Finite Group Databasebeta

← #29577943

Subgroups — #29577943

Order 15616.

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

normal subgroups only

This group has 928 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#21448401C211yesmaximal, characteristic
2#21445893C212yesmaximal, complemented
3#21445734C211yesmaximal, characteristic
4#21443723C212yesmaximal, complemented
5C61×#2405C211yesmaximal, characteristic
6C2×#14519736—21no—
7C2×#14519692—22no—
8#16839665—24no—
9#16839618—21no—
10#16839618C2212yes—
11#16839618C2212yes—
12#16839529—24no—
13C23×D4×C61C2211yescharacteristic
14C2×#123×C61C412yescomplemented
15C22×C22⋊C4×C61C2211yescharacteristic
16C61×#394—22no—
17C61×#386C412yescomplemented
18C61×#385—21no—
19C61×#379—21no—
20C61×#379C2211yescharacteristic
21#14519736—44no—
22C2×C4⋊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—22no—
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61D414yes—
31C2×C4×Dic61—22no—
32#14519692—44no—
33C25×C61D411yescharacteristic
34C25×C61C2311yescharacteristic
35C22×D4×C61—22no—
36C22×D4×C61—22no—
37C22×D4×C61—44no—
38C22×D4×C61C2×C412yes—
39C22×D4×C61D412yes—
40C23×C4×C61C2×C411yescharacteristic
41C23×C4×C61C2×C411yescharacteristic
42C23×C4×C61D411yescharacteristic
43#123×C61—44no—
44C2×C4⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61C2×C412yes—
54#95×C61—22no—
55C24×C61—21no—
56C24×C61—21no—
57C24×C61—42no—
58C24×C61—42no—
59C24×C61—42no—
60C24×C61—42no—
61C24×C61—81no—
62C24×C61—81no—
63C24×C61C22⋊C412yes—
64C24×C61C22×C411yescharacteristic
65C24×C61C2×D411yescharacteristic
66C24×C61C2×D411yescharacteristic
67C24×C61C2×D412yes—
68C24×C61#4112yes—
69C2×D4×C61—24no—
70C2×D4×C61—24no—
71C2×D4×C61—24no—
72C2×D4×C61—24no—
73C2×D4×C61—42no—
74C2×D4×C61—42no—
75C2×D4×C61—44no—
76C2×D4×C61—44no—
77C2×D4×C61—48no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—24no—
81C22×C4×C61—21no—
82C22×C4×C61—22no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—42no—
87C22×C4×C61—81no—
88C22×C4×C61C22⋊C412yes—
89C22×Dic61—24no—
90C22×Dic61—42no—
91C4⋊C4×C61—44no—
92C22⋊C4×C61—42no—
93C22⋊C4×C61—44no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—81no—
99C22⋊C4×C61—82no—
100C22⋊Dic61—44no—
101C22⋊Dic61—44no—
102C22⋊Dic61—44no—
103C22⋊Dic61—44no—
104C22⋊Dic61—82no—
105C22⋊Dic61—82no—
106C22⋊Dic61—82no—
107C22⋊Dic61—82no—
108C4⋊Dic61—44no—
109C4×Dic61—44no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—22no—
113C23×C61—22no—
114C23×C61—24no—
115C23×C61—22no—
116C23×C61—22no—
117C23×C61—22no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—41no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—41no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—81no—
135C23×C61—81no—
136C23×C61—88no—
137C23×C61—88no—
138C23×C61C2×C22⋊C411yescharacteristic
139C23×C61C4×D412yes—
140C23×C61#12011yescharacteristic
141C23×C61#12112yes—
142C23×C61#12311yescharacteristic
143D4×C61—48no—
144D4×C61—48no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—24no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61#9914yes—
160C2×Dic61—44no—
161C2×Dic61—44no—
162C2×Dic61—44no—
163C2×Dic61—82no—
164C2×Dic61—82no—
165C2×Dic61—82no—
166#9501—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—22no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—22no—
175C22×C61—21no—
176C22×C61—22no—
177C22×C61—22no—
178C22×C61—22no—
179C22×C61—21no—
180C22×C61—44no—
181C22×C61—44no—
182C22×C61—44no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61—84no—
187C22×C61—82no—
188C22×C61—82no—
189C22×C61—84no—
190C22×C61—84no—
191C22×C61—84no—
192C22×C61#38611yescharacteristic
193C22×C61C2×#9912yes—
194C22×C61#45812yes—
195C4×C61—24no—
196C4×C61—82no—
197C4×C61—81no—
198C4×C61—81no—
199Dic61—84no—
200Dic61—162no—
201#2405D6111yescharacteristic
202#2309—611no—
203#1819—612no—
204#1781—611no—
205#1304—612no—
206C2×C61—22no—
207C2×C61—22no—
208C2×C61—42no—
209C2×C61—42no—
210C2×C61—41no—
211C2×C61—82no—
212C2×C61#189011yescharacteristic
213C2×C61#189811yescharacteristic
214C2×C61#191311yescharacteristic
215C23×D4C2×D6111yescharacteristic
216C2×#123Dic6112yescomplemented
217C2×#117—1221no—
218C22×C22⋊C4C2×D6111yescharacteristic
219C2×#99—1222no—
220#394—22no—
221#390—1224no—
222#386Dic6112yescomplemented
223#385—21no—
224#379—21no—
225#379—612no—
226#379—612no—
227#379—1221no—
228#379C2×D6111yescharacteristic
229#342—1224no—
230C61#950111yescharacteristic, complemented, Sylow
231C25C22⋊D6111yescharacteristic
232C25C22×D6111yescharacteristic
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—44no—
236C22×D4C2×Dic6112yes—
237C22×D4C22⋊D6112yes—
238C23×C4C2×Dic6111yescharacteristic
239C23×C4C2×Dic6111yescharacteristic
240C23×C4C22⋊D6111yescharacteristic
241#123—44no—
242#117—2444no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—1222no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—614no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1222no—
256C2×C22⋊C4—1222no—
257C2×C22⋊C4—1222no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4C2×Dic6112yes—
262C2×C42—1222no—
263#99—2444no—
264#95—22no—
265C24—21no—
266C24—21no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—81no—
272C24—81no—
273C24C22⋊Dic6112yes—
274C24D4×D6112yes—
275C24#1175822412yes—
276C24C22×Dic6111yescharacteristic
277C24C2×C22⋊D6111yescharacteristic
278C24C2×C22⋊D6111yescharacteristic
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—48no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—1224no—
299C22×C4—2442no—
300C22×C4C22⋊Dic6112yes—
301C4⋊C4—44no—
302C4⋊C4—2444no—
303C22⋊C4—44no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4—81no—
310C22⋊C4—82no—
311C22⋊C4—4882no—
312C22⋊C4—4882no—
313C22⋊C4—4882no—
314C22⋊C4—4882no—
315C22⋊C4—2444no—
316C22⋊C4—2444no—
317C22⋊C4—2444no—
318C22⋊C4—2444no—
319C42—2444no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—24no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—88no—
347C23—88no—
348C23C2×C22⋊Dic6111yescharacteristic
349C23D4×Dic6112yes—
350C23#1451974511yescharacteristic
351C23#1451974912yes—
352C23#1451975811yescharacteristic
353D4—48no—
354D4—48no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—4882no—
370C2×C4—4882no—
371C2×C4—4882no—
372C2×C4—2444no—
373C2×C4—2444no—
374C2×C4—2444no—
375C2×C4#1451969214yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—82no—
396C22—84no—
397C22—82no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22#1683964211yescharacteristic
402C22C2×#1451969212yes—
403C22#1683995112yes—
404C4—24no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—9762no—
409C4—4884no—
410C2—22no—
411C2—22no—
412C2—42no—
413C2—41no—
414C2—42no—
415C2—82no—
416C2#2144623311yescharacteristic
417C2#2144627311yescharacteristic
418C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#2405C2
socleC22×C61#386
prime radicalC23×D4×C61C22
cosocleC25×C61C23
hypercenter#2405D61
hypocenterC61#9501
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #2405

Complements

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

NH ≅ G/Ntype
2. #21445893C2semidirect
4. #21443723C2semidirect
14. C2×#123×C61C4semidirect
17. C61×#386C4semidirect
216. C2×#123Dic61semidirect
222. #386Dic61semidirect
230. C61#9501semidirect

Maximal subgroups

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

subgroupindex
#214484012
#214458932
#214457342
#214437232
C61×#24052
#950161