Finite Group Databasebeta

← #29572622

Subgroups — #29572622

Order 15616.

1266 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C22×#14519759C211yesmaximal, characteristic
2#21445893C211yesmaximal, characteristic, complemented
3#21445891C212yesmaximal, complemented
4#21445733C212yesmaximal
5C61×#1819C211yesmaximal, characteristic, complemented
6C23×C22⋊D61C2211yescharacteristic
7C23×C4×D61C411yescharacteristic, complemented
8C2×#14519759—24no—
9C2×#14519759—22no—
10C2×#14519759—22no—
11C2×#14519759—24no—
12C22×C4⋊Dic61C411yescharacteristic, complemented
13C22×#11758196C411yescharacteristic, complemented
14C22×C22⋊Dic61C2211yescharacteristic
15C22×#11758198C411yescharacteristic, complemented
16C2×#14519691—24no—
17#16839618C2212yes—
18#16839529—22no—
19#16839528—24no—
20C22×C22⋊C4×C61C2211yescharacteristic
21C61×#379C2212yes—
22C61×#342—22no—
23C24×D61C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×C22⋊D61—22no—
26C22×C22⋊D61—21no—
27C22×C22⋊D61—44no—
28C22×C22⋊D61D414yes—
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C23×Dic61C2×C411yescharacteristic
32C22×C4×D61—22no—
33C22×C4×D61—22no—
34C22×C4×D61—22no—
35C22×C4×D61—44no—
36#14519759—216no—
37#14519759—216no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—22no—
41C2×#11758196—22no—
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—22no—
48C2×C22⋊Dic61—22no—
49C2×C22⋊Dic61—22no—
50C2×C22⋊Dic61D414yes—
51C2×#11758198—22no—
52C2×#11758198—22no—
53C2×#11758198—22no—
54#14519691—48no—
55C25×C61C2311yescharacteristic
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61—22no—
61C2×C22⋊C4×C61—22no—
62C2×C22⋊C4×C61—22no—
63C2×C22⋊C4×C61D412yes—
64C2×C22⋊C4×C61D412yes—
65C24×C61—42no—
66C24×C61—42no—
67C24×C61—42no—
68C24×C61C22×C411yescharacteristic
69C24×C61C2×D411yescharacteristic
70C24×C61C2×D412yes—
71C24×C61C2×D412yes—
72C24×C61C2×D411yescharacteristic
73C23×D61—21no—
74C23×D61—21no—
75C23×D61—24no—
76C23×D61—42no—
77C23×D61—42no—
78C23×D61C22⋊C412yes—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—42no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61C22⋊C412yes—
87C22×C4×C61C22⋊C412yes—
88C2×C22⋊D61—24no—
89C2×C22⋊D61—24no—
90C2×C22⋊D61—28no—
91C2×C22⋊D61—44no—
92C2×C22⋊D61—44no—
93C2×C22⋊D61—48no—
94C2×C22⋊D61—44no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—42no—
102C22×Dic61—42no—
103C22×Dic61—42no—
104C22×Dic61—42no—
105C22×Dic61C22⋊C412yes—
106C22×Dic61C22⋊C412yes—
107C22×Dic61C22⋊C412yes—
108C2×C4×D61—28no—
109C2×C4×D61—48no—
110C2×C4×D61—44no—
111C2×C4×D61—48no—
112C2×C4×D61—44no—
113C22⋊C4×C61—24no—
114C22⋊C4×C61—24no—
115C22⋊C4×C61—44no—
116C22⋊C4×C61—44no—
117C22⋊C4×C61—44no—
118C22⋊Dic61—24no—
119C22⋊Dic61—24no—
120C22⋊Dic61—44no—
121C22⋊Dic61—44no—
122C22⋊Dic61—44no—
123#11758198—28no—
124C4⋊Dic61—28no—
125#11758196—28no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—24no—
130C23×C61—22no—
131C23×C61—24no—
132C23×C61—22no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61—41no—
136C23×C61—42no—
137C23×C61—42no—
138C23×C61—41no—
139C23×C61—42no—
140C23×C61—41no—
141C23×C61—88no—
142C23×C61C2×C22⋊C411yescharacteristic
143C23×C61C2×C22⋊C411yescharacteristic
144C23×C61C2×C22⋊C411yescharacteristic
145C23×C61#12012yes—
146C23×C61#12012yes—
147C22×D61—24no—
148C22×D61—24no—
149C22×D61—22no—
150C22×D61—22no—
151C22×D61—416no—
152C22×D61—42no—
153C22×D61—42no—
154C22×D61—42no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61#9914yes—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—82no—
168C2×C4×C61—82no—
169C2×C4×C61—82no—
170C22⋊D61—48no—
171C22⋊D61—48no—
172C2×Dic61—22no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—24no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—82no—
185C2×Dic61—82no—
186C2×Dic61—82no—
187C4×D61—416no—
188#8543—611nomaximal, Sylow
189C22×C61—28no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—22no—
197C22×C61—22no—
198C22×C61—22no—
199C22×C61—22no—
200C22×C61—22no—
201C22×C61—22no—
202C22×C61—84no—
203C22×C61—82no—
204C22×C61—84no—
205C22×C61—82no—
206C22×C61#37911yescharacteristic
207C22×C61C2×#9912yes—
208C22×C61#41012yes—
209C22×C61#41012yes—
210C2×D61—24no—
211C2×D61—44no—
212C2×D61—48no—
213C2×D61—48no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—44no—
217C4×C61—42no—
218C4×C61—42no—
219C4×C61—162no—
220Dic61—42no—
221Dic61—42no—
222Dic61—42no—
223Dic61—162no—
224C22×#123—611no—
225#1819—612no—
226#1819—611no—
227#1819D6111yescharacteristic, complemented
228#1781—612no—
229C2×C61—22no—
230C2×C61—22no—
231C2×C61—22no—
232C2×C61—42no—
233C2×C61—42no—
234C2×C61#178111yescharacteristic
235C2×C61#178211yescharacteristic
236C2×C61#178211yescharacteristic
237D61—44no—
238C23×D4—611no—
239C24×C4—611no—
240C2×#123—1224no—
241C2×#123—1224no—
242C2×#123—1222no—
243C2×#123—1222no—
244C22×C4⋊C4—611no—
245C22×C4⋊C4—611no—
246C22×C22⋊C4—611no—
247C22×C22⋊C4—611no—
248C22×C22⋊C4C2×D6111yescharacteristic
249C2×#99—1224no—
250#379—612no—
251#379C2×D6112yes—
252#342—22no—
253#342—1224no—
254#342—1222no—
255C61#854311yescharacteristic, complemented, Sylow
256C25—611no—
257C25C22×D6111yescharacteristic
258C22×D4—614no—
259C22×D4—1221no—
260C22×D4—1222no—
261C22×D4—1221no—
262C22×D4—2444no—
263C23×C4—611no—
264C23×C4—611no—
265C23×C4—611no—
266C23×C4—1222no—
267C23×C4—1222no—
268C23×C4—1222no—
269C23×C4—2444no—
270C23×C4C4×D6111yescharacteristic
271C23×C4C4×D6111yescharacteristic
272#123—12216no—
273#123—12216no—
274C2×C4⋊C4—1222no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1222no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1222no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—614no—
286C2×C22⋊C4—1222no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1222no—
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1222no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4D24412yes—
296C2×C22⋊C4C22⋊D6112yes—
297#99—2448no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—612no—
302C24—1224no—
303C24—1221no—
304C24—1221no—
305C24—2442no—
306C24—2442no—
307C24C2×C4×D6111yescharacteristic
308C24C2×D24411yescharacteristic
309C24D4×D6112yes—
310C24D4×D6112yes—
311C24C2×C22⋊D6111yescharacteristic
312C2×D4—1228no—
313C2×D4—1224no—
314C2×D4—1224no—
315C2×D4—2444no—
316C2×D4—2444no—
317C2×D4—2448no—
318C2×D4—2444no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—612no—
327C22×C4—612no—
328C22×C4—612no—
329C22×C4—1228no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—2444no—
337C22×C4—2448no—
338C22×C4—2444no—
339C22×C4—2448no—
340C22×C4—2442no—
341C22×C4—2442no—
342C22×C4—2442no—
343C22×C4—2442no—
344C22×C4#1175819812yes—
345C22×C4#1175819812yes—
346C4⋊C4—1228no—
347C4⋊C4—1228no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—1228no—
354C22⋊C4—1224no—
355C22⋊C4—1224no—
356C22⋊C4—2444no—
357C22⋊C4—2444no—
358C22⋊C4—2444no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—88no—
375C23—614no—
376C23—1224no—
377C23—1224no—
378C23—1222no—
379C23—1222no—
380C23—2442no—
381C23—2442no—
382C23—24416no—
383C23—2442no—
384C23—2442no—
385C23—2442no—
386C23—2442no—
387C23C22⋊C4×D6111yescharacteristic
388C23C22⋊C4×D6111yescharacteristic
389C23C2×#1175819811yescharacteristic
390C23#1451974512yes—
391C23#1451974712yes—
392D4—2448no—
393D4—2448no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—1224no—
406C2×C4—1222no—
407C2×C4—1222no—
408C2×C4—1222no—
409C2×C4—4882no—
410C2×C4—4882no—
411C2×C4—4882no—
412C2×C4—24416no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2442no—
421C22—28no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—84no—
435C22—84no—
436C22—82no—
437C22—82no—
438C22—1224no—
439C22—2444no—
440C22—2444no—
441C22—2448no—
442C22—2448no—
443C22—2442no—
444C22—2442no—
445C22#1683961711yescharacteristic
446C22#99×D6112yes—
447C22#1683973412yes—
448C22#1683973512yes—
449C4—42no—
450C4—42no—
451C4—162no—
452C4—9762no—
453C4—2442no—
454C4—2442no—
455C4—2442no—
456C2—22no—
457C2—22no—
458C2—22no—
459C2—42no—
460C2—42no—
461C2—2444no—
462C2#2144573211yescharacteristic
463C2#2144573511yescharacteristic
464C2#2144573611yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#1819C2
socleC22×C61#379
prime radicalC23×C22⋊D61C22
cosocleC25×C61C23
hypercenter#1819D61
hypocenterC61#8543
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #21445893C2semidirect
3. #21445891C2semidirect
5. C61×#1819C2semidirect
7. C23×C4×D61C4semidirect
12. C22×C4⋊Dic61C4semidirect
13. C22×#11758196C4semidirect
15. C22×#11758198C4semidirect
227. #1819D61semidirect
255. C61#8543semidirect

Maximal subgroups

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

subgroupindex
C22×#145197592
#214458932
#214458912
#214457332
C61×#18192
#854361