Finite Group Databasebeta

← #29575778

Subgroups — #29575778

Order 15616.

586 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 131 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#21454158C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3C2×#16839498C212yesmaximal
4#21443698C212yesmaximal
5C2×C61×#383C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7C2×C22⋊#92418C2212yes—
8#16839668C412yescomplemented
9#16839660C412yescomplemented
10#16839646—22no—
11#16839635—21no—
12#16839625—21no—
13#16839622C2211yescharacteristic
14#16839517—22no—
15#16839498—24no—
16C22×C42×C61C2211yescharacteristic
17C61×#383—24no—
18C2×#95×C61—21no—
19C2×#95×C61—21no—
20C2×#95×C61C2212yes—
21C22×Dic122C2×C411yescharacteristic
22C22×#92418—22no—
23#14519756—22no—
24#14519756—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C412yes—
27C2×#11758196—22no—
28C2×#11758196—21no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33#14519684—22no—
34#14519684—22no—
35#14519684—22no—
36#14519684—21no—
37C23×C4×C61—22no—
38C23×C4×C61D411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C2×C42×C61—22no—
42C2×C42×C61—22no—
43C2×C42×C61C2×C412yes—
44C2×C42×C61D412yes—
45#95×C61—24no—
46#95×C61—24no—
47#95×C61—22no—
48#95×C61D414yes—
49C24×C61C2×D411yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—24no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61—44no—
60C22×C4×C61—44no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×C4×C61#4112yes—
69C22×Dic61—21no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C2×Dic122—22no—
73C2×Dic122—42no—
74C42×C61—24no—
75C42×C61—24no—
76C22⋊Dic61—41no—
77C22⋊Dic61—41no—
78C22⋊Dic61—42no—
79C4⋊Dic61—22no—
80C4⋊Dic61—42no—
81#11758196—44no—
82#11758196—42no—
83C2×#92418—44no—
84C2×#92418—42no—
85C23×C61—22no—
86C23×C61—22no—
87C23×C61C2×C22⋊C411yescharacteristic
88C23×C61#12011yescharacteristic
89C23×C61#12311yescharacteristic
90C23×C61#13512yes—
91C23×C61#13712yes—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—44no—
107C2×C4×C61—44no—
108C2×C4×C61—44no—
109C2×C4×C61—84no—
110C2×C4×C61—84no—
111C2×C4×C61C4×D412yes—
112C2×C4×C61#12112yes—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120Dic122—82no—
121#92418—82no—
122#9153—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—24no—
128C22×C61—24no—
129C22×C61#38611yescharacteristic
130C22×C61#41012yes—
131C22×C61#41611yescharacteristic
132C22×C61#41911yescharacteristic
133C22×C61#42112yes—
134C22×C61#45212yes—
135C22×C61#48512yes—
136C4×C61—22no—
137C4×C61—22no—
138C4×C61—42no—
139C4×C61—42no—
140C4×C61—84no—
141Dic61—81no—
142Dic61—81no—
143Dic61—82no—
144#2855—611no—
145C2×#383D6111yescharacteristic
146#1817—611no—
147C2×#328—612no—
148#1294—612no—
149C2×C61—22no—
150C2×C61—22no—
151C2×C61#151512yes—
152C2×C61#152112yes—
153C2×C61#187311yescharacteristic
154C2×C61#189711yescharacteristic
155C2×C61#189911yescharacteristic
156C2×C22⋊Q8—611no—
157C22×C42C2×D6111yescharacteristic
158C2×C22⋊C8—612no—
159#391—612no—
160#389—612no—
161#387—1222no—
162#385—1221no—
163#383—24no—
164#381—1221no—
165#380—611no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95C2×D6112yes—
169#336—1222no—
170#328—1224no—
171C61#915311yescharacteristic, complemented, Sylow
172C22×Q8—611no—
173C23×C4—22no—
174C23×C4C22⋊D6111yescharacteristic
175C23×C4C22⋊D6111yescharacteristic
176C23×C4C22×D6111yescharacteristic
177C22×C8—1222no—
178C22⋊Q8—1222no—
179C22⋊Q8—1222no—
180C2×C4⋊C4—611no—
181C2×C4⋊C4—612no—
182C2×C4⋊C4—1222no—
183C2×C4⋊C4—1221no—
184C2×C22⋊C4—1221no—
185C2×C22⋊C4—1221no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C4×D6112yes—
189C2×C42D24412yes—
190C22⋊C8—1222no—
191C22⋊C8—1222no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—1222no—
196#95—1222no—
197#95—1222no—
198#95—1221no—
199#95C22⋊D6114yes—
200C24C2×C22⋊D6111yescharacteristic
201C2×Q8—1222no—
202C2×Q8—2442no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—1222no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4#1175819812yes—
220C22×C4#1175819812yes—
221C22×C4C2×C4×D6111yescharacteristic
222C22×C4C2×D24411yescharacteristic
223C22×C4#1175822212yes—
224C22×C4C2×C22⋊D6112yes—
225C2×C8—2444no—
226C2×C8—2442no—
227C4⋊C4—1222no—
228C4⋊C4—2442no—
229C4⋊C4—2444no—
230C4⋊C4—2442no—
231C22⋊C4—2441no—
232C22⋊C4—2441no—
233C22⋊C4—2442no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×#1175819811yescharacteristic
239C23#1451974611yescharacteristic
240C23#1451976011yescharacteristic
241C23#1451979712yes—
242C23#1451980712yes—
243Q8—4882no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4C4×C22⋊D6112yes—
271C2×C4#1451975112yes—
272C8—4882no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1683964111yescharacteristic
280C22#1683973512yes—
281C22#1683975611yescharacteristic
282C22#1683977011yescharacteristic
283C22#1683978012yes—
284C22#1683992412yes—
285C22#1684010312yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—84no—
291C4—4882no—
292C4—4881no—
293C4—4881no—
294C2—22no—
295C2—22no—
296C2#2144436912yes—
297C2#2144439412yes—
298C2#2144613111yescharacteristic
299C2#2144626711yescharacteristic
300C2#2144627711yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#383C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#383D61
hypocenterC61#9153
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
8. #16839668C4semidirect
9. #16839660C4semidirect
171. C61#9153semidirect

Maximal subgroups

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

subgroupindex
#214541582
#214458872
C2×#168394982
#214436982
C2×C61×#3832
#915361