Finite Group Databasebeta

← #29572371

Subgroups — #29572371

Order 15616.

692 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 107 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#21444547C212yesmaximal, complemented
3#21444454C211yesmaximal, characteristic
4#21444330C212yesmaximal, complemented
5C61×#1963C211yesmaximal, characteristic
6#16840512—22no—
7#16840498C412yescomplemented
8#16840492C412yescomplemented
9#16840482—22no—
10D4×Dic122—22no—
11#16840379—24no—
12#16840333—22no—
13#16840319C2212yes—
14#16839819C2212yes—
15#16839510—22no—
16#16839509—22no—
17#16839502—22no—
18#16839498—24no—
19C61×#532C2211yescharacteristic
20C61×#423C2211yescharacteristic
21C2×#104×C61—22no—
22C61×#376C2211yescharacteristic
23C22×Dic122—22no—
24#14519777D412yes—
25#14519777D412yes—
26#14519776—24no—
27#14519775—24no—
28#14519770—28no—
29#14519769—22no—
30#14519764—24no—
31#14519764—24no—
32#14519758—24no—
33#14519758—24no—
34#14519756—24no—
35#14519756—22no—
36#14519756D414yes—
37#14519754—28no—
38C4×Dic122—22no—
39C4×Dic122C2×C414yes—
40#14519742—24no—
41D4×Dic61—24no—
42#14519736—22no—
43#14519736—22no—
44C4⋊#92418—22no—
45C22⋊#92418—22no—
46#14519688—22no—
47C2×#41×C61D412yes—
48C2×M16×C61—21no—
49C4⋊Q8×C61C2×C411yescharacteristic
50#127×C61C2×C411yescharacteristic
51#124×C61D412yes—
52#121×C61—22no—
53C4×D4×C61—22no—
54C2×C42×C61—21no—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57#104×C61—44no—
58#95×C61—22no—
59#41×C61—42no—
60#41×C61—42no—
61C2×Q8×C61C22⋊C412yes—
62C2×D4×C61—22no—
63C2×D4×C61—22no—
64C2×D4×C61C22⋊C412yes—
65C22×C4×C61—22no—
66C22×C4×C61—41no—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D412yes—
69C22×Dic61—24no—
70C2×Dic122—22no—
71C2×Dic122—24no—
72C2×Dic122—82no—
73C2×Dic122—82no—
74C2×Dic122C22⋊C414yes—
75M16×C61—42no—
76C2×C8×C61—41no—
77C4⋊C4×C61—22no—
78C22⋊C4×C61—24no—
79C42×C61—22no—
80C42×C61—41no—
81C42×C61C22×C411yescharacteristic
82C42×C61C2×D411yescharacteristic
83C42×C61C2×D412yes—
84C22⋊Dic61—24no—
85C22⋊Dic61—28no—
86C22⋊Dic61—24no—
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90C4⋊Dic61C22⋊C414yes—
91#11758196—28no—
92#11758196—24no—
93#11758196—24no—
94C4×Dic61—24no—
95C4×Dic61—24no—
96C2×#92418—42no—
97C23×C61—22no—
98C23×C61#12011yescharacteristic
99Q8×C61—42no—
100D4×C61—42no—
101D4×C61—44no—
102D4×C61—82no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—82no—
111C2×C4×C61C2×C22⋊C412yes—
112C2×C4×C61C2×C22⋊C411yescharacteristic
113C2×C4×C61#12012yes—
114C2×C4×C61#12011yescharacteristic
115C2×Dic61—24no—
116C2×Dic61—24no—
117C2×Dic61—84no—
118Dic122—42no—
119Dic122—42no—
120Dic122—44no—
121Dic122—84no—
122C8×C61—81no—
123#92418—82no—
124#8492—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—44no—
129C22×C61#37911yescharacteristic
130C4×C61—21no—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—42no—
134C4×C61—82no—
135C4×C61#45311yescharacteristic
136C4×C61#45511yescharacteristic
137Dic61—44no—
138Dic61—44no—
139#3550—611no—
140#1963D6111yescharacteristic
141#1559—612no—
142#1536—611no—
143#1506—612no—
144C2×C61—41no—
145C2×C61—82no—
146C2×C61#179211yescharacteristic
147C2×C61#179612yes—
148#563—1222no—
149#559—612no—
150Q82—612no—
151#556—1222no—
152D4×Q8—1222no—
153#542—1224no—
154#536—1222no—
155#533—612no—
156#532C2×D6111yescharacteristic
157#432—612no—
158#423C2×D6111yescharacteristic
159C2×#104—22no—
160#376C2×D6111yescharacteristic
161#333—1222no—
162#332—1222no—
163#329—1222no—
164#328—1224no—
165C61#849211yescharacteristic, complemented, Sylow
166C2×#41C22⋊D6112yes—
167C22×Q8—1222no—
168C2×M16—21no—
169C4⋊Q8—612no—
170C4⋊Q8—612no—
171C4⋊Q8—1224no—
172C4⋊Q8—1224no—
173C4⋊Q8C4×D6111yescharacteristic
174#127C4×D6111yescharacteristic
175#126—1228no—
176#125—1222no—
177#124—1224no—
178#124—1224no—
179#124D24412yes—
180#123—1224no—
181#123—1224no—
182C22⋊Q8—614no—
183C22⋊Q8—1228no—
184C22⋊Q8—1224no—
185C22⋊Q8—1222no—
186#121—22no—
187C4×Q8—614no—
188C4×Q8—1224no—
189C4×Q8—1222no—
190C4×D4—22no—
191C4×D4—1224no—
192#117—1222no—
193#117—1222no—
194C2×C42—21no—
195C2×C42C22×D6111yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—1222no—
198#104—44no—
199C22⋊C8—1222no—
200#97—1222no—
201#95—22no—
202#41—42no—
203#41—42no—
204C2×Q8—614no—
205C2×Q8—1224no—
206C2×Q8—1222no—
207C2×Q8—4882no—
208C2×Q8—4882no—
209C2×Q8#1175819812yes—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4#1175819812yes—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—1224no—
216C22×C4D4×D6112yes—
217C22×C4C2×C22⋊D6111yescharacteristic
218M16—42no—
219C2×C8—41no—
220C2×C8—2442no—
221C4⋊C4—22no—
222C4⋊C4—614no—
223C4⋊C4—1228no—
224C4⋊C4—1224no—
225C4⋊C4—1222no—
226C4⋊C4—1224no—
227C4⋊C4—1222no—
228C4⋊C4—1222no—
229C22⋊C4—24no—
230C22⋊C4—1224no—
231C22⋊C4—1228no—
232C22⋊C4—1224no—
233C42—22no—
234C42—41no—
235C42—1224no—
236C42—1224no—
237C42C2×C4×D6111yescharacteristic
238C42C2×D24411yescharacteristic
239C42D4×D6112yes—
240C23—22no—
241C23#1451974511yescharacteristic
242Q8—42no—
243Q8—4884no—
244Q8—2444no—
245Q8—2442no—
246Q8—2442no—
247D4—42no—
248D4—44no—
249D4—82no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—1224no—
259C2×C4—1224no—
260C2×C4—4884no—
261C2×C4C22⋊C4×D6112yes—
262C2×C4C2×#1175819811yescharacteristic
263C2×C4#1451974511yescharacteristic
264C2×C4#1451974712yes—
265C8—81no—
266C8—4882no—
267C22—21no—
268C22—22no—
269C22—42no—
270C22—44no—
271C22#1683961711yescharacteristic
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—82no—
277C4—2444no—
278C4—2444no—
279C4#1683992911yescharacteristic
280C4#1683993911yescharacteristic
281C2—41no—
282C2—82no—
283C2#2144577211yescharacteristic
284C2#2144579312yes—
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1963C2
socleC22×C61#379
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#1963D61
hypocenterC61#8492
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #21444547C2semidirect
4. #21444330C2semidirect
7. #16840498C4semidirect
8. #16840492C4semidirect
165. C61#8492semidirect

Maximal subgroups

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

subgroupindex
#214602332
#214445472
#214444542
#214443302
C61×#19632
#849261