Finite Group Databasebeta

← #29575317

Subgroups — #29575317

Order 15616.

570 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 147 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#21448909C211yesmaximal, characteristic
3#21446766C212yesmaximal
4#21443658C212yesmaximal
5C2×C61×#333C211yesmaximal, characteristic
6C22×#11758202—22no—
7C2×#14519777C2211yescharacteristic
8#16840141—22no—
9#16840141C414yescomplemented
10C2×C4×#92418C2211yescharacteristic
11#16839669—22no—
12#16839661—22no—
13C4⋊C4×Dic61—22no—
14C4×C4⋊Dic61C2212yes—
15#16839525—22no—
16#16839525—22no—
17C2×C4⋊Q8×C61C2211yescharacteristic
18C2×C4⋊C8×C61C2212yes—
19C61×#333—22no—
20C22×Dic122—21no—
21C2×#11758202—28no—
22C2×#11758202—44no—
23C2×#11758202—44no—
24C22×#92418—21no—
25#14519777—22no—
26#14519777—21no—
27#14519777C2×C412yes—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61—22no—
30C2×C4⋊Dic61D412yes—
31C2×C4⋊Dic61D412yes—
32C2×C4×Dic61—22no—
33C2×C4×Dic61—22no—
34C4×#92418—21no—
35C4×#92418C2×C412yes—
36#14519684—24no—
37#14519684—22no—
38C22×Q8×C61D412yes—
39C22×C8×C61—22no—
40C4⋊Q8×C61—22no—
41C4⋊Q8×C61C2×C412yes—
42C4⋊Q8×C61D412yes—
43C2×C4⋊C4×C61—22no—
44C2×C42×C61C2311yescharacteristic
45C4⋊C8×C61—22no—
46C4⋊C8×C61—22no—
47C2×Q8×C61—22no—
48C2×Q8×C61—42no—
49C2×Q8×C61—42no—
50C2×Q8×C61C22⋊C414yes—
51C22×C4×C61—22no—
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61C2×D412yes—
54C22×Dic61—22no—
55C22×Dic61—22no—
56C22×Dic61—41no—
57C2×Dic122—22no—
58C2×Dic122—42no—
59C2×Dic122—42no—
60C2×Dic122—41no—
61C2×C8×C61—42no—
62C2×C8×C61—42no—
63C2×C8×C61—42no—
64C4⋊C4×C61—24no—
65C4⋊C4×C61—24no—
66C42×C61C22×C411yescharacteristic
67C42×C61C2×D411yescharacteristic
68C42×C61#4112yes—
69#11758202—416no—
70C4⋊Dic61—22no—
71C4⋊Dic61—24no—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74C4⋊Dic61—42no—
75C4⋊Dic61—82no—
76C4⋊Dic61Q1614yescomplemented
77C4×Dic61—42no—
78C4×Dic61—42no—
79C4×Dic61—82no—
80C2×#92418—22no—
81C2×#92418—22no—
82C2×#92418—22no—
83C23×C61#12011yescharacteristic
84Q8×C61—44no—
85Q8×C61—82no—
86C2×C4×C61—21no—
87C2×C4×C61—21no—
88C2×C4×C61—24no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61C2×C22⋊C411yescharacteristic
92C2×C4×C61C4×D412yes—
93C2×C4×C61#12112yes—
94C2×C4×C61#12311yescharacteristic
95C2×C4×C61C2×Q1612yes—
96C2×C4×C61#13512yes—
97C2×C4×C61#13712yes—
98C2×C4×C61#13712yes—
99C2×Dic61—24no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—81no—
107C2×Dic61—81no—
108Dic122—44no—
109Dic122—82no—
110C8×C61—82no—
111#92418—24no—
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×C61C4×Q1612yes—
122C4×C61#44112yes—
123C4×C61#46212yes—
124C4×C61#46412yes—
125Dic61—44no—
126Dic61—82no—
127Dic61—82no—
128C2×#494—611no—
129#2369—611no—
130#1987—612no—
131C2×#333D6111yescharacteristic
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⋊C8C2×D6112yes—
144C2×C4×C8—611no—
145#391—1222no—
146#389—1222no—
147C4×C4⋊C4—612no—
148C4×C4⋊C4—1222no—
149#340—1222no—
150#340—1222no—
151#333—22no—
152C61#906611yescharacteristic, complemented, Sylow
153C22×Q8—1221no—
154C22×Q8D24412yes—
155C2×Q16—1228no—
156C2×Q16—2444no—
157C2×Q16—2444no—
158C22×C8—22no—
159C22×C8—1221no—
160C4⋊Q8—22no—
161C4⋊Q8—612no—
162C4⋊Q8—1222no—
163C4⋊Q8—1221no—
164C4⋊Q8C4×D6112yes—
165C4⋊Q8C22⋊D6112yes—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—612no—
168C2×C4⋊C4—612no—
169C2×C4⋊C4—1222no—
170C2×C4⋊C4—1222no—
171C2×C42—1222no—
172C2×C42—1222no—
173C2×C42C22×D6111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4×C8—612no—
177C4×C8—1221no—
178#95—1224no—
179#95—1222no—
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—1222no—
190C22×C4—1222no—
191C22×C4—2441no—
192C22×C4C2×D24411yescharacteristic
193C22×C4D4×D6112yes—
194Q16—24416no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—1222no—
199C2×C8—1222no—
200C2×C8—1222no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—614no—
204C4⋊C4—1222no—
205C4⋊C4—1222no—
206C4⋊C4—1224no—
207C4⋊C4—1222no—
208C4⋊C4—4882no—
209C4⋊C4—2442no—
210C42—4882no—
211C42—2442no—
212C42—2442no—
213C42C2×C4×D6111yescharacteristic
214C42#1175822412yes—
215C42C2×C22⋊D6111yescharacteristic
216C23#1451974711yescharacteristic
217Q8—44no—
218Q8—82no—
219Q8—4882no—
220Q8—2444no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—1224no—
227C2×C4—4881no—
228C2×C4—4881no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4C2×#1175819811yescharacteristic
236C2×C4#1451973712yes—
237C2×C4#1451974912yes—
238C2×C4#1451976111yescharacteristic
239C2×C4Q16×D6112yes—
240C2×C4#1451979812yes—
241C2×C4#1451980612yes—
242C2×C4#1451980912yes—
243C8—82no—
244C8—1224no—
245C22#1683963911yescharacteristic
246C22#1683991912yes—
247C22#1683992712yes—
248C22#1683993211yescharacteristic
249C22#1683993211yescharacteristic
250C4—21no—
251C4—21no—
252C4—44no—
253C4—4882no—
254C4—4882no—
255C4—2444no—
256C4#1683985412yes—
257C4#1683986612yes—
258C4#1683997012yes—
259C4#1683998212yes—
260C2#2144495412yes—
261C2#2144496011yescharacteristic
262C2#2144496511yescharacteristic
263C2#2144615812yes—
264C2#2144621511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#333C2
socle, prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenterC2×#333D61
hypocenterC61#9066
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
9. #16840141C4semidirect
76. C4⋊Dic61Q16semidirect
152. C61#9066semidirect

Maximal subgroups

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

subgroupindex
C2×#168401412
#214489092
#214467662
#214436582
C2×C61×#3332
#906661