Finite Group Databasebeta

← #29592544

Subgroups — #29592544

Order 15616.

340 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 59 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#21447769C212yesmaximal
2#21447106C211yesmaximal, characteristic
3#21443813C212yesmaximal, complemented
4#21443770C211yesmaximal, characteristic, complemented
5C61×#3255C211yesmaximal, characteristic, complemented
6#16840562C2211yescharacteristic
7#16840203—22no—
8C2×#14519705—21no—
9#16839744—21no—
10#16839571—22no—
11#16839544C2212yes—
12#16839520C2211yescharacteristic
13C61×#533—21no—
14C61×#475—22no—
15C61×#470—21no—
16C61×#470C2212yes—
17C61×#462—22no—
18C61×#423C2211yescharacteristic
19C2×#11758222—21no—
20#14519810—41no—
21#14519805—41no—
22#14519799—22no—
23C2×#11758194—21no—
24#14519723—42no—
25#14519720—42no—
26#14519717—22no—
27#14519705—41no—
28#14519705—41no—
29#14519695—41no—
30#14519693—41no—
31C2×Q16×C61—21no—
32C2×Q16×C61D412yes—
33C2×M16×C61D411yescharacteristic
34C2×M16×C61C2311yescharacteristic
35C4⋊Q8×C61—21no—
36C4⋊Q8×C61—21no—
37#125×C61—41no—
38C22⋊Q8×C61—22no—
39C22⋊Q8×C61—41no—
40C4×Q8×C61—22no—
41C4×Q8×C61—41no—
42#117×C61—41no—
43C2×C42×C61D411yescharacteristic
44#106×C61—21no—
45#106×C61D412yes—
46C4⋊C8×C61—22no—
47#103×C61—22no—
48#103×C61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—41no—
51C22×C4×C61—21no—
52C22×C4×C61C2×D411yescharacteristic
53C2×C22⋊D61—41no—
54#11758222—41no—
55#11758222—41no—
56#11758222—41no—
57#11758222—41no—
58C2×D244—21no—
59C2×C4×D61—41no—
60C2×Dic122—21no—
61Q16×C61—42no—
62Q16×C61—81no—
63M16×C61—41no—
64M16×C61—41no—
65C2×C8×C61—21no—
66C2×C8×C61C2×D412yes—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61—41no—
70C4⋊C4×C61—42no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61—42no—
73C22⋊C4×C61—42no—
74C42×C61—21no—
75C42×C61—42no—
76C42×C61D811yescharacteristic
77C42×C61D811yescharacteristic
78#11758194—41no—
79#11758194—41no—
80C2×#92418—41no—
81Dic244—22no—
82D488—22no—
83#11758190—42no—
84#11758185—42no—
85C23×C61AGL(1,ℤ/8)11yescharacteristic
86C22×D61—41no—
87Q8×C61—42no—
88Q8×C61—42no—
89Q8×C61—82no—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61#12011yescharacteristic
98C2×C4×C61C2×D811yescharacteristic
99C22⋊D61—81no—
100C22⋊D61—81no—
101C2×Dic61—41no—
102D244—41no—
103D244—41no—
104C4×D61—81no—
105C4×D61—81no—
106Dic122—41no—
107Dic122—41no—
108C8×C61—22no—
109C8×C61—41no—
110#92418—81no—
111#12174—611nomaximal, Sylow
112C22×C61—21no—
113C22×C61—41no—
114C22×C61#44711yescharacteristic
115C2×D61—81no—
116C2×D61—161no—
117C4×C61—22no—
118C4×C61—21no—
119C4×C61—41no—
120C4×C61—82no—
121C4×C61—82no—
122C4×C61#45411yescharacteristic
123C4×C61B(4,𝔽2)11yescharacteristic
124Dic61—81no—
125#3255D6111yescharacteristic, complemented
126#2191—612no—
127#2050—611no—
128#1342—612no—
129#1327—611no—
130C2×C61—21no—
131C2×C61—41no—
132C2×C61#159811yescharacteristic
133D61—161no—
134#575—611no—
135#533—21no—
136#510—1222no—
137#475—22no—
138#470—21no—
139#470C2×D6112yes—
140#462—22no—
141#423C2×D6111yescharacteristic
142C2×#104—1221no—
143#413—1221no—
144#360—1222no—
145#349—612no—
146#338—611no—
147C61#1217411yescharacteristic, complemented, Sylow
148C2×#41—1221no—
149#137—2441no—
150AGL(1,ℤ/8)—2441no—
151#135—1222no—
152C2×Q16—21no—
153C2×Q16C22⋊D6112yes—
154C2×M16—1221no—
155C2×M16C22⋊D6111yescharacteristic
156C2×M16C22×D6111yescharacteristic
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159#125—41no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C4×Q8—22no—
163C4×Q8—41no—
164#117—41no—
165C2×C42C22⋊D6111yescharacteristic
166Q32—2442no—
167SD32—2442no—
168M32—1222no—
169#106—21no—
170#106C22⋊D6112yes—
171C4⋊C8—22no—
172#104—2441no—
173#104—2441no—
174#103—22no—
175#103—22no—
176#101—2441no—
177#100—2441no—
178#41—2441no—
179#41—2441no—
180#41—2441no—
181#41—2441no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8—1221no—
185C2×D4—1221no—
186C2×D4—2441no—
187C22×C4—21no—
188C22×C4—2441no—
189C22×C4C2×C22⋊D6111yescharacteristic
190Q16—42no—
191Q16—81no—
192Q16—1222no—
193SD16—2442no—
194D8—1222no—
195M16—41no—
196M16—41no—
197M16—2441no—
198M16—2441no—
199C2×C8—21no—
200C2×C8—2441no—
201C2×C8C2×C22⋊D6112yes—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C22⋊C4—42no—
209C42—21no—
210C42—42no—
211C42#1175819911yescharacteristic
212C42#1175819911yescharacteristic
213C16—2442no—
214C23—2441no—
215C23#1451980311yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—82no—
219Q8—2441no—
220Q8—2441no—
221D4—4881no—
222D4—4881no—
223D4—2441no—
224D4—2441no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—4881no—
233C2×C4—4881no—
234C2×C4—2441no—
235C2×C4#1451974611yescharacteristic
236C2×C4C2×#1175819911yescharacteristic
237C8—22no—
238C8—41no—
239C8—4881no—
240C22—21no—
241C22—41no—
242C22—9761no—
243C22—4881no—
244C22#1683989611yescharacteristic
245C4—22no—
246C4—21no—
247C4—41no—
248C4—82no—
249C4—82no—
250C4—4881no—
251C4#1683993511yescharacteristic
252C4#1683994911yescharacteristic
253C2—21no—
254C2—41no—
255C2—9761no—
256C2#2144475411yescharacteristic
subgroupG/H
centerC2#21444754
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#3255C2
socleC2×C61#1598
prime radical#16840562C22
hypercenter#3255D61
hypocenterC61#12174
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3255

Complements

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

NH ≅ G/Ntype
3. #21443813C2semidirect
4. #21443770C2semidirect
5. C61×#3255C2semidirect
125. #3255D61semidirect
147. C61#12174semidirect

Maximal subgroups

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

subgroupindex
#214477692
#214471062
#214438132
#214437702
C61×#32552
#1217461