Finite Group Databasebeta

← #29577913

Subgroups — #29577913

Order 15616.

1212 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1212 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#21448397C211yesmaximal, characteristic, complemented
2#21445734C212yesmaximal, complemented
3#21445734C211yesmaximal, characteristic, complemented
4#21443723C212yesmaximal, complemented
5C61×#2403C211yesmaximal, characteristic
6C22×C22⋊Dic61—21no—
7C2×#14519692—24no—
8C2×#14519692—22no—
9#16839642—24no—
10#16839625—21no—
11#16839618C2212yes—
12#16839618C2212yes—
13C23×D4×C61C2211yescharacteristic
14C23×D4×C61C2211yescharacteristic
15C2×#120×C61—21no—
16C2×#120×C61C412yescomplemented
17C61×#392—22no—
18C61×#386—21no—
19C61×#386C412yescomplemented
20C61×#379C2211yescharacteristic
21C23×Dic61—22no—
22C2×C22⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—44no—
29C2×C22⋊Dic61D414yescomplemented
30#14519692—48no—
31#14519692—44no—
32#14519684—24no—
33C25×C61C2×C411yescharacteristic, complemented
34C25×C61D411yescharacteristic
35C25×C61C2311yescharacteristic
36C22×D4×C61—22no—
37C22×D4×C61—21no—
38C22×D4×C61—22no—
39C22×D4×C61—22no—
40C22×D4×C61—21no—
41C22×D4×C61—22no—
42C22×D4×C61—22no—
43C22×D4×C61—44no—
44C22×D4×C61—44no—
45C22×D4×C61C2×C412yes—
46C22×D4×C61D412yes—
47C23×C4×C61C2×C411yescharacteristic, complemented
48C23×C4×C61D411yescharacteristic
49#120×C61—44no—
50#120×C61—44no—
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61C2×C412yescomplemented
56#95×C61—22no—
57C24×C61—21no—
58C24×C61—24no—
59C24×C61—21no—
60C24×C61—21no—
61C24×C61—22no—
62C24×C61—42no—
63C24×C61—42no—
64C24×C61—42no—
65C24×C61—42no—
66C24×C61—42no—
67C24×C61—81no—
68C24×C61—81no—
69C24×C61—81no—
70C24×C61C22⋊C412yes—
71C24×C61C22×C411yescharacteristic
72C24×C61C2×D411yescharacteristic
73C24×C61C2×D411yescharacteristic
74C24×C61C2×D412yes—
75C24×C61#4112yes—
76C2×D4×C61—24no—
77C2×D4×C61—28no—
78C2×D4×C61—24no—
79C2×D4×C61—24no—
80C2×D4×C61—24no—
81C2×D4×C61—24no—
82C2×D4×C61—42no—
83C2×D4×C61—42no—
84C2×D4×C61—42no—
85C2×D4×C61—44no—
86C2×D4×C61—44no—
87C2×D4×C61—44no—
88C2×D4×C61—48no—
89C2×D4×C61—42no—
90C2×D4×C61—44no—
91C2×D4×C61—42no—
92C2×D4×C61—44no—
93C2×D4×C61—41no—
94C2×D4×C61—42no—
95C2×D4×C61—41no—
96C2×D4×C61—44no—
97C2×D4×C61—44no—
98C2×D4×C61—82no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61C22⋊C412yes—
107C22×Dic61—24no—
108C22×Dic61—44no—
109C22×Dic61—42no—
110C22×Dic61—82no—
111C22×Dic61—82no—
112C22⋊C4×C61—44no—
113C22⋊C4×C61—44no—
114C22⋊C4×C61—42no—
115C22⋊C4×C61—42no—
116C22⋊C4×C61—81no—
117C22⋊Dic61—44no—
118C22⋊Dic61—44no—
119C22⋊Dic61—44no—
120C22⋊Dic61—44no—
121C22⋊Dic61—44no—
122C22⋊Dic61—82no—
123C22⋊Dic61—82no—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61—24no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—22no—
130C23×C61—22no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—24no—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—42no—
140C23×C61—42no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—42no—
144C23×C61—42no—
145C23×C61—42no—
146C23×C61—42no—
147C23×C61—42no—
148C23×C61—42no—
149C23×C61—41no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—42no—
153C23×C61—41no—
154C23×C61—42no—
155C23×C61—81no—
156C23×C61—81no—
157C23×C61—82no—
158C23×C61—82no—
159C23×C61—88no—
160C23×C61—88no—
161C23×C61—88no—
162C23×C61#9914yescomplemented
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61C4×D412yes—
165C23×C61#12011yescharacteristic
166C23×C61#12112yes—
167C23×C61#12311yescharacteristic
168D4×C61—48no—
169D4×C61—48no—
170D4×C61—48no—
171D4×C61—44no—
172D4×C61—44no—
173D4×C61—88no—
174C2×C4×C61—22no—
175C2×C4×C61—24no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×Dic61—44no—
188C2×Dic61—44no—
189C2×Dic61—44no—
190C2×Dic61—88no—
191C2×Dic61—82no—
192C2×Dic61—82no—
193C2×Dic61—82no—
194#9495—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—22no—
200C22×C61—22no—
201C22×C61—21no—
202C22×C61—22no—
203C22×C61—21no—
204C22×C61—24no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—21no—
209C22×C61—44no—
210C22×C61—44no—
211C22×C61—44no—
212C22×C61—44no—
213C22×C61—44no—
214C22×C61—44no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—84no—
219C22×C61—82no—
220C22×C61—82no—
221C22×C61—82no—
222C22×C61—84no—
223C22×C61—84no—
224C22×C61—84no—
225C22×C61—84no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61#38611yescharacteristic
229C22×C61C2×#9912yes—
230C22×C61B(4,𝔽2)12yes—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—82no—
234Dic61—84no—
235Dic61—162no—
236#2403D6111yescharacteristic
237#2308—611no—
238#1781—612no—
239#1781—611no—
240#1304—612no—
241C2×C61—22no—
242C2×C61—22no—
243C2×C61—42no—
244C2×C61—42no—
245C2×C61—41no—
246C2×C61—44no—
247C2×C61—82no—
248C2×C61#188911yescharacteristic
249C2×C61#189811yescharacteristic
250C2×C61#191311yescharacteristic
251C23×D4C2×D6111yescharacteristic
252C23×D4C2×D6111yescharacteristic
253C2×#120—21no—
254C2×#120Dic6112yescomplemented
255C22×C22⋊C4—1221no—
256C2×#99—1224no—
257C2×#99—1222no—
258#392—22no—
259#386—21no—
260#386—1224no—
261#386Dic6112yescomplemented
262#381—1221no—
263#379—612no—
264#379—612no—
265#379C2×D6111yescharacteristic
266C61#949511yescharacteristic, complemented, Sylow
267C25C2×Dic6111yescharacteristic, complemented
268C25C22⋊D6111yescharacteristic
269C25C22×D6111yescharacteristic
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—22no—
275C22×D4—21no—
276C22×D4—22no—
277C22×D4—44no—
278C22×D4—44no—
279C22×D4C2×Dic6112yes—
280C22×D4C22⋊D6112yes—
281C23×C4—1222no—
282C23×C4C2×Dic6111yescharacteristic, complemented
283C23×C4C22⋊D6111yescharacteristic
284#120—44no—
285#120—44no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—614no—
291C2×C22⋊C4—1222no—
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1222no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—2444no—
298C2×C22⋊C4C2×Dic6112yescomplemented
299#99—2444no—
300#99—2448no—
301#95—22no—
302#95—1224no—
303C24—21no—
304C24—24no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—81no—
314C24—81no—
315C24—81no—
316C24C22⋊Dic6112yes—
317C24D4×D6112yes—
318C24#1175822412yes—
319C24C22×Dic6111yescharacteristic
320C24C2×C22⋊D6111yescharacteristic
321C24C2×C22⋊D6111yescharacteristic
322C2×D4—28no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—82no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—1224no—
353C22×C4—4882no—
354C22×C4—4882no—
355C22×C4—2442no—
356C22×C4—2444no—
357C22×C4C22⋊Dic6112yes—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—81no—
363C22⋊C4—4882no—
364C22⋊C4—4882no—
365C22⋊C4—2444no—
366C22⋊C4—2444no—
367C22⋊C4—2444no—
368C22⋊C4—2444no—
369C22⋊C4—2444no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—88no—
406C23—88no—
407C23—88no—
408C23#1451969214yescomplemented
409C23C2×C22⋊Dic6111yescharacteristic
410C23D4×Dic6112yes—
411C23#1451974511yescharacteristic
412C23#1451974912yes—
413C23#1451975811yescharacteristic
414D4—48no—
415D4—48no—
416D4—48no—
417D4—44no—
418D4—44no—
419D4—88no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—4882no—
434C2×C4—4882no—
435C2×C4—4882no—
436C2×C4—4888no—
437C2×C4—2444no—
438C2×C4—2444no—
439C2×C4—2444no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—21no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22—82no—
465C22—84no—
466C22—82no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—84no—
471C22—81no—
472C22—81no—
473C22#1683964211yescharacteristic
474C22C2×#1451969212yes—
475C22#1683994812yes—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—9762no—
480C4—4884no—
481C2—22no—
482C2—22no—
483C2—42no—
484C2—42no—
485C2—41no—
486C2—44no—
487C2—82no—
488C2#2144622811yescharacteristic
489C2#2144627311yescharacteristic
490C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC24×C61C22×C4
FrattiniC25C22×D61
Fitting, prime radicalC61×#2403C2
socleC22×C61#386
cosocleC25×C61C23
hypercenter#2403D61
hypocenterC61#9495
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #2403

Complements

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

NH ≅ G/Ntype
1. #21448397C2semidirect
2. #21445734C2semidirect
3. #21445734C2semidirect
4. #21443723C2semidirect
16. C2×#120×C61C4semidirect
19. C61×#386C4semidirect
29. C2×C22⋊Dic61D4semidirect
33. C25×C61C2×C4semidirect
47. C23×C4×C61C2×C4semidirect
55. C2×C22⋊C4×C61C2×C4semidirect
162. C23×C61#99semidirect
254. C2×#120Dic61semidirect
261. #386Dic61semidirect
266. C61#9495semidirect
267. C25C2×Dic61semidirect
282. C23×C4C2×Dic61semidirect
298. C2×C22⋊C4C2×Dic61semidirect
408. C23#14519692semidirect

Maximal subgroups

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

subgroupindex
#214483972
#214457342
#214437232
C61×#24032
#949561