Finite Group Databasebeta

← #29575322

Subgroups — #29575322

Order 15616.

444 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456141C211yesmaximal, characteristic
2#21454886C211yesmaximal, characteristic
3#21444241C212yesmaximal, complemented
4#21443657C212yesmaximal, complemented
5C61×#1525C211yesmaximal, characteristic
6#16840319C2211yescharacteristic
7#16840116—22no—
8#16840112C412yes—
9#16840108C412yes—
10#16839840—22no—
11D4×#92418—22no—
12#16839815C2212yes—
13#16839758—22no—
14#16839729—22no—
15C2×C4×#92418C2211yescharacteristic
16#16839494—22no—
17#16839494—22no—
18C61×#532C2211yescharacteristic
19C4×M16×C61C2212yes—
20C61×#329—22no—
21C2×#11758194—22no—
22C22×#92418—21no—
23C22×#92418—22no—
24#14519777C2×C412yes—
25#14519769—21no—
26#14519756—22no—
27C4×Dic122—22no—
28#14519736—21no—
29C4⋊#92418D412yes—
30C4⋊#92418D412yes—
31#14519701—24no—
32#14519698—24no—
33C22⋊#92418—24no—
34C22⋊#92418—22no—
35C4×#92418—21no—
36C4×#92418—22no—
37C4×#92418C2×C412yes—
38C2×#41×C61D412yes—
39C2×M16×C61—22no—
40C4⋊Q8×C61C2×C411yescharacteristic
41#127×C61C2×C411yescharacteristic
42#124×C61D412yes—
43#121×C61—22no—
44C4×D4×C61—22no—
45C2×C42×C61C2311yescharacteristic
46#97×C61—22no—
47C4×C8×C61—22no—
48#41×C61—42no—
49#41×C61—42no—
50C2×Q8×C61C22⋊C412yes—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61C22⋊C412yes—
54C22×C4×C61—22no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61#4112yes—
57C2×Dic122—22no—
58M16×C61—24no—
59M16×C61—42no—
60C2×C8×C61—42no—
61C4⋊C4×C61—22no—
62C22⋊C4×C61—24no—
63C42×C61C22×C411yescharacteristic
64C42×C61C2×D411yescharacteristic
65C42×C61C2×D412yes—
66C22⋊Dic61—41no—
67C4⋊Dic61—22no—
68C4⋊Dic61—22no—
69#11758196—42no—
70C4×Dic61—41no—
71#11758194—82no—
72C2×#92418—22no—
73C2×#92418—22no—
74C2×#92418—22no—
75C2×#92418—41no—
76C2×#92418—41no—
77C2×#92418—82no—
78C23×C61—22no—
79C23×C61#12311yescharacteristic
80Q8×C61—42no—
81D4×C61—42no—
82D4×C61—42no—
83D4×C61—42no—
84D4×C61—82no—
85C2×C4×C61—22no—
86C2×C4×C61—22no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—22no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C2×C4×C61C4×D412yes—
94C2×C4×C61#12011yescharacteristic
95C2×C4×C61#12112yes—
96C2×Dic61—42no—
97Dic122—42no—
98Dic122—42no—
99C8×C61—44no—
100#92418—42no—
101#92418—42no—
102#92418—42no—
103#9067—611nomaximal, Sylow
104C22×C61—22no—
105C22×C61—21no—
106C22×C61—44no—
107C22×C61—42no—
108C22×C61#38611yescharacteristic
109C4×C61—22no—
110C4×C61—21no—
111C4×C61—42no—
112C4×C61#44312yes—
113C4×C61#44412yes—
114C4×C61#45511yescharacteristic
115C4×C61#45511yescharacteristic
116Dic61—82no—
117#3131—611no—
118#2979—611no—
119#1525D6111yescharacteristic
120#1485—612no—
121#1278—612no—
122C2×C61—41no—
123C2×C61—82no—
124C2×C61#187612yes—
125C2×C61#188511yescharacteristic
126#533—611no—
127#532C2×D6111yescharacteristic
128#488—1222no—
129#487—612no—
130#486—612no—
131#436—1222no—
132C8×D4—1222no—
133#431—612no—
134#416—1222no—
135#408—1222no—
136C4×M16C2×D6112yes—
137C2×C4×C8—611no—
138#329—22no—
139Q8⋊C8—1222no—
140Q8⋊C8—1222no—
141C61#906711yescharacteristic, complemented, Sylow
142C2×#41D24412yes—
143C2×M16—22no—
144C2×M16—1222no—
145C22×C8—1222no—
146C22×C8—1221no—
147C4⋊Q8—612no—
148C4⋊Q8C4×D6111yescharacteristic
149#127C4×D6111yescharacteristic
150#125—1221no—
151#124C22⋊D6112yes—
152C22⋊Q8—1222no—
153#121—22no—
154C4×Q8—1222no—
155C4×D4—22no—
156#117—1221no—
157C2×C42C22×D6111yescharacteristic
158C4⋊C8—612no—
159C4⋊C8—612no—
160#103—1224no—
161#102—1224no—
162C22⋊C8—1224no—
163C22⋊C8—1222no—
164#97—22no—
165C4×C8—22no—
166C4×C8—612no—
167C4×C8—1222no—
168C4×C8—1221no—
169#41—42no—
170#41—42no—
171C2×Q8—1222no—
172C2×Q8#1175819812yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4#1175819812yes—
176C22×C4—22no—
177C22×C4C2×D24411yescharacteristic
178C22×C4#1175822412yes—
179M16—24no—
180M16—42no—
181M16—4882no—
182C2×C8—42no—
183C2×C8—1222no—
184C2×C8—1222no—
185C2×C8—1222no—
186C2×C8—4882no—
187C2×C8—2441no—
188C2×C8—2441no—
189C4⋊C4—22no—
190C4⋊C4—1222no—
191C4⋊C4—1222no—
192C4⋊C4—2442no—
193C22⋊C4—24no—
194C22⋊C4—2441no—
195C42—2441no—
196C42C2×C4×D6111yescharacteristic
197C42D4×D6112yes—
198C42C2×C22⋊D6111yescharacteristic
199C23—22no—
200C23#1451976111yescharacteristic
201Q8—42no—
202Q8—2442no—
203Q8—2442no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—2442no—
216C2×C4C2×#1175819811yescharacteristic
217C2×C4#1451973712yes—
218C2×C4#1451974711yescharacteristic
219C2×C4#1451974912yes—
220C8—44no—
221C8—2442no—
222C8—2442no—
223C8—2442no—
224C22—22no—
225C22—21no—
226C22—44no—
227C22—42no—
228C22#1683963911yescharacteristic
229C4—22no—
230C4—21no—
231C4—42no—
232C4—4882no—
233C4#1683987612yes—
234C4#1683988112yes—
235C4#1683994211yescharacteristic
236C4#1683994211yescharacteristic
237C2—41no—
238C2—82no—
239C2#2144614412yes—
240C2#2144620511yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1525C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#1525D61
hypocenterC61#9067
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
3. #21444241C2semidirect
4. #21443657C2semidirect
141. C61#9067semidirect

Maximal subgroups

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

subgroupindex
#214561412
#214548862
#214442412
#214436572
C61×#15252
#906761