Finite Group Databasebeta

← #29575340

Subgroups — #29575340

Order 15616.

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

normal subgroups only

This group has 426 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#21454921C211yesmaximal, characteristic, complemented
2#21444414C211yesmaximal, characteristic, complemented
3#21444236C212yesmaximal
4#21443656C212yesmaximal, complemented
5C61×#3131C211yesmaximal, characteristic
6#16839893—22no—
7Q8×#92418—22no—
8#16839823—22no—
9#16839815C2212yes—
10C4×#11758194C2212yes—
11#16839504—22no—
12D4⋊#92418—22no—
13D4⋊#92418—22no—
14C61×#533C2211yescharacteristic
15C61×#532C2211yescharacteristic
16C61×#488—22no—
17C61×#487C412yes—
18C61×#486C412yes—
19C61×#416—22no—
20C2×C4×C8×C61C2211yescharacteristic
21C2×#11758194—22no—
22C4⋊#92418—24no—
23C4⋊#92418D412yes—
24C4⋊#92418D412yes—
25C22⋊#92418—22no—
26#14519688—22no—
27#14519688—22no—
28C4×#92418—22no—
29C4×#92418—22no—
30C4×#92418—22no—
31C2×#41×C61—21no—
32C22×C8×C61—21no—
33C4⋊Q8×C61C2×C411yescharacteristic
34C4⋊Q8×C61C2×C412yes—
35#127×C61C2×C411yescharacteristic
36#125×C61D412yes—
37#124×C61—21no—
38C22⋊Q8×C61—22no—
39#121×C61—22no—
40C4×Q8×C61—22no—
41C4×D4×C61—22no—
42#117×C61D412yes—
43C2×C42×C61C2311yescharacteristic
44#103×C61—24no—
45#102×C61—24no—
46C4×C8×C61—21no—
47C4×C8×C61C2×C412yes—
48#41×C61—44no—
49C2×Q8×C61—22no—
50C2×Q8×C61—21no—
51C2×D4×C61—21no—
52C2×D4×C61—22no—
53C2×D4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61#4112yes—
57C2×C8×C61—22no—
58C2×C8×C61—42no—
59C4⋊C4×C61—22no—
60C4⋊C4×C61—24no—
61C4⋊C4×C61—22no—
62C4⋊C4×C61C22⋊C414yes—
63C22⋊C4×C61—22no—
64C22⋊C4×C61—42no—
65C42×C61—22no—
66C42×C61C22×C411yescharacteristic
67C42×C61C2×D411yescharacteristic
68C42×C61C2×D412yes—
69#11758194—24no—
70#11758194—42no—
71C2×#92418—22no—
72C2×#92418—22no—
73C2×#92418—42no—
74C23×C61—41no—
75C23×C61#12311yescharacteristic
76Q8×C61—42no—
77Q8×C61—42no—
78Q8×C61—42no—
79D4×C61—42no—
80D4×C61—42no—
81D4×C61—42no—
82D4×C61—82no—
83C2×C4×C61—22no—
84C2×C4×C61—22no—
85C2×C4×C61—22no—
86C2×C4×C61—24no—
87C2×C4×C61—41no—
88C2×C4×C61—82no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C2×C4×C61C4×D412yes—
91C2×C4×C61#12011yescharacteristic
92C2×C4×C61#12112yes—
93C8×C61—42no—
94#92418—44no—
95#92418—42no—
96#92418—42no—
97#9070—611nomaximal, Sylow
98C22×C61—22no—
99C22×C61—21no—
100C22×C61—82no—
101C22×C61—81no—
102C22×C61#38611yescharacteristic
103C4×C61—22no—
104C4×C61—21no—
105C4×C61—44no—
106C4×C61—81no—
107C4×C61#44312yes—
108C4×C61#44412yes—
109C4×C61#45412yes—
110#3131D6111yescharacteristic
111#2986—611no—
112#1526—611no—
113#1484—612no—
114#1278—612no—
115C2×C61—41no—
116C2×C61—161no—
117C2×C61#187612yes—
118C2×C61#188811yescharacteristic
119#533C2×D6111yescharacteristic
120#532C2×D6111yescharacteristic
121#488—22no—
122#487Dic6112yes—
123#486Dic6112yes—
124#446—1222no—
125C8×Q8—1222no—
126#433—1222no—
127#431—612no—
128#416—22no—
129C4×M16—612no—
130C2×C4×C8C2×D6111yescharacteristic
131#330—1222no—
132D4⋊C8—1222no—
133D4⋊C8—1222no—
134C61#907011yescharacteristic, complemented, Sylow
135C2×#41—21no—
136C2×M16—1222no—
137C22×C8—21no—
138C4⋊Q8C2×Dic6111yescharacteristic
139C4⋊Q8C2×Dic6112yes—
140#127C2×Dic6111yescharacteristic
141#125C22⋊D6112yes—
142#124—21no—
143C22⋊Q8—22no—
144#121—22no—
145C4×Q8—22no—
146C4×D4—22no—
147#117C22⋊D6112yes—
148C2×C42C22×D6111yescharacteristic
149C4⋊C8—612no—
150C4⋊C8—612no—
151C4⋊C8—1224no—
152#103—24no—
153#102—24no—
154C22⋊C8—1222no—
155#97—1222no—
156#97—1222no—
157C4×C8—21no—
158C4×C8—1222no—
159C4×C8—1222no—
160C4×C8—1222no—
161C4×C8C2×Dic6112yes—
162#41—44no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—41no—
169C22×C4#1175822412yes—
170C22×C4C2×C22⋊D6111yescharacteristic
171M16—1224no—
172M16—2442no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—1222no—
176C2×C8—1222no—
177C2×C8—2442no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4C22⋊Dic6114yes—
182C22⋊C4—22no—
183C22⋊C4—42no—
184C42—22no—
185C42D4×D6112yes—
186C42C22×Dic6111yescharacteristic
187C42C2×C22⋊D6111yescharacteristic
188C23—41no—
189C23#1451975811yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—41no—
202C2×C4—82no—
203C2×C4C2×C22⋊Dic6111yescharacteristic
204C2×C4D4×Dic6112yes—
205C2×C4#1451974511yescharacteristic
206C2×C4#1451974912yes—
207C8—42no—
208C8—2444no—
209C8—2442no—
210C8—2442no—
211C22—22no—
212C22—21no—
213C22—82no—
214C22—81no—
215C22#1683964211yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—81no—
220C4#1683987912yes—
221C4#1683988412yes—
222C4#1683993412yes—
223C2—41no—
224C2—161no—
225C2#2144614812yes—
226C2#2144622311yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3131C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3131D61
hypocenterC61#9070
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #21454921C2semidirect
2. #21444414C2semidirect
4. #21443656C2semidirect
134. C61#9070semidirect

Maximal subgroups

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

subgroupindex
#214549212
#214444142
#214442362
#214436562
C61×#31312
#907061