Finite Group Databasebeta

← #29657908

Subgroups — #29657908

Order 15616.

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

normal subgroups only

This group has 868 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#21457210C212yesmaximal, complemented
3#21445037C214yesmaximal, complemented
4#21445002C214yesmaximal, complemented
5#21444554C211yesmaximal, characteristic
6#21444275C212yesmaximal, complemented
7C61×#3131C211yesmaximal, characteristic
8#16840512—22no—
9#16840498C2212yes—
10#16840492C2212yes—
11#16840482—22no—
12D4×Dic122—22no—
13#16840379—24no—
14#16840333—22no—
15#16840319C2212yes—
16#16840047—22no—
17#16840044C2214yes—
18#16840018—24no—
19#16839972—22no—
20#16839969C2214yes—
21#16839786C2212yes—
22#16839511C2212yes—
23#16839506C2212yes—
24#16839494C2218yes—
25C61×#533C2211yescharacteristic
26C61×#532C2211yescharacteristic
27C61×#488—22no—
28C61×#487C2212yes—
29C61×#486C2212yes—
30C61×#416—22no—
31C2×C4×C8×C61C2211yescharacteristic
32C22×Dic122—22no—
33C2×#11758202—24no—
34C2×#11758194—22no—
35#14519777D412yes—
36#14519777D412yes—
37#14519776—24no—
38#14519775—24no—
39#14519770—28no—
40#14519769—22no—
41#14519764—24no—
42#14519764—24no—
43#14519758—24no—
44#14519758—24no—
45#14519756—24no—
46#14519756—22no—
47#14519756D414yes—
48#14519754—28no—
49C4×Dic122—22no—
50C4×Dic122C2314yes—
51#14519742—24no—
52D4×Dic61—24no—
53#14519736—22no—
54#14519736—22no—
55#14519708—24no—
56C4⋊#92418—22no—
57C4⋊#92418C2314yes—
58#14519701—24no—
59#14519700—24no—
60C2×#41×C61D412yes—
61C22×C8×C61—21no—
62C4⋊Q8×C61C2312yes—
63C4⋊Q8×C61C2311yescharacteristic
64#127×C61C2311yescharacteristic
65#125×C61—21no—
66#124×C61D412yes—
67C22⋊Q8×C61—22no—
68#121×C61—22no—
69C4×Q8×C61—22no—
70C4×D4×C61—22no—
71#117×C61—21no—
72C2×C42×C61C2311yescharacteristic
73#103×C61—24no—
74#102×C61—24no—
75C4×C8×C61—21no—
76C4×C8×C61C2312yes—
77#41×C61—44no—
78C2×Q8×C61—22no—
79C2×Q8×C61C2×D412yes—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61C2×D412yes—
83C22×C4×C61—22no—
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×Dic61—24no—
87C2×Dic122—22no—
88C2×Dic122—24no—
89C2×Dic122—84no—
90C2×Dic122C2×D414yes—
91C2×C8×C61—22no—
92C2×C8×C61—42no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—42no—
97C22⋊C4×C61—24no—
98C22⋊C4×C61—41no—
99C42×C61—41no—
100C42×C61C2×D412yes—
101C42×C61C2×D411yescharacteristic
102C42×C61C2411yescharacteristic
103C22⋊Dic61—24no—
104C22⋊Dic61—28no—
105C22⋊Dic61—24no—
106#11758202—84no—
107C4⋊Dic61—22no—
108C4⋊Dic61—22no—
109C4⋊Dic61—22no—
110C4⋊Dic61C2×D414yes—
111#11758196—28no—
112#11758196—24no—
113#11758196—24no—
114C4×Dic61—24no—
115C4×Dic61—24no—
116#11758194—82no—
117C2×#92418—24no—
118C23×C61—22no—
119C23×C61#12011yescharacteristic
120Q8×C61—42no—
121Q8×C61—42no—
122Q8×C61—42no—
123D4×C61—42no—
124D4×C61—42no—
125D4×C61—42no—
126D4×C61—82no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—44no—
132C2×C4×C61—42no—
133C2×C4×C61#12012yes—
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61C22×D411yescharacteristic
136C2×C4×C61C22×D412yes—
137C2×Dic61—24no—
138C2×Dic61—24no—
139C2×Dic61—84no—
140Dic122—24no—
141Dic122—24no—
142Dic122—44no—
143Dic122—84no—
144C8×C61—42no—
145#92418—44no—
146#18693—611nomaximal, Sylow
147C22×C61—22no—
148C22×C61—21no—
149C22×C61—44no—
150C22×C61—42no—
151C22×C61C2×#12011yescharacteristic
152C4×C61—22no—
153C4×C61—21no—
154C4×C61—42no—
155C4×C61—82no—
156C4×C61#45512yes—
157C4×C61#57712yes—
158C4×C61#57812yes—
159Dic61—44no—
160Dic61—44no—
161#3550—611no—
162#3255—612no—
163#3131D6111yescharacteristic
164#1645—614no—
165#1640—614no—
166#1560—611no—
167#1493—612no—
168C2×C61—41no—
169C2×C61—82no—
170C2×C61#301312yes—
171C2×C61C2×#45511yescharacteristic
172#563—1222no—
173#559—612no—
174Q82—612no—
175#556—1222no—
176D4×Q8—1222no—
177#542—1224no—
178#536—1222no—
179#533—612no—
180#533C2×D6111yescharacteristic
181#532C2×D6111yescharacteristic
182#488—22no—
183#487C2×D6112yes—
184#486C2×D6112yes—
185#475—614no—
186#475—1222no—
187#470—1224no—
188#462—614no—
189#462—1222no—
190#423—612no—
191#416—22no—
192C2×C4×C8C2×D6111yescharacteristic
193#333—612no—
194#331—612no—
195Q8⋊C8—618no—
196C61#1869311yescharacteristic, complemented, Sylow
197C2×#41C22⋊D6112yes—
198C22×Q8—1222no—
199C2×Q16—1224no—
200C2×M16—1222no—
201C22×C8—21no—
202C4⋊Q8—612no—
203C4⋊Q8—612no—
204C4⋊Q8—1224no—
205C4⋊Q8—1224no—
206C4⋊Q8C22×D6112yes—
207C4⋊Q8C22×D6111yescharacteristic
208#127C22×D6111yescharacteristic
209#126—1228no—
210#125—21no—
211#125—1222no—
212#124—1224no—
213#124—1224no—
214#124C22⋊D6112yes—
215#123—1224no—
216#123—1224no—
217C22⋊Q8—22no—
218C22⋊Q8—614no—
219C22⋊Q8—1228no—
220C22⋊Q8—1224no—
221C22⋊Q8—1222no—
222#121—22no—
223C4×Q8—22no—
224C4×Q8—614no—
225C4×Q8—1224no—
226C4×Q8—1222no—
227C4×D4—22no—
228C4×D4—1224no—
229#117—21no—
230#117—1222no—
231#117—1222no—
232C2×C42C22×D6111yescharacteristic
233#106—1224no—
234C4⋊C8—614no—
235C4⋊C8—1222no—
236#103—24no—
237#103—1224no—
238#103—1224no—
239#102—24no—
240C4×C8—21no—
241C4×C8C22×D6112yes—
242#41—44no—
243C2×Q8—22no—
244C2×Q8—614no—
245C2×Q8—1224no—
246C2×Q8—1222no—
247C2×Q8—4884no—
248C2×Q8C2×C22⋊D6112yes—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4C2×C22⋊D6112yes—
252C22×C4—22no—
253C22×C4—1224no—
254C22×C4D4×D6112yes—
255C22×C4C2×C22⋊D6111yescharacteristic
256Q16—4884no—
257M16—4882no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—1224no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—614no—
266C4⋊C4—1228no—
267C4⋊C4—1224no—
268C4⋊C4—1224no—
269C4⋊C4—1222no—
270C4⋊C4—1222no—
271C4⋊C4—1222no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C22⋊C4—1224no—
275C22⋊C4—1228no—
276C22⋊C4—1224no—
277C42—41no—
278C42—1224no—
279C42—1224no—
280C42D4×D6112yes—
281C42C2×C22⋊D6111yescharacteristic
282C42C23×D6111yescharacteristic
283C23—22no—
284C23#1451974511yescharacteristic
285Q8—42no—
286Q8—42no—
287Q8—42no—
288Q8—1224no—
289Q8—1224no—
290Q8—4884no—
291Q8—2444no—
292D4—42no—
293D4—42no—
294D4—42no—
295D4—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—1224no—
303C2×C4—1224no—
304C2×C4—4884no—
305C2×C4#1451974512yes—
306C2×C4#1451974511yescharacteristic
307C2×C4C2×D4×D6112yes—
308C2×C4C22×C22⋊D6111yescharacteristic
309C8—42no—
310C8—2444no—
311C22—22no—
312C22—21no—
313C22—44no—
314C22—42no—
315C22C2×#1451974511yescharacteristic
316C4—22no—
317C4—21no—
318C4—42no—
319C4—82no—
320C4—2444no—
321C4—2444no—
322C4#1683993912yes—
323C4#1684056912yes—
324C4#1684057612yes—
325C2—41no—
326C2—82no—
327C2#2145513012yes—
328C2C2×#1683993911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3131C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3131D61
hypocenterC61#18693
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
2. #21457210C2semidirect
3. #21445037C2semidirect
4. #21445002C2semidirect
6. #21444275C2semidirect
196. C61#18693semidirect

Maximal subgroups

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

subgroupindex
#214602332
#214572102
#214450372
#214450022
#214445542
#214442752
C61×#31312
#1869361