Finite Group Databasebeta

← #29577862

Subgroups — #29577862

Order 15616.

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

normal subgroups only

This group has 626 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#21450716C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic, complemented
3#21444245C212yesmaximal, complemented
4#21443678C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic, complemented
6C2×#14519748—21no—
7C2×C4×D244C2211yescharacteristic
8#16839786C2212yes—
9C2×#14519694—21no—
10#16839687C412yes—
11#16839671C412yes—
12#16839641—22no—
13#16839628C2211yescharacteristic
14#16839619—21no—
15#16839531—21no—
16#16839491—24no—
17C2×C4×D4×C61C2211yescharacteristic
18C8×D4×C61—24no—
19C61×#431C2212yes—
20C2×C22⋊C8×C61—22no—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—22no—
23C22×C4×D61—21no—
24C2×#11758194—22no—
25#14519748—42no—
26#14519748—42no—
27C4×D244—22no—
28C4×D244—22no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×C22⋊Dic61—22no—
32C2×#11758198—21no—
33C2×#11758198—22no—
34C4⋊#92418—22no—
35C4⋊#92418—22no—
36#14519694—42no—
37#14519684—22no—
38#14519684—21no—
39#14519684C2×C412yes—
40C22×D4×C61D411yescharacteristic
41C23×C4×C61D412yes—
42C22×C8×C61—24no—
43C4×D4×C61—22no—
44C4×D4×C61—22no—
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C22⋊C4×C61C2×C412yes—
47C2×C42×C61C2311yescharacteristic
48C4⋊C8×C61—22no—
49C22⋊C8×C61—24no—
50C22⋊C8×C61—22no—
51C4×C8×C61D414yescomplemented
52C24×C61C22⋊C412yes—
53C23×D61—21no—
54C2×D4×C61—22no—
55C2×D4×C61—22no—
56C2×D4×C61D812yes—
57C2×D4×C61SD1612yes—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—82no—
61C22×C4×C61C22⋊C412yes—
62C22×C4×C61C22×C411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C2×C22⋊D61—44no—
66C2×C22⋊D61—44no—
67C2×C22⋊D61—82no—
68C22×Dic61—21no—
69C22×Dic61—22no—
70C2×D244—22no—
71C2×D244—42no—
72C2×C4×D61—42no—
73C2×C4×D61—41no—
74C2×C4×D61—41no—
75C2×C4×D61—42no—
76C2×C8×C61—24no—
77C2×C8×C61—84no—
78C4⋊C4×C61—22no—
79C22⋊C4×C61—24no—
80C42×C61C2×D412yes—
81C42×C61#4112yes—
82C22⋊Dic61—44no—
83#11758198—41no—
84#11758198—41no—
85#11758198—82no—
86C4⋊Dic61—22no—
87C4⋊Dic61—41no—
88C4⋊Dic61—41no—
89#11758194—42no—
90#11758194—42no—
91C2×#92418—42no—
92C23×C61—22no—
93C23×C61—22no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61#10214yes—
97C23×C61C2×C22⋊C411yescharacteristic
98C22×D61—22no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—41no—
105D4×C61—24no—
106D4×C61—24no—
107C2×C4×C61—22no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—44no—
111C2×C4×C61—42no—
112C2×C4×C61—82no—
113C2×C4×C61C4×D412yes—
114C2×C4×C61#12011yescharacteristic
115C2×C4×C61#12112yes—
116C2×C4×C61#12311yescharacteristic
117C2×C4×C61C2×D811yescharacteristic
118C2×C4×C61C2×SD1611yescharacteristic
119C2×C4×C61#13512yes—
120C22⋊D61—88no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127D244—24no—
128D244—42no—
129C4×D61—82no—
130C4×D61—82no—
131C8×C61—44no—
132#92418—82no—
133#9485—611nomaximal, Sylow
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—24no—
137C22×C61—44no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—82no—
141C22×C61#38611yescharacteristic
142C22×C61C2×#10211yescharacteristic
143C22×C61#41611yescharacteristic
144C2×D61—42no—
145C2×D61—42no—
146C2×D61—42no—
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C2×D61—81no—
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—42no—
155C4×C61#46512yes—
156C4×C61#46612yes—
157Dic61—82no—
158Dic61—81no—
159#2972D6111yescharacteristic, complemented
160#2535—611no—
161#1800—611no—
162#1486—612no—
163#1286—612no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61—44no—
167C2×C61#189111yescharacteristic
168C2×C61#189311yescharacteristic
169C2×C61#191111yescharacteristic
170D61—82no—
171C2×#121—1221no—
172C2×C4×D4—611no—
173C2×C4×D4C2×D6111yescharacteristic
174C8×D4—24no—
175#431C2×D6112yes—
176#423—612no—
177C2×#100—1221no—
178C2×C22⋊C8—22no—
179#397—612no—
180#392—612no—
181#386—1222no—
182#382—611no—
183#380—1221no—
184#343—1221no—
185D4⋊C8—1224no—
186C61#948511yescharacteristic, complemented, Sylow
187C22×D4—611no—
188C22×D4—1222no—
189C22×D4D24411yescharacteristic
190C23×C4—1221no—
191C23×C4C22⋊D6112yes—
192C2×M16—1222no—
193C22×C8—24no—
194#121—2442no—
195#121—2442no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—1222no—
199C4×D4—1222no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4D24411yescharacteristic
203C2×C22⋊C4—1222no—
204C2×C22⋊C4—1222no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4C4×D6112yes—
207C2×C42C22×D6111yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—1222no—
210C4⋊C8—1222no—
211#100—2442no—
212C22⋊C8—24no—
213C22⋊C8—22no—
214C4×C8C22⋊D6114yescomplemented
215#95—612no—
216#95—1222no—
217#95—1221no—
218C24—1221no—
219C24#1175819812yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—1222no—
223C2×D4—4882no—
224C2×D4—2442no—
225C2×D4—2444no—
226C2×D4—2444no—
227C2×D4#1175819012yes—
228C2×D4D48812yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—1222no—
233C22×C4—1221no—
234C22×C4—2442no—
235C22×C4—2441no—
236C22×C4—2441no—
237C22×C4—2442no—
238C22×C4#1175819812yes—
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4C2×D24411yescharacteristic
241C22×C4C2×C22⋊D6111yescharacteristic
242M16—2442no—
243M16—2442no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—2442no—
247C4⋊C4—22no—
248C4⋊C4—1222no—
249C4⋊C4—2441no—
250C4⋊C4—2441no—
251C22⋊C4—24no—
252C22⋊C4—4882no—
253C22⋊C4—2444no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C42#1175822212yes—
257C42C2×C22⋊D6112yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—1222no—
263C23—2441no—
264C23—2441no—
265C23—2441no—
266C23—2441no—
267C23—2441no—
268C23—2441no—
269C23#1451969914yes—
270C23C2×#1175819811yescharacteristic
271D4—24no—
272D4—24no—
273D4—1224no—
274D4—4888no—
275D4—2442no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—4882no—
283C2×C4—4882no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4—2441no—
287C2×C4—2442no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4C4×C22⋊D6112yes—
291C2×C4#1451974611yescharacteristic
292C2×C4#1451975112yes—
293C2×C4#1451976011yescharacteristic
294C2×C4C2×D48811yescharacteristic
295C2×C4C2×#1175819011yescharacteristic
296C2×C4#1451979912yes—
297C8—44no—
298C8—4882no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—4881no—
307C22—4881no—
308C22—4881no—
309C22—4881no—
310C22—2442no—
311C22—2442no—
312C22—2442no—
313C22#1683964111yescharacteristic
314C22C2×#1451969911yescharacteristic
315C22#1683975911yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—4882no—
321C4—4881no—
322C4#1683999112yes—
323C4#1683999812yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—4882no—
328C2#2144623711yescharacteristic
329C2#2144624711yescharacteristic
330C2#2144635111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radical#21450716C2
cosocleC2×C42×C61C23
hypercenter#2972D61
hypocenterC61#9485
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21445811C2semidirect
3. #21444245C2semidirect
4. #21443678C2semidirect
5. C61×#2972C2semidirect
51. C4×C8×C61D4semidirect
159. #2972D61semidirect
186. C61#9485semidirect
214. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507162
#214458112
#214442452
#214436782
C61×#29722
#948561