Finite Group Databasebeta

← #30599773

Subgroups — #30599773

Order 15616.

644 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 106 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#21462349C211yesmaximal, characteristic
2#21461947C211yesmaximal, characteristic
3#21461931C212yesmaximal, complemented
4#21460653C212yesmaximal, complemented
5#331×D61C211yesmaximal, characteristic
6C2×D4×#6923—22no—
7C2×#14519845—22no—
8C2×#14519839—22no—
9C4×D61⋊C8C2211yescharacteristic
10#16840714—22no—
11#16840710—22no—
12#16840708C2211yescharacteristic
13C4×C4⋊#6923C2212yes—
14#16840636—22no—
15#127×D61C2211yescharacteristic
16C4⋊C8×D61C2212yes—
17#16839506C411yescharacteristic, complemented
18#16839505C412yescomplemented
19C61×#331C411yescharacteristic, complemented
20C23×#6923—22no—
21C2×D61⋊C8—21no—
22D4×#6923—44no—
23D4×#6923—44no—
24C2×C4⋊#6923—21no—
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923D412yes—
27C2×C4⋊#6923D412yes—
28C2×C22⋊#6923—24no—
29C2×C22⋊#6923—22no—
30C2×C4×#6923—22no—
31#14519845—44no—
32#14519839—44no—
33C4×#92420—22no—
34(C2×#6923)⋊C4—22no—
35C2×D4×D61—22no—
36C2×D4×D61D412yes—
37C2×C8×D61—22no—
38#14519774C2×C411yescharacteristic
39#14519773—22no—
40#14519773Q812yes—
41C42×D61C2311yescharacteristic
42C4⋊#92418C2×C412yescomplemented
43#14519706—22no—
44#127×C61C2×C411yescharacteristic
45C4⋊C8×C61C2×C412yescomplemented
46C23×D61—22no—
47C22×#6923—22no—
48C22×#6923—22no—
49C22×#6923—48no—
50C22×#6923—41no—
51C22×#6923—82no—
52C2×D4×C61—22no—
53C2×D4×C61C4⋊C412yes—
54C2×C22⋊D61—24no—
55C2×C22⋊D61—22no—
56D4×D61—42no—
57D4×D61—42no—
58D4×D61—42no—
59D4×D61—42no—
60D4×D61—42no—
61D4×D61—42no—
62C2×D244—22no—
63C2×D244C4⋊C412yes—
64C2×C4×D61C2×D411yescharacteristic
65C2×C4×D61C2×D412yes—
66C22⋊#6923—48no—
67C22⋊#6923—82no—
68C2×#92420—41no—
69C4⋊#6923—22no—
70C4⋊#6923—22no—
71C4⋊#6923—22no—
72C4⋊#6923—42no—
73C4⋊#6923—42no—
74C4⋊#6923—42no—
75C4⋊#6923SD1614yes—
76C4×#6923—42no—
77C4×#6923—42no—
78D61⋊C8—22no—
79D61⋊C8—41no—
80C2×C8×C61—22no—
81C42×C61C22×C411yescharacteristic
82C4×Dic61C2×Q811yescharacteristic
83C4×Dic61#4112yes—
84C2×#92418—22no—
85C8×D61—42no—
86C8×D61—42no—
87C23×C61—22no—
88C22×D61—22no—
89C22×D61—44no—
90C22×D61—44no—
91C22×D61—42no—
92C22×D61—42no—
93C22×D61#12011yescharacteristic
94C2×#6923—24no—
95C2×#6923—42no—
96C2×#6923—42no—
97C2×#6923—42no—
98C2×#6923—42no—
99C2×#6923—42no—
100C2×#6923—88no—
101C2×#6923—82no—
102C2×#6923—81no—
103D4×C61—42no—
104D4×C61—42no—
105D4×C61—42no—
106C2×C4×C61C2×C4⋊C411yescharacteristic
107C2×C4×C61C4×D412yes—
108C22⋊D61—44no—
109C22⋊D61—44no—
110C22⋊D61—82no—
111C2×Dic61C22⋊Q812yes—
112C2×Dic61#12311yescharacteristic
113D244—42no—
114D244—42no—
115D244—42no—
116C4×D61—21no—
117C4×D61—21no—
118C4×D61C2×SD1612yes—
119C4×D61#13512yes—
120C4×D61AGL(1,ℤ/8)12yes—
121C4×D61AGL(1,ℤ/8)12yes—
122#92420—42no—
123C8×C61—42no—
124#92418—42no—
125#8924—611nomaximal, Sylow
126C22×C61—42no—
127C22×C61—44no—
128C22×C61#38511yescharacteristic
129C2×D61—42no—
130C2×D61—44no—
131C2×D61—82no—
132C2×D61—82no—
133C2×D61—84no—
134C2×D61#44912yes—
135C2×D61#45012yes—
136C2×D61#45311yescharacteristic
137C2×D61#45511yescharacteristic
138#6923—44no—
139#6923—82no—
140#6923—161no—
141C4×C61—21no—
142C4×C61C4×SD1612yes—
143C4×C61#44212yes—
144Dic61—21no—
145Dic61#47612yes—
146Dic61#47812yes—
147#2366—611no—
148#1957—611no—
149#1948—612no—
150C2×#331—611no—
151#1279—612no—
152C2×C61—82no—
153C2×C61#185611yescharacteristic
154C2×C61#186412yes—
155D61—82no—
156D61#162011yescharacteristic
157D61#162911yescharacteristic
158D61#164212yes—
159C2×#127—611no—
160C2×C4×D4—1222no—
161C2×#106—1222no—
162C2×C4⋊C8—612no—
163C2×#102—1222no—
164C2×C4×C8—611no—
165#390—1222no—
166#386—1222no—
167#384—611no—
168C4×C4⋊C4—612no—
169#340—1222no—
170#331—611no—
171#331—612no—
172#331#692311yescharacteristic, complemented
173C61#892411yescharacteristic, complemented, Sylow
174C22×D4—612no—
175C22×D4—1222no—
176C23×C4—1222no—
177C22×C8—1222no—
178C22×C8—1221no—
179#127—612no—
180#127—611no—
181#127—1222no—
182#127C2×#692311yescharacteristic
183C4×D4—2444no—
184C4×D4—2444no—
185C2×C4⋊C4—612no—
186C2×C4⋊C4—612no—
187C2×C4⋊C4—1222no—
188C2×C4⋊C4—1221no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1224no—
191C2×C42—611no—
192C2×C42—1222no—
193#106—2444no—
194C4⋊C8—612no—
195C4⋊C8—1222no—
196C4⋊C8C2×#692312yescomplemented
197#102—2444no—
198C4×C8—1222no—
199#95—1222no—
200C24—1222no—
201C2×D4—22no—
202C2×D4—612no—
203C2×D4—1222no—
204C2×D4—1224no—
205C2×D4—1222no—
206C2×D4—2442no—
207C2×D4—2442no—
208C2×D4—2442no—
209C2×D4—2442no—
210C2×D4—2442no—
211C2×D4—2442no—
212C2×D4C4⋊#692312yes—
213C22×C4—612no—
214C22×C4—611no—
215C22×C4—1222no—
216C22×C4—1222no—
217C22×C4—4882no—
218C22×C4—2448no—
219C22×C4—2441no—
220C2×C8—22no—
221C2×C8—1222no—
222C2×C8—1222no—
223C2×C8—2442no—
224C2×C8—2442no—
225C2×C8—2441no—
226C2×C8—2441no—
227C4⋊C4—614no—
228C4⋊C4—1222no—
229C4⋊C4—1222no—
230C4⋊C4—1222no—
231C4⋊C4—2442no—
232C4⋊C4—2442no—
233C4⋊C4—2442no—
234C22⋊C4—4882no—
235C22⋊C4—2448no—
236C42—611no—
237C42—612no—
238C42—2442no—
239C42—2442no—
240C42C22×#692311yescharacteristic
241C23—22no—
242C23—611no—
243C23—1222no—
244C23—2442no—
245C23—2444no—
246C23—2442no—
247C23—2444no—
248D4—42no—
249D4—42no—
250D4—42no—
251D4—4882no—
252D4—2442no—
253D4—2444no—
254D4—2444no—
255D4—2442no—
256D4—2442no—
257C2×C4—612no—
258C2×C4—612no—
259C2×C4—612no—
260C2×C4—612no—
261C2×C4—612no—
262C2×C4—611no—
263C2×C4—1221no—
264C2×C4—1221no—
265C2×C4—1224no—
266C2×C4—4881no—
267C2×C4—4882no—
268C2×C4—4888no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4—2442no—
272C2×C4—2442no—
273C2×C4—2442no—
274C2×C4C2×C4⋊#692311yescharacteristic
275C2×C4D4×#692312yes—
276C8—42no—
277C8—2442no—
278C8—2442no—
279C22—42no—
280C22—44no—
281C22—611no—
282C22—611no—
283C22—612no—
284C22—612no—
285C22—4882no—
286C22—4882no—
287C22—4884no—
288C22—2442no—
289C22—2444no—
290C22#1684070911yescharacteristic
291C4—21no—
292C4—612no—
293C4—612no—
294C4—1221no—
295C4—9761no—
296C4—4882no—
297C4—2444no—
298C4SD16×#692312yes—
299C4#1684080312yes—
300C2—82no—
301C2—611no—
302C2—612no—
303C2—611no—
304C2—4882no—
305C2#2146177411yescharacteristic
306C2#2146178412yes—
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#331C4
socleC22×C61#385
prime radical#127×D61C22
cosocleC42×D61C23
hypercenter#331#6923
hypocenterC61#8924
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #21461931C2semidirect
4. #21460653C2semidirect
17. #16839506C4semidirect
18. #16839505C4semidirect
19. C61×#331C4semidirect
42. C4⋊#92418C2×C4semidirect
45. C4⋊C8×C61C2×C4semidirect
172. #331#6923semidirect
173. C61#8924semidirect
196. C4⋊C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623492
#214619472
#214619312
#214606532
#331×D612
#892461