Finite Group Databasebeta

← #29661316

Subgroups — #29661316

Order 15616.

808 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 171 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#21457009C212yesmaximal
2#21456222C211yesmaximal, characteristic
3#21445147C214yesmaximal, complemented
4#21445121C214yesmaximal, complemented
5#21444446C211yesmaximal, characteristic, complemented
6#21444421C212yesmaximal, complemented
7C2×C61×#564C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840172C2214yes—
10#16840132C2212yes—
11#16840128C2212yes—
12#16840128C2214yes—
13#16840119—22no—
14#16840080—22no—
15#16840075—22no—
16#16840059—24no—
17#16839819C2212yes—
18#16839761—22no—
19C4×#11758194C2211yescharacteristic
20#16839504C2218yes—
21#16839504C2214yes—
22C61×#564—24no—
23C61×#564—22no—
24C61×#564C2214yes—
25C2×#125×C61—22no—
26C2×#125×C61—22no—
27C2×#125×C61—22no—
28C2×#125×C61C2212yes—
29C2×#125×C61C2211yescharacteristic
30C2×#11758222—21no—
31C2×#11758194—21no—
32#14519777C2311yescharacteristic
33#14519774C2311yescharacteristic
34#14519765—21no—
35#14519751—22no—
36C4×D244—22no—
37#14519710—24no—
38#14519708—24no—
39C4⋊#92418—22no—
40#14519700—24no—
41#14519697—24no—
42#14519697—24no—
43#14519697—24no—
44C22⋊#92418—22no—
45#14519688C2312yes—
46#14519688C2314yes—
47C4×#92418—21no—
48#125×C61—28no—
49#125×C61—28no—
50#125×C61—28no—
51#125×C61—28no—
52#125×C61—28no—
53#125×C61—28no—
54#125×C61—22no—
55#125×C61—22no—
56#125×C61—22no—
57#125×C61D414yes—
58#125×C61D412yes—
59#125×C61C2312yes—
60#125×C61C2314yes—
61C2×C4⋊C4×C61—22no—
62C2×C4⋊C4×C61—22no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61D412yes—
68C2×C4⋊C4×C61D412yes—
69C2×C4⋊C4×C61D412yes—
70C2×C42×C61—22no—
71C2×C42×C61C2311yescharacteristic
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D412yes—
77C2×C22⋊D61—41no—
78#11758222—44no—
79C2×D244—21no—
80C2×D244—22no—
81C2×C4×D61—41no—
82C2×Dic122—21no—
83C4⋊C4×C61—28no—
84C4⋊C4×C61—28no—
85C4⋊C4×C61—28no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61C2×D414yes—
96C4⋊C4×C61C2×D414yes—
97C4⋊C4×C61C2×D414yes—
98C42×C61—28no—
99C42×C61C2×D411yescharacteristic
100C42×C61C2×D412yes—
101C42×C61C2411yescharacteristic
102#11758198—42no—
103C4⋊Dic61—22no—
104#11758194—42no—
105C2×#92418—22no—
106C2×#92418—24no—
107C23×C61#12011yescharacteristic
108C22×D61—41no—
109C2×C4×C61—22no—
110C2×C4×C61—24no—
111C2×C4×C61—24no—
112C2×C4×C61—24no—
113C2×C4×C61—42no—
114C2×C4×C61—44no—
115C2×C4×C61—44no—
116C2×C4×C61—44no—
117C2×C4×C61#12012yes—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61C22×D412yes—
120C2×C4×C61C22×D411yescharacteristic
121C22⋊D61—82no—
122C2×Dic61—41no—
123D244—44no—
124D244—42no—
125C4×D61—82no—
126Dic122—42no—
127#92418—44no—
128#92418—42no—
129#19003—611nomaximal, Sylow
130C22×C61—22no—
131C22×C61#45412yes—
132C22×C61C2×#12011yescharacteristic
133C2×D61—81no—
134C2×D61—82no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—44no—
140C4×C61—44no—
141C4×C61—44no—
142Dic61—81no—
143C2×#564D6111yescharacteristic, complemented
144#3234—612no—
145#3142—611no—
146#1663—614no—
147#1658—614no—
148#1534—611no—
149#1528—612no—
150C2×C61—22no—
151C2×C61C2×#45411yescharacteristic
152C2×C61#302012yes—
153D61—161no—
154#564—24no—
155#564—22no—
156#564C2×D6114yes—
157#532—611no—
158C2×#125—22no—
159C2×#125—22no—
160C2×#125—22no—
161C2×#125C2×D6111yescharacteristic
162C2×#125C2×D6112yes—
163#501—614no—
164#491—612no—
165#490—612no—
166#490—614no—
167#488—1222no—
168#481—1222no—
169#480—1222no—
170#478—1224no—
171#432—612no—
172#417—1222no—
173C4×M16—611no—
174#330—618no—
175#330—614no—
176C61#1900311yescharacteristic, complemented, Sylow
177C2×#41—1221no—
178C2×M16—1221no—
179C4⋊Q8—611no—
180#127—611no—
181#125—28no—
182#125—28no—
183#125—28no—
184#125—28no—
185#125—28no—
186#125—28no—
187#125—22no—
188#125—22no—
189#125—22no—
190#125C22⋊D6114yes—
191#125C22⋊D6112yes—
192#125C22×D6112yes—
193#125C22×D6114yes—
194#124—1221no—
195#121—1222no—
196C4×D4—1222no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4C22⋊D6112yes—
204C2×C4⋊C4C22⋊D6112yes—
205C2×C4⋊C4C22⋊D6112yes—
206C2×C42—22no—
207C2×C42C22×D6111yescharacteristic
208#107—1224no—
209#106—1224no—
210C4⋊C8—1222no—
211#103—1224no—
212#102—1224no—
213#102—1224no—
214#102—1224no—
215C22⋊C8—1222no—
216#97—612no—
217#97—614no—
218C4×C8—1221no—
219#41—2444no—
220C2×Q8—1221no—
221C2×D4—1222no—
222C2×D4—1221no—
223C2×D4—2441no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—2441no—
228C22×C4C2×C22⋊D6111yescharacteristic
229C22×C4C2×C22⋊D6112yes—
230M16—2442no—
231C2×C8—1222no—
232C2×C8—1224no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—1222no—
246C4⋊C4C2×C22⋊D6114yes—
247C4⋊C4C2×C22⋊D6114yes—
248C4⋊C4C2×C22⋊D6114yes—
249C22⋊C4—2442no—
250C42—28no—
251C42C2×C22⋊D6111yescharacteristic
252C42C2×C22⋊D6112yes—
253C42C23×D6111yescharacteristic
254C23—2441no—
255C23#1451974611yescharacteristic
256Q8—2442no—
257D4—4882no—
258D4—2444no—
259D4—2442no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—4882no—
269C2×C4—2441no—
270C2×C4#1451974612yes—
271C2×C4#1451974611yescharacteristic
272C2×C4C22×C22⋊D6112yes—
273C2×C4C22×C22⋊D6111yescharacteristic
274C8—2444no—
275C8—2442no—
276C22—22no—
277C22—4881no—
278C22—4882no—
279C22#1683993512yes—
280C22C2×#1451974611yescharacteristic
281C4—21no—
282C4—21no—
283C4—22no—
284C4—22no—
285C4—44no—
286C4—44no—
287C4—44no—
288C4—4881no—
289C2—22no—
290C2—9761no—
291C2C2×#1683993511yescharacteristic
292C2#2145518312yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#564C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenterC2×#564D61
hypocenterC61#19003
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#564

Complements

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

NH ≅ G/Ntype
3. #21445147C2semidirect
4. #21445121C2semidirect
5. #21444446C2semidirect
6. #21444421C2semidirect
7. C2×C61×#564C2semidirect
143. C2×#564D61semidirect
176. C61#19003semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214562222
#214451472
#214451212
#214444462
#214444212
C2×C61×#5642
#1900361