Finite Group Databasebeta

← #30598938

Subgroups — #30598938

Order 15616.

644 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 116 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#21461735C212yesmaximal, complemented
2C2×#16840620C212yesmaximal
3#21460697C212yesmaximal, complemented
4#21448383C211yesmaximal, characteristic
5C2×#14519844C2214yes—
6#16840628—22no—
7#16840628—22no—
8#16840620—24no—
9C2×#14519773C2211yescharacteristic
10C2×D4×Dic61C411yescharacteristic
11C2×#14519739C411yescharacteristic
12C2×#14519737—22no—
13#16839664C411yescharacteristic
14#16839662—22no—
15#16839631C2211yescharacteristic
16C42×Dic61C2211yescharacteristic
17C61×#390C411yescharacteristic
18C22×#92420—24no—
19#14519844—24no—
20#14519844—24no—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—22no—
23C23×Dic61—21no—
24C22×C4×D61—21no—
25#14519773—22no—
26#14519773D412yes—
27#14519773Q812yes—
28D4×Dic61—42no—
29#14519739—42no—
30#14519737—48no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—22no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—22no—
35C2×#11758198—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D412yes—
38C2×C4×Dic61D412yes—
39C2×C4×Dic61C2311yescharacteristic
40C22×D4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C42×C61C2×C411yescharacteristic
44C24×C61—21no—
45C23×D61—21no—
46C2×D4×C61—21no—
47C2×D4×C61—41no—
48C2×D4×C61—41no—
49C2×D4×C61C4⋊C412yes—
50C22×C4×C61—21no—
51C22×C4×C61C22⋊C412yes—
52C22×C4×C61C22×C411yescharacteristic
53C2×C22⋊D61—24no—
54C2×C22⋊D61—24no—
55C2×C22⋊D61—44no—
56C2×C22⋊D61—44no—
57C22×Dic61—21no—
58C22×Dic61—44no—
59C22×Dic61—42no—
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C22⋊C411yescharacteristic
62C22×Dic61C2×D412yes—
63C2×D244—21no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244C4⋊C412yes—
67C2×C4×D61—44no—
68C2×C4×D61—42no—
69C2×#92420—44no—
70C2×#92420—44no—
71C2×#92420—44no—
72C4⋊C4×C61—41no—
73C22⋊C4×C61—44no—
74C42×C61—41no—
75C22⋊Dic61—42no—
76#11758198—42no—
77#11758198—44no—
78C4⋊Dic61—41no—
79#11758196—44no—
80C4×Dic61—22no—
81C4×Dic61—24no—
82C4×Dic61—24no—
83C4×Dic61C2×D411yescharacteristic
84C4×Dic61C2×Q811yescharacteristic
85C4×Dic61#4112yes—
86C23×C61—22no—
87C23×C61—41no—
88C23×C61—41no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—22no—
93C22×D61—41no—
94C22×D61—41no—
95C22×D61—42no—
96C22×D61—42no—
97D4×C61—42no—
98D4×C61—42no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61C2×C4⋊C411yescharacteristic
107C2×C4×C61#11711yescharacteristic
108C22⋊D61—48no—
109C22⋊D61—48no—
110C2×Dic61—22no—
111C2×Dic61—21no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—42no—
115C2×Dic61—84no—
116C2×Dic61#10012yes—
117C2×Dic61#12112yes—
118C2×Dic61C22⋊Q812yes—
119C2×Dic61#13514yes—
120C2×Dic61AGL(1,ℤ/8)14yes—
121D244—42no—
122D244—42no—
123C4×D61—84no—
124#92420—84no—
125#8592—611nomaximal, Sylow
126C22×C61—42no—
127C22×C61—44no—
128C22×C61—81no—
129C22×C61—82no—
130C22×C61—81no—
131C22×C61#38011yescharacteristic
132C22×C61C2×#10011yescharacteristic
133C22×C61#41311yescharacteristic
134C22×C61#41612yes—
135C22×C61#41812yes—
136C2×D61—42no—
137C2×D61—44no—
138C2×D61—81no—
139C2×D61—82no—
140C2×D61—81no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—42no—
144C4×C61—81no—
145Dic61—22no—
146Dic61—41no—
147Dic61—81no—
148Dic61#46314yes—
149Dic61#47814yes—
150C4×#127—611no—
151#1831—612no—
152C2×#331—612no—
153#1296—612no—
154C2×C61—82no—
155C2×C61#154311yescharacteristic
156C2×C61#155312yes—
157C2×C61#155511yescharacteristic
158C2×C61#180011yescharacteristic
159C2×C61#180412yes—
160D61—82no—
161C2×#127—611no—
162C2×C4×D4—611no—
163C2×C4×D4—611no—
164C2×C4×D4—1222no—
165C2×C4⋊C8—614no—
166#390—611no—
167#390—1222no—
168#390#692311yescharacteristic
169#384—611no—
170C43—611no—
171#336—1222no—
172#336—1222no—
173#331—1224no—
174C61#859211yescharacteristic, complemented, Sylow
175C22×D4—611no—
176C22×D4—1222no—
177C22×D4C2×#692311yescharacteristic
178C23×C4—1221no—
179C23×C4—1221no—
180C22×C8—1224no—
181#127—612no—
182#127—612no—
183#127—1222no—
184C4×D4—2448no—
185C4×D4—2442no—
186C4×D4—2442no—
187C2×C4⋊C4—611no—
188C2×C4⋊C4—1222no—
189C2×C4⋊C4C2×#692311yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—1221no—
192C2×C22⋊C4—1221no—
193C2×C22⋊C4—1222no—
194C2×C42—611no—
195C2×C42—612no—
196C2×C42—612no—
197C2×C42—611no—
198C2×C42C2×#692311yescharacteristic
199C4⋊C8—1224no—
200C4⋊C8—1224no—
201C24—21no—
202C24—1221no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—612no—
207C2×D4—1224no—
208C2×D4—1224no—
209C2×D4—1221no—
210C2×D4—2441no—
211C2×D4—2441no—
212C2×D4—2444no—
213C2×D4—2444no—
214C2×D4C4⋊#692312yes—
215C22×C4—21no—
216C22×C4—612no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4—1221no—
220C22×C4—2442no—
221C22×C4—2444no—
222C22×C4—2442no—
223C22×C4—2444no—
224C22×C4C22⋊#692312yes—
225C22×C4C22×#692311yescharacteristic
226C2×C8—2444no—
227C2×C8—2444no—
228C2×C8—2444no—
229C4⋊C4—41no—
230C4⋊C4—2441no—
231C4⋊C4—2444no—
232C22⋊C4—44no—
233C22⋊C4—2444no—
234C22⋊C4—2442no—
235C22⋊C4—2442no—
236C42—41no—
237C42—611no—
238C42—611no—
239C42—612no—
240C42—1224no—
241C42—1222no—
242C42—1224no—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23—1222no—
249C23—2441no—
250C23—2441no—
251C23—2442no—
252C23—2442no—
253C23C2×C22⋊#692311yescharacteristic
254D4—42no—
255D4—42no—
256D4—2442no—
257D4—2442no—
258D4—2448no—
259D4—2448no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—612no—
268C2×C4—612no—
269C2×C4—614no—
270C2×C4—614no—
271C2×C4—612no—
272C2×C4—1222no—
273C2×C4—1221no—
274C2×C4—1222no—
275C2×C4—4884no—
276C2×C4—4884no—
277C2×C4—2441no—
278C2×C4—2442no—
279C2×C4C2×C4⋊#692311yescharacteristic
280C2×C4#1451985611yescharacteristic
281C8—4884no—
282C22—42no—
283C22—44no—
284C22—81no—
285C22—82no—
286C22—81no—
287C22—4881no—
288C22—4882no—
289C22—4881no—
290C22—2442no—
291C22—2444no—
292C22#1684070411yescharacteristic
293C22D61⋊#10011yescharacteristic
294C22#1684074911yescharacteristic
295C22#1684075212yes—
296C22#1684075412yes—
297C4—21no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—614no—
302C4—614no—
303C4—1222no—
304C4—4881no—
305C4—2441no—
306C2—82no—
307C2—4882no—
308C2#2146132311yescharacteristic
309C2#2146134312yes—
310C2#2146134711yescharacteristic
311C2#2146168411yescharacteristic
312C2#2146168912yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C22
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#8592
LCSG ▷ C22×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. #21461735C2semidirect
3. #21460697C2semidirect
174. C61#8592semidirect

Maximal subgroups

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

subgroupindex
#214617352
C2×#168406202
#214606972
#214483832
#859261