Finite Group Databasebeta

← #30601869

Subgroups — #30601869

Order 15616.

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

normal subgroups only

This group has 518 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#21461998C212yesmaximal
2C8×C4⋊#6923C211yesmaximal, characteristic
3D61⋊#335C211yesmaximal, characteristic
4D61⋊#332C212yesmaximal
5#494×D61C211yesmaximal, characteristic
6D61⋊(C4⋊C8)C2212yes—
7D61⋊(C4⋊C8)C2211yescharacteristic
8C2×C8×#6923—21no—
9C4×C4⋊#6923C2211yescharacteristic
10#16840635—24no—
11(C2×#6923)⋊C8—21no—
12#16840626—21no—
13#16840625—21no—
14#16840621—24no—
15C2×Q16×D61—21no—
16C2×Q16×D61—21no—
17C4⋊Q8×D61C2212yes—
18#16840142C411yescharacteristic, complemented
19#16840141C412yescomplemented
20#16840140—21no—
21#16840140—21no—
22C4×C8×D61C2211yescharacteristic
23C61×#494C411yescharacteristic, complemented
24C2×D61⋊C8—21no—
25C2×D61⋊C8—22no—
26C2×C4⋊#6923D412yes—
27C2×C4×#6923—21no—
28#14519844—22no—
29#14519844D412yes—
30C4⋊#92420—22no—
31C4⋊#92420—21no—
32C8×#6923—42no—
33(C2×#6923)⋊C4—21no—
34C2×Q8×D61—22no—
35C2×Dic244—21no—
36C2×Dic244—21no—
37C2×#11758202—22no—
38C2×#11758202—22no—
39Q16×D61—42no—
40Q16×D61—42no—
41Q16×D61—42no—
42Q16×D61—42no—
43C2×C8×D61D411yescharacteristic
44C2×C8×D61D411yescharacteristic
45#14519777C2×C412yes—
46#14519776—22no—
47#14519776—22no—
48#14519775—22no—
49C4⋊C4×D61—22no—
50C4⋊C4×D61—22no—
51C42×D61C2311yescharacteristic
52C4×#92418C2×C411yescharacteristic
53C8×Dic61D411yescharacteristic
54C8×Dic61D411yescharacteristic
55C2×Q16×C61—21no—
56C2×Q16×C61—21no—
57C4⋊Q8×C61C2×C412yes—
58C4×C8×C61C2×C411yescharacteristic
59C22×#6923—22no—
60C2×Q8×C61—22no—
61Q8×D61—42no—
62Q8×D61—42no—
63Q8×D61—42no—
64Q8×D61—42no—
65C2×C4×D61—42no—
66C2×C4×D61C2×D411yescharacteristic
67C2×C4×D61C2×D411yescharacteristic
68C2×C4×D61#4111yescharacteristic
69C2×Dic122—22no—
70C2×Dic122—42no—
71C2×#92420—22no—
72C2×#92420—42no—
73C4⋊#6923—22no—
74C4⋊#6923Q1614yescomplemented
75C4×#6923—41no—
76C4×#6923—41no—
77D61⋊C8—41no—
78D61⋊C8—41no—
79D61⋊C8—44no—
80Q16×C61—42no—
81Q16×C61—42no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22⋊C411yescharacteristic
84C4⋊C4×C61—22no—
85C4⋊C4×C61—22no—
86C42×C61C22×C411yescharacteristic
87#11758202—44no—
88#11758202—44no—
89C4⋊Dic61—22no—
90C4⋊Dic61—22no—
91#11758196—42no—
92#11758196—42no—
93C4×Dic61C2×D411yescharacteristic
94C4×Dic61C2×D411yescharacteristic
95C4×Dic61#4111yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97C2×#92418C22⋊C411yescharacteristic
98Dic244—42no—
99Dic244—42no—
100C8×D61—21no—
101C8×D61—21no—
102C8×D61D812yes—
103C8×D61SD1612yes—
104C22×D61#12111yescharacteristic
105C2×#6923—24no—
106C2×#6923—42no—
107C2×#6923—42no—
108Q8×C61—42no—
109Q8×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61C4×D411yescharacteristic
113C2×C4×C61C4×D411yescharacteristic
114C2×Dic61—42no—
115C2×Dic61#12111yescharacteristic
116C2×Dic61#12411yescharacteristic
117C2×Dic61#12711yescharacteristic
118C4×D61—22no—
119C4×D61—82no—
120C4×D61—82no—
121C4×D61C2×D811yescharacteristic
122C4×D61C2×SD1611yescharacteristic
123C4×D61C2×Q1612yes—
124C4×D61#13512yes—
125C4×D61AGL(1,ℤ/8)11yescharacteristic
126C4×D61AGL(1,ℤ/8)11yescharacteristic
127Dic122—42no—
128Dic122—42no—
129Dic122—82no—
130Dic122—82no—
131#92420—42no—
132#92420—82no—
133C8×C61—21no—
134C8×C61#10212yes—
135#92418—21no—
136#92418#10212yes—
137#9733—611nomaximal, Sylow
138C22×C61#39011yescharacteristic
139C2×D61#45911yescharacteristic
140C2×D61#46011yescharacteristic
141C2×D61#46712yes—
142C2×D61#47312yes—
143#6923—44no—
144C4×C61—21no—
145C4×C61—82no—
146C4×C61C2×#10211yescharacteristic
147C4×C61#41811yescharacteristic
148C4×C61C4×Q1612yes—
149Dic61—21no—
150Dic61—82no—
151Dic61#45911yescharacteristic
152Dic61#46011yescharacteristic
153Dic61#49411yescharacteristic
154Dic61#49511yescharacteristic
155C2×#494—611no—
156#1988—612no—
157C8×C4⋊C4—611no—
158C2×#335—611no—
159C2×#332—612no—
160C2×C61#196911yescharacteristic
161C2×C61#197911yescharacteristic
162C2×C61#198211yescharacteristic
163D61#166912yes—
164D61#167012yes—
165C22×Q16—1221no—
166C22×Q16—1221no—
167C2×C4⋊Q8—612no—
168#494—612no—
169#494—611no—
170#494—1221no—
171#494—1221no—
172#494#692311yescharacteristic, complemented
173C2×C4⋊C8—611no—
174C2×C4⋊C8—612no—
175C2×C4×C8—611no—
176C2×C4×C8—1221no—
177C4×C4⋊C4—611no—
178#340—1224no—
179#336—1221no—
180#335—1221no—
181#335—1221no—
182#332—1224no—
183C61#973311yescharacteristic, complemented, Sylow
184C22×Q8—1222no—
185C2×Q16—21no—
186C2×Q16—21no—
187C2×Q16—1222no—
188C2×Q16—1222no—
189C2×Q16—1221no—
190C2×Q16—1221no—
191C2×Q16—2442no—
192C2×Q16—2442no—
193C2×Q16—2442no—
194C2×Q16—2442no—
195C22×C8—611no—
196C22×C8—611no—
197C22×C8—1221no—
198C22×C8—1222no—
199C4⋊Q8—612no—
200C4⋊Q8—1222no—
201C4⋊Q8—1222no—
202C4⋊Q8—1222no—
203C4⋊Q8C2×#692312yes—
204C2×C4⋊C4—612no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4—1222no—
207C2×C42—611no—
208C2×C42—1221no—
209C4⋊C8—612no—
210C4⋊C8—1221no—
211C4⋊C8—1222no—
212C4⋊C8—1222no—
213C4×C8—611no—
214C4×C8—611no—
215C4×C8—611no—
216C4×C8—2442no—
217C4×C8C2×#692311yescharacteristic
218#95—1221no—
219C2×Q8—22no—
220C2×Q8—1222no—
221C2×Q8—2442no—
222C2×Q8—2442no—
223C2×Q8—2442no—
224C2×Q8—2442no—
225C2×Q8—2442no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1222no—
230C22×C4—2442no—
231Q16—42no—
232Q16—42no—
233Q16—2442no—
234Q16—2444no—
235Q16—2444no—
236Q16—2442no—
237C2×C8—611no—
238C2×C8—612no—
239C2×C8—612no—
240C2×C8—611no—
241C2×C8—1221no—
242C2×C8—1221no—
243C2×C8—1222no—
244C2×C8—2442no—
245C2×C8—2444no—
246C2×C8—2441no—
247C2×C8—2441no—
248C2×C8C22⋊#692311yescharacteristic
249C2×C8C22⋊#692311yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—614no—
253C4⋊C4—1222no—
254C4⋊C4—1222no—
255C4⋊C4—1222no—
256C4⋊C4—2442no—
257C4⋊C4—2442no—
258C42—611no—
259C42—611no—
260C42—611no—
261C42—2441no—
262C42—2441no—
263C42C22×#692311yescharacteristic
264C23—611no—
265Q8—42no—
266Q8—42no—
267Q8—4882no—
268Q8—4882no—
269Q8—2442no—
270Q8—2442no—
271C2×C4—42no—
272C2×C4—611no—
273C2×C4—611no—
274C2×C4—611no—
275C2×C4—611no—
276C2×C4—611no—
277C2×C4—611no—
278C2×C4—612no—
279C2×C4—612no—
280C2×C4—611no—
281C2×C4—1222no—
282C2×C4—1224no—
283C2×C4—4882no—
284C2×C4—4882no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4—2442no—
288C2×C4C2×C22⋊#692311yescharacteristic
289C2×C4D4×#692311yescharacteristic
290C2×C4D4×#692311yescharacteristic
291C8—21no—
292C8—612no—
293C8—1221no—
294C8—4882no—
295C8—2442no—
296C8#1451983912yes—
297C22—612no—
298C22—612no—
299C22—611no—
300C22—611no—
301C22#1684071411yescharacteristic
302C4—21no—
303C4—82no—
304C4—611no—
305C4—611no—
306C4—611no—
307C4—611no—
308C4—1221no—
309C4—4882no—
310C4—2444no—
311C4C2×#1451983911yescharacteristic
312C4#1684075411yescharacteristic
313C4Q16×#692312yes—
314C2—612no—
315C2—612no—
316C2#2146196511yescharacteristic
317C2#2146198111yescharacteristic
318C2#2146198611yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9733
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
18. #16840142C4semidirect
19. #16840141C4semidirect
23. C61×#494C4semidirect
74. C4⋊#6923Q16semidirect
172. #494#6923semidirect
183. C61#9733semidirect

Maximal subgroups

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

subgroupindex
#214619982
C8×C4⋊#69232
D61⋊#3352
D61⋊#3322
#494×D612
#973361