Finite Group Databasebeta

← #30598923

Subgroups — #30598923

Order 15616.

684 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 146 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#21461752C212yesmaximal, complemented
2C2×#16840620C212yesmaximal
3#21460697C212yesmaximal, complemented
4C4×#14519773C211yesmaximal, characteristic
5C2×#14519844—22no—
6C2×#14519844C2214yes—
7C2×C4⋊#92420—22no—
8#16840620—24no—
9C2×#14519773C2211yescharacteristic
10C2×C4×D244C411yescharacteristic
11C2×C4×C22⋊D61—22no—
12#16839665C411yescharacteristic
13#16839663C411yescharacteristic
14#16839662—22no—
15#16839631C2211yescharacteristic
16C42×Dic61C2211yescharacteristic
17C2×C4×D4×C61C411yescharacteristic
18C22×#92420—24no—
19#14519844—24no—
20#14519844—24no—
21#14519844—44no—
22C4⋊#92420—44no—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—22no—
25C22×C4×D61—21no—
26#14519773—22no—
27#14519773D412yes—
28#14519773Q812yes—
29C4×D244—42no—
30C4×C22⋊D61—44no—
31C4×C22⋊D61—44no—
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×#11758196—22no—
34C2×C22⋊Dic61—22no—
35C2×#11758198—21no—
36C2×#11758198—22no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61D412yes—
39C2×C4×Dic61D412yes—
40C2×C4×Dic61C2311yescharacteristic
41C22×D4×C61C2×C411yescharacteristic
42C23×C4×C61—21no—
43C4×D4×C61—42no—
44C2×C4⋊C4×C61C2×C411yescharacteristic
45C2×C22⋊C4×C61—21no—
46C2×C42×C61C2×C411yescharacteristic
47C24×C61—21no—
48C23×D61—21no—
49C2×D4×C61—21no—
50C2×D4×C61—41no—
51C2×D4×C61—41no—
52C2×D4×C61C4⋊C412yes—
53C22×C4×C61—21no—
54C22×C4×C61—42no—
55C22×C4×C61—44no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—24no—
60C2×C22⋊D61—24no—
61C2×C22⋊D61—42no—
62C2×C22⋊D61—42no—
63C2×C22⋊D61—42no—
64C2×C22⋊D61—42no—
65C22×Dic61—21no—
66C22×Dic61C22⋊C412yes—
67C22×Dic61C2×D412yes—
68C2×D244—21no—
69C2×D244—41no—
70C2×D244—41no—
71C2×D244C4⋊C412yes—
72C2×C4×D61—42no—
73C2×C4×D61—44no—
74C2×#92420—44no—
75C2×#92420—44no—
76C2×#92420—44no—
77C4⋊C4×C61—41no—
78C22⋊C4×C61—42no—
79C42×C61—22no—
80C22⋊Dic61—44no—
81#11758198—42no—
82#11758198—44no—
83C4⋊Dic61—41no—
84#11758196—42no—
85#11758196—42no—
86C4×Dic61—22no—
87C4×Dic61—22no—
88C4×Dic61—41no—
89C4×Dic61D814yes—
90C4×Dic61SD1614yes—
91C4×Dic61C2×D411yescharacteristic
92C4×Dic61C2×Q811yescharacteristic
93C4×Dic61#4112yes—
94C23×C61—22no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—22no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61—42no—
104C22×D61—42no—
105D4×C61—42no—
106D4×C61—42no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61—84no—
114C2×C4×C61#10012yes—
115C2×C4×C61#10214yes—
116C2×C4×C61C2×C4⋊C411yescharacteristic
117C2×C4×C61#11711yescharacteristic
118C22⋊D61—44no—
119C22⋊D61—44no—
120C22⋊D61—44no—
121C22⋊D61—44no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—41no—
125C2×Dic61—42no—
126C2×Dic61#10214yes—
127C2×Dic61#12112yes—
128C2×Dic61C22⋊Q812yes—
129C2×Dic61C2×D812yes—
130C2×Dic61C2×SD1612yes—
131C2×Dic61AGL(1,ℤ/8)12yes—
132C2×Dic61#13712yes—
133D244—42no—
134D244—42no—
135C4×D61—84no—
136#92420—84no—
137#8585—611nomaximal, Sylow
138C22×C61—42no—
139C22×C61—44no—
140C22×C61—81no—
141C22×C61—82no—
142C22×C61—81no—
143C22×C61#38011yescharacteristic
144C22×C61C2×#10011yescharacteristic
145C22×C61#41311yescharacteristic
146C22×C61C2×#10212yes—
147C22×C61#41712yes—
148C2×D61—42no—
149C2×D61—44no—
150C2×D61—81no—
151C2×D61—82no—
152C2×D61—81no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—41no—
156C4×C61—81no—
157C4×C61#33114yescomplemented
158Dic61—24no—
159Dic61—81no—
160Dic61#45912yes—
161Dic61#46112yes—
162Dic61#47412yes—
163Dic61#47612yes—
164C4×#127—611no—
165#1842—612no—
166C2×#331—612no—
167#1296—612no—
168C2×C61—82no—
169C2×C61C2×#33111yescharacteristic
170C2×C61#154712yes—
171C2×C61#156011yescharacteristic
172C2×C61#180011yescharacteristic
173C2×C61#180212yes—
174D61—82no—
175C2×#127—611no—
176C2×C4×D4—611no—
177C2×C4×D4—1222no—
178C2×C4×D4#692311yescharacteristic
179C2×C4⋊C8—614no—
180C2×C4⋊C8—1222no—
181C2×C4⋊C8—1222no—
182#390—611no—
183#390—611no—
184#390—1222no—
185#384—611no—
186C43—611no—
187#331—1224no—
188C61#858511yescharacteristic, complemented, Sylow
189C22×D4—611no—
190C22×D4—1222no—
191C22×D4C2×#692311yescharacteristic
192C23×C4—21no—
193C23×C4—1221no—
194C22×C8—1224no—
195#127—612no—
196#127—612no—
197#127—1222no—
198C4×D4—42no—
199C4×D4—2442no—
200C4×D4—2444no—
201C4×D4—2444no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4C2×#692311yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4—1222no—
208C2×C22⋊C4—1221no—
209C2×C42—612no—
210C2×C42—611no—
211C2×C42—612no—
212C2×C42—611no—
213C2×C42C2×#692311yescharacteristic
214C4⋊C8—1224no—
215C4⋊C8—1224no—
216C4⋊C8—2444no—
217C4⋊C8—2444no—
218C24—21no—
219C24—1221no—
220C2×D4—21no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—612no—
224C2×D4—1224no—
225C2×D4—1224no—
226C2×D4—1221no—
227C2×D4—2441no—
228C2×D4—2441no—
229C2×D4—2442no—
230C2×D4—2442no—
231C2×D4—2442no—
232C2×D4—2442no—
233C2×D4C4⋊#692312yes—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—1221no—
240C22×C4—2442no—
241C22×C4—2444no—
242C22×C4C22⋊#692311yescharacteristic
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22×#692311yescharacteristic
245C2×C8—2444no—
246C2×C8—2444no—
247C2×C8—2444no—
248C4⋊C4—41no—
249C4⋊C4—2441no—
250C4⋊C4—2442no—
251C4⋊C4—2442no—
252C22⋊C4—42no—
253C22⋊C4—2442no—
254C22⋊C4—2444no—
255C22⋊C4—2444no—
256C42—22no—
257C42—614no—
258C42—614no—
259C42—611no—
260C42—611no—
261C42—612no—
262C42—1222no—
263C42—1222no—
264C42—2441no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—1222no—
271C23—2441no—
272C23—2441no—
273C23—2442no—
274C23—2442no—
275C23C2×C22⋊#692311yescharacteristic
276D4—42no—
277D4—42no—
278D4—2442no—
279D4—2442no—
280D4—2444no—
281D4—2444no—
282D4—2444no—
283D4—2444no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4—612no—
292C2×C4—612no—
293C2×C4—612no—
294C2×C4—612no—
295C2×C4—612no—
296C2×C4—612no—
297C2×C4—614no—
298C2×C4—1222no—
299C2×C4—1222no—
300C2×C4—4884no—
301C2×C4—2441no—
302C2×C4—2442no—
303C2×C4#1451983712yes—
304C2×C4#1451983914yes—
305C2×C4C2×C4⋊#692311yescharacteristic
306C2×C4#1451985611yescharacteristic
307C8—4884no—
308C22—42no—
309C22—44no—
310C22—81no—
311C22—82no—
312C22—81no—
313C22—4881no—
314C22—4882no—
315C22—4881no—
316C22—2442no—
317C22—2444no—
318C22#1684070411yescharacteristic
319C22C2×#1451983711yescharacteristic
320C22#1684074811yescharacteristic
321C22C2×#1451983912yes—
322C22#1684075312yes—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—81no—
327C4—612no—
328C4—612no—
329C4—612no—
330C4—612no—
331C4—1224no—
332C4—4881no—
333C4#1684062014yescomplemented
334C2—82no—
335C2—4882no—
336C2C2×#1684062011yescharacteristic
337C2#2146133012yes—
338C2#2146135611yescharacteristic
339C2#2146168311yescharacteristic
340C2#2146168712yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C22
cosocleC2×C4×Dic61C23
hypercenterC2×C4×D4#6923
hypocenterC61#8585
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461752C2semidirect
3. #21460697C2semidirect
157. C4×C61#331semidirect
188. C61#8585semidirect
333. C4#16840620semidirect

Maximal subgroups

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

subgroupindex
#214617522
C2×#168406202
#214606972
C4×#145197732
#858561