Finite Group Databasebeta

← #30602898

Subgroups — #30602898

Order 15616.

762 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 170 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#21462385C211yesmaximal, characteristic
2#21461519C211yesmaximal, characteristic
3#21461498C212yesmaximal
4C2×#16840633C212yesmaximal
5#21448224C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8#16840727—24no—
9C4×#11758217—22no—
10#16840640—22no—
11#16840633C418yescomplemented
12#16840632—24no—
13C2×#14519735C2211yescharacteristic
14C2×#14519735C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839632C411yescharacteristic, complemented
17#16839627C411yescharacteristic, complemented
18#16839627C411yescharacteristic, complemented
19#16839619—22no—
20C4⋊C4×Dic61—22no—
21C61×#384C411yescharacteristic, complemented
22C2×#11758213D414yes—
23D61⋊M16—22no—
24D61⋊M16—22no—
25D61⋊M16C2×C414yes—
26D61⋊M16Q814yes—
27C2×#11758217—22no—
28C2×D61⋊C8—22no—
29#14519856—22no—
30#14519856—22no—
31#14519856—22no—
32#14519856C2×C414yes—
33C2×C4⋊#6923—22no—
34C2×C22⋊#6923—22no—
35C2×C4×#6923—22no—
36#14519832—24no—
37C4×#92420—24no—
38C22×C4×D61—21no—
39C22×C4×D61C2311yescharacteristic
40#14519735—22no—
41#14519735—22no—
42#14519735—22no—
43#14519735—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×#11758198—21no—
48C2×#11758198D412yes—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—21no—
51C2×C4×Dic61C2×C411yescharacteristic
52C2×C4×Dic61D412yes—
53C42×D61—22no—
54C42×D61C2×C414yes—
55#14519684—22no—
56#14519684—22no—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61C23×D61C2×D411yescharacteristic
62C22×#6923—42no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C412yes—
70C22×Dic61C2×D411yescharacteristic
71C2×C4×D61—24no—
72C2×C4×D61—42no—
73C2×C4×D61—42no—
74C2×C4×D61C22×C412yes—
75C2×C4×D61C2×D412yes—
76C2×C4×D61C2×Q812yes—
77C2×C4×D61#4112yes—
78C2×C4×D61#4112yes—
79C2×C4×D61#4112yes—
80C22⋊#6923—24no—
81C22⋊#6923—42no—
82#11758217—24no—
83#11758217—24no—
84C2×#92420—24no—
85C4⋊#6923—24no—
86C4⋊#6923—42no—
87C4×#6923—24no—
88C4×#6923—42no—
89#11758213—24no—
90#11758213C4⋊C418yescomplemented
91D61⋊C8—24no—
92D61⋊C8—24no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—44no—
96C42×C61—21no—
97C42×C61C4212yes—
98#11758198—24no—
99#11758198—42no—
100C4⋊Dic61—22no—
101C4⋊Dic61—22no—
102C4⋊Dic61—44no—
103C4×Dic61—21no—
104C4×Dic61—24no—
105C4×Dic61—22no—
106C4×Dic61—22no—
107C4×Dic61—41no—
108C4×Dic61—41no—
109C4×Dic61C4212yes—
110C23×C61C2×C22⋊C411yescharacteristic
111C22×D61—41no—
112C22×D61—41no—
113C22×D61C4×D412yes—
114C22×D61#12112yes—
115C22×D61#12411yescharacteristic
116C22×D61C4⋊Q811yescharacteristic
117C2×#6923—44no—
118C2×#6923—82no—
119C2×#6923—82no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61C2×C4211yescharacteristic
128C2×C4×C61#11711yescharacteristic
129C2×C4×C61C4×D412yes—
130C2×C4×C61C4×Q812yes—
131C2×Dic61—22no—
132C2×Dic61—22no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—41no—
138C2×Dic61C4×D412yes—
139C2×Dic61#12011yescharacteristic
140C2×Dic61C22⋊Q812yes—
141C2×Dic61#12311yescharacteristic
142C4×D61—48no—
143C4×D61C2×C4⋊C412yes—
144C4×D61#11712yes—
145C4×D61C4×D412yes—
146C4×D61C4×Q812yes—
147C4×D61C22⋊Q812yes—
148C4×D61#12312yes—
149C4×D61#12512yes—
150C4×D61#12612yes—
151#92420—28no—
152#10893—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61C4×C22⋊C411yescharacteristic
156C22×C61#38611yescharacteristic
157C22×C61#39111yescharacteristic
158C2×D61—22no—
159C2×D61—22no—
160C2×D61—42no—
161C2×D61—41no—
162C2×D61—41no—
163C2×D61#38812yes—
164C2×D61#38912yes—
165#6923—84no—
166C4×C61—24no—
167C4×C61—42no—
168C4×C61—81no—
169C4×C61C4×C4⋊C412yes—
170C4×C61#38712yes—
171Dic61—42no—
172Dic61—42no—
173Dic61—81no—
174Dic61#38512yes—
175Dic61#38712yes—
176#2873—611no—
177#2285—611no—
178#1740—611no—
179#1733—612no—
180C2×#339—612no—
181C2×C61—21no—
182C2×C61—21no—
183C2×C61#143611yescharacteristic
184C2×C61#188411yescharacteristic
185C2×C61#188411yescharacteristic
186D61—22no—
187D61—22no—
188C22×M16—612no—
189C2×#117—611no—
190C2×#117—611no—
191C2×#117—612no—
192C22×C42—611no—
193#405—1224no—
194C4×M16—1222no—
195#384—611no—
196#384#692311yescharacteristic, complemented
197#382—611no—
198#382—611no—
199#380—1222no—
200C4×C4⋊C4—1222no—
201#342—1222no—
202#339—618no—
203#339—1224no—
204C61#1089311yescharacteristic, complemented, Sylow
205C23×C4—611no—
206C23×C4—1221no—
207C2×M16—614no—
208C2×M16—614no—
209C2×M16—614no—
210C2×M16—1222no—
211C2×M16—1222no—
212C2×M16—1222no—
213C22×C8—1222no—
214#117—614no—
215#117—1222no—
216#117—1222no—
217#117—1222no—
218#117—1222no—
219#117—1222no—
220#117—1222no—
221#117—1222no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1222no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4C2×#692311yescharacteristic
228C2×C4⋊C4C2×#692311yescharacteristic
229C2×C22⋊C4—612no—
230C2×C22⋊C4—1222no—
231C2×C22⋊C4—1221no—
232C2×C42—611no—
233C2×C42—614no—
234C2×C42—612no—
235C2×C42—1222no—
236C2×C42—1221no—
237C2×C42—1222no—
238C2×C42—1221no—
239C2×C42C2×#692311yescharacteristic
240#97—1224no—
241C4×C8—1224no—
242#95—1222no—
243#95—1222no—
244C24—611no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—611no—
248C22×C4—612no—
249C22×C4—612no—
250C22×C4—612no—
251C22×C4—612no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—1224no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—2442no—
259C22×C4—2442no—
260C22×C4—2442no—
261C22×C4C22⋊#692312yes—
262C22×C4C22×#692311yescharacteristic
263M16—618no—
264M16—1224no—
265M16—1224no—
266M16—1224no—
267C2×C8—1224no—
268C2×C8—1224no—
269C2×C8—1224no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—44no—
273C4⋊C4—1224no—
274C4⋊C4—1222no—
275C4⋊C4—1222no—
276C4⋊C4—2444no—
277C4⋊C4—2442no—
278C22⋊C4—1224no—
279C22⋊C4—1224no—
280C22⋊C4—2442no—
281C22⋊C4—2442no—
282C42—21no—
283C42—612no—
284C42—1222no—
285C42—1224no—
286C42—1222no—
287C42—1224no—
288C42—1221no—
289C42—2441no—
290C42—2441no—
291C42—2442no—
292C42C4×#692312yes—
293C23—612no—
294C23—611no—
295C23—611no—
296C23—612no—
297C23—2441no—
298C23—2441no—
299C23C2×C22⋊#692311yescharacteristic
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—612no—
310C2×C4—612no—
311C2×C4—612no—
312C2×C4—612no—
313C2×C4—612no—
314C2×C4—612no—
315C2×C4—612no—
316C2×C4—612no—
317C2×C4—612no—
318C2×C4—612no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—4882no—
322C2×C4—4882no—
323C2×C4—2448no—
324C2×C4—2442no—
325C2×C4—2444no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2442no—
330C2×C4C2×C4×#692311yescharacteristic
331C2×C4#1451985611yescharacteristic
332C2×C4D4×#692312yes—
333C2×C4Q8×#692312yes—
334C8—1228no—
335C22—21no—
336C22—21no—
337C22—612no—
338C22—612no—
339C22—1222no—
340C22—1222no—
341C22—2442no—
342C22—2441no—
343C22—2441no—
344C22C4×C22⋊#692311yescharacteristic
345C22#1684071011yescharacteristic
346C22(C2×#6923)⋊Q811yescharacteristic
347C4—24no—
348C4—42no—
349C4—81no—
350C4—612no—
351C4—612no—
352C4—4884no—
353C4—4881no—
354C4—2442no—
355C4—2442no—
356C4C4⋊C4×#692312yes—
357C4#1684071112yes—
358C2—21no—
359C2—21no—
360C2—1222no—
361C2—1222no—
362C2#2146099811yescharacteristic
363C2#2146182111yescharacteristic
364C2#2146182211yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#10893
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
11. #16840633C4semidirect
16. #16839632C4semidirect
17. #16839627C4semidirect
18. #16839627C4semidirect
21. C61×#384C4semidirect
90. #11758213C4⋊C4semidirect
196. #384#6923semidirect
204. C61#10893semidirect

Maximal subgroups

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

subgroupindex
#214623852
#214615192
#214614982
C2×#168406332
#214482242
#1089361