Finite Group Databasebeta

← #49120299

Subgroups — #49120299

Order 36224.

246 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 51 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#31827042C211yesmaximal, characteristic
2#31826784C211yesmaximal, characteristic, complemented
3#31826781C211yesmaximal, characteristic, complemented
4#31826405C211yesmaximal, characteristic, complemented
5#31826198C211yesmaximal, characteristic, complemented
6#31826190C211yesmaximal, characteristic, complemented
7#327×C283C211yesmaximal, characteristic, complemented
8C2×#17985391—21no—
9#22583588—21no—
10#22583587—21no—
11#22583584C2211yescharacteristic
12#22583583—21no—
13#22583582—21no—
14#22583582—21no—
15C2×C4⋊Dic283C2211yescharacteristic
16C2×Dic283⋊C4C2211yescharacteristic
17#22583542—21no—
18#22583541—21no—
19#22583532—21no—
20#22583531—21no—
21C22⋊(C283⋊C8)C2211yescharacteristic
22C283⋊#95—21no—
23C283⋊#95—21no—
24#121×C283C2211yescharacteristic
25C22⋊C8×C283C2211yescharacteristic
26#95×C283C2211yescharacteristic
27#17985391—42no—
28#17985391—42no—
29C2×Dic566—21no—
30C2×D1132D411yescharacteristic
31C2×C22⋊D283—21no—
32C2×C22⋊D283—21no—
33C2×C22⋊D283—21no—
34C22×Dic283—21no—
35C22×Dic283—21no—
36C22×Dic283D411yescharacteristic
37C22×Dic283D411yescharacteristic
38C2×C4×D283—21no—
39C2×C283⋊C8—21no—
40C4⋊Dic283—21no—
41C4⋊Dic283—21no—
42C4⋊Dic283D411yescharacteristic
43Dic283⋊C4—21no—
44Dic283⋊C4—21no—
45Dic283⋊C4—21no—
46C22⋊Dic283—21no—
47(C2×D283)⋊C4—21no—
48(C2×D283)⋊C4—21no—
49C2×D4×C283—21no—
50C2×D4×C283D411yescharacteristic
51C22×C4×C283—21no—
52C22×C4×C283C2311yescharacteristic
53C2×C8×C283—21no—
54C4⋊C4×C283D411yescharacteristic
55C22⋊C4×C283—21no—
56C22×D283—21no—
57Dic566—42no—
58D1132—41no—
59C22⋊D283—42no—
60C22⋊D283—41no—
61C22⋊D283—41no—
62C22⋊D283—81no—
63C2×Dic283—21no—
64C2×Dic283—21no—
65C2×Dic283—21no—
66C2×Dic283—21no—
67C2×Dic283—42no—
68C2×Dic283—41no—
69C2×Dic283—42no—
70C2×Dic283—41no—
71C2×Dic283SD1612yes—
72C4×D283—42no—
73C283⋊C8—41no—
74C23×C283—21no—
75C23×C283C2×D411yescharacteristic
76D4×C283—41no—
77D4×C283—41no—
78D4×C283—41no—
79C2×C4×C283—21no—
80C2×C4×C283—41no—
81C2×C4×C283—41no—
82C2×C4×C283—41no—
83C2×C4×C283—41no—
84C2×C4×C283C2×D411yescharacteristic
85C2×C4×C283C2×D411yescharacteristic
86C8×C283—41no—
87C22×C283—21no—
88C22×C283—21no—
89C22×C283—41no—
90C22×C283—41no—
91C22×C283—41no—
92C22×C283#12011yescharacteristic
93C22×C283C2×SD1611yescharacteristic
94C22×C283AGL(1,ℤ/8)11yescharacteristic
95C2×D283—41no—
96C2×D283—41no—
97C2×D283—41no—
98C4×C283—21no—
99C4×C283—21no—
100C4×C283—41no—
101C4×C283—81no—
102Dic283—22no—
103Dic283—41no—
104Dic283—41no—
105Dic283—81no—
106C2×C283—21no—
107C2×C283—21no—
108C2×C283—81no—
109C2×C283#45011yescharacteristic
110C2×C283#45411yescharacteristic
111C2×C283#45811yescharacteristic
112D283—81no—
113C283#161611yescharacteristic, complemented, Sylow
114#1616—2831nomaximal, Sylow
115#537—2831no—
116#481—2831no—
117#481—2831no—
118#400—2831no—
119#327—2831no—
120#327D28311yescharacteristic, complemented
121#324—2831no—
122C2×#41—5661no—
123C22⋊Q8—5661no—
124C22⋊Q8—5661no—
125#121—2831no—
126#121—5661no—
127#121—5661no—
128#121—5661no—
129#121C2×D28311yescharacteristic
130C2×C4⋊C4—2831no—
131C2×C4⋊C4—2831no—
132#106—5661no—
133#106—5661no—
134#102—5661no—
135#102—5661no—
136C22⋊C8—2831no—
137C22⋊C8C2×D28311yescharacteristic
138#95—5661no—
139#95—5661no—
140#95C2×D28311yescharacteristic
141#41—11322no—
142#41—11322no—
143C2×Q8—5661no—
144C2×D4—21no—
145C2×D4—2831no—
146C2×D4—5661no—
147C2×D4—5661no—
148C2×D4—5661no—
149C2×D4D113211yescharacteristic
150C22×C4—21no—
151C22×C4—2831no—
152C22×C4—2831no—
153C22×C4—5661no—
154C22×C4—5661no—
155C22×C4—5661no—
156C22×C4C22×D28311yescharacteristic
157C2×C8—21no—
158C2×C8—5661no—
159C4⋊C4—2831no—
160C4⋊C4—5661no—
161C4⋊C4—5661no—
162C4⋊C4—5661no—
163C4⋊C4—5661no—
164C4⋊C4—5661no—
165C4⋊C4D113211yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—5661no—
168C22⋊C4—5661no—
169C22⋊C4—5661no—
170C23—21no—
171C23—5661no—
172C23D4×D28311yescharacteristic
173Q8—11322no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—11321no—
178D4—11322no—
179D4—11321no—
180D4—11321no—
181D4—22641no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—2832no—
188C2×C4—5661no—
189C2×C4—5661no—
190C2×C4—5661no—
191C2×C4—5661no—
192C2×C4—11322no—
193C2×C4—11322no—
194C2×C4—11321no—
195C2×C4—11322no—
196C2×C4—11321no—
197C2×C4D4×D28311yescharacteristic
198C2×C4C2×D113211yescharacteristic
199C8—41no—
200C8—11321no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—11321no—
207C22—11321no—
208C22—11321no—
209C22#2258358011yescharacteristic
210C22SD16×D28311yescharacteristic
211C22#2258363411yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—5662no—
217C4—11321no—
218C4—11321no—
219C4—22641no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—22641no—
224C2#3182661711yescharacteristic
225C2#3182663911yescharacteristic
226C2#3182665611yescharacteristic
subgroupG/H
centerC22#22583580
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#327×C283C2
socleC22×C283#120
prime radical#31827042C2
hypercenter#327D283
hypocenterC283#1616
LCSG ▷ C22×C4×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #31826784C2semidirect
3. #31826781C2semidirect
4. #31826405C2semidirect
5. #31826198C2semidirect
6. #31826190C2semidirect
7. #327×C283C2semidirect
113. C283#1616semidirect
120. #327D283semidirect

Maximal subgroups

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

subgroupindex
#318270422
#318267842
#318267812
#318264052
#318261982
#318261902
#327×C2832
#1616283