Finite Group Databasebeta

← #29656275

Subgroups — #29656275

Order 15616.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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#21456276C211yesmaximal, characteristic
2#21455941C212yesmaximal
3#21444959C214yesmaximal, complemented
4#21444897C214yesmaximal, complemented
5#21444545C211yesmaximal, characteristic, complemented
6#21444266C212yesmaximal, complemented
7C2×Q82×C61C211yesmaximal, characteristic, complemented
8C2×#14519808—22no—
9#16840312C2211yescharacteristic
10#16840154—22no—
11#16840141C2212yes—
12#16840136C2212yes—
13#16840075—22no—
14#16840057C2214yes—
15#16839962C2214yes—
16#16839901—22no—
17#16839860—24no—
18#16839819C2212yes—
19C4×#11758194C2211yescharacteristic
20#16839511C2214yes—
21Q8⋊#92418C2218yes—
22Q82×C61—22no—
23Q82×C61—24no—
24Q82×C61C2214yes—
25C2×C4⋊Q8×C61—22no—
26C2×C4⋊Q8×C61—22no—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C2×C4×Q8×C61—22no—
29C2×C4×Q8×C61C2212yes—
30C2×#11758222—21no—
31#14519808—48no—
32C2×#11758202—24no—
33C2×#11758201—24no—
34C2×#11758201—24no—
35C2×#11758194—21no—
36#14519777C2311yescharacteristic
37#14519774C2311yescharacteristic
38#14519765—21no—
39#14519751—22no—
40C4×D244—22no—
41#14519710—24no—
42C4⋊#92418C2314yes—
43#14519701—24no—
44#14519697—24no—
45C22⋊#92418—22no—
46#14519688—21no—
47#14519688—22no—
48C4×#92418C2312yes—
49C22×Q8×C61—22no—
50C22×Q8×C61D412yes—
51C22×Q8×C61D412yes—
52C4⋊Q8×C61—28no—
53C4⋊Q8×C61—28no—
54C4⋊Q8×C61—28no—
55C4⋊Q8×C61—28no—
56C4⋊Q8×C61—22no—
57C4⋊Q8×C61D412yes—
58C4⋊Q8×C61C2312yes—
59C4×Q8×C61—28no—
60C4×Q8×C61—28no—
61C4×Q8×C61—22no—
62C4×Q8×C61—22no—
63C4×Q8×C61D414yes—
64C4×Q8×C61C2314yes—
65C2×C4⋊C4×C61—24no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61D412yes—
69C2×C42×C61—22no—
70C2×C42×C61—22no—
71C2×C42×C61C2311yescharacteristic
72C2×Q8×C61—28no—
73C2×Q8×C61—28no—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61—44no—
77C2×Q8×C61—84no—
78C2×Q8×C61C2×D414yes—
79C2×Q8×C61C2×D414yes—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C2×C22⋊D61—41no—
86#11758222—44no—
87C2×D244—21no—
88C2×D244—22no—
89C2×C4×D61—41no—
90C2×Dic122—21no—
91C4⋊C4×C61—216no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—24no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61C2×D414yes—
98C42×C61—28no—
99C42×C61—24no—
100C42×C61—24no—
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D412yes—
103C42×C61C2411yescharacteristic
104#11758202—48no—
105#11758201—48no—
106#11758201—48no—
107#11758198—42no—
108C4⋊Dic61—22no—
109#11758194—24no—
110C2×#92418—22no—
111C2×#92418—24no—
112C23×C61#12011yescharacteristic
113C22×D61—41no—
114Q8×C61—28no—
115Q8×C61—24no—
116Q8×C61—44no—
117Q8×C61—82no—
118Q8×C61—88no—
119Q8×C61#13718yescomplemented
120C2×C4×C61—28no—
121C2×C4×C61—24no—
122C2×C4×C61—22no—
123C2×C4×C61—24no—
124C2×C4×C61—24no—
125C2×C4×C61—24no—
126C2×C4×C61—44no—
127C2×C4×C61—44no—
128C2×C4×C61#12012yes—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61C22×D411yescharacteristic
131C2×C4×C61C22×D412yes—
132C22⋊D61—82no—
133C2×Dic61—41no—
134D244—42no—
135D244—44no—
136C4×D61—82no—
137Dic122—42no—
138#92418—24no—
139#92418—44no—
140#18546—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61#45312yes—
143C22×C61C2×#12011yescharacteristic
144C2×D61—82no—
145C2×D61—81no—
146C4×C61—28no—
147C4×C61—22no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—44no—
151C4×C61—44no—
152C4×C61C2×#13714yes—
153Dic61—81no—
154C2×Q82D6111yescharacteristic, complemented
155#3152—611no—
156#3112—612no—
157#1633—614no—
158#1623—614no—
159#1558—611no—
160#1491—612no—
161C2×C61—22no—
162C2×C61#300412yes—
163C2×C61C2×#45311yescharacteristic
164D61—161no—
165C2×#137—1222no—
166Q82—22no—
167Q82—24no—
168Q82C2×D6114yes—
169#532—611no—
170C2×C4⋊Q8—22no—
171C2×C4⋊Q8—22no—
172C2×C4⋊Q8C2×D6111yescharacteristic
173C2×C4×Q8—22no—
174C2×C4×Q8C2×D6112yes—
175#497—1222no—
176#494—612no—
177#492—612no—
178#480—1222no—
179#477—614no—
180#460—614no—
181#448—1222no—
182#440—1224no—
183#432—612no—
184C4×M16—611no—
185#333—614no—
186Q8⋊C8—618no—
187C61#1854611yescharacteristic, complemented, Sylow
188C2×#41—1221no—
189C22×Q8—22no—
190C22×Q8C22⋊D6112yes—
191C22×Q8C22⋊D6112yes—
192#137—2448no—
193C2×Q16—1224no—
194C2×SD16—1224no—
195C2×SD16—1224no—
196C2×M16—1221no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—611no—
203C4⋊Q8C22⋊D6112yes—
204C4⋊Q8C22×D6112yes—
205#127—611no—
206#124—1221no—
207#121—1222no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊D6114yes—
213C4×Q8C22×D6114yes—
214C4×D4—1222no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C22⋊D6112yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×D6111yescharacteristic
222#107—1224no—
223C4⋊C8—614no—
224#103—1224no—
225#102—1224no—
226C22⋊C8—1222no—
227#97—1221no—
228#97—1222no—
229C4×C8—612no—
230#41—2444no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—1221no—
238C2×Q8C2×C22⋊D6114yes—
239C2×Q8C2×C22⋊D6114yes—
240C2×D4—1222no—
241C2×D4—1221no—
242C2×D4—2441no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—2441no—
247C22×C4C2×C22⋊D6112yes—
248C22×C4C2×C22⋊D6111yescharacteristic
249Q16—2448no—
250SD16—2448no—
251SD16—2448no—
252M16—1224no—
253C2×C8—1222no—
254C2×C8—1224no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—1222no—
262C4⋊C4C2×C22⋊D6114yes—
263C22⋊C4—2442no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D6112yes—
268C42C2×C22⋊D6111yescharacteristic
269C42C23×D6111yescharacteristic
270C23—2441no—
271C23#1451974611yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—2442no—
278Q8#1451980818yescomplemented
279D4—4882no—
280D4—2442no—
281D4—2444no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—4882no—
291C2×C4—2441no—
292C2×C4#1451974612yes—
293C2×C4#1451974611yescharacteristic
294C2×C4C22×C22⋊D6111yescharacteristic
295C2×C4C22×C22⋊D6112yes—
296C8—1224no—
297C8—2444no—
298C22—22no—
299C22—4881no—
300C22—4882no—
301C22#1683993012yes—
302C22C2×#1451974611yescharacteristic
303C4—21no—
304C4—21no—
305C4—28no—
306C4—22no—
307C4—44no—
308C4—44no—
309C4—4881no—
310C4C2×#1451980814yes—
311C2—22no—
312C2—9761no—
313C2#2145505312yes—
314C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×Q82×C61C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenterC2×Q82D61
hypocenterC61#18546
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #21444959C2semidirect
4. #21444897C2semidirect
5. #21444545C2semidirect
6. #21444266C2semidirect
7. C2×Q82×C61C2semidirect
119. Q8×C61#137semidirect
154. C2×Q82D61semidirect
187. C61#18546semidirect
278. Q8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214562762
#214559412
#214449592
#214448972
#214445452
#214442662
C2×Q82×C612
#1854661