Finite Group Databasebeta

← #29661281

Subgroups — #29661281

Order 15616.

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

normal subgroups only

This group has 850 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
1C2×#16840512C211yesmaximal, characteristic
2#21457021C212yesmaximal, complemented
3#21445152C218yesmaximal, complemented
4#21444443C211yesmaximal, characteristic
5#21444420C212yesmaximal, complemented
6C61×#3140C211yesmaximal, characteristic
7#16840512—24no—
8#16840512—22no—
9#16840512C2214yes—
10C2×#14519775—21no—
11C2×#14519770—22no—
12C2×#14519769C2212yes—
13C2×#14519758—21no—
14C2×#14519754—22no—
15#16840168C2214yescomplemented
16#16840130C2214yescomplemented
17#16840095—22no—
18#16840090—22no—
19#16840069—24no—
20#16839819C2212yes—
21#16839503C2218yescomplemented
22#16839502C2214yes—
23C61×#533C2211yescharacteristic
24C2×#124×C61C2211yescharacteristic
25C61×#489C2214yes—
26C61×#488—22no—
27C61×#419—21no—
28C61×#418—21no—
29C4×M16×C61C2211yescharacteristic
30C22×Dic122—21no—
31C23×Dic61—21no—
32#14519775—24no—
33#14519775—24no—
34#14519770—28no—
35#14519770—28no—
36#14519769—22no—
37#14519769—22no—
38#14519769D414yes—
39#14519769C2314yes—
40#14519758—24no—
41#14519758—24no—
42#14519754—28no—
43#14519754—28no—
44C2×C4⋊Dic61D412yes—
45C2×C4⋊Dic61D412yes—
46C2×#11758196—22no—
47C2×#11758196—22no—
48C2×#11758196—22no—
49C2×C22⋊Dic61—22no—
50C2×C22⋊Dic61—21no—
51C2×C4×Dic61—21no—
52#14519710—24no—
53#14519708—24no—
54C4⋊#92418—22no—
55#14519701—24no—
56#14519701—24no—
57C22⋊#92418—22no—
58#14519688C2314yes—
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61D411yescharacteristic
61C2×M16×C61—21no—
62C4⋊Q8×C61C2312yes—
63#125×C61—21no—
64#124×C61—22no—
65#124×C61D412yes—
66#124×C61C2312yes—
67C22⋊Q8×C61—22no—
68C4×Q8×C61—22no—
69#117×C61—21no—
70C2×C22⋊C4×C61—22no—
71C2×C42×C61C2311yescharacteristic
72#103×C61—24no—
73#102×C61—24no—
74#97×C61C2312yes—
75C4×C8×C61—21no—
76C24×C61—21no—
77C2×Q8×C61—21no—
78C2×Q8×C61—22no—
79C2×Q8×C61—42no—
80C2×Q8×C61C2×D412yes—
81C2×D4×C61—21no—
82C2×D4×C61—42no—
83C2×D4×C61C2×D412yes—
84C22×C4×C61—21no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D412yes—
87C22×Dic61—22no—
88C22×Dic61—22no—
89C22×Dic61—24no—
90C22×Dic61—82no—
91C2×Dic122—24no—
92C2×Dic122—82no—
93M16×C61—42no—
94C2×C8×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—42no—
98C22⋊C4×C61—24no—
99C22⋊C4×C61—24no—
100C22⋊C4×C61—41no—
101C42×C61—41no—
102C42×C61C2×D412yes—
103C42×C61C2×D411yescharacteristic
104C42×C61C2411yescharacteristic
105C22⋊Dic61—28no—
106C22⋊Dic61—24no—
107C4⋊Dic61—22no—
108C4⋊Dic61—22no—
109C4⋊Dic61C2×D414yes—
110C4⋊Dic61C2×D414yes—
111#11758196—28no—
112#11758196—24no—
113#11758196—24no—
114#11758196—24no—
115#11758196—24no—
116C4×Dic61—24no—
117C2×#92418—24no—
118C23×C61—22no—
119C23×C61—44no—
120C23×C61—42no—
121C23×C61#12011yescharacteristic
122Q8×C61—42no—
123Q8×C61—44no—
124Q8×C61—81no—
125D4×C61—42no—
126D4×C61—81no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61#12011yescharacteristic
133C2×C4×C61#12012yes—
134C2×C4×C61C22×D412yes—
135C2×C4×C61C22×D411yescharacteristic
136C2×Dic61—24no—
137C2×Dic61—24no—
138C2×Dic61—44no—
139C2×Dic61—44no—
140C2×Dic61—84no—
141Dic122—84no—
142C8×C61—42no—
143#92418—44no—
144#18999—611nomaximal, Sylow
145C22×C61—22no—
146C22×C61—42no—
147C22×C61—44no—
148C22×C61—84no—
149C22×C61#45512yes—
150C22×C61C2×#12011yescharacteristic
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—42no—
156C4×C61—82no—
157Dic61—44no—
158Dic61—44no—
159C2×#563—611no—
160#3236—612no—
161#3140D6111yescharacteristic
162#1664—618no—
163#1533—611no—
164#1528—612no—
165C2×C61—22no—
166C2×C61—82no—
167C2×C61C2×#45511yescharacteristic
168C2×C61#302012yes—
169#563—614no—
170#563—1224no—
171#563—1222no—
172#533C2×D6111yescharacteristic
173C2×C4⋊Q8—1221no—
174C2×#126—1222no—
175C2×#125—612no—
176C2×#124C2×D6111yescharacteristic
177C2×#123—1221no—
178C2×C22⋊Q8—1222no—
179#501—614no—
180#491—614no—
181#489C2×D6114yes—
182#488—22no—
183#484—1222no—
184#483—1222no—
185#479—1224no—
186#432—612no—
187#419—21no—
188#418—21no—
189C4×M16C2×D6111yescharacteristic
190#330—618no—
191#329—614no—
192C61#1899911yescharacteristic, complemented, Sylow
193C22×Q8—1221no—
194C22×Q8C22⋊D6111yescharacteristic
195C22×D4C22⋊D6111yescharacteristic
196C23×C4—1221no—
197C2×M16—21no—
198C4⋊Q8—1224no—
199C4⋊Q8—1224no—
200C4⋊Q8C22×D6112yes—
201#126—1228no—
202#126—1228no—
203#125—21no—
204#125—614no—
205#125—614no—
206#125—1222no—
207#125—1222no—
208#124—22no—
209#124C22⋊D6112yes—
210#124C22×D6112yes—
211#123—1224no—
212#123—1224no—
213C22⋊Q8—22no—
214C22⋊Q8—1228no—
215C22⋊Q8—1228no—
216C4×Q8—22no—
217#117—21no—
218C2×C4⋊C4—612no—
219C2×C4⋊C4—612no—
220C2×C4⋊C4—1222no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4—1222no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—1222no—
225C2×C22⋊C4—1221no—
226C2×C42—1221no—
227C2×C42C22×D6111yescharacteristic
228#107—1224no—
229#106—1224no—
230C4⋊C8—1222no—
231#103—24no—
232#103—1224no—
233#103—1224no—
234#102—24no—
235C22⋊C8—1222no—
236#97—614no—
237#97C22×D6112yes—
238C4×C8—21no—
239C24—21no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—1224no—
244C2×Q8—4882no—
245C2×Q8C2×C22⋊D6112yes—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C2×C22⋊D6112yes—
249C22×C4—21no—
250C22×C4—1224no—
251C22×C4—1222no—
252C22×C4—1222no—
253C22×C4—4882no—
254C22×C4D4×D6112yes—
255C22×C4C2×C22⋊D6111yescharacteristic
256M16—42no—
257C2×C8—22no—
258C2×C8—1224no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—614no—
263C4⋊C4—614no—
264C4⋊C4—1228no—
265C4⋊C4—1224no—
266C4⋊C4—1224no—
267C4⋊C4—1222no—
268C4⋊C4—1222no—
269C4⋊C4—1224no—
270C4⋊C4—1224no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C22⋊C4—1228no—
275C22⋊C4—1224no—
276C42—41no—
277C42—1224no—
278C42D4×D6112yes—
279C42C2×C22⋊D6111yescharacteristic
280C42C23×D6111yescharacteristic
281C23—22no—
282C23—44no—
283C23—42no—
284C23#1451974511yescharacteristic
285Q8—42no—
286Q8—44no—
287Q8—81no—
288Q8—4884no—
289D4—42no—
290D4—81no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—1224no—
297C2×C4—1224no—
298C2×C4—4884no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4#1451974511yescharacteristic
302C2×C4#1451974512yes—
303C2×C4C2×D4×D6112yes—
304C2×C4C22×C22⋊D6111yescharacteristic
305C8—42no—
306C8—2444no—
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22#1683993912yes—
312C22C2×#1451974511yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—42no—
318C4—82no—
319C4—2444no—
320C4—2444no—
321C2—22no—
322C2—82no—
323C2C2×#1683993911yescharacteristic
324C2#2145518112yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3140C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3140D61
hypocenterC61#18999
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
2. #21457021C2semidirect
3. #21445152C2semidirect
5. #21444420C2semidirect
15. #16840168C22semidirect
16. #16840130C22semidirect
21. #16839503C22semidirect
192. C61#18999semidirect

Maximal subgroups

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

subgroupindex
C2×#168405122
#214570212
#214451522
#214444432
#214444202
C61×#31402
#1899961