Finite Group Databasebeta

← #29577790

Subgroups — #29577790

Order 15616.

716 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 115 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#21454848C211yesmaximal, characteristic
2#21449184C211yesmaximal, characteristic
3#21445753C212yesmaximal, complemented
4#21443686C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C2×#14519756—21no—
8#16839834—24no—
9#16839819—21no—
10C2×#14519700—24no—
11#16839726—21no—
12C2×C22⋊#92418C2212yes—
13#16839682C412yescomplemented
14#16839674C412yescomplemented
15#16839668—21no—
16#16839642—21no—
17#16839634—21no—
18#16839622C2211yescharacteristic
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C61×#380C2212yes—
22C61×#342—21no—
23C22×Dic122—22no—
24C22×Dic122—21no—
25C22×Dic122—22no—
26C22×Dic122—44no—
27C22×Dic122C2×C412yes—
28C23×Dic61—21no—
29C2×#11758194—22no—
30C22×#92418D414yes—
31#14519756—44no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—22no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—21no—
36C4⋊#92418—22no—
37#14519700—48no—
38C22⋊#92418—22no—
39C22⋊#92418—22no—
40C22⋊#92418—22no—
41#14519688—22no—
42#14519684—22no—
43#14519684—22no—
44#14519684C2×C412yes—
45C22×D4×C61D411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C4×D4×C61—22no—
49C4×D4×C61—22no—
50C2×C4⋊C4×C61—22no—
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C42×C61D411yescharacteristic
56#99×C61—42no—
57#95×C61—22no—
58C24×C61C22⋊C411yescharacteristic
59C24×C61C2×D411yescharacteristic
60C2×D4×C61—22no—
61C2×D4×C61—21no—
62C2×D4×C61—21no—
63C2×D4×C61—22no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D412yes—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61#4112yes—
76C22×Dic61—22no—
77C22×Dic61—22no—
78C22×Dic61—44no—
79C22×Dic61—42no—
80C2×Dic122—28no—
81C2×Dic122—24no—
82C2×Dic122—24no—
83C2×Dic122—44no—
84C2×Dic122—48no—
85C2×Dic122—44no—
86C2×Dic122—42no—
87C2×Dic122—41no—
88C2×Dic122—41no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—44no—
91C22⋊C4×C61—22no—
92C22⋊C4×C61—22no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—42no—
95C42×C61—21no—
96C42×C61—21no—
97C22⋊Dic61—44no—
98C22⋊Dic61—41no—
99C22⋊Dic61—41no—
100C4⋊Dic61—42no—
101#11758196—44no—
102#11758194—82no—
103C2×#92418—24no—
104C2×#92418—44no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113C23×C61#12112yes—
114C23×C61AGL(1,ℤ/8)11yescharacteristic
115C23×C61#13711yescharacteristic
116C23×C61#13712yes—
117D4×C61—42no—
118D4×C61—42no—
119C2×C4×C61—24no—
120C2×C4×C61—22no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—81no—
132C2×C4×C61—82no—
133C2×C4×C61—82no—
134C2×C4×C61—82no—
135C2×C4×C61C4×D412yes—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12311yescharacteristic
138C2×Dic61—24no—
139C2×Dic61—48no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145Dic122—28no—
146Dic122—44no—
147Dic122—44no—
148Dic122—44no—
149#92418—44no—
150#9471—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—22no—
153C22×C61—21no—
154C22×C61—22no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61—41no—
162C22×C61—81no—
163C22×C61#38611yescharacteristic
164C22×C61#41711yescharacteristic
165C22×C61#41911yescharacteristic
166C22×C61#46812yes—
167C22×C61#47012yes—
168C4×C61—22no—
169C4×C61—21no—
170C4×C61—41no—
171C4×C61—41no—
172C4×C61—41no—
173C4×C61—162no—
174C4×C61B(4,𝔽2)12yes—
175Dic61—44no—
176Dic61—82no—
177#2973—611no—
178#2409—611no—
179#1799D6111yescharacteristic
180#1787—612no—
181#1289—612no—
182C2×C61—21no—
183C2×C61—22no—
184C2×C61—21no—
185C2×C61—22no—
186C2×C61—81no—
187C2×C61#188911yescharacteristic
188C2×C61#189411yescharacteristic
189C2×C61#190911yescharacteristic
190C23×Q8—611no—
191C2×C22⋊Q8—1221no—
192C2×C4×D4C2×D6111yescharacteristic
193#435—1224no—
194#432—1221no—
195C2×#103—1224no—
196C2×#99—21no—
197#407—1221no—
198C2×C22⋊C8—612no—
199#395—612no—
200#393—612no—
201#391—1221no—
202#386—1221no—
203#385—1221no—
204#380—611no—
205#380C2×D6112yes—
206#342—21no—
207C61#947111yescharacteristic, complemented, Sylow
208C22×Q8—612no—
209C22×Q8—1221no—
210C22×Q8—1222no—
211C22×Q8—1222no—
212C22×Q8—2444no—
213C22×D4C22⋊D6111yescharacteristic
214C23×C4—1221no—
215C23×C4D24411yescharacteristic
216C23×C4C22×D6111yescharacteristic
217C2×M16—1222no—
218C22×C8—614no—
219C22⋊Q8—2444no—
220C4×D4—22no—
221C4×D4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—1222no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4C4×D6111yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4C4×D6111yescharacteristic
230C2×C22⋊C4C22⋊D6111yescharacteristic
231C2×C42D24411yescharacteristic
232C4⋊C8—1222no—
233#103—2448no—
234#99—42no—
235C22⋊C8—1222no—
236C22⋊C8—1222no—
237C22⋊C8—1222no—
238#97—1222no—
239#95—22no—
240#95—612no—
241#95—1222no—
242#95—1222no—
243C24#1175819811yescharacteristic
244C24C2×C22⋊D6111yescharacteristic
245C2×Q8—1224no—
246C2×Q8—1228no—
247C2×Q8—1224no—
248C2×Q8—2444no—
249C2×Q8—2442no—
250C2×Q8—2444no—
251C2×Q8—2448no—
252C2×Q8—2441no—
253C2×Q8—2441no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—2444no—
266C22×C4—2442no—
267C22×C4#1175819811yescharacteristic
268C22×C4#1175819811yescharacteristic
269C22×C4#1175819811yescharacteristic
270C22×C4C2×C4×D6111yescharacteristic
271C22×C4C2×D24411yescharacteristic
272C22×C4D4×D6112yes—
273C22×C4#1175822412yes—
274M16—4882no—
275C2×C8—1224no—
276C2×C8—2444no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C4⋊C4—2442no—
280C4⋊C4—2444no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—2444no—
286C22⋊C4—2441no—
287C22⋊C4—2441no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×#1175819811yescharacteristic
298C23#1451974912yes—
299C23#1451980311yescharacteristic
300C23#1451980712yes—
301C23#1451980811yescharacteristic
302Q8—1228no—
303Q8—2444no—
304Q8—2444no—
305Q8—2444no—
306D4—42no—
307D4—42no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—1224no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2448no—
330C2×C4—2442no—
331C2×C4#1451973712yes—
332C2×C4#1451974711yescharacteristic
333C2×C4#1451976111yescharacteristic
334C8—2444no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1683963911yescharacteristic
348C22#1683976111yescharacteristic
349C22#1683977011yescharacteristic
350C22#1684000612yes—
351C22#1684001812yes—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—162no—
358C4—4882no—
359C4—2444no—
360C4#1683995012yes—
361C2—22no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#2144622511yescharacteristic
367C2#2144625011yescharacteristic
368C2#2144633911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#9471
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21445753C2semidirect
4. #21443686C2semidirect
13. #16839682C4semidirect
14. #16839674C4semidirect
207. C61#9471semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214491842
#214457532
#214436862
C61×#17992
#947161