Finite Group Databasebeta

← #29593884

Subgroups — #29593884

Order 15616.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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#21448042C211yesmaximal, characteristic
2#21447961C211yesmaximal, characteristic, complemented
3#21445213C211yesmaximal, characteristic, complemented
4#21443865C211yesmaximal, characteristic
5Q8⋊#11758185C211yesmaximal, characteristic, complemented
6#21443783C211yesmaximal, characteristic, complemented
7C61×#3396C211yesmaximal, characteristic
8#16840167C2211yescharacteristic
9#16839972—21no—
10#16839589—21no—
11C4⋊#11758185C2211yescharacteristic
12#16839566—21no—
13#16839563—22no—
14C4×#11758185C2211yescharacteristic
15#16839515C2211yescharacteristic
16#16839507—21no—
17C61×#559—21no—
18C61×#495—21no—
19C61×#493C2211yescharacteristic
20C61×#463—22no—
21C61×#460—21no—
22C8×Q8×C61C2211yescharacteristic
23C4×Q16×C61C2211yescharacteristic
24C4×D8×C61—21no—
25C2×#11758202—41no—
26#14519777—21no—
27C2×#11758185—21no—
28C2×#11758185—21no—
29#14519711—21no—
30#14519711—21no—
31C4⋊#92418—21no—
32#14519700—41no—
33C2×Q16×C61D411yescharacteristic
34C2×SD16×C61—22no—
35C2×D8×C61—21no—
36C2×D8×C61D412yes—
37#127×C61—21no—
38#124×C61—22no—
39#124×C61—41no—
40#123×C61—42no—
41C4×Q8×C61—21no—
42C4×Q8×C61D411yescharacteristic
43C4×D4×C61—22no—
44C4×D4×C61—41no—
45#107×C61D411yescharacteristic
46C4⋊C8×C61—21no—
47C4⋊C8×C61D411yescharacteristic
48#103×C61—21no—
49#102×C61—22no—
50C4×C8×C61—21no—
51C4×C8×C61C2311yescharacteristic
52C2×Q8×C61—21no—
53C2×Q8×C61D811yescharacteristic
54C2×D4×C61—22no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C22×C4×C61—42no—
58C2×Dic122—41no—
59Q16×C61—21no—
60Q16×C61—21no—
61SD16×C61—82no—
62D8×C61—42no—
63D8×C61—41no—
64D8×C61—41no—
65C2×C8×C61—21no—
66C2×C8×C61C2×D411yescharacteristic
67C2×C8×C61C2×D411yescharacteristic
68C4⋊C4×C61—21no—
69C4⋊C4×C61—21no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61D811yescharacteristic
72C22⋊C4×C61—42no—
73C22⋊C4×C61—42no—
74C22⋊C4×C61—42no—
75C42×C61—21no—
76C42×C61—41no—
77C42×C61C2×D411yescharacteristic
78#11758202—82no—
79C4⋊Dic61—41no—
80C4⋊Dic61—41no—
81C2×#92418—41no—
82#11758185—41no—
83#11758185—41no—
84C23×C61—42no—
85Q8×C61—21no—
86Q8×C61—21no—
87Q8×C61—41no—
88Q8×C61—41no—
89D4×C61—42no—
90D4×C61—42no—
91D4×C61—81no—
92D4×C61—81no—
93D4×C61—161no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61—162no—
99C2×C4×C61#12011yescharacteristic
100C2×C4×C61C2×D811yescharacteristic
101C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
102C2×Dic61—81no—
103Dic122—81no—
104Dic122—81no—
105C8×C61—21no—
106C8×C61—41no—
107C8×C61#13511yescharacteristic
108C8×C61#13711yescharacteristic
109#92418—81no—
110#12381—611nomaximal, Sylow
111C22×C61—82no—
112C22×C61—82no—
113C22×C61—82no—
114C22×C61#44711yescharacteristic
115C4×C61—21no—
116C4×C61—41no—
117C4×C61—41no—
118C4×C61—81no—
119C4×C61—81no—
120C4×C61#45211yescharacteristic
121C4×C61#45511yescharacteristic
122C4×C61#50811yescharacteristic
123C4×C61#50911yescharacteristic
124Dic61—161no—
125#3396D6111yescharacteristic
126#2240—611no—
127#2225—611no—
128#1674—611no—
129#1361—611no—
130Q8⋊C16—611no—
131#1332—611no—
132C2×C61—162no—
133C2×C61#164111yescharacteristic
134C2×C61#218611yescharacteristic
135C2×C61#219411yescharacteristic
136#559—21no—
137#500—611no—
138#495—21no—
139#493C2×D6111yescharacteristic
140#463—22no—
141#462—1221no—
142#460—21no—
143C8×Q8C2×D6111yescharacteristic
144C4×Q16C2×D6111yescharacteristic
145C4×D8—21no—
146#367—1221no—
147C4⋊C16—611no—
148#358—1221no—
149#357—1222no—
150C4×C16—611no—
151#335—611no—
152#332—1221no—
153C61#1238111yescharacteristic, complemented, Sylow
154C2×Q16—2441no—
155C2×Q16C22⋊D6111yescharacteristic
156C2×SD16—22no—
157C2×D8—21no—
158C2×D8C22⋊D6112yes—
159C4⋊Q8—1221no—
160#127—21no—
161#124—22no—
162#124—41no—
163#123—42no—
164C4×Q8—21no—
165C4×Q8C22⋊D6111yescharacteristic
166C4×D4—22no—
167C4×D4—41no—
168C2×C16—1221no—
169C2×C16—1221no—
170#107—1221no—
171#107—1221no—
172#107C22⋊D6111yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—1221no—
175C4⋊C8C22⋊D6111yescharacteristic
176#103—21no—
177#103—2441no—
178#102—22no—
179C4×C8—21no—
180C4×C8C22×D6111yescharacteristic
181C2×Q8—21no—
182C2×Q8—2441no—
183C2×Q8#1175819911yescharacteristic
184C2×D4—22no—
185C2×D4—41no—
186C2×D4—41no—
187C22×C4—42no—
188Q16—21no—
189Q16—21no—
190Q16—4882no—
191SD16—82no—
192D8—42no—
193D8—41no—
194D8—41no—
195C2×C8—21no—
196C2×C8—2441no—
197C2×C8C2×C22⋊D6111yescharacteristic
198C2×C8C2×C22⋊D6111yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—2441no—
203C4⋊C4—2441no—
204C4⋊C4#1175819911yescharacteristic
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—21no—
209C42—41no—
210C42C2×C22⋊D6111yescharacteristic
211C16—2441no—
212C16—2441no—
213C23—42no—
214Q8—21no—
215Q8—21no—
216Q8—41no—
217Q8—41no—
218Q8—4881no—
219Q8—4881no—
220D4—42no—
221D4—42no—
222D4—81no—
223D4—81no—
224D4—161no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—162no—
230C2×C4—4881no—
231C2×C4#1451974611yescharacteristic
232C2×C4C2×#1175819911yescharacteristic
233C2×C4#1451980311yescharacteristic
234C8—21no—
235C8—41no—
236C8—4881no—
237C8#1451979711yescharacteristic
238C8#1451980711yescharacteristic
239C22—82no—
240C22—82no—
241C22—82no—
242C22#1683989611yescharacteristic
243C4—21no—
244C4—41no—
245C4—41no—
246C4—81no—
247C4—81no—
248C4—9761no—
249C4#1683992411yescharacteristic
250C4#1683994011yescharacteristic
251C4#1684019211yescharacteristic
252C4#1684019711yescharacteristic
253C2—162no—
254C2#2144501211yescharacteristic
255C2#2144774211yescharacteristic
256C2#2144779011yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#3396C2
socleC22×C61#447
prime radicalC61×#493C22
hypercenter#3396D61
hypocenterC61#12381
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
2. #21447961C2semidirect
3. #21445213C2semidirect
5. Q8⋊#11758185C2semidirect
6. #21443783C2semidirect
153. C61#12381semidirect

Maximal subgroups

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

subgroupindex
#214480422
#214479612
#214452132
#214438652
Q8⋊#117581852
#214437832
C61×#33962
#1238161