Finite Group Databasebeta

← #29661930

Subgroups — #29661930

Order 15616.

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

normal subgroups only

This group has 920 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#21460219C211yesmaximal, characteristic
2#21456826C212yesmaximal, complemented
3#21445134C218yesmaximal, complemented
4#21444457C211yesmaximal, characteristic, complemented
5#21444429C212yesmaximal, complemented
6C61×#3162C211yesmaximal, characteristic, complemented
7#16840510C2214yes—
8#16840472—22no—
9#16840434—22no—
10#16840415—22no—
11#16840367—22no—
12#16840342—24no—
13#16840312C2212yes—
14#16840154C2214yes—
15#16840124C2214yes—
16#16839972—22no—
17#16839962—22no—
18#16839922—24no—
19#16839819C2212yes—
20#16839504C2214yes—
21#16839501C2218yes—
22C61×#533C2211yescharacteristic
23C61×#533C2211yescharacteristic
24C61×#501—22no—
25C61×#501C2214yes—
26C61×#427—22no—
27C2×#97×C61C2211yescharacteristic
28C2×#11758226—24no—
29C2×#11758222D414yes—
30C2×#11758202—24no—
31C2×#11758201—24no—
32#14519777D412yes—
33#14519774—22no—
34#14519774D412yes—
35#14519769—22no—
36#14519765C2314yes—
37#14519762—28no—
38#14519755—28no—
39#14519753—24no—
40#14519752—24no—
41#14519752—28no—
42#14519751—22no—
43#14519751—22no—
44#14519750—28no—
45C4×D244—24no—
46C4×D244—22no—
47C4×D244—22no—
48C4⋊C4×D61—24no—
49C4⋊#92418—22no—
50#14519700—24no—
51#14519697—24no—
52C22⋊#92418—22no—
53#14519688C2314yes—
54C22×C8×C61—21no—
55C4⋊Q8×C61D412yes—
56C4⋊Q8×C61C2312yes—
57#125×C61—21no—
58#125×C61C2312yes—
59C22⋊Q8×C61—22no—
60C22⋊Q8×C61—22no—
61C4×Q8×C61—22no—
62C4×Q8×C61—22no—
63#117×C61—21no—
64#117×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66#107×C61—24no—
67#106×C61—24no—
68#97×C61—21no—
69#97×C61C2312yes—
70C2×Q8×C61—22no—
71C2×Q8×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D412yes—
74C2×C22⋊D61—24no—
75#11758226—88no—
76#11758222—84no—
77C2×D244—22no—
78C2×D244—22no—
79C2×D244—28no—
80C2×D244C2×D414yes—
81C2×C4×D61—24no—
82C2×C4×D61—28no—
83C2×Dic122C2×D414yes—
84C2×C8×C61—24no—
85C2×C8×C61—22no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—42no—
91C4⋊C4×C61C2×D414yes—
92C22⋊C4×C61—22no—
93C22⋊C4×C61—41no—
94C42×C61—22no—
95C42×C61—41no—
96C42×C61C2×D412yes—
97C42×C61C2×D411yescharacteristic
98C42×C61C2411yescharacteristic
99#11758202—84no—
100#11758201—84no—
101#11758198—28no—
102#11758198—24no—
103#11758198—24no—
104C4⋊Dic61—22no—
105C4⋊Dic61—22no—
106#11758196—28no—
107C2×#92418—24no—
108C23×C61#12011yescharacteristic
109C22×D61—24no—
110Q8×C61—44no—
111Q8×C61—42no—
112Q8×C61—42no—
113C2×C4×C61—22no—
114C2×C4×C61—24no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61#12012yes—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61C22×D412yes—
120C2×C4×C61C22×D411yescharacteristic
121C22⋊D61—84no—
122C2×Dic61—24no—
123D244—44no—
124D244—44no—
125D244—44no—
126D244—88no—
127C4×D61—48no—
128C4×D61—88no—
129Dic122—44no—
130C8×C61—24no—
131#92418—44no—
132#19065—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—22no—
135C22×C61C2×#12011yescharacteristic
136C2×D61—44no—
137C2×D61—44no—
138C2×D61—44no—
139C4×C61—21no—
140C4×C61—22no—
141C4×C61—22no—
142C4×C61—44no—
143C4×C61—82no—
144C4×C61#45612yes—
145Dic61—44no—
146#3548—611no—
147#3216—612no—
148#3162D6111yescharacteristic, complemented
149#1661—618no—
150#1537—611no—
151#1530—612no—
152C2×C61—41no—
153C2×C61C2×#45611yescharacteristic
154C2×C61#301912yes—
155D61—84no—
156#562—614no—
157#554—1222no—
158#550—1222no—
159#547—1222no—
160#540—1222no—
161#537—1224no—
162#533C2×D6111yescharacteristic
163#533C2×D6111yescharacteristic
164#532—612no—
165#501—22no—
166#501C2×D6114yes—
167#497—614no—
168#489—614no—
169#462—1222no—
170#460—1222no—
171#452—1224no—
172#432—612no—
173#427—22no—
174C2×#97C2×D6111yescharacteristic
175#330—614no—
176#329—618no—
177C61#1906511yescharacteristic, complemented, Sylow
178C2×#41—614no—
179C2×#41—1224no—
180C2×Q16—1224no—
181C2×SD16—1224no—
182C22×C8—21no—
183C4⋊Q8—612no—
184C4⋊Q8C22⋊D6112yes—
185C4⋊Q8C22×D6112yes—
186#127—612no—
187#127—1222no—
188#125—21no—
189#125—1222no—
190#125C22×D6112yes—
191#124—614no—
192#123—1228no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—1228no—
196C22⋊Q8—1224no—
197#121—1228no—
198#121—1228no—
199#121—1224no—
200#121—1222no—
201#121—1222no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×D4—1224no—
205C4×D4—1222no—
206C4×D4—1222no—
207#117—21no—
208#117C22⋊D6112yes—
209C2×C4⋊C4—1224no—
210C2×C42C22×D6111yescharacteristic
211#107—24no—
212#106—24no—
213C4⋊C8—1222no—
214#103—1224no—
215#102—1224no—
216C22⋊C8—1222no—
217#97—21no—
218#97—614no—
219#97C22×D6112yes—
220#41—4884no—
221#41—4888no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—614no—
225C2×D4—614no—
226C2×D4—1228no—
227C2×D4—1224no—
228C2×D4—1222no—
229C2×D4—1222no—
230C22×C4—1228no—
231C22×C4—1224no—
232C22×C4D4×D6112yes—
233C22×C4C2×C22⋊D6111yescharacteristic
234Q16—4884no—
235SD16—4884no—
236C2×C8—24no—
237C2×C8—22no—
238C2×C8—1224no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—1228no—
245C4⋊C4—1222no—
246C4⋊C4—1222no—
247C4⋊C4C2×C22⋊D6114yes—
248C22⋊C4—22no—
249C22⋊C4—41no—
250C22⋊C4—1228no—
251C22⋊C4—1224no—
252C22⋊C4—1224no—
253C42—22no—
254C42—41no—
255C42D4×D6112yes—
256C42C2×C22⋊D6111yescharacteristic
257C42C23×D6111yescharacteristic
258C23—1224no—
259C23#1451974511yescharacteristic
260Q8—44no—
261Q8—42no—
262Q8—42no—
263Q8—2444no—
264D4—4888no—
265D4—4884no—
266D4—2444no—
267D4—2444no—
268D4—2444no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—1224no—
274C2×C4—4888no—
275C2×C4—2448no—
276C2×C4#1451974512yes—
277C2×C4#1451974511yescharacteristic
278C2×C4C2×D4×D6112yes—
279C2×C4C22×C22⋊D6111yescharacteristic
280C8—24no—
281C8—2444no—
282C22—21no—
283C22—22no—
284C22—2444no—
285C22—2444no—
286C22—2444no—
287C22C2×#1451974511yescharacteristic
288C4—21no—
289C4—22no—
290C4—22no—
291C4—44no—
292C4—82no—
293C4—2444no—
294C4#1683994412yes—
295C2—41no—
296C2—4884no—
297C2C2×#1683994411yescharacteristic
298C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3162C2
socleC22×C61C2×#120
prime radical#21460219C2
hypercenter#3162D61
hypocenterC61#19065
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
2. #21456826C2semidirect
3. #21445134C2semidirect
4. #21444457C2semidirect
5. #21444429C2semidirect
6. C61×#3162C2semidirect
148. #3162D61semidirect
177. C61#19065semidirect

Maximal subgroups

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

subgroupindex
#214602192
#214568262
#214451342
#214444572
#214444292
C61×#31622
#1906561