Finite Group Databasebeta

← #30599778

Subgroups — #30599778

Order 15616.

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

normal subgroups only

This group has 476 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#21462352C211yesmaximal, characteristic
2#21461946C211yesmaximal, characteristic
3#21461930C212yesmaximal
4#21460653C212yesmaximal
5#333×D61C211yesmaximal, characteristic
6C2×#14519846—22no—
7C2×#14519840—22no—
8C4×D61⋊C8C2211yescharacteristic
9(C2×#6923)⋊Q8—22no—
10#16840713—22no—
11#16840708C2211yescharacteristic
12C4⋊C4×#6923—22no—
13C4×C4⋊#6923C2212yes—
14#16840636—22no—
15C4⋊Q8×D61C2211yescharacteristic
16C4⋊C8×D61C2212yes—
17#16839511C411yescharacteristic, complemented
18#16839510C412yescomplemented
19C61×#333C411yescharacteristic, complemented
20C2×D61⋊C8—21no—
21C2×C4⋊#6923—21no—
22C2×C4⋊#6923—22no—
23C2×C4⋊#6923—22no—
24C2×C4⋊#6923D412yes—
25C2×C4⋊#6923D412yes—
26C2×C4×#6923—22no—
27C2×C4×#6923—22no—
28#14519846—44no—
29#14519840—44no—
30C4×#92420—22no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33C2×Q8×D61D412yes—
34C2×C8×D61—22no—
35#14519777C2×C411yescharacteristic
36#14519776—22no—
37#14519775Q812yes—
38C4⋊C4×D61—22no—
39C42×D61C2311yescharacteristic
40C4⋊#92418C2×C412yes—
41#14519706—22no—
42C4⋊Q8×C61C2×C411yescharacteristic
43C4⋊C8×C61C2×C412yes—
44C22×#6923—22no—
45C22×#6923—22no—
46C22×#6923—41no—
47C2×Q8×C61C4⋊C412yes—
48Q8×D61—42no—
49Q8×D61—42no—
50C2×C4×D61—22no—
51C2×C4×D61C2×D411yescharacteristic
52C2×C4×D61C2×D412yes—
53C2×Dic122—22no—
54C2×Dic122C4⋊C412yes—
55C2×#92420—41no—
56C4⋊#6923—22no—
57C4⋊#6923—22no—
58C4⋊#6923—22no—
59C4⋊#6923—42no—
60C4⋊#6923—42no—
61C4⋊#6923—42no—
62C4⋊#6923—82no—
63C4⋊#6923Q1614yes—
64C4×#6923—42no—
65C4×#6923—42no—
66C4×#6923—82no—
67D61⋊C8—22no—
68D61⋊C8—41no—
69C2×C8×C61—22no—
70C4⋊C4×C61—22no—
71C42×C61C22×C411yescharacteristic
72C4⋊Dic61—22no—
73#11758196—24no—
74C4×Dic61C2×Q811yescharacteristic
75C4×Dic61#4112yes—
76C2×#92418—22no—
77C8×D61—42no—
78C8×D61—42no—
79C22×D61#12011yescharacteristic
80C2×#6923—24no—
81C2×#6923—42no—
82C2×#6923—42no—
83C2×#6923—42no—
84C2×#6923—42no—
85C2×#6923—42no—
86C2×#6923—82no—
87C2×#6923—81no—
88Q8×C61—42no—
89C2×C4×C61—22no—
90C2×C4×C61C2×C4⋊C411yescharacteristic
91C2×C4×C61C4×D412yes—
92C2×Dic61—22no—
93C2×Dic61C22⋊Q812yes—
94C2×Dic61#12311yescharacteristic
95C4×D61—21no—
96C4×D61—21no—
97C4×D61—42no—
98C4×D61—42no—
99C4×D61C2×Q1612yes—
100C4×D61#13512yes—
101C4×D61AGL(1,ℤ/8)12yes—
102C4×D61#13712yes—
103Dic122—42no—
104Dic122—82no—
105#92420—42no—
106C8×C61—42no—
107#92418—42no—
108#8929—611nomaximal, Sylow
109C22×C61#38511yescharacteristic
110C2×D61#44912yes—
111C2×D61#45112yes—
112C2×D61#45311yescharacteristic
113C2×D61#45511yescharacteristic
114#6923—44no—
115#6923—82no—
116#6923—161no—
117C4×C61—21no—
118C4×C61—42no—
119C4×C61C4×Q1612yes—
120C4×C61#44012yes—
121Dic61—21no—
122Dic61—42no—
123Dic61#47712yes—
124Dic61#47912yes—
125#2369—611no—
126#1956—611no—
127#1947—612no—
128C2×#333—611no—
129#1279—612no—
130C2×C61#185611yescharacteristic
131C2×C61#186312yes—
132D61#162411yescharacteristic
133D61#163311yescharacteristic
134D61#164412yes—
135C2×C4⋊Q8—611no—
136C2×#107—1222no—
137C2×C4⋊C8—612no—
138C2×#103—1222no—
139C2×C4×C8—611no—
140#391—1222no—
141#389—1222no—
142#384—611no—
143C4×C4⋊C4—612no—
144C4×C4⋊C4—1222no—
145#340—1222no—
146#333—611no—
147#333—612no—
148#333#692311yescharacteristic, complemented
149C61#892911yescharacteristic, complemented, Sylow
150C22×Q8—612no—
151C22×C8—1222no—
152C22×C8—1221no—
153C4⋊Q8—612no—
154C4⋊Q8—611no—
155C4⋊Q8—1222no—
156C4⋊Q8C2×#692311yescharacteristic
157C2×C4⋊C4—612no—
158C2×C4⋊C4—612no—
159C2×C4⋊C4—1222no—
160C2×C4⋊C4—1222no—
161C2×C4⋊C4—1222no—
162C2×C4⋊C4—1221no—
163C2×C42—611no—
164C2×C42—1222no—
165C2×C42—1222no—
166#107—2444no—
167C4⋊C8—612no—
168C4⋊C8—1222no—
169C4⋊C8C2×#692312yes—
170#103—2444no—
171C4×C8—1222no—
172#95—1222no—
173#95—1224no—
174C2×Q8—612no—
175C2×Q8—1222no—
176C2×Q8—2442no—
177C2×Q8—2442no—
178C2×Q8C4⋊#692312yes—
179C22×C4—612no—
180C22×C4—611no—
181C22×C4—1222no—
182C22×C4—1222no—
183C22×C4—1222no—
184C22×C4—2441no—
185C2×C8—22no—
186C2×C8—1222no—
187C2×C8—1222no—
188C2×C8—2442no—
189C2×C8—2442no—
190C2×C8—2441no—
191C2×C8—2441no—
192C4⋊C4—22no—
193C4⋊C4—614no—
194C4⋊C4—1224no—
195C4⋊C4—1222no—
196C4⋊C4—1222no—
197C4⋊C4—1222no—
198C4⋊C4—1222no—
199C4⋊C4—4882no—
200C4⋊C4—2442no—
201C4⋊C4—2442no—
202C4⋊C4—2442no—
203C42—611no—
204C42—612no—
205C42—4882no—
206C42—2442no—
207C42—2442no—
208C42C22×#692311yescharacteristic
209C23—611no—
210Q8—42no—
211Q8—4882no—
212Q8—2442no—
213C2×C4—22no—
214C2×C4—612no—
215C2×C4—612no—
216C2×C4—612no—
217C2×C4—612no—
218C2×C4—612no—
219C2×C4—611no—
220C2×C4—1221no—
221C2×C4—1221no—
222C2×C4—1224no—
223C2×C4—1222no—
224C2×C4—4881no—
225C2×C4—4882no—
226C2×C4—2442no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4C2×C4⋊#692311yescharacteristic
234C2×C4D4×#692312yes—
235C8—42no—
236C8—2442no—
237C8—2442no—
238C22—611no—
239C22—611no—
240C22—612no—
241C22—612no—
242C22#1684070911yescharacteristic
243C4—21no—
244C4—42no—
245C4—612no—
246C4—612no—
247C4—1221no—
248C4—9761no—
249C4—4882no—
250C4—2444no—
251C4—2442no—
252C4Q16×#692312yes—
253C4#1684080112yes—
254C2—611no—
255C2—612no—
256C2—611no—
257C2#2146177411yescharacteristic
258C2#2146178312yes—
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#333C4
socleC22×C61#385
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#333#6923
hypocenterC61#8929
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
17. #16839511C4semidirect
18. #16839510C4semidirect
19. C61×#333C4semidirect
148. #333#6923semidirect
149. C61#8929semidirect

Maximal subgroups

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

subgroupindex
#214623522
#214619462
#214619302
#214606532
#333×D612
#892961