Finite Group Databasebeta

← #29580955

Subgroups — #29580955

Order 15616.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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#21455937C212yesmaximal, complemented
2#21446758C211yesmaximal, characteristic, complemented
3#21444430C212yesmaximal, complemented
4#21443689C211yesmaximal, characteristic, complemented
5C61×#3245C211yesmaximal, characteristic
6C2×C4×Dic122C2211yescharacteristic
7#16840073—22no—
8#16840054—22no—
9#16839959—22no—
10#16839900—22no—
11#16839862—24no—
12#16839819C2212yes—
13C2×#14519688C2211yescharacteristic
14#16839516—22no—
15#16839502—24no—
16C61×#532C2212yes—
17C61×#474—22no—
18C61×#466—22no—
19C61×#459—22no—
20C2×C4⋊C8×C61C2211yescharacteristic
21C22×Dic122D411yescharacteristic
22C2×#11758200—24no—
23C22×#92418—21no—
24C4×Dic122—22no—
25C4×Dic122—22no—
26C2×C4⋊Dic61D411yescharacteristic
27C2×#11758196—21no—
28C2×C4×Dic61—21no—
29#14519710—24no—
30C4⋊#92418—22no—
31#14519700—24no—
32#14519698—24no—
33C22⋊#92418—22no—
34#14519688—22no—
35#14519688D414yes—
36C2×#41×C61—22no—
37C2×D8×C61—22no—
38C22×C8×C61D412yes—
39C4⋊Q8×C61D412yes—
40#127×C61D412yes—
41#124×C61—22no—
42#121×C61—22no—
43#121×C61—22no—
44C4×D4×C61—22no—
45C4×D4×C61—22no—
46C2×C42×C61C2311yescharacteristic
47#107×C61—22no—
48C4⋊C8×C61—22no—
49#102×C61—24no—
50#41×C61—84no—
51C2×Q8×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61—42no—
56C22×C4×C61—42no—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61#4111yescharacteristic
60C22×Dic61—21no—
61C22×Dic61—21no—
62C2×Dic122—22no—
63C2×Dic122—42no—
64D8×C61—82no—
65C2×C8×C61—22no—
66C2×C8×C61—42no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—22no—
69C22⋊C4×C61—42no—
70C22⋊C4×C61—42no—
71C42×C61C2×D412yes—
72C42×C61C2×D412yes—
73#11758200—48no—
74C4⋊Dic61—22no—
75#11758196—42no—
76C4×Dic61—41no—
77C4×Dic61—41no—
78C2×#92418—22no—
79C2×#92418—24no—
80C2×#92418—41no—
81C2×#92418—41no—
82C23×C61—42no—
83C23×C61#12111yescharacteristic
84Q8×C61—82no—
85D4×C61—44no—
86D4×C61—42no—
87D4×C61—42no—
88D4×C61—82no—
89D4×C61—84no—
90C2×C4×C61—22no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—41no—
94C2×C4×C61—42no—
95C2×C4×C61—84no—
96C2×C4×C61#12012yes—
97C2×C4×C61#12011yescharacteristic
98C2×C4×C61#12112yes—
99C2×C4×C61#12711yescharacteristic
100C2×Dic61—22no—
101C2×Dic61—41no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104C2×Dic61—42no—
105Dic122—24no—
106Dic122—42no—
107C8×C61—42no—
108#92418—42no—
109#92418—44no—
110#10028—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—21no—
113C22×C61—82no—
114C22×C61—82no—
115C22×C61—82no—
116C22×C61#39211yescharacteristic
117C22×C61#47212yes—
118C4×C61—21no—
119C4×C61—22no—
120C4×C61—22no—
121C4×C61—82no—
122C4×C61#45511yescharacteristic
123C4×C61#45611yescharacteristic
124Dic61—42no—
125Dic61—81no—
126#3245D6111yescharacteristic
127#3112—612no—
128#1985—611no—
129#1530—612no—
130#1290—611no—
131C2×C61—22no—
132C2×C61—162no—
133C2×C61#200711yescharacteristic
134C2×C61#201011yescharacteristic
135C2×C61#201811yescharacteristic
136#532C2×D6112yes—
137C2×C4×Q8—611no—
138#480—1222no—
139#477—1222no—
140#474—22no—
141#466—22no—
142#460—1222no—
143#459—22no—
144#448—1222no—
145#440—1224no—
146#432—612no—
147C2×C4⋊C8C2×D6111yescharacteristic
148C2×#97—611no—
149#336—1222no—
150#329—1224no—
151C61#1002811yescharacteristic, complemented, Sylow
152C2×#41—22no—
153C22×Q8—611no—
154C2×SD16—1224no—
155C2×D8—22no—
156C22×C8—1221no—
157C22×C8C22⋊D6112yes—
158C4⋊Q8C22⋊D6112yes—
159#127C22⋊D6112yes—
160#124—22no—
161#121—22no—
162#121—22no—
163C4×Q8—1222no—
164C4×Q8—1222no—
165C4×D4—22no—
166C4×D4—22no—
167C2×C4⋊C4—611no—
168C2×C4⋊C4—1221no—
169C2×C42—1221no—
170C2×C42C22×D6111yescharacteristic
171#107—22no—
172#107—1224no—
173C4⋊C8—22no—
174C4⋊C8—1222no—
175#103—1224no—
176#102—24no—
177#102—1224no—
178C22⋊C8—1222no—
179#97—614no—
180#97—1222no—
181#41—84no—
182C2×Q8—22no—
183C2×Q8—1222no—
184C2×Q8—2442no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—42no—
189C22×C4—42no—
190C22×C4—1221no—
191C22×C4—1221no—
192C22×C4D4×D6111yescharacteristic
193C22×C4#1175822411yescharacteristic
194C22×C4C2×C22⋊D6111yescharacteristic
195SD16—2448no—
196D8—82no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—1222no—
200C2×C8—1224no—
201C2×C8—2441no—
202C2×C8—2441no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—1222no—
206C4⋊C4—2442no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—2441no—
210C42—2441no—
211C42D4×D6112yes—
212C42C2×C22⋊D6112yes—
213C23—42no—
214C23#1451974811yescharacteristic
215Q8—82no—
216Q8—1224no—
217Q8—2442no—
218D4—44no—
219D4—42no—
220D4—42no—
221D4—82no—
222D4—84no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—1222no—
230C2×C4—2442no—
231C2×C4—2441no—
232C2×C4—2441no—
233C2×C4—2441no—
234C2×C4#1451974512yes—
235C2×C4#1451974611yescharacteristic
236C2×C4#1451974912yes—
237C2×C4#1451977311yescharacteristic
238C8—42no—
239C8—2442no—
240C8—2444no—
241C22—21no—
242C22—21no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#1683967011yescharacteristic
247C22#1684002912yes—
248C4—22no—
249C4—21no—
250C4—22no—
251C4—82no—
252C4—4881no—
253C4—2442no—
254C4#1683994011yescharacteristic
255C4#1683994511yescharacteristic
256C2—22no—
257C2—162no—
258C2#2144686611yescharacteristic
259C2#2144688311yescharacteristic
260C2#2144692611yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3245C2
socleC22×C61#392
hypercenter#3245D61
hypocenterC61#10028
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21455937C2semidirect
2. #21446758C2semidirect
3. #21444430C2semidirect
4. #21443689C2semidirect
151. C61#10028semidirect

Maximal subgroups

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

subgroupindex
#214559372
#214467582
#214444302
#214436892
C61×#32452
#1002861