Finite Group Databasebeta

← #44549403

Subgroups — #44549403

Order 32128.

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#30632264C211yesmaximal, characteristic
2#30632006C211yesmaximal, characteristic, complemented
3#30632003C211yesmaximal, characteristic, complemented
4#30631627C211yesmaximal, characteristic, complemented
5#30631420C211yesmaximal, characteristic, complemented
6#30631412C211yesmaximal, characteristic, complemented
7#327×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9#22548693—21no—
10#22548692—21no—
11#22548689C2211yescharacteristic
12#22548688—21no—
13#22548687—21no—
14#22548687—21no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×Dic251⋊C4C2211yescharacteristic
17#22548647—21no—
18#22548646—21no—
19#22548637—21no—
20#22548636—21no—
21C22⋊(C251⋊C8)C2211yescharacteristic
22C251⋊#95—21no—
23C251⋊#95—21no—
24#121×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27#16860809—42no—
28#16860809—42no—
29C2×Dic502—21no—
30C2×D1004D411yescharacteristic
31C2×C22⋊D251—21no—
32C2×C22⋊D251—21no—
33C2×C22⋊D251—21no—
34C22×Dic251—21no—
35C22×Dic251—21no—
36C22×Dic251D411yescharacteristic
37C22×Dic251D411yescharacteristic
38C2×C4×D251—21no—
39C2×C251⋊C8—21no—
40C4⋊Dic251—21no—
41C4⋊Dic251—21no—
42C4⋊Dic251D411yescharacteristic
43Dic251⋊C4—21no—
44Dic251⋊C4—21no—
45Dic251⋊C4—21no—
46C22⋊Dic251—21no—
47(C2×D251)⋊C4—21no—
48(C2×D251)⋊C4—21no—
49C2×D4×C251—21no—
50C2×D4×C251D411yescharacteristic
51C22×C4×C251—21no—
52C22×C4×C251C2311yescharacteristic
53C2×C8×C251—21no—
54C4⋊C4×C251D411yescharacteristic
55C22⋊C4×C251—21no—
56C22×D251—21no—
57Dic502—42no—
58D1004—41no—
59C22⋊D251—42no—
60C22⋊D251—41no—
61C22⋊D251—41no—
62C22⋊D251—81no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—21no—
66C2×Dic251—21no—
67C2×Dic251—42no—
68C2×Dic251—41no—
69C2×Dic251—42no—
70C2×Dic251—41no—
71C2×Dic251SD1612yes—
72C4×D251—42no—
73C251⋊C8—41no—
74C23×C251—21no—
75C23×C251C2×D411yescharacteristic
76D4×C251—41no—
77D4×C251—41no—
78D4×C251—41no—
79C2×C4×C251—21no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251—41no—
83C2×C4×C251—41no—
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C8×C251—41no—
87C22×C251—21no—
88C22×C251—21no—
89C22×C251—41no—
90C22×C251—41no—
91C22×C251—41no—
92C22×C251#12011yescharacteristic
93C22×C251C2×SD1611yescharacteristic
94C22×C251AGL(1,ℤ/8)11yescharacteristic
95C2×D251—41no—
96C2×D251—41no—
97C2×D251—41no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—41no—
101C4×C251—81no—
102Dic251—22no—
103Dic251—41no—
104Dic251—41no—
105Dic251—81no—
106C2×C251—21no—
107C2×C251—21no—
108C2×C251—81no—
109C2×C251#45011yescharacteristic
110C2×C251#45411yescharacteristic
111C2×C251#45811yescharacteristic
112D251—81no—
113C251#161611yescharacteristic, complemented, Sylow
114#1616—2511nomaximal, Sylow
115#537—2511no—
116#481—2511no—
117#481—2511no—
118#400—2511no—
119#327—2511no—
120#327D25111yescharacteristic, complemented
121#324—2511no—
122C2×#41—5021no—
123C22⋊Q8—5021no—
124C22⋊Q8—5021no—
125#121—5021no—
126#121—5021no—
127#121—5021no—
128#121—2511no—
129#121C2×D25111yescharacteristic
130C2×C4⋊C4—2511no—
131C2×C4⋊C4—2511no—
132#106—5021no—
133#106—5021no—
134#102—5021no—
135#102—5021no—
136C22⋊C8—2511no—
137C22⋊C8C2×D25111yescharacteristic
138#95—5021no—
139#95—5021no—
140#95C2×D25111yescharacteristic
141#41—10042no—
142#41—10042no—
143C2×Q8—5021no—
144C2×D4—21no—
145C2×D4—5021no—
146C2×D4—5021no—
147C2×D4—5021no—
148C2×D4—2511no—
149C2×D4D100411yescharacteristic
150C22×C4—21no—
151C22×C4—5021no—
152C22×C4—5021no—
153C22×C4—5021no—
154C22×C4—2511no—
155C22×C4—2511no—
156C22×C4C22×D25111yescharacteristic
157C2×C8—21no—
158C2×C8—5021no—
159C4⋊C4—5021no—
160C4⋊C4—5021no—
161C4⋊C4—5021no—
162C4⋊C4—5021no—
163C4⋊C4—5021no—
164C4⋊C4—2511no—
165C4⋊C4D100411yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—5021no—
168C22⋊C4—5021no—
169C22⋊C4—5021no—
170C23—21no—
171C23—5021no—
172C23D4×D25111yescharacteristic
173Q8—10042no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—20081no—
178D4—10041no—
179D4—10041no—
180D4—10041no—
181D4—10042no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—10042no—
188C2×C4—10042no—
189C2×C4—10041no—
190C2×C4—10042no—
191C2×C4—10041no—
192C2×C4—5021no—
193C2×C4—5021no—
194C2×C4—5021no—
195C2×C4—5021no—
196C2×C4—2512no—
197C2×C4D4×D25111yescharacteristic
198C2×C4C2×D100411yescharacteristic
199C8—41no—
200C8—10041no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—10041no—
207C22—10041no—
208C22—10041no—
209C22#2254868511yescharacteristic
210C22SD16×D25111yescharacteristic
211C22#2254873911yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—20081no—
217C4—10041no—
218C4—10041no—
219C4—5022no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—20081no—
224C2#3063183911yescharacteristic
225C2#3063186111yescharacteristic
226C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#327×C251C2
socleC22×C251#120
prime radical#30632264C2
hypercenter#327D251
hypocenterC251#1616
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #30632006C2semidirect
3. #30632003C2semidirect
4. #30631627C2semidirect
5. #30631420C2semidirect
6. #30631412C2semidirect
7. #327×C251C2semidirect
113. C251#1616semidirect
120. #327D251semidirect

Maximal subgroups

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

subgroupindex
#306322642
#306320062
#306320032
#306316272
#306314202
#306314122
#327×C2512
#1616251