Finite Group Databasebeta

← #29577479

Subgroups — #29577479

Order 15616.

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

normal subgroups only

This group has 720 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#21444335C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21443705C212yesmaximal, complemented
5C61×#2539C211yesmaximal, characteristic
6#16839726C2212yes—
7#16839655—24no—
8#16839622—21no—
9#16839618—21no—
10C4×C22⋊Dic61C2212yes—
11#16839496—24no—
12#16839486—22no—
13C2×#123×C61—21no—
14C2×#121×C61C2211yescharacteristic
15C2×#120×C61—21no—
16C22×C22⋊C4×C61C2211yescharacteristic
17C61×#399C412yescomplemented
18C61×#392C412yescomplemented
19C61×#390—22no—
20C4×C22⋊C4×C61C2211yescharacteristic
21C2×#11758194—22no—
22C2×C4⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C4×Dic61—22no—
27C22⋊#92418—22no—
28C22⋊#92418—22no—
29#14519684—22no—
30#14519684D414yescomplemented
31C25×C61D411yescharacteristic
32C22×D4×C61—21no—
33C22×D4×C61—22no—
34C22×D4×C61C2×C411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C23×C4×C61C2311yescharacteristic
37#123×C61—44no—
38#121×C61—22no—
39#121×C61—22no—
40#120×C61—44no—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61C2×C412yes—
50C2×C22⋊C4×C61D412yes—
51C2×C42×C61—21no—
52#95×C61C2×C412yes—
53C24×C61—21no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—42no—
57C24×C61—42no—
58C24×C61—81no—
59C24×C61C22⋊C412yes—
60C24×C61C2×D411yescharacteristic
61C2×D4×C61—22no—
62C2×D4×C61—44no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C2×D4×C61—44no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C2×D4×C61—82no—
69C2×D4×C61—81no—
70C22×C4×C61—22no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—81no—
76C22×C4×C61C22⋊C412yes—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D412yes—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4112yes—
81C22×Dic61—24no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—44no—
84C22⋊C4×C61—24no—
85C22⋊C4×C61—24no—
86C22⋊C4×C61—24no—
87C22⋊C4×C61—44no—
88C22⋊C4×C61—44no—
89C22⋊C4×C61—42no—
90C42×C61—81no—
91C22⋊Dic61—44no—
92C22⋊Dic61—82no—
93C22⋊Dic61—82no—
94C4⋊Dic61—82no—
95C4×Dic61—82no—
96#11758194—82no—
97C2×#92418—42no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—41no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—42no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—41no—
114C23×C61—42no—
115C23×C61—81no—
116C23×C61—81no—
117C23×C61—88no—
118C23×C61C2×C22⋊C411yescharacteristic
119C23×C61#12011yescharacteristic
120C23×C61#12311yescharacteristic
121D4×C61—82no—
122D4×C61—88no—
123D4×C61—82no—
124D4×C61—82no—
125C2×C4×C61—24no—
126C2×C4×C61—22no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—81no—
139C2×C4×C61C4×D412yes—
140C2×C4×C61#12112yes—
141C2×Dic61—44no—
142C2×Dic61—44no—
143C2×Dic61—44no—
144#92418—82no—
145#9413—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—44no—
154C22×C61—44no—
155C22×C61—42no—
156C22×C61—42no—
157C22×C61—42no—
158C22×C61—82no—
159C22×C61—84no—
160C22×C61—82no—
161C22×C61—82no—
162C22×C61—84no—
163C22×C61—81no—
164C22×C61—81no—
165C22×C61—81no—
166C22×C61#38611yescharacteristic
167C22×C61#42112yes—
168C22×C61#45312yes—
169C4×C61—22no—
170C4×C61—22no—
171C4×C61—44no—
172C4×C61—42no—
173C4×C61—81no—
174Dic61—84no—
175#2539D6111yescharacteristic
176#2312—611no—
177#1507—612no—
178#1462—611no—
179#1297—612no—
180C2×C61—22no—
181C2×C61—22no—
182C2×C61—41no—
183C2×C61—82no—
184C2×C61—161no—
185C2×C61#188711yescharacteristic
186C2×C61#189711yescharacteristic
187C2×C61#191311yescharacteristic
188C2×#123—21no—
189C2×#121C2×D6111yescharacteristic
190C2×#120—21no—
191C22×C22⋊C4C2×D6111yescharacteristic
192#407—612no—
193#399Dic6112yescomplemented
194#392Dic6112yescomplemented
195#390—22no—
196#388—1224no—
197#380—1221no—
198#379—1221no—
199C4×C22⋊C4—612no—
200C4×C22⋊C4C2×D6111yescharacteristic
201#327—1224no—
202#323—1222no—
203C61#941311yescharacteristic, complemented, Sylow
204C25C22⋊D6111yescharacteristic
205C22×D4—21no—
206C22×D4—22no—
207C22×D4C2×Dic6111yescharacteristic
208C23×C4C22⋊D6111yescharacteristic
209C23×C4C22×D6111yescharacteristic
210C2×M16—1222no—
211#123—44no—
212#121—22no—
213#121—22no—
214#120—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—1222no—
217C2×C4⋊C4C2×Dic6111yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—1222no—
225C2×C22⋊C4—1222no—
226C2×C22⋊C4—1222no—
227C2×C22⋊C4C2×Dic6112yes—
228C2×C22⋊C4C22⋊D6112yes—
229C2×C42—21no—
230C2×C42—1222no—
231C22⋊C8—1222no—
232C22⋊C8—1222no—
233#95—614no—
234#95—1222no—
235#95C2×Dic6112yes—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—42no—
240C24—42no—
241C24—81no—
242C24C22⋊Dic6112yes—
243C24C2×C22⋊D6111yescharacteristic
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—82no—
252C2×D4—81no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—1224no—
260C22×C4C22⋊Dic6112yes—
261C22×C4D4×D6112yes—
262C22×C4#1175822412yes—
263C22×C4C22×Dic6111yescharacteristic
264C22×C4C2×C22⋊D6111yescharacteristic
265M16—4882no—
266C2×C8—2442no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—4882no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—4882no—
277C22⋊C4—4882no—
278C22⋊C4—2444no—
279C42—81no—
280C42—4882no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—41no—
298C23—81no—
299C23—81no—
300C23—88no—
301C23C2×C22⋊Dic6111yescharacteristic
302C23#1451974511yescharacteristic
303C23#1451975811yescharacteristic
304D4—82no—
305D4—88no—
306D4—82no—
307D4—82no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—2444no—
323C2×C4—2444no—
324C2×C4—2444no—
325C2×C4D4×Dic6112yes—
326C2×C4#1451974912yes—
327C8—4882no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1683964211yescharacteristic
349C22#1683977912yes—
350C22#1683992912yes—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—42no—
355C4—81no—
356C4—4884no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—161no—
362C2#2144621811yescharacteristic
363C2#2144626811yescharacteristic
364C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#2539C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#2539D61
hypocenterC61#9413
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. #21448417C2semidirect
2. #21444335C2semidirect
3. #21444146C2semidirect
4. #21443705C2semidirect
17. C61×#399C4semidirect
18. C61×#392C4semidirect
30. #14519684D4semidirect
193. #399Dic61semidirect
194. #392Dic61semidirect
203. C61#9413semidirect

Maximal subgroups

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

subgroupindex
#214484172
#214443352
#214441462
#214437052
C61×#25392
#941361