Finite Group Databasebeta

← #29577928

Subgroups — #29577928

Order 15616.

900 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 107 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#21448417C211yesmaximal, characteristic, complemented
2#21445893C211yesmaximal, characteristic, complemented
3#21445734C212yesmaximal
4#21443723C212yesmaximal, complemented
5C61×#2405C211yesmaximal, characteristic
6C2×#14519692—24no—
7#16839655—24no—
8#16839622—21no—
9#16839618C2212yes—
10#16839618C2212yes—
11C4×C22⋊Dic61—21no—
12#16839529—22no—
13C23×D4×C61C2211yescharacteristic
14C2×#123×C61—21no—
15C22×C22⋊C4×C61C2211yescharacteristic
16C61×#394—22no—
17C61×#386—21no—
18C61×#385C412yescomplemented
19C61×#379C412yescomplemented
20C61×#379C2211yescharacteristic
21C2×C4⋊Dic61—22no—
22C2×C22⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61D414yes—
28C2×C4×Dic61—22no—
29#14519692—48no—
30#14519684—22no—
31#14519684—22no—
32C25×C61C2×C411yescharacteristic
33C25×C61C2311yescharacteristic
34C22×D4×C61—22no—
35C22×D4×C61—21no—
36C22×D4×C61—21no—
37C22×D4×C61—22no—
38C22×D4×C61—44no—
39C23×C4×C61C2×C411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42#123×C61—44no—
43C2×C4⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61—22no—
50C2×C22⋊C4×C61C2×C412yes—
51C2×C22⋊C4×C61C2×C412yes—
52C2×C22⋊C4×C61D412yes—
53#95×C61—22no—
54C24×C61—21no—
55C24×C61—22no—
56C24×C61—42no—
57C24×C61—42no—
58C24×C61—42no—
59C24×C61—42no—
60C24×C61—81no—
61C24×C61—81no—
62C24×C61C22×C411yescharacteristic
63C24×C61C2×D411yescharacteristic
64C24×C61C2×D411yescharacteristic
65C24×C61C2×D412yes—
66C24×C61#4112yes—
67C2×D4×C61—24no—
68C2×D4×C61—28no—
69C2×D4×C61—42no—
70C2×D4×C61—44no—
71C2×D4×C61—42no—
72C2×D4×C61—44no—
73C2×D4×C61—42no—
74C2×D4×C61—44no—
75C2×D4×C61—41no—
76C2×D4×C61—42no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—42no—
86C22×C4×C61—42no—
87C22×C4×C61—81no—
88C22×C4×C61C22⋊C412yes—
89C22×C4×C61C22⋊C412yes—
90C22×Dic61—24no—
91C22×Dic61—42no—
92C4⋊C4×C61—44no—
93C22⋊C4×C61—24no—
94C22⋊C4×C61—24no—
95C22⋊C4×C61—44no—
96C22⋊C4×C61—42no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—82no—
99C22⋊C4×C61—81no—
100C22⋊Dic61—44no—
101C22⋊Dic61—44no—
102C22⋊Dic61—44no—
103C22⋊Dic61—44no—
104C22⋊Dic61—82no—
105C22⋊Dic61—82no—
106C4⋊Dic61—82no—
107C4×Dic61—82no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—24no—
113C23×C61—22no—
114C23×C61—24no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—41no—
127C23×C61—42no—
128C23×C61—41no—
129C23×C61—42no—
130C23×C61—82no—
131C23×C61—82no—
132C23×C61—88no—
133C23×C61—88no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61C4×D412yes—
136C23×C61#12011yescharacteristic
137C23×C61#12112yes—
138C23×C61#12311yescharacteristic
139D4×C61—48no—
140D4×C61—44no—
141D4×C61—44no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×Dic61—44no—
159C2×Dic61—44no—
160C2×Dic61—44no—
161C2×Dic61—82no—
162C2×Dic61—82no—
163C2×Dic61—82no—
164#9498—611nomaximal, Sylow
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—22no—
170C22×C61—22no—
171C22×C61—21no—
172C22×C61—22no—
173C22×C61—22no—
174C22×C61—21no—
175C22×C61—22no—
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—44no—
179C22×C61—44no—
180C22×C61—44no—
181C22×C61—42no—
182C22×C61—84no—
183C22×C61—82no—
184C22×C61—82no—
185C22×C61—84no—
186C22×C61—84no—
187C22×C61—84no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#38611yescharacteristic
191C22×C61#41012yes—
192C22×C61B(4,𝔽2)12yes—
193C4×C61—42no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61—82no—
197Dic61—84no—
198Dic61—162no—
199#2405D6111yescharacteristic
200#2312—611no—
201#1819—611no—
202#1781—612no—
203#1304—612no—
204C2×C61—22no—
205C2×C61—22no—
206C2×C61—42no—
207C2×C61—41no—
208C2×C61—42no—
209C2×C61—82no—
210C2×C61#188911yescharacteristic
211C2×C61#189911yescharacteristic
212C2×C61#191311yescharacteristic
213C23×D4C2×D6111yescharacteristic
214C2×#123—21no—
215C22×C22⋊C4C2×D6111yescharacteristic
216C2×#99—1224no—
217#394—22no—
218#388—1224no—
219#386—21no—
220#385Dic6112yescomplemented
221#380—1221no—
222#379—612no—
223#379—612no—
224#379Dic6112yescomplemented
225#379C2×D6111yescharacteristic
226C4×C22⋊C4—1221no—
227#342—1222no—
228C61#949811yescharacteristic, complemented, Sylow
229C25C2×Dic6111yescharacteristic
230C25C22×D6111yescharacteristic
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—44no—
236C23×C4C2×Dic6111yescharacteristic
237C23×C4C22⋊D6111yescharacteristic
238C23×C4C22⋊D6111yescharacteristic
239#123—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—1222no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—614no—
249C2×C22⋊C4—1222no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1222no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4C2×Dic6112yes—
255C2×C22⋊C4C2×Dic6112yes—
256C2×C22⋊C4C22⋊D6112yes—
257C2×C42—1222no—
258#99—2448no—
259#95—22no—
260#95—1222no—
261#95—1222no—
262C24—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—81no—
269C24—81no—
270C24D4×D6112yes—
271C24#1175822412yes—
272C24C22×Dic6111yescharacteristic
273C24C2×C22⋊D6111yescharacteristic
274C24C2×C22⋊D6111yescharacteristic
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4—1224no—
297C22×C4—2442no—
298C22×C4C22⋊Dic6112yes—
299C22×C4C22⋊Dic6112yes—
300C4⋊C4—44no—
301C4⋊C4—4882no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—82no—
308C22⋊C4—81no—
309C22⋊C4—4882no—
310C22⋊C4—4882no—
311C22⋊C4—2444no—
312C22⋊C4—2444no—
313C22⋊C4—2444no—
314C22⋊C4—2444no—
315C42—4882no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—82no—
339C23—82no—
340C23—88no—
341C23—88no—
342C23C2×C22⋊Dic6111yescharacteristic
343C23D4×Dic6112yes—
344C23#1451974511yescharacteristic
345C23#1451974912yes—
346C23#1451975811yescharacteristic
347D4—44no—
348D4—48no—
349D4—44no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—4882no—
367C2×C4—4882no—
368C2×C4—4882no—
369C2×C4—2444no—
370C2×C4—2444no—
371C2×C4—2444no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—82no—
390C22—84no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—81no—
396C22—81no—
397C22#1683964211yescharacteristic
398C22#1683973612yes—
399C22#1683994812yes—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—82no—
404C4—9762no—
405C4—4884no—
406C2—22no—
407C2—22no—
408C2—42no—
409C2—42no—
410C2—41no—
411C2—82no—
412C2#2144622811yescharacteristic
413C2#2144627811yescharacteristic
414C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#2405C2
socleC22×C61#386
prime radicalC23×D4×C61C22
cosocleC25×C61C23
hypercenter#2405D61
hypocenterC61#9498
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #2405

Complements

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

NH ≅ G/Ntype
1. #21448417C2semidirect
2. #21445893C2semidirect
4. #21443723C2semidirect
18. C61×#385C4semidirect
19. C61×#379C4semidirect
220. #385Dic61semidirect
224. #379Dic61semidirect
228. C61#9498semidirect

Maximal subgroups

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

subgroupindex
#214484172
#214458932
#214457342
#214437232
C61×#24052
#949861