Finite Group Databasebeta

← #29589483

Subgroups — #29589483

Order 15616.

444 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C4×#14519766C211yesmaximal, characteristic
2#21444163C211yesmaximal, characteristic, complemented
3#21443762C212yesmaximal, complemented
4#21443750C212yesmaximal
5C61×#1467C211yesmaximal, characteristic, complemented
6C2×#14519766C411yescharacteristic
7C2×C4×D244C411yescharacteristic
8#16839819—21no—
9#16839669C411yescharacteristic
10#16839663C2211yescharacteristic
11#16839652—21no—
12#16839652—21no—
13#16839627C2211yescharacteristic
14C4×#11758198—21no—
15C4×#11758198—21no—
16C42×Dic61C411yescharacteristic
17#16839488—21no—
18#16839488C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#432—21no—
21C61×#324—21no—
22C61×#324C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C4×D61—21no—
25#14519766—22no—
26#14519766—22no—
27C4×D244—41no—
28C4×D244—41no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758198—21no—
31C2×#11758198—21no—
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61C2×C411yescharacteristic
39C4⋊#92418—41no—
40C22⋊#92418—22no—
41#14519688—41no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C22×Q8×C61C2×C411yescharacteristic
47C4×Q8×C61—41no—
48C4×Q8×C61—41no—
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61C2311yescharacteristic
52C2×C42×C61C2×C411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C4⋊C8×C61—41no—
56C22⋊C8×C61—22no—
57#97×C61—41no—
58C23×D61—21no—
59C2×Q8×C61—21no—
60C2×Q8×C61—21no—
61C2×Q8×C61—41no—
62C2×Q8×C61—41no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C2×D244—21no—
74C2×D244—41no—
75C2×D244—41no—
76C2×D244M1612yes—
77C2×C4×D61—42no—
78C2×C4×D61—42no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×C8×C61—42no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88C42×C61—21no—
89C42×C61—21no—
90C42×C61—21no—
91#11758198—24no—
92#11758198—24no—
93#11758198—42no—
94#11758198—42no—
95#11758198—42no—
96C4⋊Dic61—41no—
97C4×Dic61—21no—
98C4×Dic61—42no—
99C4×Dic61—42no—
100C4×Dic61—41no—
101C4×Dic61M1612yes—
102C2×#92418—42no—
103C23×C61C2×C22⋊C411yescharacteristic
104C22×D61—22no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61—41no—
111Q8×C61—42no—
112Q8×C61—42no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61#9911yescharacteristic
129C2×C4×C61#9911yescharacteristic
130C2×C4×C61C2×M1611yescharacteristic
131C2×C4×C61C2×M1611yescharacteristic
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141D244—42no—
142D244—42no—
143C4×D61—82no—
144C4×D61—82no—
145C8×C61—82no—
146#92418—82no—
147#11530—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#40711yescharacteristic
151C22×C61C2×#9911yescharacteristic
152C22×C61#41311yescharacteristic
153C2×D61—42no—
154C2×D61—42no—
155C2×D61—42no—
156C2×D61—81no—
157C2×D61—81no—
158C2×D61—81no—
159C2×D61—81no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—41no—
164C4×C61—41no—
165C4×C61—41no—
166C4×C61—41no—
167C4×C61—41no—
168C4×C61#35512yes—
169Dic61—42no—
170Dic61—42no—
171Dic61—81no—
172C4×#124—611no—
173#1467—611no—
174#1467D6111yescharacteristic, complemented
175#1324—612no—
176#1318—612no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61#146511yescharacteristic
180C2×C61#212611yescharacteristic
181C2×C61C2×#35511yescharacteristic
182D61—82no—
183C2×#124—611no—
184C2×C4×Q8C2×D6111yescharacteristic
185C2×C4×D4—611no—
186#432—21no—
187#432—1221no—
188#391—611no—
189#390—611no—
190#388—1221no—
191#388—1221no—
192#382—611no—
193C4×C22⋊C4—1221no—
194C4×C22⋊C4—1221no—
195C43—611no—
196#324—21no—
197#324—612no—
198#324—1221no—
199#324C2×D6112yes—
200C61#1153011yescharacteristic, complemented, Sylow
201C22×Q8C4×D6111yescharacteristic
202C22×D4—611no—
203C23×C4—1221no—
204#124—1222no—
205#124—1222no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×D4—2441no—
209C4×D4—2441no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4D24411yescharacteristic
212C2×C4⋊C4C22⋊D6111yescharacteristic
213C2×C4⋊C4C22×D6111yescharacteristic
214C2×C22⋊C4—1221no—
215C2×C22⋊C4—1221no—
216C2×C22⋊C4—1221no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1221no—
219C2×C42—611no—
220C2×C42—611no—
221C2×C42—1221no—
222C2×C42—1221no—
223C2×C42C4×D6111yescharacteristic
224C2×C42D24411yescharacteristic
225C2×C42C22⋊D6111yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—2441no—
228C22⋊C8—22no—
229C22⋊C8—1222no—
230#97—41no—
231#97—2441no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95—1221no—
236C24—1221no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—612no—
242C2×D4—1221no—
243C2×D4—2441no—
244C2×D4—2441no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—2442no—
249C22×C4—2442no—
250C22×C4—2441no—
251C22×C4—2441no—
252C22×C4#1175819811yescharacteristic
253C22×C4#1175819811yescharacteristic
254C22×C4#1175819811yescharacteristic
255C22×C4#1175819811yescharacteristic
256C22×C4C2×C4×D6111yescharacteristic
257C22×C4C2×D24411yescharacteristic
258C22×C4C2×C22⋊D6111yescharacteristic
259C2×C8—42no—
260C2×C8—2442no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2441no—
265C22⋊C4—1224no—
266C22⋊C4—1224no—
267C22⋊C4—2442no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—612no—
277C42—1221no—
278C42—2441no—
279C42—2442no—
280C42—2442no—
281C23—1222no—
282C23—2441no—
283C23—2441no—
284C23—2441no—
285C23—2441no—
286C23—2441no—
287C23—2441no—
288C23C2×#1175819811yescharacteristic
289Q8—42no—
290Q8—42no—
291D4—2442no—
292D4—2442no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—1222no—
309C2×C4—1222no—
310C2×C4—4882no—
311C2×C4—4882no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4#1451969111yescharacteristic
320C2×C4#1451969111yescharacteristic
321C2×C4M16×D6111yescharacteristic
322C2×C4M16×D6111yescharacteristic
323C8—82no—
324C8—4882no—
325C22—21no—
326C22—21no—
327C22—4881no—
328C22—4881no—
329C22—4881no—
330C22—4881no—
331C22—2442no—
332C22—2442no—
333C22—2442no—
334C22#1683972511yescharacteristic
335C22C2×#1451969111yescharacteristic
336C22#1683974411yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—4881no—
346C4—2442no—
347C4—2442no—
348C4#1683955812yes—
349C2—21no—
350C2—21no—
351C2—4882no—
352C2#2144415511yescharacteristic
353C2#2144748211yescharacteristic
354C2C2×#1683955811yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1467C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1467D61
hypocenterC61#11530
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #21444163C2semidirect
3. #21443762C2semidirect
5. C61×#1467C2semidirect
174. #1467D61semidirect
200. C61#11530semidirect

Maximal subgroups

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

subgroupindex
C4×#145197662
#214441632
#214437622
#214437502
C61×#14672
#1153061