Finite Group Databasebeta

← #16858425

Subgroups — #16858425

Order 3968.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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
1Q8×D124C211yesmaximal, characteristic, complemented
2#14521408C212yesmaximal, complemented
3#14521399C211yesmaximal, characteristic, complemented
4#14521344C212yesmaximal, complemented
5#14521329C211yesmaximal, characteristic, complemented
6#14521261C212yesmaximal
7#14521251C211yesmaximal, characteristic, complemented
8#14521180C211yesmaximal, characteristic, complemented
9#14521147C212yesmaximal, complemented
10C4×#92479C211yesmaximal, characteristic
11C31×#556C211yesmaximal, characteristic, complemented
12C2×Q8×D31—21no—
13C31×C4⋊Q8C2211yescharacteristic
14C31×#126—22no—
15C31×#124C2212yes—
16C31×C22⋊Q8—21no—
17C4×Q8×C31C2212yes—
18C4×Q8×C31C2211yescharacteristic
19C4×D4×C31C2211yescharacteristic
20C31×#117—21no—
21C2×#92481C2212yes—
22C2×#92479C2211yescharacteristic
23#11758474—21no—
24#11758474—22no—
25C4×D124C2211yescharacteristic
26C4×D124C2212yes—
27#11758446C2212yes—
28#11758446C2211yescharacteristic
29#11758423C2212yes—
30#11758423C2212yes—
31#11758423C2212yes—
32#11758420C2211yescharacteristic
33#11758420C2212yes—
34#11758397C2211yescharacteristic
35#11758395C2211yescharacteristic
36#11758395C2212yes—
37C4⋊(C31⋊C8)C2212yes—
38C4⋊(C31⋊C8)C2211yescharacteristic
39#11758391C2212yes—
40C4×C31⋊C8C2211yescharacteristic
41C2×Q8×C31D411yescharacteristic
42C2×Q8×C31C2312yes—
43C2×D4×C31C2311yescharacteristic
44C22×C4×C31—21no—
45Q8×D31—44no—
46C2×D124C2311yescharacteristic
47C2×C4×D31—22no—
48C2×C4×D31—21no—
49C2×Dic62—22no—
50C2×Dic62—21no—
51C4⋊C4×C31—21no—
52C4⋊C4×C31—22no—
53C4⋊C4×C31—21no—
54C4⋊C4×C31D411yescharacteristic
55C4⋊C4×C31D412yes—
56C4⋊C4×C31C2311yescharacteristic
57C22⋊C4×C31—22no—
58C22⋊C4×C31—21no—
59C22⋊C4×C31—22no—
60C42×C31—22no—
61C42×C31C2311yescharacteristic
62C42×C31C2312yes—
63#92481—24no—
64#92479—22no—
65(C2×D31)⋊C4—22no—
66(C2×D31)⋊C4—21no—
67C4⋊Dic31—22no—
68C4⋊Dic31—21no—
69C4⋊Dic31C2311yescharacteristic
70C4⋊Dic31C2312yes—
71C2×C31⋊C8C2311yescharacteristic
72C2×C31⋊C8C2311yescharacteristic
73C2×C31⋊C8C2312yes—
74C23×C31—21no—
75C22×D31—21no—
76Q8×C31—22no—
77Q8×C31—22no—
78Q8×C31—22no—
79D4×C31—21no—
80D4×C31—21no—
81C2×C4×C31—21no—
82C2×C4×C31—22no—
83C2×C4×C31—42no—
84C2×C4×C31C2×D412yes—
85C2×C4×C31C2×D411yescharacteristic
86C2×C4×C31C2×D412yes—
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2411yescharacteristic
89C2×Dic31—21no—
90C2×Dic31—22no—
91D124—21no—
92D124#4112yes—
93C4×D31—44no—
94C4×D31—42no—
95Dic62—44no—
96Dic62—42no—
97C31⋊C8—21no—
98C31⋊C8—22no—
99C31⋊C8—21no—
100#3367—311nomaximal, Sylow
101C22×C31—41no—
102C22×C31—41no—
103C22×C31—41no—
104C22×C31C22×D411yescharacteristic
105C2×D31—22no—
106C2×D31—41no—
107C2×D31—41no—
108C4×C31—22no—
109C4×C31—22no—
110C4×C31—21no—
111C4×C31—42no—
112C4×C31—41no—
113C4×C31AGL(1,ℤ/8)12yes—
114C4×C31C2×#4111yescharacteristic
115C4×C312−1+411yescharacteristic
116Dic31—41no—
117Dic31—42no—
118#556D3111yescharacteristic, complemented
119D4×Q8—311no—
120#489—312no—
121#487—311no—
122#477—312no—
123#474—311no—
124#463—312no—
125#461—311no—
126#446—311no—
127#440—312no—
128C4×D8—311no—
129C2×C31—81no—
130C2×C31#54811yescharacteristic
131C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C31#57811yescharacteristic
133D31—42no—
134C22×Q8—621no—
135C2×SD16—312no—
136C2×D8—311no—
137C4⋊Q8—311no—
138C4⋊Q8—312no—
139C4⋊Q8C2×D3111yescharacteristic
140#126—22no—
141#124C2×D3112yes—
142C22⋊Q8—21no—
143C22⋊Q8—621no—
144C22⋊Q8—622no—
145C4×Q8C2×D3112yes—
146C4×Q8C2×D3111yescharacteristic
147C4×D4—311no—
148C4×D4—312no—
149C4×D4C2×D3111yescharacteristic
150#117—21no—
151#107—311no—
152#107—312no—
153C4⋊C8—311no—
154C4⋊C8—312no—
155#103—312no—
156#103—312no—
157#103—312no—
158#102—311no—
159#102—312no—
160#102—311no—
161#97—312no—
162C4×C8—311no—
163C31#336711yescharacteristic, complemented, Sylow
164C2×Q8—622no—
165C2×Q8—621no—
166C2×Q8—1244no—
167C2×Q8C22⋊D3111yescharacteristic
168C2×Q8C22×D3112yes—
169C2×D4—311no—
170C2×D4C22×D3111yescharacteristic
171C22×C4—21no—
172C22×C4—622no—
173C22×C4—621no—
174SD16—624no—
175D8—622no—
176C2×C8—311no—
177C2×C8—311no—
178C2×C8—312no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—311no—
183C4⋊C4—312no—
184C4⋊C4—622no—
185C4⋊C4—621no—
186C4⋊C4C22⋊D3111yescharacteristic
187C4⋊C4C22⋊D3112yes—
188C4⋊C4C22×D3111yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—622no—
193C22⋊C4—621no—
194C42—22no—
195C42C22×D3111yescharacteristic
196C42C22×D3112yes—
197C23—21no—
198C23—621no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—1244no—
203Q8—1242no—
204D4—21no—
205D4—21no—
206D4—312no—
207D4—621no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—622no—
212C2×C4—621no—
213C2×C4—1244no—
214C2×C4—1242no—
215C2×C4C2×C22⋊D3112yes—
216C2×C4C2×C22⋊D3111yescharacteristic
217C2×C4C2×C22⋊D3112yes—
218C2×C4C2×C22⋊D3111yescharacteristic
219C2×C4C23×D3111yescharacteristic
220C8—621no—
221C8—622no—
222C8—621no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—622no—
227C22—1241no—
228C22—1241no—
229C22C22×C22⋊D3111yescharacteristic
230C4—22no—
231C4—21no—
232C4—22no—
233C4—42no—
234C4—41no—
235C4—1242no—
236C4—1241no—
237C4#1175852612yes—
238C4#41×D3111yescharacteristic
239C4#1175856711yescharacteristic
240C2—81no—
241C2—1242no—
242C2#1452170611yescharacteristic
243C2C2×#1175852611yescharacteristic
244C2#1452186411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#556C2
socleC22×C31C22×D4
prime radicalC2×#92479C22
hypercenter#556D31
hypocenterC31#3367
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D124C2semidirect
2. #14521408C2semidirect
3. #14521399C2semidirect
4. #14521344C2semidirect
5. #14521329C2semidirect
7. #14521251C2semidirect
8. #14521180C2semidirect
9. #14521147C2semidirect
11. C31×#556C2semidirect
118. #556D31semidirect
163. C31#3367semidirect

Maximal subgroups

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

subgroupindex
Q8×D1242
#145214082
#145213992
#145213442
#145213292
#145212612
#145212512
#145211802
#145211472
C4×#924792
C31×#5562
#336731