172
C2×C31×AGL(1,ℤ/8)2
#200631