Finite Group Databasebeta

← #29593719

Subgroups — #29593719

Order 15616.

322 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 73 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×D488C211yesmaximal, characteristic
2#21448047C211yesmaximal, characteristic, complemented
3#21447865C211yesmaximal, characteristic
4#21445189C211yesmaximal, characteristic, complemented
5#21443873C211yesmaximal, characteristic, complemented
6#21443781C211yesmaximal, characteristic
7C61×Q8⋊C16C211yesmaximal, characteristic, complemented
8Q8×D244—21no—
9C2×#14519722—21no—
10#16840167—21no—
11#16840167C2211yescharacteristic
12#16840142C2211yescharacteristic
13#16840047—21no—
14#16840043—22no—
15#16840043—21no—
16C4×D488—21no—
17C4×D488C2211yescharacteristic
18#16839567—22no—
19#16839567—21no—
20#16839515C2211yescharacteristic
21#16839509—21no—
22C8×Q8×C61C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C4×C16×C61C2211yescharacteristic
25C2×Q8×D61—41no—
26C2×Dic244—21no—
27C2×Dic244—21no—
28C2×D488D411yescharacteristic
29#14519777—22no—
30#14519777—21no—
31#14519777—21no—
32#14519757—42no—
33#14519757—41no—
34C4×D244—22no—
35C4×D244—21no—
36#14519722—42no—
37#14519711—22no—
38#14519711—21no—
39#14519711D412yes—
40#14519711D411yescharacteristic
41#14519708—41no—
42C4⋊#92418—21no—
43#14519701—41no—
44#14519699—22no—
45#14519699—21no—
46C4×Q8×C61D411yescharacteristic
47C2×C16×C61—21no—
48C2×C16×C61—21no—
49C4⋊C8×C61—21no—
50C4⋊C8×C61D411yescharacteristic
51C4×C8×C61—21no—
52C4×C8×C61C2311yescharacteristic
53C2×Q8×C61D811yescharacteristic
54Q8×D61—82no—
55Q8×D61—82no—
56C2×D244—21no—
57C2×C4×D61—42no—
58C2×C4×D61—41no—
59C2×Dic122—42no—
60C2×Dic122—41no—
61C2×Dic122—41no—
62C2×C8×C61—21no—
63C2×C8×C61C2×D411yescharacteristic
64C2×C8×C61C2×D411yescharacteristic
65C4⋊C4×C61—21no—
66C4⋊C4×C61D811yescharacteristic
67C42×C61—21no—
68C42×C61C2×D411yescharacteristic
69#11758198—42no—
70#11758198—41no—
71C4⋊Dic61—22no—
72C4⋊Dic61—21no—
73C4⋊Dic61—42no—
74C4⋊Dic61—41no—
75C4⋊Dic61—41no—
76C4⋊Dic61—41no—
77C2×#92418—41no—
78Dic244—42no—
79Dic244—41no—
80Dic244—41no—
81D488—21no—
82D488SD1612yes—
83C16×C61—41no—
84C16×C61—41no—
85C22×D61—41no—
86Q8×C61—21no—
87Q8×C61SD3212yescomplemented
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61#12011yescharacteristic
91C2×C4×C61C2×D811yescharacteristic
92C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
93C2×Dic61—41no—
94C2×Dic61—42no—
95C2×Dic61—81no—
96D244—22no—
97D244—41no—
98C4×D61—84no—
99C4×D61—81no—
100C4×D61—81no—
101Dic122—84no—
102Dic122—81no—
103Dic122—81no—
104Dic122—81no—
105Dic122—81no—
106C8×C61—21no—
107C8×C61—41no—
108C8×C61C2×SD1611yescharacteristic
109C8×C61#13711yescharacteristic
110#92418—81no—
111#12356—611nomaximal, Sylow
112C22×C61#44711yescharacteristic
113C2×D61—42no—
114C2×D61—81no—
115C2×D61—81no—
116C4×C61—22no—
117C4×C61—21no—
118C4×C61—41no—
119C4×C61#44811yescharacteristic
120C4×C61#45411yescharacteristic
121C4×C61C2×SD3211yescharacteristic
122C4×C61#50911yescharacteristic
123Dic61—81no—
124Dic61—82no—
125Dic61—161no—
126Q8×D8—611no—
127#2241—611no—
128#2208—611no—
129#1670—611no—
130#1364—611no—
131Q8⋊C16D6111yescharacteristic, complemented
132#1331—611no—
133C2×C61#162711yescharacteristic
134C2×C61#218711yescharacteristic
135C2×C61#219511yescharacteristic
136D61—82no—
137D4×Q8—1221no—
138C2×SD32—1221no—
139#500—611no—
140#500—1221no—
141#494—611no—
142#475—1221no—
143#474—1222no—
144#474—1221no—
145C8×Q8C2×D6111yescharacteristic
146C4×D8—611no—
147C4×D8—1221no—
148C4⋊C16C2×D6111yescharacteristic
149#358—1221no—
150#358—1222no—
151C4×C16C2×D6111yescharacteristic
152#335—611no—
153#332—1221no—
154C61#1235611yescharacteristic, complemented, Sylow
155C22×Q8—2441no—
156C2×Q16—1221no—
157C2×Q16—1221no—
158C2×D8—611no—
159C4⋊Q8—1222no—
160C4⋊Q8—1221no—
161C4⋊Q8—1221no—
162C22⋊Q8—2442no—
163C22⋊Q8—2441no—
164C4×Q8D24411yescharacteristic
165C4×D4—1222no—
166C4×D4—1221no—
167SD32—2442no—
168C2×C16—21no—
169C2×C16—21no—
170#107—611no—
171#107—612no—
172#107—1222no—
173#107—1221no—
174#106—2441no—
175C4⋊C8—21no—
176C4⋊C8—1221no—
177C4⋊C8D24411yescharacteristic
178#103—2441no—
179#102—1222no—
180#102—1221no—
181C4×C8—21no—
182C4×C8C22×D6111yescharacteristic
183C2×Q8—4882no—
184C2×Q8—4882no—
185C2×Q8—2442no—
186C2×Q8—2441no—
187C2×Q8—2441no—
188C2×Q8D48811yescharacteristic
189C2×D4—1221no—
190C22×C4—2442no—
191C22×C4—2441no—
192Q16—2442no—
193Q16—2441no—
194Q16—2441no—
195D8—612no—
196D8—1221no—
197C2×C8—21no—
198C2×C8—2441no—
199C2×C8D4×D6111yescharacteristic
200C2×C8D4×D6111yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—1222no—
203C4⋊C4—1221no—
204C4⋊C4—2442no—
205C4⋊C4—2441no—
206C4⋊C4—2441no—
207C4⋊C4—2441no—
208C4⋊C4D48811yescharacteristic
209C22⋊C4—2442no—
210C22⋊C4—2441no—
211C42—21no—
212C42C2×D24411yescharacteristic
213C16—41no—
214C16—41no—
215C23—2441no—
216Q8—21no—
217Q8—4884no—
218Q8—4881no—
219Q8—4881no—
220Q8—4881no—
221Q8—4881no—
222Q8#1451972212yescomplemented
223D4—1222no—
224D4—2441no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—4884no—
228C2×C4—4881no—
229C2×C4—4881no—
230C2×C4—4881no—
231C2×C4—2442no—
232C2×C4—2441no—
233C2×C4#1451974711yescharacteristic
234C2×C4C2×D48811yescharacteristic
235C2×C4#1451980511yescharacteristic
236C8—21no—
237C8—41no—
238C8—4881no—
239C8SD16×D6111yescharacteristic
240C8#1451980911yescharacteristic
241C22—4881no—
242C22—4881no—
243C22—2442no—
244C22#1683989811yescharacteristic
245C4—22no—
246C4—21no—
247C4—41no—
248C4—9761no—
249C4—4882no—
250C4—4881no—
251C4#1683990211yescharacteristic
252C4#1683993611yescharacteristic
253C4C2×#1451972211yescharacteristic
254C4#1684020011yescharacteristic
255C2—4882no—
256C2#2144492411yescharacteristic
257C2#2144775011yescharacteristic
258C2#2144779911yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×Q8⋊C16C2
socleC22×C61#447
prime radicalC2×D488D4
hypercenterQ8⋊C16D61
hypocenterC61#12356
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
2. #21448047C2semidirect
4. #21445189C2semidirect
5. #21443873C2semidirect
7. C61×Q8⋊C16C2semidirect
87. Q8×C61SD32semidirect
131. Q8⋊C16D61semidirect
154. C61#12356semidirect
222. Q8#14519722semidirect

Maximal subgroups

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

subgroupindex
Q8×D4882
#214480472
#214478652
#214451892
#214438732
#214437812
C61×Q8⋊C162
#1235661