Finite Group Databasebeta

← #29577812

Subgroups — #29577812

Order 15616.

958 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 111 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#21449138C211yesmaximal, characteristic
2#21445764C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21443672C212yesmaximal, complemented
5C61×#2273C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519746—21no—
8C2×#14519705—24no—
9#16839726C2212yes—
10#16839674C412yescomplemented
11#16839671C412yescomplemented
12#16839641—21no—
13#16839638—22no—
14#16839625C2211yescharacteristic
15#16839486—22no—
16C22×C22⋊C4×C61—21no—
17C22×C22⋊C4×C61C2211yescharacteristic
18C4×C22⋊C4×C61—21no—
19C4×C22⋊C4×C61—24no—
20C4×C22⋊C4×C61C2212yes—
21C2×#11758222—22no—
22C2×#11758222—22no—
23C2×#11758222—44no—
24C22×Dic122C2×C411yescharacteristic, complemented
25C22×D244C2×C411yescharacteristic, complemented
26C22×C22⋊D61—21no—
27C22×C22⋊D61—22no—
28C22×C4×D61—21no—
29C2×#11758194—22no—
30#14519746—44no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—21no—
33C2×#11758198—22no—
34#14519705—48no—
35C22⋊#92418—22no—
36C22⋊#92418—22no—
37#14519684—22no—
38#14519684—21no—
39#14519684C2×C412yescomplemented
40C25×C61D411yescharacteristic
41C23×C4×C61—22no—
42C23×C4×C61D411yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—28no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61C2×C412yes—
50C2×C22⋊C4×C61D412yes—
51C2×C42×C61—22no—
52C2×C42×C61D414yescomplemented
53#95×C61—22no—
54#95×C61—22no—
55C24×C61—21no—
56C24×C61—24no—
57C24×C61—21no—
58C24×C61—42no—
59C24×C61—81no—
60C24×C61C22⋊C412yes—
61C24×C61C2×D411yescharacteristic
62C23×D61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—24no—
65C22×C4×C61—28no—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61—81no—
69C22×C4×C61—82no—
70C22×C4×C61—82no—
71C22×C4×C61C22⋊C412yes—
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D412yes—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61#4112yes—
76C2×C22⋊D61—24no—
77C2×C22⋊D61—42no—
78C2×C22⋊D61—41no—
79C2×C22⋊D61—41no—
80C2×C22⋊D61—44no—
81C2×C22⋊D61—44no—
82C2×C22⋊D61—82no—
83C22×Dic61—21no—
84C22×Dic61—22no—
85#11758222—28no—
86#11758222—44no—
87#11758222—44no—
88#11758222—44no—
89C2×D244—22no—
90C2×D244—42no—
91C2×C4×D61—42no—
92C2×C4×D61—42no—
93C2×C4×D61—42no—
94C2×Dic122—22no—
95C2×Dic122—42no—
96C22⋊C4×C61—28no—
97C22⋊C4×C61—28no—
98C22⋊C4×C61—24no—
99C22⋊C4×C61—24no—
100C42×C61—44no—
101C42×C61—82no—
102C22⋊Dic61—44no—
103C22⋊Dic61—42no—
104#11758198—82no—
105#11758194—44no—
106C2×#92418—42no—
107C23×C61—22no—
108C23×C61—22no—
109C23×C61—24no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—22no—
113C23×C61—22no—
114C23×C61—41no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—42no—
119C23×C61—81no—
120C23×C61—81no—
121C23×C61—88no—
122C23×C61#10414yescomplemented
123C23×C61C2×C22⋊C411yescharacteristic
124C23×C61#12011yescharacteristic
125C23×C61#12311yescharacteristic
126C22×D61—22no—
127C22×D61—42no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—42no—
131C2×C4×C61—24no—
132C2×C4×C61—28no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61—44no—
138C2×C4×C61—42no—
139C2×C4×C61—44no—
140C2×C4×C61—42no—
141C2×C4×C61—81no—
142C2×C4×C61—88no—
143C2×C4×C61C4×D412yes—
144C2×C4×C61#12112yes—
145C22⋊D61—44no—
146C22⋊D61—44no—
147C22⋊D61—88no—
148C2×Dic61—22no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154D244—24no—
155D244—42no—
156C4×D61—48no—
157Dic122—24no—
158Dic122—42no—
159#92418—82no—
160#9475—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—24no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—44no—
170C22×C61—44no—
171C22×C61—44no—
172C22×C61—44no—
173C22×C61—42no—
174C22×C61—42no—
175C22×C61—82no—
176C22×C61—84no—
177C22×C61#38611yescharacteristic
178C22×C61C2×#10412yes—
179C22×C61B(4,𝔽2)12yes—
180C2×D61—44no—
181C2×D61—42no—
182C2×D61—82no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—22no—
186C4×C61—22no—
187C4×C61—48no—
188C4×C61—42no—
189Dic61—42no—
190Dic61—82no—
191#2404—611no—
192#2273D6111yescharacteristic, complemented
193#1789—612no—
194#1462—611no—
195#1284—612no—
196C2×C61—22no—
197C2×C61—22no—
198C2×C61—41no—
199C2×C61—44no—
200C2×C61#188911yescharacteristic
201C2×C61#189611yescharacteristic
202C2×C61#191111yescharacteristic
203D61—82no—
204C22×#41—611no—
205C2×#120—1221no—
206C22×C22⋊C4—21no—
207C22×C22⋊C4C2×D6111yescharacteristic
208C2×#104—1224no—
209#407—612no—
210#393—612no—
211#392—612no—
212#386—1221no—
213#386—1222no—
214#381—611no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4C2×D6112yes—
218#323—1222no—
219C61#947511yescharacteristic, complemented, Sylow
220C25C22⋊D6111yescharacteristic
221C2×#41—1222no—
222C2×#41—1222no—
223C2×#41—2444no—
224C22×Q8—611no—
225C22×D4—611no—
226C22×D4—1221no—
227C22×D4—1222no—
228C23×C4—22no—
229C23×C4—1221no—
230C23×C4C22⋊D6111yescharacteristic
231C23×C4C22×D6111yescharacteristic
232C2×M16—1222no—
233#120—2444no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4C4×D6112yes—
243C2×C22⋊C4D24412yes—
244C2×C42—22no—
245C2×C42C22⋊D6114yescomplemented
246#104—2448no—
247C22⋊C8—1222no—
248C22⋊C8—1222no—
249#95—22no—
250#95—22no—
251#95—612no—
252#95—1222no—
253#95—1221no—
254C24—21no—
255C24—24no—
256C24—21no—
257C24—42no—
258C24—81no—
259C24—1221no—
260C24#1175819812yes—
261C24C2×C22⋊D6111yescharacteristic
262#41—1228no—
263#41—2444no—
264#41—2444no—
265#41—2444no—
266C2×Q8—1222no—
267C2×Q8—2442no—
268C2×D4—1224no—
269C2×D4—1222no—
270C2×D4—4882no—
271C2×D4—2442no—
272C2×D4—2444no—
273C2×D4—2444no—
274C2×D4—2442no—
275C2×D4—2441no—
276C2×D4—2441no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—81no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—1221no—
286C22×C4—1222no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4#1175819812yes—
291C22×C4C2×C4×D6111yescharacteristic
292C22×C4C2×D24411yescharacteristic
293C22×C4#1175822212yes—
294C22×C4C2×C22⋊D6112yes—
295M16—2444no—
296C2×C8—2442no—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—4882no—
302C22⋊C4—2444no—
303C22⋊C4—2442no—
304C42—44no—
305C42—82no—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—88no—
321C23—1222no—
322C23—2442no—
323C23—2441no—
324C23—2442no—
325C23—2441no—
326C23#1451970514yescomplemented
327C23C2×#1175819811yescharacteristic
328C23#1451974611yescharacteristic
329C23#1451976011yescharacteristic
330Q8—1224no—
331Q8—2442no—
332D4—1224no—
333D4—4888no—
334D4—2442no—
335D4—2444no—
336D4—2444no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—88no—
349C2×C4—1222no—
350C2×C4—2448no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4C4×C22⋊D6112yes—
357C2×C4#1451975112yes—
358C8—4882no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—82no—
374C22—84no—
375C22—4882no—
376C22—4881no—
377C22—4881no—
378C22—2444no—
379C22—2442no—
380C22#1683964111yescharacteristic
381C22C2×#1451970512yes—
382C22#1683994912yes—
383C4—22no—
384C4—22no—
385C4—48no—
386C4—42no—
387C4—4882no—
388C4—2442no—
389C2—22no—
390C2—22no—
391C2—41no—
392C2—44no—
393C2—4882no—
394C2#2144622711yescharacteristic
395C2#2144626211yescharacteristic
396C2#2144635111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2273C2
socleC22×C61#386
prime radical#21449138C2
cosocleC23×C4×C61C23
hypercenter#2273D61
hypocenterC61#9475
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
2. #21445764C2semidirect
3. #21444146C2semidirect
4. #21443672C2semidirect
5. C61×#2273C2semidirect
10. #16839674C4semidirect
11. #16839671C4semidirect
24. C22×Dic122C2×C4semidirect
25. C22×D244C2×C4semidirect
39. #14519684C2×C4semidirect
52. C2×C42×C61D4semidirect
122. C23×C61#104semidirect
192. #2273D61semidirect
219. C61#9475semidirect
245. C2×C42C22⋊D61semidirect
326. C23#14519705semidirect

Maximal subgroups

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

subgroupindex
#214491382
#214457642
#214441462
#214436722
C61×#22732
#947561