Finite Group Databasebeta

← #30595402

Subgroups — #30595402

Order 15616.

500 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461747C211yesmaximal, characteristic, complemented
2#21461534C211yesmaximal, characteristic, complemented
3#21460638C212yesmaximal, complemented
4#21460635C212yesmaximal
5#21448281C211yesmaximal, characteristic
6C2×C4×#92420—21no—
7#16840628—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×D4×Dic61C411yescharacteristic
13C2×#14519733—21no—
14#16839665C2211yescharacteristic
15#16839660C411yescharacteristic
16#16839658C2211yescharacteristic
17#16839656—21no—
18#16839650—21no—
19C4⋊C4×Dic61C411yescharacteristic
20C22⋊C4×Dic61—21no—
21C42×Dic61C2211yescharacteristic
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C4×#92420—24no—
26C23×Dic61—21no—
27D4×Dic61—22no—
28D4×Dic61C814yescomplemented
29#14519733—44no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61C2311yescharacteristic
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47C22×D4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—21no—
53C2×D4×C61—21no—
54C2×D4×C61—41no—
55C2×D4×C61—41no—
56C2×D4×C61C2×C812yes—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61—21no—
62C22×Dic61—24no—
63C22×Dic61—21no—
64C22×Dic61—42no—
65C22×Dic61—42no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×#92420—24no—
71C2×#92420—42no—
72C2×#92420—42no—
73C2×#92420—42no—
74C2×#92420—42no—
75C2×#92420—42no—
76C4⋊C4×C61—41no—
77C22⋊C4×C61—42no—
78C22⋊C4×C61—42no—
79C42×C61—41no—
80C22⋊Dic61—24no—
81C22⋊Dic61—42no—
82C22⋊Dic61—42no—
83C4⋊Dic61—21no—
84C4⋊Dic61—41no—
85C4⋊Dic61C2×C812yes—
86#11758196—42no—
87#11758196—42no—
88C4×Dic61—22no—
89C4×Dic61—22no—
90C4×Dic61—22no—
91C4×Dic61—41no—
92C4×Dic61—41no—
93C4×Dic61—42no—
94C4×Dic61C2×C812yes—
95C4×Dic61M1612yes—
96C23×C61—22no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61C2×C22⋊C411yescharacteristic
104D4×C61—42no—
105D4×C61—42no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#10412yes—
117C2×C4×C61C22×C811yescharacteristic
118C2×C4×C61C2×M1611yescharacteristic
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—22no—
128C2×Dic61—48no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61C22⋊C812yes—
135C2×Dic61C22⋊C812yes—
136C2×Dic61#13512yes—
137C2×Dic61AGL(1,ℤ/8)12yes—
138#92420—44no—
139#92420—82no—
140#7580—611nomaximal, Sylow
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61—81no—
145C22×C61—81no—
146C22×C61—81no—
147C22×C61—81no—
148C22×C61C2×C22⋊C811yescharacteristic
149C22×C61#41011yescharacteristic
150C22×C61#41311yescharacteristic
151C22×C61#41611yescharacteristic
152C22×C61#41811yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C22×C61#42111yescharacteristic
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—81no—
160Dic61—22no—
161Dic61—22no—
162Dic61—22no—
163Dic61—22no—
164Dic61—42no—
165Dic61—81no—
166#2293—611no—
167#1839—611no—
168#1744—611no—
169#1271—612no—
170#1270—612no—
171C2×C61—82no—
172C2×C61#145811yescharacteristic
173C2×C61#147611yescharacteristic
174C2×C61#147811yescharacteristic
175C2×C61#150511yescharacteristic
176C2×C61#152111yescharacteristic
177C2×C61#153211yescharacteristic
178C2×C61#155311yescharacteristic
179C2×C4×D4—611no—
180C2×#117—1221no—
181C2×C4×C8—1221no—
182#390—611no—
183#390#692311yescharacteristic
184#389—611no—
185#389—611no—
186#388—1221no—
187#388—1221no—
188C4×C4⋊C4—611no—
189C4×C22⋊C4—1221no—
190C43—611no—
191#336—612no—
192#336—612no—
193#336—1221no—
194#336—1221no—
195#336—1221no—
196C61#758011yescharacteristic, complemented, Sylow
197C22×D4C2×#692311yescharacteristic
198C23×C4—1221no—
199C22×C8—1222no—
200C22×C8—1222no—
201C4×D4—614no—
202C4×D4—1222no—
203#117—2444no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4C2×#692311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C42—611no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C2×C42—1221no—
221C2×C42C2×#692311yescharacteristic
222C4×C8—1224no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226C24—21no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4D61⋊C812yes—
231C22×C4—21no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—1224no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—2442no—
240C22×C4—2442no—
241C22×C4C22⋊#692311yescharacteristic
242C22×C4C22⋊#692311yescharacteristic
243C22×C4C22×#692311yescharacteristic
244C2×C8—1224no—
245C2×C8—2442no—
246C2×C8—2442no—
247C2×C8—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C4⋊C4—41no—
251C4⋊C4—612no—
252C4⋊C4—1221no—
253C4⋊C4—2441no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—1224no—
259C22⋊C4—2442no—
260C22⋊C4—2442no—
261C42—41no—
262C42—612no—
263C42—612no—
264C42—1222no—
265C42—1222no—
266C42—1222no—
267C42—2441no—
268C42—2441no—
269C42—2442no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#692311yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—612no—
291C2×C4—612no—
292C2×C4—612no—
293C2×C4—612no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1222no—
303C2×C4—2448no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4#1451984112yes—
310C2×C4C2×D61⋊C811yescharacteristic
311C2×C4D61⋊M1611yescharacteristic
312C8—4882no—
313C8—2444no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22D61⋊(C22⋊C8)11yescharacteristic
322C22#1684074211yescharacteristic
323C22#1684074911yescharacteristic
324C22#1684075211yescharacteristic
325C22#1684075411yescharacteristic
326C22C2×#1451984111yescharacteristic
327C22#1684075911yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—81no—
333C4—1222no—
334C4—1222no—
335C4—1222no—
336C4—1222no—
337C4—4881no—
338C4—2442no—
339C2—82no—
340C2#2146105111yescharacteristic
341C2#2146111111yescharacteristic
342C2#2146111911yescharacteristic
343C2#2146120811yescharacteristic
344C2#2146126011yescharacteristic
345C2#2146129511yescharacteristic
346C2#2146134311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7580
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461747C2semidirect
2. #21461534C2semidirect
3. #21460638C2semidirect
28. D4×Dic61C8semidirect
196. C61#7580semidirect

Maximal subgroups

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

subgroupindex
#214617472
#214615342
#214606382
#214606352
#214482812
#758061