Finite Group Databasebeta

← #29584988

Subgroups — #29584988

Order 15616.

372 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 93 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#21446616C211yesmaximal, characteristic
2#21445993C211yesmaximal, characteristic, complemented
3#21444593C211yesmaximal, characteristic, complemented
4#21444245C211yesmaximal, characteristic
5#21444243C211yesmaximal, characteristic
6#21443649C211yesmaximal, characteristic, complemented
7C61×#1571C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16839815C2211yescharacteristic
10C2×#14519712—22no—
11#16839786C2211yescharacteristic
12C2×#14519689—22no—
13C2×#14519688C2211yescharacteristic
14#16839514—22no—
15#16839491—22no—
16#16839489—22no—
17C61×#431C2211yescharacteristic
18C61×#423C2211yescharacteristic
19C2×C4⋊C8×C61C2211yescharacteristic
20C61×#335—22no—
21C22×D244Q811yescharacteristic
22C22×C4×D61—21no—
23C2×#11758194—21no—
24C22×#92418D412yes—
25C4×D244—21no—
26C4×D244—21no—
27C4×D244—21no—
28C4×D244—21no—
29C2×C4⋊Dic61Q811yescharacteristic
30C2×#11758198—21no—
31#14519712—44no—
32C4⋊#92418—21no—
33C4⋊#92418—21no—
34C4⋊#92418—21no—
35C22⋊#92418—21no—
36#14519689—44no—
37#14519688—21no—
38#14519688—21no—
39C4×#92418D412yes—
40C2×M16×C61—21no—
41C22×C8×C61—21no—
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61—21no—
44C4⋊C8×C61—21no—
45C4⋊C8×C61—21no—
46C4⋊C8×C61—21no—
47C4⋊C8×C61—21no—
48C22⋊C8×C61—21no—
49C4×C8×C61D412yes—
50C23×D61—21no—
51C22×C4×C61C2×D411yescharacteristic
52C22×C4×C61C2×Q811yescharacteristic
53C22×C4×C61#4111yescharacteristic
54C22×Dic61—21no—
55C2×D244—21no—
56C2×D244—21no—
57C2×D244—42no—
58C2×C4×D61—41no—
59C2×C4×D61—41no—
60C2×C4×D61—42no—
61C2×C4×D61—42no—
62M16×C61—42no—
63C2×C8×C61—22no—
64C2×C8×C61—41no—
65C2×C8×C61—41no—
66C2×C8×C61—42no—
67C42×C61C2×D411yescharacteristic
68C42×C61C2×D411yescharacteristic
69C42×C61#4111yescharacteristic
70C42×C61#4111yescharacteristic
71#11758198—41no—
72#11758198—41no—
73C4⋊Dic61—21no—
74C4⋊Dic61—21no—
75#11758194—42no—
76C2×#92418—22no—
77C2×#92418—22no—
78C2×#92418—22no—
79C2×#92418—41no—
80C2×#92418D814yes—
81C23×C61C22⋊Q811yescharacteristic
82C22×D61—22no—
83C22×D61—41no—
84C22×D61—42no—
85C22×D61—41no—
86C22×D61—42no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61C22⋊Q811yescharacteristic
90C2×C4×C61C22⋊Q811yescharacteristic
91C2×C4×C61#12311yescharacteristic
92C2×C4×C61#12311yescharacteristic
93C2×C4×C61#12311yescharacteristic
94C2×C4×C61#12711yescharacteristic
95C2×C4×C61C2×D812yes—
96C2×C4×C61#13712yes—
97C2×Dic61—41no—
98C2×Dic61—42no—
99D244—22no—
100D244—22no—
101D244—41no—
102D244—41no—
103C4×D61—82no—
104C4×D61—81no—
105C4×D61—81no—
106C8×C61—42no—
107C8×C61—81no—
108C8×C61—81no—
109#92418—24no—
110#92418—42no—
111#92418—81no—
112#10660—611nomaximal, Sylow
113C22×C61—21no—
114C22×C61—21no—
115C22×C61#39911yescharacteristic
116C22×C61#47412yes—
117C2×D61—44no—
118C2×D61—42no—
119C2×D61—82no—
120C2×D61—81no—
121C2×D61—81no—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—21no—
125C4×C61—21no—
126C4×C61#48012yes—
127C4×C61#49311yescharacteristic
128C4×C61#49711yescharacteristic
129Dic61—81no—
130#1955—611no—
131#1849—611no—
132#1571—611no—
133#1571D6111yescharacteristic, complemented
134#1486—611no—
135#1486—611no—
136#1275—611no—
137C2×C61—22no—
138C2×C61#209111yescharacteristic
139C2×C61#209311yescharacteristic
140C2×C61#209811yescharacteristic
141D61—82no—
142C2×C4×D4—611no—
143#431—611no—
144#431C2×D6111yescharacteristic
145C2×#108—1222no—
146#423—611no—
147#423C2×D6111yescharacteristic
148C2×C4⋊C8C2×D6111yescharacteristic
149C2×C22⋊C8—1222no—
150C2×#97—611no—
151#335—22no—
152#335—1222no—
153D4⋊C8—1222no—
154D4⋊C8—1222no—
155C61#1066011yescharacteristic, complemented, Sylow
156C22×D4—611no—
157C23×C4—1221no—
158C2×M16—21no—
159C2×M16—1221no—
160C22×C8—21no—
161C22×C8—612no—
162C4×D4—1221no—
163C4×D4—1221no—
164C4×D4—1221no—
165C4×D4—1221no—
166C2×C4⋊C4—611no—
167C2×C22⋊C4—1221no—
168C2×C42C22×D6111yescharacteristic
169#108—2444no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—1221no—
176C4⋊C8—1221no—
177C4⋊C8—1221no—
178C22⋊C8—21no—
179C22⋊C8—1221no—
180C22⋊C8—2444no—
181#97—1221no—
182#97—1221no—
183C4×C8—612no—
184C4×C8C22⋊D6112yes—
185C24—1221no—
186C2×D4—1221no—
187C2×D4—1221no—
188C2×D4—2442no—
189C22×C4—1221no—
190C22×C4—2442no—
191C22×C4—2441no—
192C22×C4—2441no—
193C22×C4—2442no—
194C22×C4#1175822211yescharacteristic
195C22×C4D4×D6111yescharacteristic
196C22×C4Q8×D6111yescharacteristic
197M16—42no—
198M16—2442no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—614no—
204C2×C8—1222no—
205C2×C8—1222no—
206C2×C8—1222no—
207C2×C8—2441no—
208C4⋊C4—1221no—
209C4⋊C4—1221no—
210C22⋊C4—2441no—
211C22⋊C4—2441no—
212C42#1175822211yescharacteristic
213C42D4×D6111yescharacteristic
214C42#1175822611yescharacteristic
215C42C2×C22⋊D6111yescharacteristic
216C23—1222no—
217C23—2441no—
218C23—2442no—
219C23—2441no—
220C23—2442no—
221C23#1451975511yescharacteristic
222D4—1222no—
223D4—1222no—
224D4—2441no—
225D4—2441no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—4882no—
229C2×C4—4881no—
230C2×C4—4881no—
231C2×C4—2441no—
232C2×C4—2442no—
233C2×C4#1451975311yescharacteristic
234C2×C4#1451975511yescharacteristic
235C2×C4#1451976011yescharacteristic
236C2×C4#1451976211yescharacteristic
237C2×C4#1451976211yescharacteristic
238C2×C4#1451977311yescharacteristic
239C2×C4D8×D6112yes—
240C2×C4#1451980912yes—
241C8—42no—
242C8—81no—
243C8—81no—
244C8—1224no—
245C8—4881no—
246C8—2442no—
247C22—21no—
248C22—21no—
249C22—4882no—
250C22—4881no—
251C22—4881no—
252C22—2444no—
253C22—2442no—
254C22#1683969711yescharacteristic
255C22#1684004112yes—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—4881no—
261C4#1684007712yes—
262C4#1684013711yescharacteristic
263C4#1684015511yescharacteristic
264C2—22no—
265C2—4882no—
266C2#2144733411yescharacteristic
267C2#2144734411yescharacteristic
268C2#2144737511yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1571C2
socleC22×C61#399
prime radicalC22×D244Q8
hypercenter#1571D61
hypocenterC61#10660
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
2. #21445993C2semidirect
3. #21444593C2semidirect
6. #21443649C2semidirect
7. C61×#1571C2semidirect
133. #1571D61semidirect
155. C61#10660semidirect

Maximal subgroups

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

subgroupindex
#214466162
#214459932
#214445932
#214442452
#214442432
#214436492
C61×#15712
#1066061