Finite Group Databasebeta

← #29575319

Subgroups — #29575319

Order 15616.

570 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×#16840141C211yesmaximal, characteristic
2#21446767C212yesmaximal
3C2×#16839511C211yesmaximal, characteristic
4#21443660C212yesmaximal
5C61×#2369C211yesmaximal, characteristic
6C22×#11758202—22no—
7C2×#14519777C2211yescharacteristic
8#16840141—22no—
9#16840141C414yescomplemented
10C2×C4⋊#92418C2212yes—
11C2×C4×#92418C2211yescharacteristic
12#16839524—22no—
13#16839524—22no—
14#16839511—22no—
15C2×C4⋊Q8×C61C2211yescharacteristic
16C61×#391—22no—
17C61×#389—22no—
18C4×C4⋊C4×C61—22no—
19C4×C4⋊C4×C61C2212yes—
20C22×Dic122—21no—
21C2×#11758202—28no—
22C2×#11758202—44no—
23C2×#11758202—44no—
24C22×#92418—21no—
25C22×#92418—22no—
26#14519777—22no—
27#14519777—21no—
28#14519777C2×C412yes—
29C2×C4⋊Dic61—22no—
30C4⋊#92418—22no—
31C4⋊#92418—22no—
32C4×#92418—21no—
33C4×#92418C2×C412yes—
34C22×Q8×C61D412yes—
35C4⋊Q8×C61—22no—
36C4⋊Q8×C61C2×C412yes—
37C4⋊Q8×C61D412yes—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61D412yes—
41C2×C4⋊C4×C61D412yes—
42C2×C42×C61—22no—
43C2×C42×C61—22no—
44C2×C42×C61C2311yescharacteristic
45#95×C61—24no—
46#95×C61—22no—
47C2×Q8×C61—22no—
48C2×Q8×C61—42no—
49C2×Q8×C61—42no—
50C2×Q8×C61C22⋊C414yes—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61C2×D412yes—
56C22×Dic61—41no—
57C2×Dic122—22no—
58C2×Dic122—42no—
59C2×Dic122—42no—
60C2×Dic122—41no—
61C4⋊C4×C61—24no—
62C4⋊C4×C61—24no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—22no—
66C4⋊C4×C61—82no—
67C4⋊C4×C61Q1614yescomplemented
68C42×C61—42no—
69C42×C61—42no—
70C42×C61—82no—
71C42×C61C22×C411yescharacteristic
72C42×C61C2×D411yescharacteristic
73C42×C61#4112yes—
74#11758202—416no—
75C4⋊Dic61—24no—
76C4⋊Dic61—42no—
77C2×#92418—22no—
78C2×#92418—22no—
79C2×#92418—22no—
80C2×#92418—42no—
81C2×#92418—42no—
82C2×#92418—42no—
83C23×C61#12011yescharacteristic
84Q8×C61—44no—
85Q8×C61—82no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—24no—
89C2×C4×C61—24no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61—42no—
93C2×C4×C61—42no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61C2×C22⋊C411yescharacteristic
98C2×C4×C61C4×D412yes—
99C2×C4×C61#12112yes—
100C2×C4×C61#12311yescharacteristic
101C2×C4×C61C2×Q1612yes—
102C2×C4×C61#13512yes—
103C2×C4×C61#13712yes—
104C2×C4×C61#13712yes—
105C2×Dic61—42no—
106C2×Dic61—81no—
107C2×Dic61—81no—
108Dic122—44no—
109Dic122—82no—
110#92418—24no—
111#92418—82no—
112#9066—611nomaximal, Sylow
113C22×C61#38611yescharacteristic
114C22×C61#45112yes—
115C22×C61#45212yes—
116C22×C61#45311yescharacteristic
117C22×C61#45311yescharacteristic
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—44no—
121C4×C61—44no—
122C4×C61—82no—
123C4×C61C4×Q1612yes—
124C4×C61#44112yes—
125C4×C61#46212yes—
126C4×C61#46412yes—
127Dic61—82no—
128C2×#494—611no—
129#2369D6111yescharacteristic
130#1987—612no—
131C2×#333—611no—
132#1279—612no—
133C2×C61#163212yes—
134C2×C61#163311yescharacteristic
135C2×C61#163411yescharacteristic
136C2×C61#187812yes—
137C2×C61#188711yescharacteristic
138C22×Q16—1222no—
139C2×C4⋊Q8—611no—
140C2×C4⋊Q8C2×D6111yescharacteristic
141#494—614no—
142#494—1222no—
143C2×C4⋊C8—612no—
144C2×C4×C8—611no—
145#391—22no—
146#389—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4C2×D6112yes—
149#340—1222no—
150#340—1222no—
151#333—1222no—
152C61#906611yescharacteristic, complemented, Sylow
153C22×Q8—1221no—
154C22×Q8C22⋊D6112yes—
155C2×Q16—1228no—
156C2×Q16—2444no—
157C2×Q16—2444no—
158C22×C8—1221no—
159C22×C8—1222no—
160C4⋊Q8—22no—
161C4⋊Q8—612no—
162C4⋊Q8—1221no—
163C4⋊Q8—1222no—
164C4⋊Q8C4×D6112yes—
165C4⋊Q8D24412yes—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—1222no—
169C2×C4⋊C4C22⋊D6112yes—
170C2×C4⋊C4C22⋊D6112yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C22×D6111yescharacteristic
174C4⋊C8—1222no—
175C4⋊C8—1222no—
176C4×C8—612no—
177C4×C8—1221no—
178#95—24no—
179#95—22no—
180C2×Q8—22no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—1222no—
184C2×Q8—2442no—
185C2×Q8—2442no—
186C2×Q8—2441no—
187C2×Q8#1175819814yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—2441no—
192C22×C4C2×C22⋊D6111yescharacteristic
193C22×C4C2×C22⋊D6112yes—
194Q16—24416no—
195C2×C8—1222no—
196C2×C8—1222no—
197C2×C8—1222no—
198C2×C8—2442no—
199C2×C8—2442no—
200C2×C8—2442no—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—82no—
207C4⋊C4—1224no—
208C4⋊C4—2442no—
209C4⋊C4#1175820214yescomplemented
210C42—42no—
211C42—42no—
212C42—82no—
213C42C2×C4×D6111yescharacteristic
214C42C2×D24411yescharacteristic
215C42#1175822212yes—
216C23#1451974611yescharacteristic
217Q8—44no—
218Q8—82no—
219Q8—4882no—
220Q8—2444no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—4881no—
233C2×C4—4881no—
234C2×C4—2442no—
235C2×C4C2×#1175819811yescharacteristic
236C2×C4C4×C22⋊D6112yes—
237C2×C4#1451975112yes—
238C2×C4#1451976011yescharacteristic
239C2×C4C2×#1175820212yes—
240C2×C4#1451979712yes—
241C2×C4#1451980712yes—
242C2×C4#1451980812yes—
243C8—1224no—
244C8—4882no—
245C22#1683964111yescharacteristic
246C22#1683991812yes—
247C22#1683992412yes—
248C22#1683993011yescharacteristic
249C22#1683993011yescharacteristic
250C4—21no—
251C4—21no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—4882no—
256C4C4×#1175820212yes—
257C4#1683986712yes—
258C4#1683997212yes—
259C4#1683998412yes—
260C2#2144495312yes—
261C2#2144495911yescharacteristic
262C2#2144496411yescharacteristic
263C2#2144616012yes—
264C2#2144621711yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2369C2
socle, prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#2369D61
hypocenterC61#9066
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2369

Complements

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

NH ≅ G/Ntype
9. #16840141C4semidirect
67. C4⋊C4×C61Q16semidirect
152. C61#9066semidirect
209. C4⋊C4#11758202semidirect

Maximal subgroups

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

subgroupindex
C2×#168401412
#214467672
C2×#168395112
#214436602
C61×#23692
#906661