Finite Group Databasebeta

← #44558845

Subgroups — #44558845

Order 32128.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#22548763C211yesmaximal, characteristic
2#30632453C212yesmaximal, complemented
3#30631670C214yesmaximal, complemented
4#30631669C212yesmaximal, complemented
5C2×#22548634C211yesmaximal, characteristic, complemented
6#101×D251C214yesmaximal, complemented
7C2×#101×C251C211yesmaximal, characteristic, complemented
8#22548763—24no—
9#22548763—24no—
10C2×#16860811C412yes—
11C2×#16860811C2212yes—
12C2×#16860810C414yes—
13C2×#16860810C2211yescharacteristic
14C2×#16860810C2211yescharacteristic
15C2×Q8×D251C412yes—
16C2×Q8×D251C2212yes—
17#22548723—24no—
18#22548721—24no—
19C2×#16860794C2212yes—
20M16×D251C2218yes—
21#22548634C2214yes—
22#22548633C2214yes—
23#22548631C2214yes—
24C22×Q8×C251C2211yescharacteristic
25C2×M16×C251C2212yes—
26#101×C251C2214yes—
27#16860811—28no—
28#16860811—28no—
29#16860810—28no—
30#16860810—28no—
31#16860810—22no—
32#16860810—22no—
33#16860810D414yes—
34#16860810D414yes—
35C2×Dic502C2×C414yes—
36C2×Dic502C2×C411yescharacteristic
37C2×Dic502C2311yescharacteristic
38Q8×D251—28no—
39Q8×D251—28no—
40C2×D1004C2×C414yes—
41C2×D1004C2×C411yescharacteristic
42C2×D1004C2311yescharacteristic
43C2×C22⋊D251C2×C412yes—
44C2×C22⋊D251C2×C412yes—
45C2×C4×D251C2×C414yes—
46C2×C4×D251C2×C412yes—
47C2×C4×D251C2×C414yes—
48C2×C4×D251C2312yes—
49#16860794—22no—
50#16860794C2314yes—
51#16860793—28no—
52C2×C251⋊C8—22no—
53C8×D251—28no—
54C2×Q8×C251—22no—
55C2×Q8×C251—22no—
56C2×Q8×C251C2×C412yes—
57C2×Q8×C251C2312yes—
58C22×C4×C251C2×C412yes—
59C22×C4×C251C2311yescharacteristic
60M16×C251—22no—
61M16×C251C2314yes—
62C2×C8×C251—22no—
63C22×D251C22×C412yes—
64C22×D251C22×C412yes—
65Dic502—22no—
66Dic502—28no—
67Dic502C22⋊C414yes—
68D1004—28no—
69D1004—22no—
70D1004C22⋊C414yes—
71C22⋊D251—22no—
72C22⋊D251—22no—
73C22⋊D251C22⋊C414yes—
74C22⋊D251C22⋊C414yes—
75C2×Dic251C22×C412yes—
76C2×Dic251C22×C412yes—
77C4×D251—28no—
78C4×D251—24no—
79C4×D251—28no—
80C4×D251C2×D418yes—
81C251⋊C8—24no—
82C23×C251C22×C411yescharacteristic
83Q8×C251—24no—
84Q8×C251—24no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251C22×C414yes—
88C2×C4×C251C22×C411yescharacteristic
89C2×C4×C251C2×D412yes—
90C2×C4×C251C2×D412yes—
91C2×C4×C251C2411yescharacteristic
92C8×C251—24no—
93C22×C251—41no—
94C22×C251C2×C22⋊C411yescharacteristic
95C22×C251C2×C22⋊C411yescharacteristic
96C22×C251C23×C411yescharacteristic
97C2×D251—22no—
98C2×D251—42no—
99C2×D251—42no—
100C2×D251C2×C22⋊C414yes—
101C4×C251—24no—
102C4×C251C2×C22⋊C412yes—
103C4×C251C22×D412yes—
104Dic251—22no—
105Dic251C2×C22⋊C414yes—
106C2×C251—21no—
107C2×C251—21no—
108C2×C251—21no—
109C2×C251C22×C22⋊C411yescharacteristic
110D251—24no—
111D251—42no—
112C251#288911yescharacteristic, complemented, Sylow
113#2889—2511nomaximal, Sylow
114C2×2−1+4—2511no—
115#568—2512no—
116#413—2512no—
117#413—2514no—
118C2×#101—2511no—
119C2×#101—2514no—
120C2×#101D25111yescharacteristic, complemented
1212−1+4—5024no—
1222−1+4—5024no—
123C2×#41—2512no—
124C2×#41—2512no—
125C2×#41—2514no—
126C2×#41—2511no—
127C2×#41—2511no—
128C22×Q8—2512no—
129C22×Q8—2512no—
130C22×Q8C2×D25111yescharacteristic
131#131—5024no—
132#131—5024no—
133C2×M16—2512no—
134C2×M16—2518no—
135C2×M16C2×D25112yes—
136#101—2514no—
137#101—2514no—
138#101C2×D25114yes—
139#100—2514no—
140#41—5028no—
141#41—5028no—
142#41—5028no—
143#41—5028no—
144#41—5022no—
145#41—5022no—
146#41—2514no—
147#41—2514no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—5028no—
151C2×Q8—5028no—
152C2×Q8—2514no—
153C2×Q8—2511no—
154C2×Q8—2511no—
155C2×Q8C4×D25112yes—
156C2×Q8C22×D25112yes—
157C2×D4—2512no—
158C2×D4—2512no—
159C2×D4—2514no—
160C2×D4—2511no—
161C2×D4—2511no—
162C22×C4—2514no—
163C22×C4—2514no—
164C22×C4—2512no—
165C22×C4—2512no—
166C22×C4C4×D25112yes—
167C22×C4C22×D25111yescharacteristic
168M16—22no—
169M16—5028no—
170M16—5022no—
171M16—2514no—
172M16C22×D25114yes—
173C2×C8—22no—
174C2×C8—5028no—
175C2×C8—5022no—
176C23—2512no—
177C23—2512no—
178C23C2×C4×D25111yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—5028no—
182Q8—5022no—
183Q8—2514no—
184D4—5028no—
185D4—5022no—
186D4—5022no—
187D4—5022no—
188D4—2514no—
189D4—2514no—
190D4—2514no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—5028no—
194C2×C4—5028no—
195C2×C4—5024no—
196C2×C4—2512no—
197C2×C4—2512no—
198C2×C4—2518no—
199C2×C4C2×C4×D25114yes—
200C2×C4C2×C4×D25111yescharacteristic
201C2×C4D4×D25112yes—
202C2×C4D4×D25112yes—
203C2×C4C23×D25111yescharacteristic
204C8—24no—
205C8—5024no—
206C22—41no—
207C22—10042no—
208C22—10042no—
209C22—5022no—
210C22—2514no—
211C22C22⋊C4×D25111yescharacteristic
212C22C22⋊C4×D25111yescharacteristic
213C22C22×C4×D25111yescharacteristic
214C4—24no—
215C4—5022no—
216C4—2514no—
217C4C22⋊C4×D25112yes—
218C4C2×D4×D25112yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—10042no—
223C2—5024no—
224C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC2×C4C23×D251
FittingC2×#101×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548763C2
cosocleC2×C4×C251C24
hypercenterC2×#101D251
hypocenterC251#2889
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #30632453C2semidirect
3. #30631670C2semidirect
4. #30631669C2semidirect
5. C2×#22548634C2semidirect
6. #101×D251C2semidirect
7. C2×#101×C251C2semidirect
112. C251#2889semidirect
120. C2×#101D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487632
#306324532
#306316702
#306316692
C2×#225486342
#101×D2512
C2×#101×C2512
#2889251