Finite Group Databasebeta

← #29577948

Subgroups — #29577948

Order 15616.

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

normal subgroups only

This group has 660 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
2#21445893C212yesmaximal
3#21445893C211yesmaximal, characteristic
4#21443723C212yesmaximal
5C61×#2483C211yesmaximal, characteristic
6#16839655—24no—
7#16839622—21no—
8#16839618C2212yes—
9#16839618C2212yes—
10C4×C22⋊Dic61—21no—
11#16839529—24no—
12#16839529—22no—
13C22×C22⋊C4×C61C2211yescharacteristic
14C22×C22⋊C4×C61C2211yescharacteristic
15C61×#397—22no—
16C61×#385—21no—
17C61×#385C412yescomplemented
18C61×#381—21no—
19C61×#381C412yescomplemented
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#14519684—22no—
30#14519684—22no—
31C25×C61C2311yescharacteristic
32C23×C4×C61C2×C411yescharacteristic
33C23×C4×C61C2×C411yescharacteristic
34C23×C4×C61D411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C2×C4⋊C4×C61—22no—
37C2×C22⋊C4×C61—22no—
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61C2×C412yes—
46C2×C22⋊C4×C61C2×C412yes—
47C2×C22⋊C4×C61D412yes—
48#95×C61—22no—
49#95×C61—22no—
50#95×C61—22no—
51C24×C61—42no—
52C24×C61—42no—
53C24×C61—81no—
54C24×C61C22×C411yescharacteristic
55C24×C61C2×D411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C24×C61C2×D412yes—
58C24×C61#4112yes—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61—81no—
71C22×C4×C61—81no—
72C22×C4×C61C22⋊C412yes—
73C22×C4×C61C22⋊C412yes—
74C22×Dic61—24no—
75C22×Dic61—42no—
76C4⋊C4×C61—82no—
77C22⋊C4×C61—24no—
78C22⋊C4×C61—24no—
79C22⋊C4×C61—42no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—81no—
85C22⋊Dic61—44no—
86C22⋊Dic61—44no—
87C22⋊Dic61—44no—
88C22⋊Dic61—44no—
89C22⋊Dic61—82no—
90C22⋊Dic61—82no—
91C4⋊Dic61—82no—
92C4×Dic61—82no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—22no—
97C23×C61—24no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—88no—
109C23×C61C2×C22⋊C411yescharacteristic
110C23×C61C4×D412yes—
111C23×C61#12011yescharacteristic
112C23×C61#12112yes—
113C23×C61#12311yescharacteristic
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—81no—
128C2×C4×C61—81no—
129C2×C4×C61—82no—
130C2×C4×C61—82no—
131C2×Dic61—44no—
132C2×Dic61—44no—
133C2×Dic61—44no—
134C2×Dic61—82no—
135C2×Dic61—82no—
136C2×Dic61—82no—
137#9502—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—21no—
148C22×C61—22no—
149C22×C61—22no—
150C22×C61—21no—
151C22×C61—44no—
152C22×C61—44no—
153C22×C61—84no—
154C22×C61—82no—
155C22×C61—84no—
156C22×C61#38611yescharacteristic
157C22×C61#41012yes—
158C22×C61#45812yes—
159C4×C61—42no—
160C4×C61—42no—
161C4×C61—82no—
162C4×C61—81no—
163C4×C61—81no—
164Dic61—84no—
165Dic61—162no—
166#2483D6111yescharacteristic
167#2312—611no—
168#1819—612no—
169#1819—611no—
170#1304—612no—
171C2×C61—22no—
172C2×C61—22no—
173C2×C61—42no—
174C2×C61—41no—
175C2×C61—42no—
176C2×C61#189011yescharacteristic
177C2×C61#189911yescharacteristic
178C2×C61#191311yescharacteristic
179C22×C22⋊C4C2×D6111yescharacteristic
180C22×C22⋊C4C2×D6111yescharacteristic
181#397—22no—
182#388—1224no—
183#385—21no—
184#385Dic6112yescomplemented
185#381—21no—
186#381Dic6112yescomplemented
187#380—1221no—
188#379—612no—
189#379—612no—
190#379C2×D6111yescharacteristic
191C4×C22⋊C4—1221no—
192#342—1224no—
193#342—1222no—
194C61#950211yescharacteristic, complemented, Sylow
195C25C22×D6111yescharacteristic
196C23×C4C2×Dic6111yescharacteristic
197C23×C4C2×Dic6111yescharacteristic
198C23×C4C22⋊D6111yescharacteristic
199C23×C4C22⋊D6111yescharacteristic
200C2×C4⋊C4—22no—
201C2×C4⋊C4—1222no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—614no—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1222no—
216C2×C22⋊C4C2×Dic6112yes—
217C2×C22⋊C4C2×Dic6112yes—
218C2×C22⋊C4C22⋊D6112yes—
219C2×C42—1222no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—1222no—
224#95—1222no—
225C24—42no—
226C24—42no—
227C24—81no—
228C24D4×D6112yes—
229C24#1175822412yes—
230C24C22×Dic6111yescharacteristic
231C24C2×C22⋊D6111yescharacteristic
232C24C2×C22⋊D6111yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—1224no—
247C22×C4—2442no—
248C22×C4C22⋊Dic6112yes—
249C22×C4C22⋊Dic6112yes—
250C4⋊C4—82no—
251C4⋊C4—4882no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—4882no—
261C22⋊C4—4882no—
262C22⋊C4—2444no—
263C22⋊C4—2444no—
264C22⋊C4—2444no—
265C22⋊C4—2444no—
266C42—4882no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—41no—
282C23—88no—
283C23C2×C22⋊Dic6111yescharacteristic
284C23D4×Dic6112yes—
285C23#1451974511yescharacteristic
286C23#1451974912yes—
287C23#1451975811yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—4882no—
308C2×C4—2444no—
309C2×C4—2444no—
310C2×C4—2444no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—22no—
324C22—44no—
325C22—44no—
326C22—82no—
327C22—84no—
328C22—84no—
329C22#1683964211yescharacteristic
330C22#1683973612yes—
331C22#1683995112yes—
332C4—42no—
333C4—42no—
334C4—82no—
335C4—81no—
336C4—81no—
337C4—9762no—
338C4—4884no—
339C2—22no—
340C2—22no—
341C2—42no—
342C2—41no—
343C2—42no—
344C2#2144623311yescharacteristic
345C2#2144627811yescharacteristic
346C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC24×C61C22×C4
FrattiniC25C22×D61
FittingC61×#2483C2
socleC22×C61#386
prime radical, cosocleC25×C61C23
hypercenter#2483D61
hypocenterC61#9502
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #2483

Complements

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

NH ≅ G/Ntype
17. C61×#385C4semidirect
19. C61×#381C4semidirect
184. #385Dic61semidirect
186. #381Dic61semidirect
194. C61#9502semidirect

Maximal subgroups

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

subgroupindex
#214484172
#214458932
#214437232
C61×#24832
#950261