Finite Group Databasebeta

← #30607242

Subgroups — #30607242

Order 15616.

628 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461655C211yesmaximal, characteristic
2#21461648C211yesmaximal, characteristic
3#21460904C214yesmaximal, complemented
4#21448407C211yesmaximal, characteristic
5C2×#14519838—21no—
6C2×#14519837C2212yes—
7C2×#14519836—21no—
8C2×#14519835C2212yes—
9#16840734—21no—
10#16840705—21no—
11C22×#11758222C2211yescharacteristic
12C22×C4⋊Dic61C411yescharacteristic, complemented
13#16839668—21no—
14#16839664—21no—
15#16839639C412yescomplemented
16#16839639C2212yes—
17#16839619—21no—
18#16839619—21no—
19C2×#117×C61C411yescharacteristic, complemented
20C2×#11758217—22no—
21C2×C22⋊#6923—22no—
22#14519838—42no—
23#14519837—24no—
24#14519836—42no—
25#14519835—24no—
26#14519834—42no—
27(C2×#6923)⋊C4—42no—
28C2×#11758222—24no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61C2×C411yescharacteristic
34C22×C22⋊D61C2311yescharacteristic
35C23×Dic61C2×C412yes—
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—42no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43#14519684—22no—
44#14519684—22no—
45C23×C4×C61C2×C411yescharacteristic
46#117×C61—42no—
47C2×C4⋊C4×C61—21no—
48C2×C22⋊C4×C61C2×C412yes—
49C2×C42×C61—21no—
50C24×C61C22×C411yescharacteristic
51C23×D61C22⋊C411yescharacteristic
52C23×D61C2×D411yescharacteristic
53C22×#6923—42no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C2×C22⋊D61—22no—
61C2×C22⋊D61—41no—
62C2×C22⋊D61—41no—
63C2×C22⋊D61—41no—
64C2×C22⋊D61—41no—
65C2×C22⋊D61#4114yes—
66C22×Dic61—22no—
67C22×Dic61—42no—
68C22×Dic61—82no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71#11758222—416no—
72C2×D244—22no—
73C2×D244—22no—
74C2×D244—41no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×Dic122—22no—
82C2×Dic122—22no—
83C2×Dic122—41no—
84C22⋊#6923—42no—
85C22⋊#6923—42no—
86#11758217—42no—
87#11758217—42no—
88C2×#92420—42no—
89C4⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C42×C61—42no—
92#11758198—82no—
93#11758198—82no—
94C4⋊Dic61—42no—
95C4⋊Dic61—42no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61#11712yes—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61#12312yes—
112C2×#6923—82no—
113C2×#6923—82no—
114C2×#6923—82no—
115C2×C4×C61—28no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61#9912yes—
124C2×C4×C61#9912yes—
125C22⋊D61—44no—
126C22⋊D61—44no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—84no—
136C2×Dic61#12312yes—
137D244—44no—
138C4×D61—44no—
139C4×D61—44no—
140Dic122—44no—
141#92420—82no—
142#12855—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61#38111yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#9911yescharacteristic
153C2×D61—22no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—42no—
157C2×D61—42no—
158C2×D61—41no—
159C2×D61—41no—
160C2×D61—41no—
161C2×D61—81no—
162C2×D61—81no—
163#6923—162no—
164C4×C61—22no—
165C4×C61—22no—
166C4×C61—82no—
167Dic61—22no—
168Dic61—41no—
169Dic61—82no—
170#2311—611no—
171#1785—611no—
172#1782—611no—
173#1390—614no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61—42no—
177C2×C61#181911yescharacteristic
178C2×C61#212211yescharacteristic
179C2×C61#212311yescharacteristic
180D61—42no—
181D61—81no—
182C22×#41—611no—
183C2×#117#692311yescharacteristic, complemented
184C22×C4⋊C4—611no—
185C2×#101—1221no—
186C2×#100—612no—
187C2×#99—612no—
188C2×#99—1221no—
189#407—1221no—
190#391—1221no—
191#390—1221no—
192#386—612no—
193#386—612no—
194#381—1221no—
195#380—1221no—
196#380—1221no—
197C61#1285511yescharacteristic, complemented, Sylow
198C2×#41—1224no—
199C2×#41—2442no—
200C2×#41—2442no—
201C22×Q8—611no—
202C22×D4—611no—
203C22×D4—611no—
204C22×D4—611no—
205C23×C4—611no—
206C23×C4—611no—
207C23×C4—612no—
208C23×C4C2×#692311yescharacteristic
209C2×M16—1222no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—2442no—
215C2×C22⋊C4—1222no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4C2×#692312yes—
219C2×C42—21no—
220#101—2442no—
221#100—1224no—
222#99—1224no—
223#99—2442no—
224C22⋊C8—2442no—
225#95—1222no—
226#95—1222no—
227#95—2442no—
228C24—611no—
229C24—611no—
230C24C22×#692311yescharacteristic
231#41—24416no—
232C2×Q8—1222no—
233C2×Q8—1222no—
234C2×Q8—2441no—
235C2×D4—614no—
236C2×D4—1222no—
237C2×D4—1222no—
238C2×D4—1222no—
239C2×D4—2441no—
240C2×D4—2441no—
241C2×D4—2441no—
242C2×D4—2441no—
243C2×D4—2441no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—1222no—
251C22×C4—1222no—
252C22×C4—1222no—
253C22×C4—1222no—
254C22×C4—1222no—
255C22×C4—4882no—
256C22×C4—2441no—
257C22×C4—2441no—
258C22×C4—2442no—
259C22×C4—2442no—
260C22×C4C22⋊#692311yescharacteristic
261C22×C4C22⋊#692311yescharacteristic
262M16—2442no—
263M16—2442no—
264C2×C8—2442no—
265C4⋊C4—42no—
266C4⋊C4—2442no—
267C4⋊C4—2442no—
268C22⋊C4—42no—
269C22⋊C4—4882no—
270C22⋊C4—4882no—
271C22⋊C4—2442no—
272C22⋊C4—2442no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—612no—
279C23—1221no—
280C23—1221no—
281C23—1221no—
282C23—1221no—
283C23—1221no—
284C23—2441no—
285C23—2441no—
286C23—2441no—
287C23—2441no—
288C23C2×C22⋊#692311yescharacteristic
289C23#1451985612yes—
290Q8—2444no—
291D4—2444no—
292D4—2444no—
293D4—2444no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—612no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—1221no—
308C2×C4—4882no—
309C2×C4—4882no—
310C2×C4—4882no—
311C2×C4—4884no—
312C2×C4—2444no—
313C2×C4—2444no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4#1451983512yes—
318C2×C4#1451983612yes—
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—1222no—
328C22—4881no—
329C22—4881no—
330C22—2442no—
331C22—2442no—
332C22—2441no—
333C22—2441no—
334C22—2441no—
335C22—2441no—
336C22—2441no—
337C22#1684070511yescharacteristic
338C22C2×#1451983511yescharacteristic
339C22C2×#1451983611yescharacteristic
340C4—22no—
341C4—22no—
342C4—82no—
343C4—1222no—
344C4—9762no—
345C4—4882no—
346C4—2441no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—4881no—
351C2—2442no—
352C2#2146171411yescharacteristic
353C2#2146208011yescharacteristic
354C2#2146208311yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×C22⋊D61C23
hypercenterC2×#117#6923
hypocenterC61#12855
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21460904C2semidirect
12. C22×C4⋊Dic61C4semidirect
15. #16839639C4semidirect
19. C2×#117×C61C4semidirect
183. C2×#117#6923semidirect
197. C61#12855semidirect

Maximal subgroups

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

subgroupindex
#214616552
#214616482
#214609042
#214484072
#1285561