Finite Group Databasebeta

← #29580693

Subgroups — #29580693

Order 15616.

750 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 87 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#21456619C211yesmaximal, characteristic
2C2×#16839901C212yesmaximal, complemented
3#21444180C212yesmaximal, complemented
4#21443706C211yesmaximal, characteristic, complemented
5C61×#1784C211yesmaximal, characteristic, complemented
6C22×#11758201—22no—
7C2×#14519751C2212yes—
8#16840081—22no—
9#16840015—22no—
10#16839901—24no—
11#16839901—24no—
12#16839898—22no—
13C2×#14519701—22no—
14C2×C22⋊#92418C2212yes—
15#16839485—24no—
16C23×Q8×C61C2211yescharacteristic
17C2×#101×C61—22no—
18C61×#407C2211yescharacteristic
19C61×#407C2211yescharacteristic
20C22×D244D412yes—
21C22×C22⋊D61—22no—
22C2×#11758201—28no—
23C2×#11758201—48no—
24C2×D488—22no—
25C22×#92418—22no—
26#14519751—22no—
27#14519751—22no—
28#14519751—22no—
29#14519751—22no—
30C2×C4⋊Dic61D412yes—
31C2×#11758198—22no—
32#14519709—22no—
33#14519701—28no—
34#14519699—24no—
35C22⋊#92418—22no—
36C22⋊#92418D414yes—
37C22×Q8×C61—21no—
38C22×Q8×C61—22no—
39C22×Q8×C61—22no—
40C22×Q8×C61—44no—
41C22×Q8×C61D412yes—
42C23×C4×C61—21no—
43C23×C4×C61C2311yescharacteristic
44C2×M16×C61—21no—
45C2×M16×C61D412yes—
46#101×C61—44no—
47C22⋊C8×C61—22no—
48C22⋊C8×C61—22no—
49C24×C61#4111yescharacteristic
50C23×D61—22no—
51C2×Q8×C61—24no—
52C2×Q8×C61—28no—
53C2×Q8×C61—24no—
54C2×Q8×C61—44no—
55C2×Q8×C61—44no—
56C2×Q8×C61—42no—
57C2×Q8×C61—48no—
58C2×Q8×C61—41no—
59C2×Q8×C61—41no—
60C22×C4×C61—22no—
61C22×C4×C61—42no—
62C22×C4×C61—44no—
63C22×C4×C61—42no—
64C22×C4×C61C2×D412yes—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D412yes—
68C2×C22⋊D61—42no—
69C2×C22⋊D61—42no—
70C2×C22⋊D61—84no—
71C22×Dic61—22no—
72C2×D244—22no—
73C2×D244—22no—
74C2×D244—44no—
75M16×C61—42no—
76M16×C61—42no—
77C2×C8×C61—22no—
78C2×C8×C61—41no—
79#11758201—416no—
80#11758198—42no—
81#11758198—42no—
82C4⋊Dic61—22no—
83C4⋊Dic61—22no—
84C2×#92418—24no—
85C2×#92418—44no—
86D488—82no—
87C23×C61—24no—
88C23×C61—41no—
89C23×C61#12111yescharacteristic
90C23×C61#12112yes—
91C22×D61—42no—
92C22×D61—44no—
93C22×D61—82no—
94C22×D61—82no—
95Q8×C61—28no—
96Q8×C61—44no—
97Q8×C61—44no—
98Q8×C61—44no—
99C2×C4×C61—24no—
100C2×C4×C61—22no—
101C2×C4×C61—24no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—48no—
106C2×C4×C61—42no—
107C2×C4×C61#12012yes—
108C2×C4×C61#12011yescharacteristic
109C2×C4×C61#12711yescharacteristic
110C22⋊D61—84no—
111C22⋊D61—84no—
112C2×Dic61—42no—
113C2×Dic61—82no—
114D244—44no—
115D244—44no—
116C8×C61—42no—
117C8×C61—81no—
118#92418—44no—
119#9988—611nomaximal, Sylow
120C22×C61—22no—
121C22×C61—22no—
122C22×C61—22no—
123C22×C61—21no—
124C22×C61—44no—
125C22×C61#39211yescharacteristic
126C22×C61#47114yes—
127C2×D61—82no—
128C2×D61—82no—
129C2×D61—82no—
130C2×D61—164no—
131C4×C61—22no—
132C4×C61—21no—
133C4×C61—44no—
134C4×C61B(4,𝔽2)12yes—
135Dic61—82no—
136#3192—611no—
137C2×#448—612no—
138#1784D6111yescharacteristic, complemented
139#1471—612no—
140#1298—611no—
141C2×C61—24no—
142C2×C61—41no—
143C2×C61#201912yes—
144C2×C61#202111yescharacteristic
145D61—162no—
146C23×Q8C2×D6111yescharacteristic
147C22×SD16—1222no—
148C2×#121—612no—
149#481—1222no—
150#469—1222no—
151#448—1224no—
152#448—1224no—
153#447—1222no—
154C2×#103—1222no—
155C2×#101—22no—
156#407C2×D6111yescharacteristic
157#407C2×D6111yescharacteristic
158C2×C22⋊C8—612no—
159#323—1224no—
160C61#998811yescharacteristic, complemented, Sylow
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—44no—
165C22×Q8D24412yes—
166C22×D4—612no—
167C22×D4—1222no—
168C23×C4—21no—
169C23×C4C22×D6111yescharacteristic
170C2×SD16—1228no—
171C2×SD16—2448no—
172C2×D8—1222no—
173C2×M16—21no—
174C2×M16C22⋊D6112yes—
175C22×C8—1222no—
176#121—1222no—
177#121—1222no—
178#121—1222no—
179#121—1222no—
180C2×C4⋊C4—612no—
181C2×C22⋊C4—1222no—
182#106—1222no—
183#103—1228no—
184#102—1224no—
185#101—44no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—614no—
189C22⋊C8—1222no—
190C24—1222no—
191C24#1175822211yescharacteristic
192C2×Q8—24no—
193C2×Q8—28no—
194C2×Q8—24no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—42no—
198C2×Q8—48no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×D4—1222no—
202C2×D4—1222no—
203C2×D4—4884no—
204C2×D4—2444no—
205C2×D4—2442no—
206C2×D4—2442no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—1222no—
212C22×C4C2×D24411yescharacteristic
213C22×C4D4×D6112yes—
214C22×C4D4×D6112yes—
215C22×C4C2×C22⋊D6111yescharacteristic
216SD16—24416no—
217D8—4882no—
218M16—42no—
219M16—42no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—1224no—
223C2×C8—2444no—
224C4⋊C4—1222no—
225C4⋊C4—1222no—
226C22⋊C4—2442no—
227C22⋊C4—2442no—
228C23—24no—
229C23—41no—
230C23—4882no—
231C23—4882no—
232C23—2444no—
233C23—2442no—
234C23#1451975012yes—
235C23#1451975111yescharacteristic
236Q8—28no—
237Q8—44no—
238Q8—44no—
239Q8—44no—
240D4—4884no—
241D4—4884no—
242D4—2444no—
243D4—2444no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—48no—
251C2×C4—42no—
252C2×C4—4882no—
253C2×C4—2442no—
254C2×C4#1451974511yescharacteristic
255C2×C4#1451974712yes—
256C2×C4#1451977311yescharacteristic
257C8—42no—
258C8—81no—
259C8—2444no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—44no—
265C22—9764no—
266C22—4882no—
267C22—4882no—
268C22—4882no—
269C22#1683967111yescharacteristic
270C22#1684002814yes—
271C4—21no—
272C4—22no—
273C4—44no—
274C4—4882no—
275C4#1683994812yes—
276C2—24no—
277C2—41no—
278C2—9762no—
279C2#2144693212yes—
280C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1784C2
socleC22×C61#392
prime radical#21456619C2
hypercenter#1784D61
hypocenterC61#9988
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C2×#16839901C2semidirect
3. #21444180C2semidirect
4. #21443706C2semidirect
5. C61×#1784C2semidirect
138. #1784D61semidirect
160. C61#9988semidirect

Maximal subgroups

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

subgroupindex
#214566192
C2×#168399012
#214441802
#214437062
C61×#17842
#998861