Finite Group Databasebeta

← #29577613

Subgroups — #29577613

Order 15616.

504 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 111 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#21454886C211yesmaximal, characteristic
2#21444554C211yesmaximal, characteristic
3#21444504C212yesmaximal, complemented
4#21443709C212yesmaximal, complemented
5C61×#3152C211yesmaximal, characteristic
6#16839840—22no—
7D4×#92418—22no—
8#16839815C2212yes—
9#16839786C2212yes—
10#16839729—22no—
11#16839511—21no—
12#16839508—22no—
13#16839508—22no—
14#16839506—21no—
15C2×#137×C61—22no—
16C61×#532C2211yescharacteristic
17C2×C4⋊Q8×C61C2211yescharacteristic
18C61×#497C414yes—
19C61×#494—21no—
20C61×#492—21no—
21C4×M16×C61C2211yescharacteristic
22C2×#11758194—22no—
23C2×#11758194—22no—
24C22×#92418—22no—
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27C4⋊#92418D412yes—
28C4⋊#92418D412yes—
29C22⋊#92418—24no—
30C22⋊#92418—22no—
31C4×#92418—22no—
32C2×#41×C61D412yes—
33C22×Q8×C61—21no—
34#137×C61—44no—
35#137×C61—44no—
36C2×Q16×C61—24no—
37C2×SD16×C61—24no—
38C2×M16×C61—21no—
39C4⋊Q8×C61—22no—
40C4⋊Q8×C61—21no—
41C4⋊Q8×C61C2×C412yes—
42C4⋊Q8×C61D411yescharacteristic
43#127×C61D411yescharacteristic
44#124×C61C2×C412yes—
45#121×C61—22no—
46C4×D4×C61—22no—
47C2×C4⋊C4×C61—22no—
48C2×C42×C61C2311yescharacteristic
49#97×C61C2×C412yes—
50C4×C8×C61—21no—
51#41×C61—42no—
52#41×C61—42no—
53C2×Q8×C61—22no—
54C2×Q8×C61—42no—
55C2×Q8×C61—42no—
56C2×Q8×C61—41no—
57C2×Q8×C61C22⋊C412yes—
58C2×D4×C61—22no—
59C2×D4×C61—22no—
60C2×D4×C61C22⋊C412yes—
61C22×C4×C61—22no—
62C22×C4×C61—41no—
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61#4112yes—
65Q16×C61—48no—
66SD16×C61—48no—
67M16×C61—22no—
68M16×C61—22no—
69C2×C8×C61—22no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—24no—
72C4⋊C4×C61—42no—
73C22⋊C4×C61—24no—
74C42×C61C22×C411yescharacteristic
75C42×C61C2×D411yescharacteristic
76C42×C61C2×D412yes—
77#11758194—82no—
78#11758194—82no—
79C2×#92418—22no—
80C2×#92418—22no—
81C2×#92418—42no—
82C2×#92418—82no—
83C23×C61—22no—
84C23×C61#12311yescharacteristic
85Q8×C61—42no—
86Q8×C61—44no—
87Q8×C61—81no—
88Q8×C61—81no—
89D4×C61—42no—
90D4×C61—42no—
91D4×C61—42no—
92D4×C61—82no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—82no—
102C2×C4×C61C2×C22⋊C411yescharacteristic
103C2×C4×C61C4×D412yes—
104C2×C4×C61#12011yescharacteristic
105C2×C4×C61#12112yes—
106C8×C61—24no—
107#92418—42no—
108#92418—42no—
109#92418—82no—
110#9435—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—22no—
113C22×C61—42no—
114C22×C61—44no—
115C22×C61#38611yescharacteristic
116C4×C61—21no—
117C4×C61—22no—
118C4×C61—42no—
119C4×C61—82no—
120C4×C61#45311yescharacteristic
121C4×C61#45311yescharacteristic
122C4×C61#47112yes—
123C4×C61#47312yes—
124#3152D6111yescharacteristic
125#2979—611no—
126#1560—611no—
127#1549—612no—
128#1299—612no—
129C2×C61—41no—
130C2×C61—82no—
131C2×C61#188711yescharacteristic
132C2×C61#190812yes—
133C2×#137—22no—
134#532C2×D6111yescharacteristic
135C2×C4⋊Q8C2×D6111yescharacteristic
136#497Dic6114yes—
137#494—21no—
138#492—21no—
139#436—1222no—
140C8×D4—1222no—
141#431—612no—
142#423—612no—
143#408—1222no—
144C4×M16C2×D6111yescharacteristic
145#333—1221no—
146#332—1222no—
147#332—1222no—
148#331—1221no—
149C61#943511yescharacteristic, complemented, Sylow
150C2×#41C22⋊D6112yes—
151C22×Q8—21no—
152#137—44no—
153#137—44no—
154C2×Q16—24no—
155C2×SD16—24no—
156C2×M16—21no—
157C2×M16—1222no—
158C2×M16—1222no—
159C22×C8—1222no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8C2×Dic6112yes—
163C4⋊Q8C22⋊D6111yescharacteristic
164#127C22⋊D6111yescharacteristic
165#124C2×Dic6112yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—22no—
169C2×C42C22×D6111yescharacteristic
170C4⋊C8—612no—
171C4⋊C8—612no—
172C4⋊C8—1222no—
173C4⋊C8—1222no—
174C22⋊C8—1224no—
175C22⋊C8—1222no—
176#97C2×Dic6112yes—
177C4×C8—21no—
178C4×C8—1222no—
179#41—42no—
180#41—42no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8C22⋊Dic6112yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4C22⋊Dic6112yes—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4#1175822412yes—
192C22×C4C2×C22⋊D6111yescharacteristic
193Q16—48no—
194SD16—48no—
195M16—22no—
196M16—22no—
197M16—4882no—
198M16—4882no—
199C2×C8—22no—
200C2×C8—1222no—
201C2×C8—1222no—
202C2×C8—4882no—
203C2×C8—2442no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—42no—
207C22⋊C4—24no—
208C42D4×D6112yes—
209C42C22×Dic6111yescharacteristic
210C42C2×C22⋊D6111yescharacteristic
211C23—22no—
212C23#1451975811yescharacteristic
213Q8—42no—
214Q8—44no—
215Q8—81no—
216Q8—81no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4C2×C22⋊Dic6111yescharacteristic
231C2×C4D4×Dic6112yes—
232C2×C4#1451974511yescharacteristic
233C2×C4#1451974912yes—
234C8—24no—
235C8—4882no—
236C8—2442no—
237C8—2442no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—44no—
242C22#1683964211yescharacteristic
243C4—21no—
244C4—22no—
245C4—42no—
246C4—82no—
247C4#1683992911yescharacteristic
248C4#1683992911yescharacteristic
249C4#1684002712yes—
250C4#1684003512yes—
251C2—41no—
252C2—82no—
253C2#2144621811yescharacteristic
254C2#2144633612yes—
subgroupG/H
centerC22#16839642
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3152C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3152D61
hypocenterC61#9435
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
3. #21444504C2semidirect
4. #21443709C2semidirect
149. C61#9435semidirect

Maximal subgroups

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

subgroupindex
#214548862
#214445542
#214445042
#214437092
C61×#31522
#943561