Finite Group Databasebeta

← #29574518

Subgroups — #29574518

Order 15616.

412 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 111 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#21454886C211yesmaximal, characteristic
2#21446648C211yesmaximal, characteristic
3#21444608C212yesmaximal, complemented
4#21443651C212yesmaximal, complemented
5C61×#1543C211yesmaximal, characteristic
6#16839840—22no—
7D4×#92418—22no—
8#16839815C2212yes—
9#16839786C2211yescharacteristic
10C2×#14519704—22no—
11#16839729—22no—
12C42×Dic61C2211yescharacteristic
13#16839515—22no—
14#16839513—22no—
15C61×#532C2211yescharacteristic
16C61×#431C2212yes—
17C61×#333C412yescomplemented
18C61×#331C411yescharacteristic, complemented
19C61×#331C411yescharacteristic, complemented
20C2×#11758194—22no—
21C2×#11758194—21no—
22C22×#92418—22no—
23C2×C4×Dic61—21no—
24C2×C4×Dic61—22no—
25C4⋊#92418—22no—
26C4⋊#92418D412yes—
27C4⋊#92418D412yes—
28#14519704—44no—
29C22⋊#92418—24no—
30C22⋊#92418—22no—
31C4×#92418—22no—
32C2×#41×C61D412yes—
33C4⋊Q8×C61C2×C411yescharacteristic
34#127×C61C2×C411yescharacteristic
35#124×C61Q812yes—
36#121×C61—22no—
37C4×D4×C61—22no—
38C2×C42×C61C2311yescharacteristic
39C4⋊C8×C61C2×C412yes—
40C4⋊C8×C61C2×C412yes—
41C22⋊C8×C61—22no—
42C4×C8×C61—22no—
43#41×C61—42no—
44#41×C61—42no—
45C2×Q8×C61C4⋊C412yes—
46C2×D4×C61—22no—
47C2×D4×C61—22no—
48C2×D4×C61C4⋊C412yes—
49C22×C4×C61—22no—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61#4112yes—
52C22×Dic61—41no—
53C2×C8×C61—22no—
54C2×C8×C61—22no—
55C4⋊C4×C61—22no—
56C22⋊C4×C61—24no—
57C42×C61C22×C411yescharacteristic
58C42×C61C2×D412yes—
59C42×C61C2×Q811yescharacteristic
60C4×Dic61—22no—
61C4×Dic61—24no—
62C4×Dic61—42no—
63C4×Dic61—41no—
64#11758194—42no—
65#11758194—82no—
66C2×#92418—22no—
67C2×#92418—22no—
68C2×#92418—41no—
69C2×#92418—82no—
70C23×C61—22no—
71C23×C61#12311yescharacteristic
72Q8×C61—42no—
73D4×C61—42no—
74D4×C61—42no—
75D4×C61—42no—
76D4×C61—82no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—22no—
82C2×C4×C61—42no—
83C2×C4×C61—42no—
84C2×C4×C61C2×C4⋊C411yescharacteristic
85C2×C4×C61C4×D412yes—
86C2×C4×C61#12011yescharacteristic
87C2×C4×C61C22⋊Q812yes—
88C2×Dic61—42no—
89C2×Dic61—44no—
90C8×C61—42no—
91C8×C61—42no—
92#92418—42no—
93#92418—42no—
94#92418—81no—
95#8934—611nomaximal, Sylow
96C22×C61—21no—
97C22×C61—22no—
98C22×C61—42no—
99C22×C61—44no—
100C22×C61#38511yescharacteristic
101C4×C61—21no—
102C4×C61—22no—
103C4×C61—42no—
104C4×C61#44312yes—
105C4×C61#44412yes—
106C4×C61#45311yescharacteristic
107C4×C61#45511yescharacteristic
108Dic61—44no—
109#2979—611no—
110#1962—611no—
111#1574—612no—
112#1543D6111yescharacteristic
113#1276—612no—
114C2×C61—41no—
115C2×C61—82no—
116C2×C61#185611yescharacteristic
117C2×C61#186612yes—
118#532C2×D6111yescharacteristic
119#436—1222no—
120C8×D4—1222no—
121#431—612no—
122#431C2×D6112yes—
123#423—611no—
124C2×#104—1222no—
125#408—1222no—
126C43—611no—
127#335—1222no—
128#334—1222no—
129#333Dic6112yescomplemented
130#331Dic6111yescharacteristic, complemented
131#331Dic6111yescharacteristic, complemented
132C61#893411yescharacteristic, complemented, Sylow
133C2×#41D24412yes—
134C2×M16—1222no—
135C2×M16—1221no—
136C22×C8—1222no—
137C4⋊Q8C2×Dic6111yescharacteristic
138#127C2×Dic6111yescharacteristic
139#124Dic12212yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—1221no—
143C2×C42—1222no—
144C2×C42C22×D6111yescharacteristic
145C4⋊C8—612no—
146C4⋊C8—612no—
147C4⋊C8—1222no—
148C4⋊C8C2×Dic6112yes—
149C4⋊C8C2×Dic6112yes—
150#104—2444no—
151C22⋊C8—22no—
152C22⋊C8—1224no—
153C22⋊C8—1222no—
154C4×C8—22no—
155C4×C8—1222no—
156#41—42no—
157#41—42no—
158C2×Q8C4⋊Dic6112yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4C4⋊Dic6112yes—
162C22×C4—22no—
163C22×C4—2441no—
164C22×C4C2×D24411yescharacteristic
165C22×C4#1175822412yes—
166M16—4882no—
167M16—2442no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—1222no—
171C2×C8—1222no—
172C2×C8—4882no—
173C2×C8—2441no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—1222no—
177C42—1224no—
178C42—2441no—
179C42—2442no—
180C42C2×Dic12211yescharacteristic
181C42D4×D6112yes—
182C42C22×Dic6111yescharacteristic
183C23—22no—
184C23#1451976111yescharacteristic
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—2442no—
198C2×C4—2444no—
199C2×C4C2×C4⋊Dic6111yescharacteristic
200C2×C4D4×Dic6112yes—
201C2×C4#1451974711yescharacteristic
202C2×C4#1451975412yes—
203C8—42no—
204C8—42no—
205C8—4881no—
206C8—2442no—
207C8—2442no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22#1683963711yescharacteristic
213C4—21no—
214C4—22no—
215C4—42no—
216C4—2444no—
217C4#1683987912yes—
218C4#1683988412yes—
219C4#1683993211yescharacteristic
220C4#1683994211yescharacteristic
221C2—41no—
222C2—82no—
223C2#2144603111yescharacteristic
224C2#2144609612yes—
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1543C2
socleC22×C61#385
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#1543D61
hypocenterC61#8934
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
3. #21444608C2semidirect
4. #21443651C2semidirect
17. C61×#333C4semidirect
18. C61×#331C4semidirect
19. C61×#331C4semidirect
129. #333Dic61semidirect
130. #331Dic61semidirect
131. #331Dic61semidirect
132. C61#8934semidirect

Maximal subgroups

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

subgroupindex
#214548862
#214466482
#214446082
#214436512
C61×#15432
#893461