Finite Group Databasebeta

← #29572140

Subgroups — #29572140

Order 15616.

1330 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 123 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#21459454C211yesmaximal, characteristic
2#21445740C211yesmaximal, characteristic, complemented
3#21444321C214yesmaximal, complemented
4C61×#1438C211yesmaximal, characteristic, complemented
5C2×#14519822C411yescharacteristic, complemented
6#16840406—24no—
7D4×C22⋊D61—24no—
8#16840329C412yescomplemented
9#16840321—22no—
10#16840273C2212yes—
11C2×#14519746—22no—
12#16840237C411yescharacteristic, complemented
13C2×#14519694—22no—
14#16839726C2212yescomplemented
15#16839498—24no—
16#16839495—24no—
17C23×D4×C61C2211yescharacteristic
18C22×C22⋊C4×C61—22no—
19C2×#95×C61C2212yescomplemented
20#14519822—44no—
21C2×#11758222C2×C412yes—
22C2×#11758224—22no—
23C22×C22⋊D61—22no—
24C22×C22⋊D61—24no—
25C2×D4×D61—22no—
26C2×#11758194—22no—
27#14519773—22no—
28#14519764—22no—
29#14519760—22no—
30#14519758—24no—
31#14519756D414yes—
32#14519751D414yes—
33#14519749—28no—
34#14519748—24no—
35#14519746—28no—
36#14519746—24no—
37#14519746—22no—
38#14519745—24no—
39D4×Dic61—24no—
40C4×C22⋊D61—24no—
41#14519736C2×C412yes—
42C2×C22⋊Dic61—24no—
43C2×C22⋊Dic61—22no—
44#14519694—44no—
45C22⋊#92418—24no—
46C25×C61D412yes—
47C22×D4×C61—24no—
48C22×D4×C61—44no—
49C22×D4×C61C2×C411yescharacteristic
50C22×D4×C61C2×C411yescharacteristic
51C22×D4×C61D412yes—
52C23×C4×C61—22no—
53C23×C4×C61C2311yescharacteristic
54C2×C22⋊C4×C61—48no—
55C2×C22⋊C4×C61—44no—
56#95×C61—24no—
57C24×C61—24no—
58C24×C61—24no—
59C24×C61—42no—
60C24×C61—44no—
61C24×C61—44no—
62C24×C61C22⋊C412yes—
63C24×C61C22⋊C412yes—
64C24×C61C2×D411yescharacteristic
65C23×D61—22no—
66C2×D4×C61—24no—
67C2×D4×C61—24no—
68C2×D4×C61—22no—
69C2×D4×C61—22no—
70C2×D4×C61—24no—
71C2×D4×C61—42no—
72C2×D4×C61—42no—
73C2×D4×C61—416no—
74C22×C4×C61—24no—
75C22×C4×C61—42no—
76C22×C4×C61—44no—
77C22×C4×C61—44no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D414yes—
83C2×C22⋊D61—24no—
84C2×C22⋊D61—22no—
85C2×C22⋊D61—24no—
86C2×C22⋊D61—28no—
87C2×C22⋊D61—44no—
88C2×C22⋊D61—44no—
89C2×C22⋊D61—84no—
90C2×C22⋊D61—84no—
91C22×Dic61—22no—
92C22×Dic61—24no—
93#11758224—48no—
94D4×D61—48no—
95#11758222—44no—
96C2×D244C22⋊C412yescomplemented
97C2×C4×D61—22no—
98C2×Dic122C22⋊C412yescomplemented
99C22⋊C4×C61—416no—
100C22⋊Dic61—24no—
101C22⋊Dic61—28no—
102C22⋊Dic61—22no—
103C22⋊Dic61—24no—
104#11758198—24no—
105C4⋊Dic61C22⋊C414yescomplemented
106#11758196—24no—
107C4×Dic61—22no—
108#11758194—44no—
109C2×#92418—42no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—22no—
113C23×C61—22no—
114C23×C61—22no—
115C23×C61—22no—
116C23×C61—22no—
117C23×C61—22no—
118C23×C61—48no—
119C23×C61—44no—
120C23×C61—44no—
121C23×C61—44no—
122C23×C61—48no—
123C23×C61—44no—
124C23×C61—44no—
125C23×C61—44no—
126C23×C61—84no—
127C23×C61—82no—
128C23×C61—84no—
129C23×C61—82no—
130C23×C61#9914yes—
131C23×C61C2×C22⋊C411yescharacteristic
132C23×C61#12012yes—
133C22×D61—22no—
134C22×D61—44no—
135C22×D61—84no—
136D4×C61—28no—
137D4×C61—48no—
138D4×C61—44no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—44no—
142C2×C4×C61—48no—
143C2×C4×C61—44no—
144C2×C4×C61—84no—
145C2×C4×C61—82no—
146C2×C4×C61—82no—
147C2×C4×C61C2×C22⋊C412yes—
148C2×C4×C61#12012yes—
149C22⋊D61—48no—
150C22⋊D61—44no—
151C22⋊D61—44no—
152C22⋊D61—44no—
153C22⋊D61—88no—
154C2×Dic61—24no—
155C2×Dic61—22no—
156C2×Dic61—44no—
157C2×Dic61—84no—
158D244—42no—
159C4×D61—44no—
160Dic122—42no—
161#92418—82no—
162#8453—611nomaximal, Sylow
163C22×C61—22no—
164C22×C61—24no—
165C22×C61—22no—
166C22×C61—22no—
167C22×C61—22no—
168C22×C61—22no—
169C22×C61—24no—
170C22×C61—44no—
171C22×C61—44no—
172C22×C61—44no—
173C22×C61—44no—
174C22×C61—42no—
175C22×C61—82no—
176C22×C61—88no—
177C22×C61—88no—
178C22×C61—82no—
179C22×C61#37911yescharacteristic
180C22×C61C2×#9912yes—
181C22×C61#45312yes—
182C22×C61#45512yes—
183C2×D61—44no—
184C2×D61—42no—
185C2×D61—164no—
186C4×C61—22no—
187C4×C61—22no—
188C4×C61—84no—
189Dic61—44no—
190Dic61—42no—
191#3484—611no—
192#1783—611no—
193#1504—614no—
194#1438D6111yescharacteristic, complemented
195C2×C61—22no—
196C2×C61—22no—
197C2×C61—22no—
198C2×C61—44no—
199C2×C61—82no—
200C2×C61#178111yescharacteristic
201C2×C61#179212yes—
202D61—82no—
203C2×H(5,𝔽2)—611no—
204C23×D4C2×D6111yescharacteristic
205#546—1224no—
206D42—1224no—
207#535—612no—
208#534—1222no—
209#525—612no—
210C2×#120—1222no—
211#518—611no—
212C22×C22⋊C4—22no—
213C2×#100—1222no—
214#407—612no—
215C2×#95C2×D6112yescomplemented
216#328—1224no—
217#327—1224no—
218C61#845311yescharacteristic, complemented, Sylow
219C25C22⋊D6112yes—
220H(5,𝔽2)—2444no—
221C2×#41—612no—
222C2×#41—1222no—
223C22×D4—24no—
224C22×D4—44no—
225C22×D4—1222no—
226C22×D4—1224no—
227C22×D4—1222no—
228C22×D4C4×D6111yescharacteristic
229C22×D4C4×D6111yescharacteristic
230C22×D4D24412yes—
231C23×C4—22no—
232C23×C4C22×D6111yescharacteristic
233C2×M16—1222no—
234#127—1222no—
235#124—1222no—
236#123—1224no—
237#123—1222no—
238C22⋊Q8—614no—
239#121—614no—
240#121—1224no—
241#121—1228no—
242#120—1224no—
243#120—1228no—
244#120—1224no—
245#120—1222no—
246C4×D4—1224no—
247C4×D4—1224no—
248#117—612no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—1224no—
252C2×C22⋊C4—1222no—
253#100—2444no—
254C22⋊C8—1224no—
255#95—24no—
256C24—24no—
257C24—24no—
258C24—42no—
259C24—44no—
260C24—44no—
261C24—1222no—
262C24#1175819812yes—
263C24#1175819812yes—
264C24C2×C22⋊D6111yescharacteristic
265#41—2444no—
266#41—2448no—
267C2×Q8—612no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—416no—
276C2×D4—612no—
277C2×D4—1228no—
278C2×D4—1224no—
279C2×D4—1222no—
280C2×D4—1224no—
281C2×D4—4884no—
282C2×D4—4884no—
283C2×D4—2448no—
284C2×D4—2444no—
285C2×D4—2444no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—1224no—
293C22×C4—1222no—
294C22×C4—1222no—
295C22×C4C2×C4×D6111yescharacteristic
296C22×C4C2×D24411yescharacteristic
297C22×C4D4×D6114yes—
298M16—2444no—
299C2×C8—2442no—
300C4⋊C4—614no—
301C4⋊C4—1224no—
302C22⋊C4—416no—
303C22⋊C4—1224no—
304C22⋊C4—1228no—
305C22⋊C4—1222no—
306C22⋊C4—1224no—
307C22⋊C4—1224no—
308C42—1222no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—48no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—48no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—84no—
326C23—82no—
327C23—84no—
328C23—82no—
329C23—1222no—
330C23—4884no—
331C23—2444no—
332C23#1451969114yes—
333C23C2×#1175819811yescharacteristic
334C23#1451974512yes—
335Q8—2442no—
336D4—28no—
337D4—48no—
338D4—44no—
339D4—4888no—
340D4—2442no—
341D4—2448no—
342D4—2444no—
343D4—2444no—
344D4—2444no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—1224no—
354C2×C4—1222no—
355C2×C4—4884no—
356C2×C4—2444no—
357C2×C4—2444no—
358C2×C4C22⋊C4×D6112yes—
359C2×C4#1451974712yes—
360C8—4882no—
361C22—22no—
362C22—24no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—24no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—44no—
373C22—82no—
374C22—88no—
375C22—88no—
376C22—82no—
377C22—9764no—
378C22—2442no—
379C22—2444no—
380C22#1683961711yescharacteristic
381C22C2×#1451969112yes—
382C22#1683992912yes—
383C22#1683993912yes—
384C4—22no—
385C4—22no—
386C4—84no—
387C4—2442no—
388C4—2444no—
389C2—22no—
390C2—22no—
391C2—22no—
392C2—44no—
393C2—82no—
394C2—4882no—
395C2#2144573311yescharacteristic
396C2#2144577212yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1438C2
socleC22×C61#379
prime radical#21459454C2
cosocleC23×C4×C61C23
hypercenter#1438D61
hypocenterC61#8453
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #21445740C2semidirect
3. #21444321C2semidirect
4. C61×#1438C2semidirect
5. C2×#14519822C4semidirect
8. #16840329C4semidirect
12. #16840237C4semidirect
14. #16839726C22semidirect
19. C2×#95×C61C22semidirect
96. C2×D244C22⋊C4semidirect
98. C2×Dic122C22⋊C4semidirect
105. C4⋊Dic61C22⋊C4semidirect
194. #1438D61semidirect
215. C2×#95C2×D61semidirect
218. C61#8453semidirect

Maximal subgroups

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

subgroupindex
#214594542
#214457402
#214443212
C61×#14382
#845361