Finite Group Databasebeta

← #29661874

Subgroups — #29661874

Order 15616.

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

normal subgroups only

This group has 840 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
1#21460233C211yesmaximal, characteristic
2#21457029C212yesmaximal, complemented
3#21445152C218yesmaximal, complemented
4#21444454C211yesmaximal, characteristic
5#21444433C212yesmaximal, complemented
6C61×#3135C211yesmaximal, characteristic
7#16840512C2214yes—
8#16840498—21no—
9#16840492—21no—
10#16840482—22no—
11D4×Dic122—22no—
12#16840379—24no—
13#16840333—22no—
14#16840319C2212yes—
15#16840168C2214yes—
16#16840130C2214yes—
17#16840100—24no—
18#16840057—22no—
19#16840047—22no—
20#16839819C2212yes—
21#16839503C2218yes—
22#16839502C2214yes—
23C61×#533C2211yescharacteristic
24C61×#532C2211yescharacteristic
25C61×#491—21no—
26C61×#490—21no—
27C61×#489C2214yes—
28C61×#416—22no—
29C2×#97×C61C2211yescharacteristic
30C22×Dic122—22no—
31#14519777D412yes—
32#14519777D412yes—
33#14519776—24no—
34#14519775—24no—
35#14519770—28no—
36#14519769C2314yes—
37#14519764—24no—
38#14519764—24no—
39#14519758—24no—
40#14519758—24no—
41#14519756—24no—
42#14519756—22no—
43#14519756—22no—
44#14519754—28no—
45C4×Dic122—22no—
46C4×Dic122—22no—
47#14519742—24no—
48D4×Dic61—24no—
49#14519736—22no—
50#14519736D414yes—
51#14519710—24no—
52#14519708—24no—
53C4⋊#92418—22no—
54#14519701—24no—
55#14519701—24no—
56C22⋊#92418—22no—
57#14519688C2314yes—
58C2×#41×C61D412yes—
59C22×C8×C61—21no—
60C4⋊Q8×C61D411yescharacteristic
61C4⋊Q8×C61C2312yes—
62#127×C61D411yescharacteristic
63#125×C61—21no—
64#124×C61C2312yes—
65C22⋊Q8×C61—22no—
66#121×C61—22no—
67C4×Q8×C61—22no—
68C4×D4×C61—22no—
69#117×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#103×C61—24no—
72#102×C61—24no—
73#97×C61—21no—
74#97×C61C2312yes—
75#41×C61—44no—
76C2×Q8×C61—22no—
77C2×Q8×C61C2×D412yes—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61C2×D412yes—
81C22×C4×C61—22no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D412yes—
84C22×Dic61—24no—
85C2×Dic122—22no—
86C2×Dic122—22no—
87C2×Dic122—24no—
88C2×Dic122—84no—
89C2×C8×C61—22no—
90C2×C8×C61—42no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—42no—
95C22⋊C4×C61—24no—
96C22⋊C4×C61—41no—
97C42×C61—41no—
98C42×C61C2×D412yes—
99C42×C61C2×D411yescharacteristic
100C42×C61C2411yescharacteristic
101C22⋊Dic61—24no—
102C22⋊Dic61—28no—
103C22⋊Dic61—24no—
104C4⋊Dic61—22no—
105C4⋊Dic61—22no—
106C4⋊Dic61C2×D414yes—
107C4⋊Dic61C2×D414yes—
108#11758196—28no—
109#11758196—24no—
110#11758196—24no—
111C4×Dic61—24no—
112C4×Dic61—24no—
113C2×#92418—24no—
114C23×C61—22no—
115C23×C61#12011yescharacteristic
116Q8×C61—42no—
117Q8×C61—44no—
118D4×C61—42no—
119D4×C61—44no—
120D4×C61—82no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—42no—
124C2×C4×C61—44no—
125C2×C4×C61—42no—
126C2×C4×C61#12012yes—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61C22×D412yes—
129C2×C4×C61C22×D411yescharacteristic
130C2×Dic61—24no—
131C2×Dic61—24no—
132C2×Dic61—84no—
133Dic122—44no—
134Dic122—44no—
135Dic122—84no—
136C8×C61—42no—
137#92418—44no—
138#19058—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—22no—
141C22×C61—42no—
142C22×C61—44no—
143C22×C61C2×#12011yescharacteristic
144C4×C61—21no—
145C4×C61—22no—
146C4×C61—22no—
147C4×C61—42no—
148C4×C61—82no—
149C4×C61#45512yes—
150Dic61—44no—
151Dic61—44no—
152#3550—611no—
153#3237—612no—
154#3135D6111yescharacteristic
155#1664—618no—
156#1536—611no—
157#1531—612no—
158C2×C61—41no—
159C2×C61—82no—
160C2×C61C2×#45511yescharacteristic
161C2×C61#302012yes—
162#563—614no—
163#559—1221no—
164Q82—1221no—
165#556—1222no—
166D4×Q8—1222no—
167#542—1224no—
168#536—1222no—
169#533—612no—
170#533C2×D6111yescharacteristic
171#532C2×D6111yescharacteristic
172#501—614no—
173#491—21no—
174#491—614no—
175#490—21no—
176#489C2×D6114yes—
177#485—1224no—
178#477—1222no—
179#475—1222no—
180#432—612no—
181#416—22no—
182C2×#97C2×D6111yescharacteristic
183#330—618no—
184#329—614no—
185C61#1905811yescharacteristic, complemented, Sylow
186C2×#41C22⋊D6112yes—
187C22×Q8—1222no—
188C22×C8—21no—
189C4⋊Q8—612no—
190C4⋊Q8—612no—
191C4⋊Q8—1224no—
192C4⋊Q8—1224no—
193C4⋊Q8C22⋊D6111yescharacteristic
194C4⋊Q8C22×D6112yes—
195#127C22⋊D6111yescharacteristic
196#126—1228no—
197#125—21no—
198#125—614no—
199#124—1224no—
200#124—1224no—
201#124C22×D6112yes—
202#123—1224no—
203#123—1224no—
204C22⋊Q8—22no—
205C22⋊Q8—1228no—
206C22⋊Q8—1224no—
207C22⋊Q8—1222no—
208C22⋊Q8—1222no—
209#121—22no—
210C4×Q8—22no—
211C4×Q8—1224no—
212C4×Q8—1222no—
213C4×Q8—1222no—
214C4×D4—22no—
215C4×D4—1224no—
216#117—21no—
217#117—614no—
218#117—1222no—
219C2×C42C22×D6111yescharacteristic
220#107—1224no—
221#106—1224no—
222C4⋊C8—1222no—
223#103—24no—
224#103—1224no—
225#103—1224no—
226#102—24no—
227C22⋊C8—1222no—
228#97—21no—
229#97—614no—
230#97C22×D6112yes—
231#41—44no—
232C2×Q8—22no—
233C2×Q8—1224no—
234C2×Q8—1222no—
235C2×Q8—1222no—
236C2×Q8—4884no—
237C2×Q8C2×C22⋊D6112yes—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4C2×C22⋊D6112yes—
241C22×C4—22no—
242C22×C4—1224no—
243C22×C4D4×D6112yes—
244C22×C4C2×C22⋊D6111yescharacteristic
245C2×C8—22no—
246C2×C8—42no—
247C2×C8—1224no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—614no—
253C4⋊C4—614no—
254C4⋊C4—1228no—
255C4⋊C4—1224no—
256C4⋊C4—1224no—
257C4⋊C4—1222no—
258C4⋊C4—1222no—
259C22⋊C4—24no—
260C22⋊C4—41no—
261C22⋊C4—1224no—
262C22⋊C4—1228no—
263C22⋊C4—1224no—
264C42—41no—
265C42—1224no—
266C42—1224no—
267C42D4×D6112yes—
268C42C2×C22⋊D6111yescharacteristic
269C42C23×D6111yescharacteristic
270C23—22no—
271C23#1451974511yescharacteristic
272Q8—42no—
273Q8—44no—
274Q8—4884no—
275Q8—2444no—
276Q8—2444no—
277D4—42no—
278D4—44no—
279D4—82no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—1224no—
286C2×C4—1224no—
287C2×C4—4884no—
288C2×C4#1451974512yes—
289C2×C4#1451974511yescharacteristic
290C2×C4C2×D4×D6112yes—
291C2×C4C22×C22⋊D6111yescharacteristic
292C8—42no—
293C8—2444no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—44no—
298C22C2×#1451974511yescharacteristic
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—82no—
304C4—2444no—
305C4—2444no—
306C4#1683993912yes—
307C2—41no—
308C2—82no—
309C2C2×#1683993911yescharacteristic
310C2#2145518112yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3135C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3135D61
hypocenterC61#19058
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
2. #21457029C2semidirect
3. #21445152C2semidirect
5. #21444433C2semidirect
185. C61#19058semidirect

Maximal subgroups

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

subgroupindex
#214602332
#214570292
#214451522
#214444542
#214444332
C61×#31352
#1905861