Finite Group Databasebeta

← #29658190

Subgroups — #29658190

Order 15616.

928 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 163 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#21456970C212yesmaximal, complemented
3#21444957C214yesmaximal, complemented
4#21444938C214yesmaximal, complemented
5#21444481C211yesmaximal, characteristic
6#21444241C212yesmaximal, complemented
7C61×#3152C211yesmaximal, characteristic
8#16840512—22no—
9#16840498C2212yes—
10#16840492C2212yes—
11#16840482—22no—
12D4×Dic122—22no—
13#16840379—24no—
14#16840333—22no—
15#16840319C2212yes—
16#16840165—22no—
17#16840114—22no—
18#16840097—24no—
19#16840054C2218yes—
20#16839815C2212yes—
21#16839511C2212yes—
22#16839506C2212yes—
23#16839494C2218yes—
24C2×#137×C61—22no—
25C61×#532C2211yescharacteristic
26C2×C4⋊Q8×C61C2211yescharacteristic
27C61×#497—22no—
28C61×#494C2212yes—
29C61×#492C2212yes—
30C4×M16×C61C2211yescharacteristic
31C22×Dic122—22no—
32#14519777D412yes—
33#14519777D412yes—
34#14519776—24no—
35#14519775—24no—
36#14519770—28no—
37#14519769—22no—
38#14519764—24no—
39#14519764—24no—
40#14519758—24no—
41#14519758—24no—
42#14519756—24no—
43#14519756—22no—
44#14519756D414yes—
45#14519754—28no—
46C4×Dic122—22no—
47C4×Dic122C2314yes—
48#14519742—24no—
49D4×Dic61—24no—
50#14519736—22no—
51#14519736—22no—
52#14519710—28no—
53C4⋊#92418C2314yes—
54#14519700—28no—
55C22⋊#92418—22no—
56C4×#92418—22no—
57C2×#41×C61D412yes—
58C22×Q8×C61—21no—
59#137×C61—48no—
60C2×Q16×C61—24no—
61C2×SD16×C61—24no—
62C2×M16×C61—21no—
63C4⋊Q8×C61—22no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61C2312yes—
66C4⋊Q8×C61C2311yescharacteristic
67#127×C61C2311yescharacteristic
68#124×C61D412yes—
69#121×C61—22no—
70C4×D4×C61—22no—
71C2×C4⋊C4×C61—22no—
72C2×C42×C61C2311yescharacteristic
73#97×C61—21no—
74C4×C8×C61C2312yes—
75#41×C61—44no—
76C2×Q8×C61—22no—
77C2×Q8×C61—44no—
78C2×Q8×C61—41no—
79C2×Q8×C61C2×D412yes—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61C2×D412yes—
83C22×C4×C61—22no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—24no—
88C2×Dic122—22no—
89C2×Dic122—24no—
90C2×Dic122—84no—
91C2×Dic122C2×D414yes—
92Q16×C61—48no—
93SD16×C61—48no—
94M16×C61—24no—
95C2×C8×C61—22no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61—42no—
99C22⋊C4×C61—24no—
100C42×C61C2×D412yes—
101C42×C61C2×D411yescharacteristic
102C42×C61C2411yescharacteristic
103C22⋊Dic61—24no—
104C22⋊Dic61—28no—
105C22⋊Dic61—24no—
106C4⋊Dic61—22no—
107C4⋊Dic61—22no—
108C4⋊Dic61—22no—
109C4⋊Dic61C2×D414yes—
110#11758196—28no—
111#11758196—24no—
112#11758196—24no—
113C4×Dic61—24no—
114C4×Dic61—24no—
115C2×#92418—24no—
116C23×C61—22no—
117C23×C61#12011yescharacteristic
118Q8×C61—42no—
119Q8×C61—44no—
120Q8×C61—82no—
121D4×C61—42no—
122D4×C61—44no—
123D4×C61—82no—
124C2×C4×C61—24no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—44no—
128C2×C4×C61—42no—
129C2×C4×C61—82no—
130C2×C4×C61#12012yes—
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61C22×D411yescharacteristic
133C2×C4×C61C22×D412yes—
134C2×Dic61—24no—
135C2×Dic61—24no—
136C2×Dic61—84no—
137Dic122—28no—
138Dic122—44no—
139Dic122—84no—
140C8×C61—24no—
141#92418—44no—
142#18719—611nomaximal, Sylow
143C22×C61—22no—
144C22×C61—21no—
145C22×C61—44no—
146C22×C61—42no—
147C22×C61C2×#12011yescharacteristic
148C4×C61—22no—
149C4×C61—21no—
150C4×C61—42no—
151C4×C61—82no—
152C4×C61#45312yes—
153C4×C61#57814yes—
154Dic61—44no—
155Dic61—44no—
156#3550—611no—
157#3230—612no—
158#3152D6111yescharacteristic
159#1633—614no—
160#1630—614no—
161#1543—611no—
162#1485—612no—
163C2×C61—41no—
164C2×C61—82no—
165C2×C61#301112yes—
166C2×C61C2×#45311yescharacteristic
167C2×#137—22no—
168#563—1222no—
169#559—612no—
170Q82—612no—
171#556—1222no—
172D4×Q8—1222no—
173#542—1224no—
174#536—1222no—
175#533—612no—
176#532C2×D6111yescharacteristic
177C2×C4⋊Q8C2×D6111yescharacteristic
178#500—1222no—
179#497—22no—
180#494C2×D6112yes—
181#492C2×D6112yes—
182#487—1222no—
183#484—1224no—
184#477—618no—
185#431—612no—
186C4×M16C2×D6111yescharacteristic
187#333—612no—
188#331—612no—
189Q8⋊C8—618no—
190C61#1871911yescharacteristic, complemented, Sylow
191C2×#41C22⋊D6112yes—
192C22×Q8—21no—
193C22×Q8—1222no—
194#137—48no—
195C2×Q16—24no—
196C2×SD16—24no—
197C2×M16—21no—
198C4⋊Q8—22no—
199C4⋊Q8—21no—
200C4⋊Q8—612no—
201C4⋊Q8—612no—
202C4⋊Q8—1224no—
203C4⋊Q8—1224no—
204C4⋊Q8C22×D6112yes—
205C4⋊Q8C22×D6111yescharacteristic
206#127C22×D6111yescharacteristic
207#126—1228no—
208#125—1222no—
209#124—1224no—
210#124—1224no—
211#124C22⋊D6112yes—
212#123—1224no—
213#123—1224no—
214C22⋊Q8—614no—
215C22⋊Q8—1228no—
216C22⋊Q8—1224no—
217C22⋊Q8—1222no—
218#121—22no—
219C4×Q8—614no—
220C4×Q8—1224no—
221C4×Q8—1222no—
222C4×D4—22no—
223C4×D4—1224no—
224#117—1222no—
225#117—1222no—
226C2×C4⋊C4—22no—
227C2×C42C22×D6111yescharacteristic
228#107—1228no—
229C4⋊C8—614no—
230#103—1228no—
231C22⋊C8—1222no—
232#97—21no—
233C4×C8—1222no—
234C4×C8C22×D6112yes—
235#41—44no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—614no—
240C2×Q8—1224no—
241C2×Q8—1222no—
242C2×Q8—4884no—
243C2×Q8C2×C22⋊D6112yes—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4C2×C22⋊D6112yes—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—1224no—
250C22×C4D4×D6112yes—
251C22×C4C2×C22⋊D6111yescharacteristic
252Q16—48no—
253SD16—48no—
254M16—24no—
255C2×C8—22no—
256C2×C8—1224no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—42no—
260C4⋊C4—614no—
261C4⋊C4—1228no—
262C4⋊C4—1224no—
263C4⋊C4—1224no—
264C4⋊C4—1222no—
265C4⋊C4—1222no—
266C4⋊C4—1222no—
267C22⋊C4—24no—
268C22⋊C4—1224no—
269C22⋊C4—1228no—
270C22⋊C4—1224no—
271C42—1224no—
272C42—1224no—
273C42D4×D6112yes—
274C42C2×C22⋊D6111yescharacteristic
275C42C23×D6111yescharacteristic
276C23—22no—
277C23#1451974511yescharacteristic
278Q8—42no—
279Q8—44no—
280Q8—82no—
281Q8—1228no—
282Q8—4884no—
283Q8—2444no—
284D4—42no—
285D4—44no—
286D4—82no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—82no—
293C2×C4—1224no—
294C2×C4—1224no—
295C2×C4—4884no—
296C2×C4#1451974512yes—
297C2×C4#1451974511yescharacteristic
298C2×C4C2×D4×D6112yes—
299C2×C4C22×C22⋊D6111yescharacteristic
300C8—24no—
301C8—2444no—
302C22—22no—
303C22—21no—
304C22—44no—
305C22—42no—
306C22C2×#1451974511yescharacteristic
307C4—22no—
308C4—21no—
309C4—42no—
310C4—82no—
311C4—2444no—
312C4—2444no—
313C4#1683992912yes—
314C4#1684057814yes—
315C2—41no—
316C2—82no—
317C2#2145511312yes—
318C2C2×#1683992911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3152C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3152D61
hypocenterC61#18719
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #21456970C2semidirect
3. #21444957C2semidirect
4. #21444938C2semidirect
6. #21444241C2semidirect
190. C61#18719semidirect

Maximal subgroups

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

subgroupindex
#214602332
#214569702
#214449572
#214449382
#214444812
#214442412
C61×#31522
#1871961