Finite Group Databasebeta

← #29575324

Subgroups — #29575324

Order 15616.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#21444411C211yesmaximal, characteristic
3#21444239C212yesmaximal, complemented
4#21443655C212yesmaximal, complemented
5C61×#2979C211yesmaximal, characteristic
6#16840319C2211yescharacteristic
7#16840116—22no—
8#16840112C412yes—
9#16840108C412yes—
10#16839758—22no—
11C4×#11758194C2212yes—
12C2×C4×#92418C2211yescharacteristic
13#16839502—22no—
14#16839492—22no—
15#16839492—22no—
16C61×#532C2211yescharacteristic
17C61×#436—22no—
18C8×D4×C61—22no—
19C61×#431C2212yes—
20C61×#408—22no—
21C2×#11758194—22no—
22C22×#92418—21no—
23#14519777C2×C412yes—
24#14519769—21no—
25#14519756—22no—
26C4×Dic122—22no—
27#14519736—21no—
28#14519701—24no—
29#14519698—24no—
30#14519688—22no—
31C4×#92418—21no—
32C4×#92418—22no—
33C4×#92418C2×C412yes—
34C2×#41×C61D412yes—
35C2×M16×C61—22no—
36C22×C8×C61—22no—
37C4⋊Q8×C61C2×C411yescharacteristic
38#127×C61C2×C411yescharacteristic
39#124×C61D412yes—
40#121×C61—22no—
41C4×D4×C61—22no—
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61D412yes—
44C4⋊C8×C61D412yes—
45C22⋊C8×C61—24no—
46C22⋊C8×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—82no—
59C2×C8×C61—22no—
60C2×C8×C61—22no—
61C2×C8×C61—82no—
62C4⋊C4×C61—22no—
63C22⋊C4×C61—24no—
64C42×C61C22×C411yescharacteristic
65C42×C61C2×D411yescharacteristic
66C42×C61C2×D412yes—
67C22⋊Dic61—41no—
68C4⋊Dic61—22no—
69C4⋊Dic61—22no—
70#11758196—42no—
71C4×Dic61—41no—
72#11758194—24no—
73#11758194—42no—
74C2×#92418—22no—
75C2×#92418—41no—
76C2×#92418—41no—
77C2×#92418—42no—
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—42no—
100C8×C61—42no—
101#92418—42no—
102#92418—44no—
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#2979D6111yescharacteristic
119#1525—611no—
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—22no—
132C8×D4—22no—
133#431C2×D6112yes—
134#416—1222no—
135#408—22no—
136C4×M16—612no—
137C2×C4×C8—611no—
138#329—1222no—
139Q8⋊C8—1222no—
140Q8⋊C8—1222no—
141C61#906711yescharacteristic, complemented, Sylow
142C2×#41C22⋊D6112yes—
143C2×M16—22no—
144C2×M16—1222no—
145C22×C8—22no—
146C22×C8—1221no—
147C4⋊Q8—612no—
148C4⋊Q8C4×D6111yescharacteristic
149#127C4×D6111yescharacteristic
150#125—1221no—
151#124D24412yes—
152C22⋊Q8—1222no—
153#121—22no—
154C4×Q8—1222no—
155C4×D4—22no—
156#117—1221no—
157C2×C42C22×D6111yescharacteristic
158C4⋊C8C22⋊D6112yes—
159C4⋊C8C22⋊D6112yes—
160#103—1224no—
161#102—1224no—
162C22⋊C8—24no—
163C22⋊C8—22no—
164#97—1222no—
165C4×C8—22no—
166C4×C8—612no—
167C4×C8—1221no—
168C4×C8—1222no—
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×C4#1175822212yes—
178C22×C4C2×C22⋊D6111yescharacteristic
179M16—82no—
180M16—1224no—
181M16—2442no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—82no—
185C2×C8—1222no—
186C2×C8—2441no—
187C2×C8—2441no—
188C2×C8—2442no—
189C4⋊C4—22no—
190C4⋊C4—1222no—
191C4⋊C4—1222no—
192C4⋊C4—2442no—
193C22⋊C4—24no—
194C22⋊C4—2441no—
195C42—2441no—
196C42C2×C4×D6111yescharacteristic
197C42C2×D24411yescharacteristic
198C42C2×C22⋊D6112yes—
199C23—22no—
200C23#1451976011yescharacteristic
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×C4C4×C22⋊D6112yes—
218C2×C4#1451974611yescharacteristic
219C2×C4#1451975112yes—
220C8—42no—
221C8—42no—
222C8—2442no—
223C8—2444no—
224C22—22no—
225C22—21no—
226C22—44no—
227C22—42no—
228C22#1683964111yescharacteristic
229C4—22no—
230C4—21no—
231C4—42no—
232C4—4882no—
233C4#1683987712yes—
234C4#1683988212yes—
235C4#1683994011yescharacteristic
236C4#1683994011yescharacteristic
237C2—41no—
238C2—82no—
239C2#2144614612yes—
240C2#2144620711yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2979C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#2979D61
hypocenterC61#9067
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #21444239C2semidirect
4. #21443655C2semidirect
141. C61#9067semidirect

Maximal subgroups

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

subgroupindex
#214561412
#214444112
#214442392
#214436552
C61×#29792
#906761