Finite Group Databasebeta

← #29575804

Subgroups — #29575804

Order 15616.

980 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#16839625C211yesmaximal, characteristic, complemented
2#21445729C211yesmaximal, characteristic, complemented
3C2×#16839496C212yesmaximal, complemented
4#21443697C212yesmaximal, complemented
5C61×#2847C211yesmaximal, characteristic
6C22×C22⋊Dic61—21no—
7C2×C22⋊#92418—22no—
8C2×C22⋊#92418C2212yes—
9#16839625—24no—
10C2×#14519684—21no—
11C2×#14519684C2212yes—
12#16839496—24no—
13C24×C4×C61C2211yescharacteristic
14C2×#121×C61C412yescomplemented
15C2×#121×C61C2211yescharacteristic
16C2×#120×C61—21no—
17C61×#386—21no—
18C61×#386—22no—
19C61×#380C412yescomplemented
20C61×#380C2211yescharacteristic
21C23×Dic61—22no—
22C22×#92418—22no—
23C2×C22⋊Dic61—24no—
24C2×C22⋊Dic61—44no—
25C22⋊#92418—22no—
26C22⋊#92418—22no—
27C22⋊#92418—44no—
28#14519684—24no—
29#14519684—22no—
30#14519684D414yescomplemented
31C25×C61D411yescharacteristic
32C22×D4×C61—22no—
33C22×D4×C61—21no—
34C22×D4×C61C2×C411yescharacteristic
35C23×C4×C61—22no—
36C23×C4×C61—22no—
37C23×C4×C61—44no—
38C23×C4×C61C2×C412yes—
39C23×C4×C61D412yes—
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42#121×C61—22no—
43#121×C61—22no—
44#121×C61—44no—
45#120×C61—44no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61C2×C412yes—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61—21no—
52#95×C61—22no—
53#95×C61—21no—
54C24×C61—24no—
55C24×C61—21no—
56C24×C61—21no—
57C24×C61—22no—
58C24×C61—22no—
59C24×C61—21no—
60C24×C61—42no—
61C24×C61C22⋊C412yes—
62C24×C61C2×D411yescharacteristic
63C2×D4×C61—22no—
64C2×D4×C61—42no—
65C2×D4×C61—44no—
66C2×D4×C61—44no—
67C2×D4×C61—44no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61—42no—
71C2×D4×C61—42no—
72C22×C4×C61—24no—
73C22×C4×C61—24no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—24no—
77C22×C4×C61—24no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—42no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61—42no—
84C22×C4×C61—48no—
85C22×C4×C61—48no—
86C22×C4×C61C22⋊C412yes—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D412yes—
90C22×C4×C61#4112yes—
91C22×Dic61—24no—
92C22×Dic61—44no—
93C22×Dic61—44no—
94C22×Dic61—42no—
95C22×Dic61—42no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—42no—
98C22⋊C4×C61—44no—
99C22⋊C4×C61—44no—
100C22⋊C4×C61—42no—
101C22⋊C4×C61—42no—
102C22⋊Dic61—48no—
103C2×#92418—44no—
104C2×#92418—42no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—24no—
108C23×C61—24no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—22no—
113C23×C61—22no—
114C23×C61—22no—
115C23×C61—48no—
116C23×C61—48no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—41no—
122C23×C61—42no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—42no—
126C23×C61#9914yes—
127C23×C61C2×C22⋊C411yescharacteristic
128C23×C61#12011yescharacteristic
129C23×C61#12311yescharacteristic
130C23×C61#13512yes—
131C23×C61AGL(1,ℤ/8)12yes—
132D4×C61—82no—
133D4×C61—88no—
134D4×C61—82no—
135D4×C61—82no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—48no—
144C2×C4×C61—48no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61C4×D412yes—
154C2×C4×C61#12112yes—
155C2×Dic61—44no—
156C2×Dic61—44no—
157C2×Dic61—44no—
158C2×Dic61—84no—
159C2×Dic61—84no—
160#92418—82no—
161#9158—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—24no—
167C22×C61—24no—
168C22×C61—24no—
169C22×C61—24no—
170C22×C61—24no—
171C22×C61—48no—
172C22×C61—48no—
173C22×C61—42no—
174C22×C61—42no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—82no—
181C22×C61—82no—
182C22×C61#38611yescharacteristic
183C22×C61C2×#9912yes—
184C22×C61#41611yescharacteristic
185C22×C61#41811yescharacteristic
186C22×C61#42112yes—
187C22×C61#44912yes—
188C22×C61#48212yes—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—42no—
192C4×C61—81no—
193C4×C61—82no—
194Dic61—84no—
195#2847D6111yescharacteristic
196C2×#381—611no—
197#1779—611no—
198C2×#327—612no—
199#1294—612no—
200C2×C61—22no—
201C2×C61—22no—
202C2×C61—44no—
203C2×C61—161no—
204C2×C61#151012yes—
205C2×C61#151612yes—
206C2×C61#187511yescharacteristic
207C2×C61#189711yescharacteristic
208C2×C61#189811yescharacteristic
209C24×C4C2×D6111yescharacteristic
210C2×#121Dic6112yescomplemented
211C2×#121C2×D6111yescharacteristic
212C2×#120—21no—
213C22×C22⋊C4—1221no—
214C2×C22⋊C8—612no—
215C2×C22⋊C8—1222no—
216#386—21no—
217#386—22no—
218#381—1224no—
219#380Dic6112yescomplemented
220#380C2×D6111yescharacteristic
221C2×#95—612no—
222C2×#95—1221no—
223#327—1224no—
224C61#915811yescharacteristic, complemented, Sylow
225C25C22⋊D6111yescharacteristic
226C22×D4—22no—
227C22×D4—21no—
228C22×D4C2×Dic6111yescharacteristic
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—1222no—
233C23×C4C2×Dic6112yes—
234C23×C4C22⋊D6112yes—
235C23×C4C22⋊D6111yescharacteristic
236C23×C4C22×D6111yescharacteristic
237C22×C8—1222no—
238#121—22no—
239#121—22no—
240#121—44no—
241#120—44no—
242C2×C4⋊C4C2×Dic6111yescharacteristic
243C2×C4⋊C4C2×Dic6112yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1224no—
249C2×C22⋊C4—2444no—
250C22⋊C8—1222no—
251C22⋊C8—1222no—
252C22⋊C8—2444no—
253#95—22no—
254#95—21no—
255#95—614no—
256#95—1224no—
257#95—1222no—
258C24—24no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—42no—
265C24C22⋊Dic6112yes—
266C24C2×C22⋊D6111yescharacteristic
267C2×D4—22no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—1224no—
291C22×C4—2444no—
292C22×C4—2444no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4C22⋊Dic6112yes—
296C22×C4D4×D6112yes—
297C22×C4#1175822412yes—
298C22×C4C22×Dic6111yescharacteristic
299C22×C4C2×C22⋊D6111yescharacteristic
300C2×C8—2444no—
301C2×C8—2442no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—2448no—
309C23—22no—
310C23—22no—
311C23—24no—
312C23—24no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—48no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23#1451969214yes—
331C23C2×C22⋊Dic6111yescharacteristic
332C23#1451974511yescharacteristic
333C23#1451975811yescharacteristic
334C23#1451979712yes—
335C23#1451980312yes—
336D4—82no—
337D4—88no—
338D4—82no—
339D4—82no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—4884no—
358C2×C4—4884no—
359C2×C4—2444no—
360C2×C4—2444no—
361C2×C4—2444no—
362C2×C4D4×Dic6112yes—
363C2×C4#1451974912yes—
364C8—4882no—
365C22—24no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—24no—
374C22—48no—
375C22—48no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22—82no—
384C22—81no—
385C22#1683964211yescharacteristic
386C22C2×#1451969212yes—
387C22#1683975811yescharacteristic
388C22#1683976711yescharacteristic
389C22#1683977912yes—
390C22#1683990512yes—
391C22#1684008312yes—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—82no—
396C4—81no—
397C4—4884no—
398C2—22no—
399C2—22no—
400C2—44no—
401C2—161no—
402C2#2144434812yes—
403C2#2144437512yes—
404C2#2144614211yescharacteristic
405C2#2144626811yescharacteristic
406C2#2144627311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#2847C2
socleC23×C61C2×C22⋊C4
cosocleC23×C4×C61C23
hypercenter#2847D61
hypocenterC61#9158
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C2×#16839625C2semidirect
2. #21445729C2semidirect
3. C2×#16839496C2semidirect
4. #21443697C2semidirect
14. C2×#121×C61C4semidirect
19. C61×#380C4semidirect
30. #14519684D4semidirect
210. C2×#121Dic61semidirect
219. #380Dic61semidirect
224. C61#9158semidirect

Maximal subgroups

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

subgroupindex
C2×#168396252
#214457292
C2×#168394962
#214436972
C61×#28472
#915861