Finite Group Databasebeta

← #29580691

Subgroups — #29580691

Order 15616.

750 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16839900C212yesmaximal, complemented
2#21445742C211yesmaximal, characteristic, complemented
3#21444179C212yesmaximal, complemented
4#21443706C211yesmaximal, characteristic, complemented
5C61×#3192C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C22×#11758200—22no—
8#16839900—24no—
9#16839900—24no—
10C2×#14519700—22no—
11C2×#14519695—22no—
12#16839726C2211yescharacteristic
13C2×C22⋊#92418C2212yes—
14#16839486—24no—
15C2×#121×C61C2212yes—
16C61×#481—22no—
17C61×#469—22no—
18C61×#447—22no—
19C61×#407C2211yescharacteristic
20C22×Dic122—22no—
21C22×Dic122—21no—
22C22×Dic122—22no—
23C22×Dic122—44no—
24C22×Dic122D412yes—
25C23×Dic61—21no—
26C2×#11758200—28no—
27C2×#11758200—48no—
28C2×#11758194—21no—
29C22×#92418—22no—
30#14519700—28no—
31#14519695—44no—
32C22⋊#92418—22no—
33C22⋊#92418—22no—
34C22⋊#92418D414yes—
35C22×D4×C61—22no—
36C22×D4×C61D412yes—
37C23×C4×C61C2311yescharacteristic
38C2×D8×C61—22no—
39C2×M16×C61D412yes—
40#121×C61—22no—
41#121×C61—22no—
42#121×C61—22no—
43#121×C61—22no—
44C2×C4⋊C4×C61D412yes—
45C2×C22⋊C4×C61—22no—
46#106×C61—22no—
47#102×C61—24no—
48C22⋊C8×C61—22no—
49C24×C61—22no—
50C24×C61#4111yescharacteristic
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61—44no—
54C2×D4×C61—42no—
55C2×D4×C61—42no—
56C2×D4×C61—84no—
57C22×C4×C61—22no—
58C22×C4×C61—42no—
59C22×C4×C61C2×D412yes—
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63C22×Dic61—22no—
64C22×Dic61—44no—
65C22×Dic61—42no—
66C2×Dic122—28no—
67C2×Dic122—24no—
68C2×Dic122—24no—
69C2×Dic122—44no—
70C2×Dic122—48no—
71C2×Dic122—44no—
72C2×Dic122—42no—
73C2×Dic122—41no—
74C2×Dic122—41no—
75D8×C61—82no—
76M16×C61—42no—
77C2×C8×C61—22no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C22⋊C4×C61—42no—
81C22⋊C4×C61—42no—
82#11758200—416no—
83#11758194—42no—
84C2×#92418—24no—
85C2×#92418—44no—
86C2×#92418—41no—
87C23×C61—24no—
88C23×C61—41no—
89C23×C61—42no—
90C23×C61—44no—
91C23×C61—82no—
92C23×C61—82no—
93C23×C61#12111yescharacteristic
94C23×C61#12112yes—
95D4×C61—44no—
96D4×C61—44no—
97D4×C61—84no—
98D4×C61—84no—
99C2×C4×C61—24no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—82no—
105C2×C4×C61#12012yes—
106C2×C4×C61#12011yescharacteristic
107C2×C4×C61#12711yescharacteristic
108C2×Dic61—24no—
109C2×Dic61—48no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112Dic122—28no—
113Dic122—44no—
114Dic122—44no—
115Dic122—44no—
116C8×C61—42no—
117#92418—44no—
118#92418—81no—
119#9988—611nomaximal, Sylow
120C22×C61—22no—
121C22×C61—22no—
122C22×C61—22no—
123C22×C61—21no—
124C22×C61—44no—
125C22×C61—82no—
126C22×C61—82no—
127C22×C61—82no—
128C22×C61—164no—
129C22×C61#39211yescharacteristic
130C22×C61#47114yes—
131C4×C61—22no—
132C4×C61—21no—
133C4×C61—82no—
134C4×C61B(4,𝔽2)12yes—
135Dic61—44no—
136#3192D6111yescharacteristic
137C2×#448—612no—
138#1784—611no—
139#1471—612no—
140#1298—611no—
141C2×C61—24no—
142C2×C61—41no—
143C2×C61—162no—
144C2×C61#201912yes—
145C2×C61#202111yescharacteristic
146C23×Q8—611no—
147C22×SD16—1222no—
148C2×#121C2×D6112yes—
149#481—22no—
150#469—22no—
151#448—1224no—
152#448—1224no—
153#447—22no—
154C2×#103—1222no—
155C2×#101—1222no—
156#407—611no—
157#407C2×D6111yescharacteristic
158C2×C22⋊C8—612no—
159#323—1224no—
160C61#998811yescharacteristic, complemented, Sylow
161C22×Q8—612no—
162C22×Q8—1221no—
163C22×Q8—1222no—
164C22×Q8—1222no—
165C22×Q8—2444no—
166C22×D4—22no—
167C22×D4C22⋊D6112yes—
168C23×C4—1221no—
169C23×C4C22×D6111yescharacteristic
170C2×SD16—1228no—
171C2×SD16—2448no—
172C2×D8—22no—
173C2×M16—1221no—
174C2×M16C22⋊D6112yes—
175C22×C8—1222no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180C2×C4⋊C4C22⋊D6112yes—
181C2×C22⋊C4—22no—
182#106—22no—
183#103—1228no—
184#102—24no—
185#101—2444no—
186C22⋊C8—22no—
187C22⋊C8—614no—
188C22⋊C8—1222no—
189C22⋊C8—1222no—
190C24—22no—
191C24#1175822411yescharacteristic
192C2×Q8—1224no—
193C2×Q8—1228no—
194C2×Q8—1224no—
195C2×Q8—2444no—
196C2×Q8—2442no—
197C2×Q8—2444no—
198C2×Q8—2448no—
199C2×Q8—2441no—
200C2×Q8—2441no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—44no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—84no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—1222no—
210C22×C4—2444no—
211C22×C4—2442no—
212C22×C4D4×D6111yescharacteristic
213C22×C4D4×D6112yes—
214C22×C4C2×C22⋊D6112yes—
215C22×C4C2×C22⋊D6111yescharacteristic
216SD16—24416no—
217D8—82no—
218M16—42no—
219M16—2442no—
220C2×C8—22no—
221C2×C8—1224no—
222C2×C8—2441no—
223C2×C8—2444no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C23—24no—
229C23—41no—
230C23—42no—
231C23—44no—
232C23—82no—
233C23—82no—
234C23#1451974811yescharacteristic
235C23#1451974912yes—
236Q8—1228no—
237Q8—2444no—
238Q8—2444no—
239Q8—2444no—
240D4—44no—
241D4—44no—
242D4—84no—
243D4—84no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—1224no—
251C2×C4—2442no—
252C2×C4—2448no—
253C2×C4—2442no—
254C2×C4#1451974512yes—
255C2×C4#1451974611yescharacteristic
256C2×C4#1451977311yescharacteristic
257C8—42no—
258C8—4881no—
259C8—2444no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—44no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—164no—
269C22#1683967011yescharacteristic
270C22#1684002314yes—
271C4—21no—
272C4—22no—
273C4—82no—
274C4—2444no—
275C4#1683994912yes—
276C2—24no—
277C2—41no—
278C2—162no—
279C2#2144693312yes—
280C2#2144694311yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3192C2
socleC22×C61#392
hypercenter#3192D61
hypocenterC61#9988
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
1. C2×#16839900C2semidirect
2. #21445742C2semidirect
3. #21444179C2semidirect
4. #21443706C2semidirect
160. C61#9988semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
#214457422
#214441792
#214437062
C61×#31922
#998861