Finite Group Databasebeta

← #29593696

Subgroups — #29593696

Order 15616.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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#21448044C211yesmaximal, characteristic
2#21447859C211yesmaximal, characteristic
3#21445179C211yesmaximal, characteristic
4#21443874C211yesmaximal, characteristic, complemented
5Q8⋊#11758185C211yesmaximal, characteristic, complemented
6#21443783C211yesmaximal, characteristic, complemented
7C61×#3383C211yesmaximal, characteristic, complemented
8C2×#14519720—21no—
9#16840139C2211yescharacteristic
10#16839962—21no—
11C4⋊#11758185C2211yescharacteristic
12#16839562—22no—
13#16839562—21no—
14C4×#11758185C2211yescharacteristic
15#16839515C2211yescharacteristic
16#16839505—21no—
17C61×#557—21no—
18C61×#500C2211yescharacteristic
19C61×#499—21no—
20C61×#479—22no—
21C61×#475—21no—
22C8×Q8×C61C2211yescharacteristic
23C4×Q16×C61—21no—
24C4×Q16×C61C2211yescharacteristic
25C2×#11758201—41no—
26C2×D488—21no—
27C2×D488—21no—
28#14519774—21no—
29#14519720—42no—
30C2×#11758185—21no—
31C2×#11758185—21no—
32C4⋊#92418—21no—
33#14519697—41no—
34C2×Q16×C61D411yescharacteristic
35C4⋊Q8×C61—21no—
36C4⋊Q8×C61—41no—
37#125×C61—22no—
38#125×C61—42no—
39C4×Q8×C61—22no—
40C4×Q8×C61—21no—
41C4×Q8×C61—41no—
42C4×Q8×C61D411yescharacteristic
43#107×C61—21no—
44#107×C61D411yescharacteristic
45#107×C61D412yes—
46#106×C61—22no—
47C4⋊C8×C61—21no—
48C4⋊C8×C61D411yescharacteristic
49#103×C61—22no—
50#103×C61—21no—
51C4×C8×C61—21no—
52C4×C8×C61C2311yescharacteristic
53C2×Q8×C61—21no—
54C2×Q8×C61—41no—
55C2×Q8×C61D811yescharacteristic
56C2×D244—41no—
57C2×D244—41no—
58C2×D244—41no—
59Q16×C61—21no—
60Q16×C61SD1612yescomplemented
61C2×C8×C61—21no—
62C2×C8×C61C2×D411yescharacteristic
63C2×C8×C61C2×D411yescharacteristic
64C4⋊C4×C61—21no—
65C4⋊C4×C61—21no—
66C4⋊C4×C61—22no—
67C4⋊C4×C61—42no—
68C4⋊C4×C61—42no—
69C4⋊C4×C61—42no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61D811yescharacteristic
73C42×C61—21no—
74C42×C61—42no—
75C42×C61—41no—
76C42×C61C2×D411yescharacteristic
77#11758201—82no—
78C2×#92418—41no—
79D488—42no—
80D488—41no—
81D488—41no—
82#11758185—41no—
83#11758185—41no—
84C22×D61—81no—
85Q8×C61—22no—
86Q8×C61—21no—
87Q8×C61—21no—
88Q8×C61—41no—
89Q8×C61—81no—
90Q8×C61—81no—
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—41no—
94C2×C4×C61—41no—
95C2×C4×C61—42no—
96C2×C4×C61#12011yescharacteristic
97C2×C4×C61C2×D811yescharacteristic
98C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
99D244—81no—
100D244—81no—
101D244—81no—
102D244—81no—
103D244—161no—
104C8×C61—21no—
105C8×C61—41no—
106C8×C61C2×SD1611yescharacteristic
107C8×C61#13711yescharacteristic
108#92418—81no—
109#12353—611nomaximal, Sylow
110C22×C61#44711yescharacteristic
111C2×D61—161no—
112C2×D61—161no—
113C2×D61—161no—
114C4×C61—21no—
115C4×C61—42no—
116C4×C61—41no—
117C4×C61—41no—
118C4×C61—82no—
119C4×C61—81no—
120C4×C61#44811yescharacteristic
121C4×C61#45411yescharacteristic
122C4×C61#50811yescharacteristic
123C4×C61#50911yescharacteristic
124#3383D6111yescharacteristic, complemented
125#2240—611no—
126#2207—611no—
127#1668—611no—
128#1364—611no—
129Q8⋊C16—611no—
130#1332—611no—
131C2×C61#162711yescharacteristic
132C2×C61#218611yescharacteristic
133C2×C61#219311yescharacteristic
134D61—321no—
135#557—21no—
136C2×SD32—1221no—
137#500C2×D6111yescharacteristic
138#499—21no—
139#493—611no—
140#479—22no—
141#475—21no—
142#460—1221no—
143C8×Q8C2×D6111yescharacteristic
144C4×Q16—21no—
145C4×Q16C2×D6111yescharacteristic
146C4⋊C16—611no—
147#357—1221no—
148#357—1222no—
149C4×C16—611no—
150#335—611no—
151#331—1221no—
152C61#1235311yescharacteristic, complemented, Sylow
153C2×Q16C22⋊D6111yescharacteristic
154C2×SD16—2441no—
155C2×D8—1221no—
156C2×D8—1221no—
157C4⋊Q8—21no—
158C4⋊Q8—41no—
159#127—1221no—
160#125—22no—
161#125—42no—
162C4×Q8—22no—
163C4×Q8—21no—
164C4×Q8—41no—
165C4×Q8C22⋊D6111yescharacteristic
166SD32—2442no—
167C2×C16—1221no—
168C2×C16—1221no—
169#107—21no—
170#107C22⋊D6111yescharacteristic
171#107C22⋊D6112yes—
172#106—22no—
173C4⋊C8—21no—
174C4⋊C8—1221no—
175C4⋊C8C22⋊D6111yescharacteristic
176#103—22no—
177#103—21no—
178#102—2441no—
179C4×C8—21no—
180C4×C8C22×D6111yescharacteristic
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8#1175819911yescharacteristic
184C2×D4—2441no—
185C2×D4—2441no—
186C2×D4—2441no—
187Q16—21no—
188Q16#1175820112yescomplemented
189SD16—4882no—
190D8—2442no—
191D8—2441no—
192D8—2441no—
193C2×C8—21no—
194C2×C8—2441no—
195C2×C8C2×C22⋊D6111yescharacteristic
196C2×C8C2×C22⋊D6111yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4#1175819911yescharacteristic
206C42—21no—
207C42—42no—
208C42—41no—
209C42C2×C22⋊D6111yescharacteristic
210C16—2441no—
211C16—2441no—
212C23—4881no—
213Q8—21no—
214Q8—21no—
215Q8—22no—
216Q8—41no—
217Q8—81no—
218Q8—81no—
219D4—9761no—
220D4—4881no—
221D4—4881no—
222D4—4881no—
223D4—4881no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4#1451974611yescharacteristic
230C2×C4C2×#1175819911yescharacteristic
231C2×C4#1451980311yescharacteristic
232C8—21no—
233C8—41no—
234C8—4881no—
235C8C2×#1175820111yescharacteristic
236C8#1451980811yescharacteristic
237C22—9761no—
238C22—9761no—
239C22—9761no—
240C22#1683989611yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—81no—
247C4#1683990111yescharacteristic
248C4#1683993511yescharacteristic
249C4#1684019211yescharacteristic
250C4#1684019711yescharacteristic
251C2—19521no—
252C2#2144492211yescharacteristic
253C2#2144774211yescharacteristic
254C2#2144778311yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#3383C2
socleC22×C61#447
prime radical#16840139C22
hypercenter#3383D61
hypocenterC61#12353
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
4. #21443874C2semidirect
5. Q8⋊#11758185C2semidirect
6. #21443783C2semidirect
7. C61×#3383C2semidirect
60. Q16×C61SD16semidirect
124. #3383D61semidirect
152. C61#12353semidirect
188. Q16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214480442
#214478592
#214451792
#214438742
Q8⋊#117581852
#214437832
C61×#33832
#1235361