Finite Group Databasebeta

← #16855025

Subgroups — #16855025

Order 3968.

712 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758564C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3#14521627C211yesmaximal, characteristic, complemented
4#14521210C212yesmaximal, complemented
5#14521203C212yesmaximal, complemented
6#14521193C212yesmaximal, complemented
7#14521186C212yesmaximal, complemented
8#14521051C211yesmaximal, characteristic, complemented
9C2×#11758423C211yesmaximal, characteristic, complemented
10#14521017C211yesmaximal, characteristic, complemented
11C2×C31×#137C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×Q8×C31C2211yescharacteristic
14#11758564—24no—
15#11758564—24no—
16C2×#92499—21no—
17C2×#92499C2212yes—
18C2×Q8×D31—21no—
19C2×Q8×D31C2212yes—
20C2×#92495—22no—
21C2×#92495C2211yescharacteristic
22C2×#92495C2211yescharacteristic
23C31×#137—24no—
24C2×Q16×C31C2212yes—
25C2×SD16×C31C2212yes—
26C2×M16×C31C2211yescharacteristic
27#11758527—24no—
28#11758523C2212yes—
29C2×#92482C2211yescharacteristic
30C2×#92481C2211yescharacteristic
31#11758516—22no—
32#11758515C2212yes—
33C2×#92480C2211yescharacteristic
34C2×#92479C2211yescharacteristic
35#11758507—21no—
36C2×C4⋊Dic31C2211yescharacteristic
37#11758501—21no—
38C22×C31⋊C8C2211yescharacteristic
39#11758423C2212yes—
40#11758423C2212yes—
41#11758409C2212yes—
42(C2×D31)⋊C8C2212yes—
43(C2×D31)⋊C8C2212yes—
44#11758404C2212yes—
45#41×C31—22no—
46#41×C31—22no—
47C2×Q8×C31—21no—
48C2×Q8×C31—22no—
49C2×Q8×C31D414yescomplemented
50C2×Q8×C31C2311yescharacteristic
51C2×Q8×C31C2312yes—
52C2×D4×C31—21no—
53C2×D4×C31C2311yescharacteristic
54C22×C4×C31—21no—
55C22×C4×C31—21no—
56C22×C4×C31C2311yescharacteristic
57C2×C22⋊D31—21no—
58C2×C22⋊D31—22no—
59C2×C22⋊D31D412yes—
60C22×Dic31—21no—
61#92499—28no—
62#92499—24no—
63#92499—24no—
64Q8×D31—28no—
65Q8×D31—24no—
66Q8×D31—24no—
67#92495—216no—
68#92495—22no—
69#92495—22no—
70#92495—22no—
71#92495D414yescomplemented
72C2×D124—22no—
73C2×D124D411yescharacteristic
74C2×D124C2311yescharacteristic
75C2×C4×D31—22no—
76C2×C4×D31—21no—
77C2×C4×D31—22no—
78C2×C4×D31C2312yes—
79C2×Dic62—22no—
80C2×Dic62D411yescharacteristic
81C2×Dic62C2311yescharacteristic
82Q16×C31—24no—
83SD16×C31—24no—
84M16×C31—22no—
85C2×C8×C31C2312yes—
86C22⋊Dic31—22no—
87#92482—22no—
88#92481—22no—
89#92480—22no—
90#92479—22no—
91(C2×D31)⋊C4—22no—
92C4⋊Dic31—21no—
93C4⋊Dic31C2312yes—
94Dic31⋊C4—22no—
95C2×C31⋊C8—22no—
96C2×C31⋊C8C2311yescharacteristic
97C2×C31⋊C8C2311yescharacteristic
98C23×C31—21no—
99C23×C31C2×D411yescharacteristic
100C22×D31—21no—
101C22×D31C2×D412yes—
102Q8×C31—24no—
103Q8×C31—22no—
104Q8×C31—21no—
105Q8×C31—21no—
106Q8×C31C2×D414yes—
107D4×C31—21no—
108D4×C31—21no—
109D4×C31—42no—
110C2×C4×C31—24no—
111C2×C4×C31—22no—
112C2×C4×C31—21no—
113C2×C4×C31—42no—
114C2×C4×C31C2×D411yescharacteristic
115C2×C4×C31C2×D412yes—
116C2×C4×C31C2×D412yes—
117C2×C4×C31C2411yescharacteristic
118C22⋊D31—24no—
119C22⋊D31—24no—
120C22⋊D31—82no—
121C2×Dic31—21no—
122C2×Dic31—22no—
123C2×Dic31—81no—
124C2×Dic31C2×D412yes—
125D124—28no—
126D124—22no—
127D124—21no—
128D124C2×D412yes—
129C4×D31—28no—
130C4×D31—28no—
131C4×D31—24no—
132C4×D31—22no—
133C4×D31—22no—
134Dic62—28no—
135Dic62—22no—
136Dic62—21no—
137Dic62C2×D412yes—
138C8×C31—22no—
139C31⋊C8—21no—
140C31⋊C8—21no—
141#3013—311nomaximal, Sylow
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—41no—
145C22×C31—41no—
146C22×C31—41no—
147C22×C31#12012yes—
148C22×C31C22×D411yescharacteristic
149C2×D31—22no—
150C2×D31—22no—
151C2×D31—22no—
152C2×D31—22no—
153C2×D31—41no—
154C2×D31—41no—
155C4×C31—24no—
156C4×C31—41no—
157C4×C31#12012yes—
158C4×C31C22×D411yescharacteristic
159C4×C31C22×D411yescharacteristic
160Dic31—22no—
161Dic31—22no—
162Dic31—42no—
163C2×2−1+4—311no—
164C2×#137D3111yescharacteristic, complemented
165C2×#135—311no—
166#537—311no—
167#452—312no—
168#451—312no—
169#449—312no—
170#448—312no—
171#417—311no—
172C2×#103—311no—
173#408—311no—
174C2×C31—22no—
175C2×C31—81no—
176C2×C31C2×#12011yescharacteristic
177C2×C31#57711yescharacteristic
178C2×C31#57811yescharacteristic
179D31—42no—
180D31—42no—
1812−1+4—624no—
1822−1+4—624no—
183C2×#41—311no—
184C2×#41—312no—
185C2×#41—311no—
186C2×#41—622no—
187C2×#41—621no—
188C2×#41C2×D3111yescharacteristic
189C22×Q8—312no—
190C22×Q8—621no—
191C22×Q8C2×D3111yescharacteristic
192#137—24no—
193#135—624no—
194C2×Q16—311no—
195C2×Q16C2×D3112yes—
196C2×SD16—311no—
197C2×SD16—311no—
198C2×SD16C2×D3112yes—
199C2×D8—311no—
200C2×M16C2×D3111yescharacteristic
201C22×C8—311no—
202C22⋊Q8—312no—
203C22⋊Q8—621no—
204#121—312no—
205#121—621no—
206#121—622no—
207C2×C4⋊C4—311no—
208#103—312no—
209#103—312no—
210#103—312no—
211#102—312no—
212C22⋊C8—312no—
213C22⋊C8—312no—
214C31#301311yescharacteristic, complemented, Sylow
215#41—22no—
216#41—22no—
217#41—314no—
218#41—6216no—
219#41—624no—
220#41—628no—
221#41—624no—
222#41—622no—
223#41—622no—
224#41—622no—
225C2×Q8—21no—
226C2×Q8—22no—
227C2×Q8—311no—
228C2×Q8—311no—
229C2×Q8—622no—
230C2×Q8—624no—
231C2×Q8—628no—
232C2×Q8—624no—
233C2×Q8C22⋊D3114yescomplemented
234C2×Q8C22×D3111yescharacteristic
235C2×Q8C22×D3112yes—
236C2×D4—21no—
237C2×D4—312no—
238C2×D4—311no—
239C2×D4—311no—
240C2×D4—622no—
241C2×D4—621no—
242C2×D4—622no—
243C2×D4C22×D3111yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—312no—
247C22×C4—622no—
248C22×C4—622no—
249C22×C4—621no—
250C22×C4—621no—
251C22×C4C22×D3111yescharacteristic
252Q16—24no—
253Q16—622no—
254SD16—24no—
255SD16—622no—
256SD16—622no—
257D8—622no—
258M16—22no—
259C2×C8—311no—
260C2×C8—311no—
261C2×C8—622no—
262C2×C8C22×D3112yes—
263C4⋊C4—312no—
264C4⋊C4—622no—
265C4⋊C4—621no—
266C22⋊C4—622no—
267C22⋊C4—622no—
268C23—21no—
269C23—312no—
270C23—621no—
271C23D4×D3111yescharacteristic
272Q8—24no—
273Q8—22no—
274Q8—21no—
275Q8—21no—
276Q8—312no—
277Q8—628no—
278Q8—622no—
279Q8—621no—
280Q8C2×C22⋊D3114yes—
281D4—21no—
282D4—21no—
283D4—42no—
284D4—312no—
285D4—628no—
286D4—624no—
287D4—624no—
288D4—622no—
289D4—621no—
290D4—2482no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—312no—
296C2×C4—628no—
297C2×C4—628no—
298C2×C4—621no—
299C2×C4—624no—
300C2×C4—622no—
301C2×C4—622no—
302C2×C4—622no—
303C2×C4—2481no—
304C2×C4D4×D3111yescharacteristic
305C2×C4D4×D3112yes—
306C2×C4C2×C22⋊D3112yes—
307C2×C4C23×D3111yescharacteristic
308C8—22no—
309C8—621no—
310C8—621no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—622no—
317C22—622no—
318C22—622no—
319C22—622no—
320C22—1241no—
321C22—1241no—
322C22#1175851412yes—
323C22C2×D4×D3111yescharacteristic
324C4—24no—
325C4—41no—
326C4—622no—
327C4—622no—
328C4—1242no—
329C4#1175851412yes—
330C4C2×D4×D3111yescharacteristic
331C4C22×C22⋊D3111yescharacteristic
332C2—22no—
333C2—81no—
334C2—1242no—
335C2—1242no—
336C2C2×#1175851411yescharacteristic
337C2#1452185811yescharacteristic
338C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#137C2
socleC22×C31C22×D4
hypercenterC2×#137D31
hypocenterC31#3013
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. C2×#11758527C2semidirect
3. #14521627C2semidirect
4. #14521210C2semidirect
5. #14521203C2semidirect
6. #14521193C2semidirect
7. #14521186C2semidirect
8. #14521051C2semidirect
9. C2×#11758423C2semidirect
10. #14521017C2semidirect
11. C2×C31×#137C2semidirect
49. C2×Q8×C31D4semidirect
71. #92495D4semidirect
164. C2×#137D31semidirect
214. C31#3013semidirect
233. C2×Q8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
C2×#117585272
#145216272
#145212102
#145212032
#145211932
#145211862
#145210512
C2×#117584232
#145210172
C2×C31×#1372
#301331