Finite Group Databasebeta

← #29661898

Subgroups — #29661898

Order 15616.

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

normal subgroups only

This group has 942 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#21456814C212yesmaximal, complemented
2#21456167C211yesmaximal, characteristic, complemented
3#21445142C214yesmaximal, complemented
4#21445112C214yesmaximal, complemented
5#21444457C211yesmaximal, characteristic, complemented
6#21444430C212yesmaximal, complemented
7C61×#3548C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840152C2214yes—
10#16840132C2212yes—
11#16840128C2212yes—
12#16840124—22no—
13#16840121C2214yes—
14#16839964—22no—
15#16839954—22no—
16#16839907—24no—
17#16839819C2212yes—
18#16839756—22no—
19C2×#14519688C2211yescharacteristic
20#16839504C2214yes—
21#16839502C2218yes—
22C61×#562C2214yes—
23C61×#554—22no—
24C61×#550—22no—
25C61×#547—22no—
26C61×#540—22no—
27C61×#537—24no—
28C61×#533C2211yescharacteristic
29C61×#532C2212yes—
30C2×#11758222—21no—
31C2×#11758201—24no—
32C2×#11758199—24no—
33C22×#92418—21no—
34#14519777C2311yescharacteristic
35#14519774C2311yescharacteristic
36#14519765—21no—
37#14519751—22no—
38C4×D244—22no—
39C4⋊#92418—22no—
40#14519701—24no—
41#14519700—24no—
42#14519698—24no—
43#14519697—24no—
44C22⋊#92418—22no—
45#14519688—21no—
46#14519688C2312yes—
47#14519688C2314yes—
48C2×#41×C61—24no—
49C2×#41×C61D414yes—
50C4⋊Q8×C61D412yes—
51C4⋊Q8×C61D412yes—
52#127×C61—22no—
53#127×C61D412yes—
54#125×C61—22no—
55#125×C61C2312yes—
56#124×C61C2314yes—
57#123×C61—28no—
58C22⋊Q8×C61—24no—
59C22⋊Q8×C61—28no—
60C22⋊Q8×C61—22no—
61#121×C61—28no—
62#121×C61—24no—
63#121×C61—28no—
64#121×C61—22no—
65#121×C61—22no—
66C4×Q8×C61—22no—
67C4×D4×C61—24no—
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70#117×C61D412yes—
71C2×C4⋊C4×C61—24no—
72C2×C42×C61C2311yescharacteristic
73#41×C61—84no—
74#41×C61—88no—
75C2×Q8×C61—22no—
76C2×Q8×C61C2×D414yes—
77C2×D4×C61—28no—
78C2×D4×C61—24no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61C2×D414yes—
82C22×C4×C61—28no—
83C22×C4×C61—24no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D412yes—
86C2×C22⋊D61—41no—
87#11758222—44no—
88C2×D244—21no—
89C2×D244—22no—
90C2×C4×D61—41no—
91C2×Dic122—21no—
92C4⋊C4×C61—28no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61C2×D414yes—
98C22⋊C4×C61—28no—
99C22⋊C4×C61—22no—
100C22⋊C4×C61—24no—
101C22⋊C4×C61—24no—
102C42×C61—22no—
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D412yes—
105C42×C61C2411yescharacteristic
106#11758201—84no—
107#11758199—84no—
108#11758198—42no—
109C4⋊Dic61—22no—
110C2×#92418—22no—
111C2×#92418—24no—
112C2×#92418—42no—
113C23×C61—24no—
114C23×C61#12011yescharacteristic
115C22×D61—41no—
116Q8×C61—44no—
117Q8×C61—44no—
118D4×C61—44no—
119D4×C61—44no—
120D4×C61—44no—
121D4×C61—84no—
122D4×C61—88no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—24no—
126C2×C4×C61—42no—
127C2×C4×C61—48no—
128C2×C4×C61—88no—
129C2×C4×C61#12012yes—
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61C22×D412yes—
132C2×C4×C61C22×D411yescharacteristic
133C22⋊D61—82no—
134C2×Dic61—41no—
135D244—42no—
136D244—42no—
137D244—42no—
138C4×D61—82no—
139Dic122—42no—
140#92418—42no—
141#92418—44no—
142#19061—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—44no—
146C22×C61—44no—
147C22×C61—44no—
148C22×C61C2×#12011yescharacteristic
149C2×D61—81no—
150C2×D61—82no—
151C4×C61—21no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—44no—
155C4×C61—44no—
156C4×C61#45412yes—
157Dic61—81no—
158#3548D6111yescharacteristic, complemented
159#3214—612no—
160#3135—611no—
161#1662—614no—
162#1656—614no—
163#1537—611no—
164#1530—612no—
165C2×C61—41no—
166C2×C61—84no—
167C2×C61C2×#45411yescharacteristic
168C2×C61#301912yes—
169D61—161no—
170#562C2×D6114yes—
171#554—22no—
172#550—22no—
173#547—22no—
174#540—22no—
175#537—24no—
176#533C2×D6111yescharacteristic
177#532—611no—
178#532C2×D6112yes—
179#496—614no—
180#491—612no—
181#490—612no—
182#489—614no—
183#489—1222no—
184#461—1222no—
185#459—1222no—
186#449—1224no—
187#432—612no—
188#416—1222no—
189C2×#97—611no—
190#330—614no—
191#329—618no—
192C61#1906111yescharacteristic, complemented, Sylow
193C2×#41—24no—
194C2×#41—1221no—
195C2×#41C22⋊D6114yes—
196C2×SD16—1224no—
197C2×D8—1224no—
198C22×C8—1221no—
199C4⋊Q8—611no—
200C4⋊Q8C22⋊D6112yes—
201C4⋊Q8C22⋊D6112yes—
202#127—22no—
203#127—611no—
204#127C22⋊D6112yes—
205#125—22no—
206#125C22×D6112yes—
207#124—1221no—
208#124C22×D6114yes—
209#123—28no—
210C22⋊Q8—24no—
211C22⋊Q8—28no—
212C22⋊Q8—22no—
213#121—28no—
214#121—24no—
215#121—28no—
216#121—22no—
217#121—22no—
218#121—1222no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—1222no—
224#117C22⋊D6112yes—
225C2×C4⋊C4—24no—
226C2×C42C22×D6111yescharacteristic
227C4⋊C8—1222no—
228#103—1224no—
229#103—1224no—
230#102—1224no—
231#102—1224no—
232C22⋊C8—1222no—
233#97—612no—
234#97—614no—
235#97—1221no—
236#41—84no—
237#41—88no—
238#41—2444no—
239C2×Q8—22no—
240C2×Q8—1221no—
241C2×Q8C2×C22⋊D6114yes—
242C2×D4—28no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—1221no—
247C2×D4—1222no—
248C2×D4—2441no—
249C2×D4C2×C22⋊D6114yes—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—2441no—
253C22×C4C2×C22⋊D6111yescharacteristic
254C22×C4C2×C22⋊D6112yes—
255SD16—4884no—
256D8—4884no—
257C2×C8—1222no—
258C2×C8—1224no—
259C2×C8—2442no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—1222no—
266C4⋊C4C2×C22⋊D6114yes—
267C22⋊C4—28no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—2442no—
272C42—22no—
273C42C2×C22⋊D6111yescharacteristic
274C42C2×C22⋊D6112yes—
275C42C23×D6111yescharacteristic
276C23—24no—
277C23—2441no—
278C23#1451974611yescharacteristic
279Q8—44no—
280Q8—44no—
281Q8—2442no—
282D4—44no—
283D4—44no—
284D4—44no—
285D4—84no—
286D4—88no—
287D4—4882no—
288D4—2442no—
289D4—2442no—
290D4—2442no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—48no—
296C2×C4—88no—
297C2×C4—4882no—
298C2×C4—2441no—
299C2×C4#1451974612yes—
300C2×C4#1451974611yescharacteristic
301C2×C4C22×C22⋊D6111yescharacteristic
302C2×C4C22×C22⋊D6112yes—
303C8—2442no—
304C8—2444no—
305C22—21no—
306C22—22no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—4881no—
311C22—4882no—
312C22C2×#1451974611yescharacteristic
313C4—21no—
314C4—22no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—4881no—
319C4#1683993512yes—
320C2—41no—
321C2—84no—
322C2—9761no—
323C2C2×#1683993511yescharacteristic
324C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#19061
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21456814C2semidirect
2. #21456167C2semidirect
3. #21445142C2semidirect
4. #21445112C2semidirect
5. #21444457C2semidirect
6. #21444430C2semidirect
7. C61×#3548C2semidirect
158. #3548D61semidirect
192. C61#19061semidirect

Maximal subgroups

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

subgroupindex
#214568142
#214561672
#214451422
#214451122
#214444572
#214444302
C61×#35482
#1906161