Finite Group Databasebeta

← #30601878

Subgroups — #30601878

Order 15616.

506 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 112 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
2#21461613C211yesmaximal, characteristic
3D61⋊#334C211yesmaximal, characteristic
4D61⋊#332C212yesmaximal
5#494×D61C211yesmaximal, characteristic
6D61⋊(C4⋊C8)C2212yes—
7D61⋊(C4⋊C8)C2211yescharacteristic
8C2×#14519831—21no—
9C4×C4⋊#6923C2211yescharacteristic
10#16840635—24no—
11(C2×#6923)⋊C8—21no—
12#16840624—21no—
13#16840623—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—
32#14519831—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—24no—
74C4×#6923—41no—
75C4×#6923—41no—
76D61⋊C8—41no—
77D61⋊C8—41no—
78D61⋊C8—44no—
79Q16×C61—42no—
80Q16×C61—42no—
81C2×C8×C61C22⋊C411yescharacteristic
82C2×C8×C61C22⋊C411yescharacteristic
83C4⋊C4×C61—22no—
84C4⋊C4×C61—22no—
85C42×C61C22×C411yescharacteristic
86#11758202—44no—
87#11758202—44no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90#11758196—42no—
91#11758196—42no—
92C4×Dic61C2×D411yescharacteristic
93C4×Dic61C2×D411yescharacteristic
94C4×Dic61#4111yescharacteristic
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97Dic244—42no—
98Dic244—42no—
99C8×D61—21no—
100C8×D61—21no—
101C8×D61D812yes—
102C8×D61SD1612yes—
103C22×D61#12111yescharacteristic
104C2×#6923—42no—
105C2×#6923—42no—
106C2×#6923—42no—
107Q8×C61—42no—
108Q8×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61C4×D411yescharacteristic
112C2×C4×C61C4×D411yescharacteristic
113C2×Dic61—42no—
114C2×Dic61#12111yescharacteristic
115C2×Dic61#12411yescharacteristic
116C2×Dic61#12711yescharacteristic
117C4×D61—22no—
118C4×D61—82no—
119C4×D61—82no—
120C4×D61C2×D811yescharacteristic
121C4×D61C2×SD1611yescharacteristic
122C4×D61AGL(1,ℤ/8)11yescharacteristic
123C4×D61AGL(1,ℤ/8)11yescharacteristic
124C4×D61#13714yes—
125Dic122—42no—
126Dic122—42no—
127Dic122—82no—
128Dic122—82no—
129#92420—42no—
130#92420—82no—
131C8×C61—21no—
132C8×C61#10212yes—
133#92418—21no—
134#92418#10212yes—
135#9742—611nomaximal, Sylow
136C22×C61#39011yescharacteristic
137C2×D61#45911yescharacteristic
138C2×D61#46011yescharacteristic
139C2×D61#47012yes—
140C2×D61#47312yes—
141#6923—82no—
142C4×C61—21no—
143C4×C61—82no—
144C4×C61C2×#10211yescharacteristic
145C4×C61#41811yescharacteristic
146C4×C61#44112yes—
147Dic61—21no—
148Dic61—82no—
149Dic61#45911yescharacteristic
150Dic61#46011yescharacteristic
151Dic61#49712yes—
152C2×#494—611no—
153#1988—612no—
154#1770—611no—
155C2×#334—611no—
156C2×#332—612no—
157C2×C61#197111yescharacteristic
158C2×C61#197911yescharacteristic
159C2×C61#198211yescharacteristic
160D61#168912yes—
161D61#169012yes—
162C22×Q16—1221no—
163C22×Q16—1221no—
164C2×C4⋊Q8—612no—
165#494—612no—
166#494—611no—
167#494—1221no—
168#494—1221no—
169#494#692311yescharacteristic, complemented
170C2×C4⋊C8—611no—
171C2×C4⋊C8—612no—
172C2×#97—1221no—
173C2×C4×C8—611no—
174C4×C4⋊C4—611no—
175#340—1224no—
176#336—1221no—
177#334—1221no—
178#334—1221no—
179#332—1224no—
180C61#974211yescharacteristic, complemented, Sylow
181C22×Q8—1222no—
182C2×Q16—21no—
183C2×Q16—21no—
184C2×Q16—1222no—
185C2×Q16—1222no—
186C2×Q16—1221no—
187C2×Q16—1221no—
188C2×Q16—2442no—
189C2×Q16—2442no—
190C2×Q16—2442no—
191C2×Q16—2442no—
192C22×C8—611no—
193C22×C8—611no—
194C22×C8—1221no—
195C22×C8—1222no—
196C4⋊Q8—612no—
197C4⋊Q8—1222no—
198C4⋊Q8—1222no—
199C4⋊Q8—1222no—
200C4⋊Q8C2×#692312yes—
201C2×C4⋊C4—612no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—1222no—
204C2×C42—611no—
205C2×C42—1221no—
206C4⋊C8—612no—
207C4⋊C8—1221no—
208C4⋊C8—1222no—
209C4⋊C8—1222no—
210#97—2442no—
211C4×C8—611no—
212C4×C8—611no—
213C4×C8—611no—
214C4×C8C2×#692311yescharacteristic
215#95—1221no—
216C2×Q8—22no—
217C2×Q8—1222no—
218C2×Q8—2442no—
219C2×Q8—2442no—
220C2×Q8—2442no—
221C2×Q8—2442no—
222C2×Q8—2442no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—1222no—
227C22×C4—2442no—
228Q16—42no—
229Q16—42no—
230Q16—2442no—
231Q16—2444no—
232Q16—2444no—
233Q16—2442no—
234C2×C8—611no—
235C2×C8—612no—
236C2×C8—612no—
237C2×C8—611no—
238C2×C8—1221no—
239C2×C8—1221no—
240C2×C8—1222no—
241C2×C8—2442no—
242C2×C8—2444no—
243C2×C8—2441no—
244C2×C8—2441no—
245C2×C8C22⋊#692311yescharacteristic
246C2×C8C22⋊#692311yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—1224no—
250C4⋊C4—1222no—
251C4⋊C4—1222no—
252C4⋊C4—2442no—
253C4⋊C4—2442no—
254C42—611no—
255C42—611no—
256C42—611no—
257C42—2441no—
258C42—2441no—
259C42C22×#692311yescharacteristic
260C23—611no—
261Q8—42no—
262Q8—42no—
263Q8—4882no—
264Q8—4882no—
265Q8—2442no—
266Q8—2442no—
267C2×C4—42no—
268C2×C4—611no—
269C2×C4—611no—
270C2×C4—611no—
271C2×C4—611no—
272C2×C4—611no—
273C2×C4—611no—
274C2×C4—614no—
275C2×C4—611no—
276C2×C4—1222no—
277C2×C4—4882no—
278C2×C4—4882no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4C2×C22⋊#692311yescharacteristic
284C2×C4D4×#692311yescharacteristic
285C2×C4D4×#692311yescharacteristic
286C8—21no—
287C8—612no—
288C8—1221no—
289C8—4882no—
290C8—2442no—
291C8#1451983912yes—
292C22—612no—
293C22—612no—
294C22—611no—
295C22—611no—
296C22#1684071411yescharacteristic
297C4—21no—
298C4—82no—
299C4—612no—
300C4—611no—
301C4—611no—
302C4—1221no—
303C4—4882no—
304C4—4882no—
305C4C2×#1451983911yescharacteristic
306C4#1684075411yescharacteristic
307C4#1684080212yes—
308C2—612no—
309C2—612no—
310C2#2146196711yescharacteristic
311C2#2146198111yescharacteristic
312C2#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#9742
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
169. #494#6923semidirect
180. C61#9742semidirect

Maximal subgroups

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

subgroupindex
#214619982
#214616132
D61⋊#3342
D61⋊#3322
#494×D612
#974261