Finite Group Databasebeta

← #29572774

Subgroups — #29572774

Order 15616.

540 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 125 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#124×Dic61C211yesmaximal, characteristic
2#21445673C212yesmaximal, complemented
3C2×#16839502C211yesmaximal, characteristic
4#21443701C212yesmaximal, complemented
5C2×C61×#329C211yesmaximal, characteristic
6C2×Q8×Dic61C411yescharacteristic
7C2×D4×Dic61C411yescharacteristic
8C2×#14519706—22no—
9C2×#14519688C2212yes—
10#16839669C411yescharacteristic
11#16839665C411yescharacteristic
12#16839655—22no—
13#16839628C2211yescharacteristic
14C22⋊C4×Dic61—22no—
15C42×Dic61C2211yescharacteristic
16#16839516—22no—
17#16839502—22no—
18C2×#124×C61C2211yescharacteristic
19C2×#97×C61C2212yes—
20C61×#329—22no—
21C23×Dic61—21no—
22C22×#92418—22no—
23Q8×Dic61—42no—
24D4×Dic61—42no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×C22⋊Dic61—21no—
29C2×C22⋊Dic61—22no—
30C2×C4×Dic61—21no—
31C2×C4×Dic61C2×C411yescharacteristic
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61D412yes—
34C2×C4×Dic61D412yes—
35#14519706—28no—
36#14519688—22no—
37#14519688—22no—
38#14519684—22no—
39#14519684—22no—
40C22×Q8×C61C2×C411yescharacteristic
41C22×D4×C61C2×C411yescharacteristic
42C22×C8×C61—22no—
43#124×C61—22no—
44#124×C61D412yes—
45#124×C61Q812yes—
46C2×C22⋊C4×C61—22no—
47C2×C42×C61C2311yescharacteristic
48#97×C61—22no—
49#97×C61—22no—
50C24×C61—21no—
51C2×Q8×C61—21no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61C4⋊C412yes—
55C2×D4×C61—21no—
56C2×D4×C61—41no—
57C2×D4×C61—41no—
58C2×D4×C61C4⋊C412yes—
59C22×C4×C61—21no—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61—42no—
65C22×Dic61—44no—
66C22×Dic61C22⋊C412yes—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C2×C8×C61—24no—
70C2×C8×C61—42no—
71C2×C8×C61—42no—
72C22⋊C4×C61—24no—
73C22⋊C4×C61—24no—
74C42×C61C2×D411yescharacteristic
75C42×C61C2×Q811yescharacteristic
76C42×C61#4112yes—
77C22⋊Dic61—44no—
78C22⋊Dic61—42no—
79C4⋊Dic61—42no—
80C4⋊Dic61—41no—
81C4⋊Dic61—41no—
82C4×Dic61—22no—
83C4×Dic61—24no—
84C4×Dic61—41no—
85C4×Dic61—42no—
86C4×Dic61—42no—
87C2×#92418—24no—
88C2×#92418—42no—
89C2×#92418—42no—
90C23×C61—22no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96Q8×C61—42no—
97Q8×C61—42no—
98D4×C61—42no—
99D4×C61—42no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61C2×C4⋊C411yescharacteristic
108C2×C4×C61#11711yescharacteristic
109C2×C4×C61#12112yes—
110C2×C4×C61C22⋊Q812yes—
111C2×Dic61—21no—
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—84no—
121C2×Dic61#10012yes—
122C2×Dic61#10414yes—
123C8×C61—44no—
124#92418—44no—
125#8579—611nomaximal, Sylow
126C22×C61—42no—
127C22×C61—44no—
128C22×C61—81no—
129C22×C61—81no—
130C22×C61—82no—
131C22×C61#38011yescharacteristic
132C22×C61C2×#10011yescharacteristic
133C22×C61#41311yescharacteristic
134C22×C61C2×#10412yes—
135C22×C61#42112yes—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—22no—
139C4×C61—22no—
140C4×C61—42no—
141Dic61—41no—
142Dic61—42no—
143Dic61—81no—
144Dic61—81no—
145Dic61#32914yescomplemented
146C4×#124—611no—
147#1767—612no—
148C2×#329—611no—
149C2×#329D6111yescharacteristic
150#1295—612no—
151C2×C61—82no—
152C2×C61C2×#32911yescharacteristic
153C2×C61#152912yes—
154C2×C61#153611yescharacteristic
155C2×C61#180011yescharacteristic
156C2×C61#180612yes—
157C2×#124C2×D6111yescharacteristic
158C2×C4×Q8—611no—
159C2×C4×D4—611no—
160C2×C4⋊C8—1222no—
161C2×#97—612no—
162C2×#97C2×D6112yes—
163#391—611no—
164#390—611no—
165#388—1222no—
166#382—611no—
167C4×C22⋊C4—1222no—
168C43—611no—
169#336—1222no—
170#329—22no—
171#329—1222no—
172C61#857911yescharacteristic, complemented, Sylow
173C22×Q8C4×D6111yescharacteristic
174C22×D4C4×D6111yescharacteristic
175C23×C4—1221no—
176C22×C8—22no—
177C22×C8—1222no—
178#124—22no—
179#124Dic12212yes—
180#124C22⋊D6112yes—
181C4×Q8—2442no—
182C4×D4—2442no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—1221no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—1221no—
188C2×C22⋊C4—1222no—
189C2×C42—611no—
190C2×C42—612no—
191C2×C42—611no—
192C2×C42—612no—
193C2×C42—1221no—
194C2×C42C22×D6111yescharacteristic
195C4⋊C8—1228no—
196#97—22no—
197#97—22no—
198#97—1222no—
199#97—1222no—
200#95—1222no—
201#95—1222no—
202C24—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8#1175819612yes—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4#1175819612yes—
211C22×C4—21no—
212C22×C4—611no—
213C22×C4—612no—
214C22×C4—611no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—2442no—
218C22×C4—2444no—
219C22×C4C2×C4×D6111yescharacteristic
220C22×C4D4×D6112yes—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—1224no—
225C2×C8—2442no—
226C2×C8—2442no—
227C4⋊C4—2441no—
228C4⋊C4—2441no—
229C4⋊C4—2442no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—2442no—
233C22⋊C4—2444no—
234C42—1224no—
235C42—1222no—
236C42—2441no—
237C42—2442no—
238C42—2442no—
239C42C2×Dic12211yescharacteristic
240C42#1175822412yes—
241C42C2×C22⋊D6111yescharacteristic
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C22⋊C4×D6111yescharacteristic
248Q8—42no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—612no—
260C2×C4—614no—
261C2×C4—1221no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—4884no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4C2×#1175819611yescharacteristic
272C2×C4#1451973311yescharacteristic
273C2×C4#1451974912yes—
274C2×C4#1451975412yes—
275C8—44no—
276C8—2444no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22#1683962011yescharacteristic
283C22#100×D6111yescharacteristic
284C22#1683974311yescharacteristic
285C22#104×D6112yes—
286C22#1683977712yes—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—614no—
293C4—4881no—
294C4—4881no—
295C4—2442no—
296C4—2441no—
297C2—82no—
298C2#329×D6111yescharacteristic
299C2#2144442312yes—
300C2#2144445211yescharacteristic
301C2#2144580911yescharacteristic
302C2#2144583712yes—
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#329C2
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenterC2×#329D61
hypocenterC61#8579
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. #21445673C2semidirect
4. #21443701C2semidirect
145. Dic61#329semidirect
172. C61#8579semidirect

Maximal subgroups

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

subgroupindex
#124×Dic612
#214456732
C2×#168395022
#214437012
C2×C61×#3292
#857961