Finite Group Databasebeta

← #30606574

Subgroups — #30606574

Order 15616.

460 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462110C211yesmaximal, characteristic
2#21462090C211yesmaximal, characteristic
3#21461575C212yesmaximal, complemented
4#21460919C212yesmaximal, complemented
5#21454527C211yesmaximal, characteristic
6#16840748C2212yes—
7#16840741C2212yes—
8#16840725—22no—
9#16840672C412yescomplemented
10#16840670C412yescomplemented
11#16840668C412yes—
12#16840666C412yes—
13#16840644—22no—
14C2×#14519785C2211yescharacteristic
15C2×#14519784C411yescharacteristic, complemented
16#16840308C2211yescharacteristic
17#16839837—22no—
18#16839825—22no—
19#16839785C411yescharacteristic, complemented
20#16839728C2211yescharacteristic
21C2×C8×Dic61C411yescharacteristic, complemented
22C61×#408C411yescharacteristic, complemented
23D61⋊M16—22no—
24#14519856—22no—
25#14519838C2×C414yes—
26#14519837—22no—
27#14519836C2×C414yes—
28#14519835—22no—
29#14519831—42no—
30C8×#6923—42no—
31C2×#11758222C2311yescharacteristic
32#14519785—41no—
33#14519785—41no—
34#14519784—41no—
35#14519784—41no—
36C2×#11758194—21no—
37C2×#11758194C2×C411yescharacteristic
38C2×#11758189D412yes—
39C22×#92418—21no—
40C22×#92418C2×C411yescharacteristic
41C2×C8×D61D412yes—
42#14519775C2×C411yescharacteristic
43#14519773C2×C411yescharacteristic
44#14519766C2×C411yescharacteristic
45#14519764C2×C411yescharacteristic
46#14519749—22no—
47C4×C22⋊D61—22no—
48C2×C4×Dic61C2×C411yescharacteristic
49#14519706—22no—
50#14519689—22no—
51C8×Dic61—21no—
52C8×Dic61—21no—
53C2×#41×C61C2×C411yescharacteristic
54C2×M16×C61C2×C411yescharacteristic
55C22×C8×C61C2×C411yescharacteristic
56C22⋊C8×C61—22no—
57#41×C61—41no—
58#41×C61—41no—
59C2×Q8×C61C4211yescharacteristic
60C2×D4×C61—21no—
61C2×D4×C61C4211yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—22no—
65C2×C22⋊D61#4112yes—
66C22×Dic61—21no—
67#11758222—41no—
68#11758222—41no—
69C2×D244C22×C411yescharacteristic
70C2×C4×D61C2×D412yes—
71C2×Dic122C22×C411yescharacteristic
72C22⋊#6923—24no—
73#11758217—24no—
74C4⋊#6923—42no—
75C4×#6923—42no—
76#11758213—42no—
77D61⋊C8—42no—
78M16×C61—41no—
79C2×C8×C61—21no—
80C2×C8×C61—41no—
81C2×C8×C61C22⋊C412yes—
82C22⋊Dic61—22no—
83#11758198—22no—
84#11758196—22no—
85C4×Dic61—21no—
86C4×Dic61C4211yescharacteristic
87C4×Dic61C4211yescharacteristic
88#11758194—42no—
89#11758194—41no—
90C2×#92418—21no—
91C2×#92418—21no—
92C2×#92418—21no—
93C2×#92418—41no—
94C2×#92418—41no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97#11758189—42no—
98C8×D61—22no—
99C8×D61—22no—
100C23×C61—21no—
101C23×C61#11711yescharacteristic
102C22×D61C4×D412yes—
103C2×#6923—44no—
104Q8×C61—41no—
105D4×C61—41no—
106D4×C61—42no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61C2×C4211yescharacteristic
113C2×C4×C61C2×C22⋊C411yescharacteristic
114C22⋊D61—42no—
115C22⋊D61—44no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—42no—
119C2×Dic61C4×D412yes—
120D244—41no—
121C4×D61—22no—
122C4×D61—22no—
123Dic122—41no—
124#92420—44no—
125C8×C61—22no—
126C8×C61—41no—
127#92418—21no—
128#92418—41no—
129#92418#9911yescharacteristic
130#92418#9911yescharacteristic
131#12633—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—41no—
134C22×C61—42no—
135C22×C61—41no—
136C22×C61C4×C22⋊C411yescharacteristic
137C2×D61—22no—
138C2×D61—42no—
139#6923—84no—
140C4×C61—21no—
141C4×C61—41no—
142C4×C61C2×#9911yescharacteristic
143C4×C61#41011yescharacteristic
144Dic61—22no—
145Dic61—22no—
146Dic61—41no—
147#2931—611no—
148#2132—611no—
149#2126—611no—
150#1757—612no—
151#1395—612no—
152C2×C61—21no—
153C2×C61—41no—
154C2×C61—81no—
155C2×C61C4×#9911yescharacteristic
156D61—42no—
157C2×#131—611no—
158C2×#131—611no—
159#532—611no—
160#436—1222no—
161C8×D4—1222no—
162#423—611no—
163#413—612no—
164#410—612no—
165#408—611no—
166#408#692311yescharacteristic, complemented
167#405—1222no—
168C2×C4×C8—611no—
169#356—612no—
170#355—612no—
171#354—612no—
172#353—612no—
173#344—1222no—
174C61#1263311yescharacteristic, complemented, Sylow
175C2×#41—611no—
176C2×#41C2×#692311yescharacteristic
177#131—2441no—
178#131—2441no—
179#131—2441no—
180#131—2441no—
181C2×M16—612no—
182C2×M16—611no—
183C2×M16—1221no—
184C2×M16—1222no—
185C2×M16C2×#692311yescharacteristic
186C22×C8—611no—
187C22×C8—612no—
188C22×C8—1221no—
189C22×C8C2×#692311yescharacteristic
190C4⋊Q8—611no—
191#127—611no—
192#124—611no—
193#124—611no—
194#121—1222no—
195C4×D4—1222no—
196#117—1222no—
197C2×C42—611no—
198C4⋊C8—1222no—
199#101—614no—
200#100—1222no—
201#99—614no—
202#99—1222no—
203C22⋊C8—22no—
204C22⋊C8—1222no—
205#97—2442no—
206C4×C8—1221no—
207C4×C8—1221no—
208C4×C8—2442no—
209#41—41no—
210#41—41no—
211#41—2441no—
212#41—2441no—
213C2×Q8—611no—
214C2×Q8C4×#692311yescharacteristic
215C2×D4—21no—
216C2×D4—612no—
217C2×D4—611no—
218C2×D4—1222no—
219C2×D4C4×#692311yescharacteristic
220C22×C4—21no—
221C22×C4—612no—
222C22×C4—1221no—
223C22×C4C22×#692311yescharacteristic
224M16—41no—
225M16—1224no—
226M16—2442no—
227M16—2442no—
228M16—2442no—
229M16—2441no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8—611no—
233C2×C8—611no—
234C2×C8—1222no—
235C2×C8—1222no—
236C2×C8—1221no—
237C2×C8—1221no—
238C2×C8—1221no—
239C2×C8—2441no—
240C2×C8—2441no—
241C2×C8—2442no—
242C2×C8C22⋊#692312yes—
243C4⋊C4—1222no—
244C4⋊C4—2442no—
245C22⋊C4—1222no—
246C22⋊C4—1222no—
247C22⋊C4—1224no—
248C42—611no—
249C42—611no—
250C42—1221no—
251C42—2442no—
252C23—21no—
253C23—612no—
254C23#1451985611yescharacteristic
255Q8—41no—
256Q8—2441no—
257D4—41no—
258D4—42no—
259D4—2441no—
260D4—2442no—
261D4—2444no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—612no—
268C2×C4—1221no—
269C2×C4—1222no—
270C2×C4—1222no—
271C2×C4—1221no—
272C2×C4—2442no—
273C2×C4—2444no—
274C2×C4C2×C4×#692311yescharacteristic
275C2×C4C2×C22⋊#692311yescharacteristic
276C8—22no—
277C8—41no—
278C8—611no—
279C8—611no—
280C8—1221no—
281C8—2444no—
282C8—2441no—
283C22—21no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—1222no—
288C22—2442no—
289C22C4×C22⋊#692311yescharacteristic
290C4—21no—
291C4—41no—
292C4—1222no—
293C4—1222no—
294C4—4884no—
295C4—2441no—
296C4D61⋊#9911yescharacteristic
297C4#1684074211yescharacteristic
298C2—21no—
299C2—41no—
300C2—81no—
301C2—2442no—
302C2#2146156611yescharacteristic
subgroupG/H
centerC4D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#408C4
socleC2×C61C4×#99
prime radical#16840308C22
cosocleC2×#11758222C23
hypercenter#408#6923
hypocenterC61#12633
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #408

Complements

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

NH ≅ G/Ntype
3. #21461575C2semidirect
4. #21460919C2semidirect
9. #16840672C4semidirect
10. #16840670C4semidirect
15. C2×#14519784C4semidirect
19. #16839785C4semidirect
21. C2×C8×Dic61C4semidirect
22. C61×#408C4semidirect
166. #408#6923semidirect
174. C61#12633semidirect

Maximal subgroups

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

subgroupindex
#214621102
#214620902
#214615752
#214609192
#214545272
#1263361