Finite Group Databasebeta

← #29593554

Subgroups — #29593554

Order 15616.

324 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#21458832C211yesmaximal, characteristic
2#21448076C211yesmaximal, characteristic
3#21447866C211yesmaximal, characteristic, complemented
4#21443864C211yesmaximal, characteristic
5Q8⋊#11758185C211yesmaximal, characteristic, complemented
6#21443782C211yesmaximal, characteristic, complemented
7C61×#1670C211yesmaximal, characteristic, complemented
8#16840497—21no—
9C2×#14519723—21no—
10C2×#14519720—22no—
11#16840146—21no—
12#16840139C2211yescharacteristic
13#16839978—22no—
14#16839963—21no—
15C4×Dic244C2211yescharacteristic
16C4×D488—21no—
17C4⋊#11758185C2211yescharacteristic
18#16839566—21no—
19C4×#11758185C2211yescharacteristic
20C61×#494C2211yescharacteristic
21C61×#475—21no—
22C8×Q8×C61C2211yescharacteristic
23C61×#335C2211yescharacteristic
24C61×#332—21no—
25C2×Dic244D411yescharacteristic
26C2×#11758190—22no—
27C2×D488—21no—
28C2×D488D412yes—
29#14519774—21no—
30#14519766—41no—
31#14519765—22no—
32#14519762—42no—
33C4×Dic122—21no—
34C4×D244—22no—
35#14519739—41no—
36#14519723—42no—
37#14519720—44no—
38C2×#11758185—21no—
39C2×#11758185—21no—
40#14519711D411yescharacteristic
41#14519702—21no—
42#14519699—22no—
43C2×Q16×C61—21no—
44C2×Q16×C61—21no—
45C4⋊Q8×C61—21no—
46C4×Q8×C61D411yescharacteristic
47#106×C61—41no—
48C4⋊C8×C61—21no—
49C4⋊C8×C61—21no—
50C4⋊C8×C61D411yescharacteristic
51#103×C61—41no—
52C4×C8×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—41no—
55C2×Q8×C61D811yescharacteristic
56C2×D244—22no—
57C2×D244—41no—
58C2×D244—41no—
59C2×C4×D61—42no—
60C2×Dic122—21no—
61Q16×C61—42no—
62Q16×C61—41no—
63Q16×C61—41no—
64C2×C8×C61—21no—
65C2×C8×C61—41no—
66C2×C8×C61C2×D411yescharacteristic
67C2×C8×C61C2×D411yescharacteristic
68C4⋊C4×C61—21no—
69C4⋊C4×C61—41no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61D811yescharacteristic
72C42×C61—21no—
73C42×C61C2×D411yescharacteristic
74#11758198—42no—
75#11758198—42no—
76#11758198—42no—
77C4⋊Dic61—21no—
78#11758196—41no—
79C4×Dic61—41no—
80Dic244—21no—
81Dic244—21no—
82D488—42no—
83D488—41no—
84D488—41no—
85#11758190—82no—
86#11758185—22no—
87#11758185—41no—
88C22×D61—42no—
89Q8×C61—21no—
90Q8×C61—21no—
91Q8×C61—81no—
92Q8×C61—81no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—81no—
96C2×C4×C61#12011yescharacteristic
97C2×C4×C61C2×D811yescharacteristic
98C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
99C2×Dic61—41no—
100C2×Dic61—41no—
101D244—42no—
102D244—42no—
103D244—81no—
104D244—81no—
105D244—161no—
106C4×D61—162no—
107Dic122—41no—
108Dic122—41no—
109C8×C61—21no—
110C8×C61—41no—
111C8×C61—81no—
112C8×C61#13511yescharacteristic
113C8×C61AGL(1,ℤ/8)11yescharacteristic
114#12333—611nomaximal, Sylow
115C22×C61#44711yescharacteristic
116C2×D61—82no—
117C2×D61—82no—
118C2×D61—82no—
119C4×C61—21no—
120C4×C61—41no—
121C4×C61—41no—
122C4×C61—161no—
123C4×C61#44911yescharacteristic
124C4×C61#45311yescharacteristic
125C4×C61#50811yescharacteristic
126C4×C61#51011yescharacteristic
127Dic61—81no—
128Dic61—81no—
129#3396—611no—
130#2248—611no—
131#2209—611no—
132#1670D6111yescharacteristic, complemented
133#1361—611no—
134Q8⋊C16—611no—
135#1332—611no—
136C2×C61#162111yescharacteristic
137C2×C61#218911yescharacteristic
138C2×C61#219411yescharacteristic
139D61—162no—
140#559—1221no—
141C2×Q32—1221no—
142C2×SD32—1222no—
143#495—1221no—
144#494C2×D6111yescharacteristic
145#493—611no—
146#475—21no—
147#463—1222no—
148#460—1221no—
149C8×Q8C2×D6111yescharacteristic
150C4×Q16—611no—
151C4×D8—1221no—
152C4⋊C16—611no—
153#358—1221no—
154C4×C16—611no—
155#335C2×D6111yescharacteristic
156#332—21no—
157C61#1233311yescharacteristic, complemented, Sylow
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—611no—
161C2×SD16—1222no—
162C2×D8—612no—
163C2×D8—1221no—
164C4⋊Q8—21no—
165#127—1221no—
166#124—1222no—
167#124—2441no—
168#123—2442no—
169C4×Q8—1221no—
170C4×Q8C22⋊D6111yescharacteristic
171C4×D4—1222no—
172C4×D4—2441no—
173Q32—2442no—
174SD32—2444no—
175C2×C16—1221no—
176C2×C16—1221no—
177#107—611no—
178#106—41no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8C22⋊D6111yescharacteristic
182#103—41no—
183#103—1221no—
184#102—1222no—
185C4×C8—21no—
186C4×C8C22×D6111yescharacteristic
187C2×Q8—41no—
188C2×Q8—1221no—
189C2×Q8#1175819911yescharacteristic
190C2×D4—1222no—
191C2×D4—2441no—
192C2×D4—2441no—
193C22×C4—2442no—
194Q16—42no—
195Q16—41no—
196Q16—41no—
197Q16—1221no—
198Q16—1221no—
199SD16—4882no—
200D8—2442no—
201D8—2441no—
202D8—2441no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8D4×D6111yescharacteristic
206C2×C8D4×D6111yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—1221no—
211C4⋊C4—2441no—
212C4⋊C4#1175819911yescharacteristic
213C22⋊C4—2442no—
214C22⋊C4—2442no—
215C22⋊C4—2442no—
216C42—21no—
217C42—2441no—
218C42C2×C22⋊D6111yescharacteristic
219C16—1222no—
220C16—2441no—
221C23—2442no—
222Q8—21no—
223Q8—21no—
224Q8—81no—
225Q8—81no—
226Q8—2441no—
227Q8—2441no—
228D4—9761no—
229D4—4881no—
230D4—4881no—
231D4—2442no—
232D4—2442no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—81no—
236C2×C4—9762no—
237C2×C4—2441no—
238C2×C4—2441no—
239C2×C4#1451974511yescharacteristic
240C2×C4C2×#1175819911yescharacteristic
241C2×C4#1451980211yescharacteristic
242C8—21no—
243C8—41no—
244C8—81no—
245C8#1451979811yescharacteristic
246C8#1451980411yescharacteristic
247C22—4882no—
248C22—4882no—
249C22—4882no—
250C22#1683989511yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—161no—
255C4—4881no—
256C4—4881no—
257C4#1683990611yescharacteristic
258C4#1683992811yescharacteristic
259C4#1684019211yescharacteristic
260C4#1684020311yescharacteristic
261C2—9762no—
262C2#2144488211yescharacteristic
263C2#2144775711yescharacteristic
264C2#2144778811yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1670C2
socleC22×C61#447
prime radical#16840139C22
hypercenter#1670D61
hypocenterC61#12333
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
3. #21447866C2semidirect
5. Q8⋊#11758185C2semidirect
6. #21443782C2semidirect
7. C61×#1670C2semidirect
132. #1670D61semidirect
157. C61#12333semidirect

Maximal subgroups

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

subgroupindex
#214588322
#214480762
#214478662
#214438642
Q8⋊#117581852
#214437822
C61×#16702
#1233361