Finite Group Databasebeta

← #44559892

Subgroups — #44559892

Order 32128.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22548746C211yesmaximal, characteristic
2#30632261C211yesmaximal, characteristic
3#30632165C211yesmaximal, characteristic, complemented
4#30631911C214yesmaximal, complemented
5#30631895C214yesmaximal, complemented
6#30631713C211yesmaximal, characteristic, complemented
7#30631696C211yesmaximal, characteristic, complemented
8#30631684C211yesmaximal, characteristic
9C22×Q16×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11#22548746—24no—
12C2×#16860800C2212yes—
13C2×#16860797C2212yes—
14C2×#16860794C2211yescharacteristic
15#22548713C2212yes—
16#22548704C2212yes—
17#22548698—22no—
18#22548691—22no—
19Q8×Dic251—22no—
20Q8×Dic251C2214yes—
21#22548674—22no—
22#22548674C2211yescharacteristic
23#22548674C2211yescharacteristic
24Dic251⋊C8C2214yes—
25#22548640C2212yes—
26#22548639C2214yes—
27#22548637C2212yes—
28C22×Q8×C251C2211yescharacteristic
29C22×Q8×C251C2211yescharacteristic
30C2×Q16×C251—24no—
31C2×Q16×C251C2214yes—
32C22×C8×C251C2211yescharacteristic
33#16860810—22no—
34#16860810—22no—
35C2×Dic502C2311yescharacteristic
36C2×D1004C2311yescharacteristic
37C2×C22⋊D251—21no—
38C2×C4×D251—21no—
39#16860800—24no—
40#16860797—24no—
41#16860794—22no—
42C2×C251⋊C8C2312yes—
43C4⋊Dic251—24no—
44C4⋊Dic251D412yes—
45C4⋊Dic251C2312yes—
46Dic251⋊C4—24no—
47C22⋊Dic251—21no—
48C22⋊Dic251D412yes—
49(C2×D251)⋊C4—24no—
50C4×Dic251—24no—
51C4×Dic251—21no—
52C4×Dic251C2312yes—
53C2×Q8×C251—21no—
54C2×Q8×C251—21no—
55C2×Q8×C251—22no—
56C2×Q8×C251—22no—
57C2×Q8×C251—22no—
58C2×Q8×C251D414yes—
59C2×Q8×C251C2312yes—
60C2×Q8×C251C2312yes—
61C22×C4×C251—21no—
62C22×C4×C251—21no—
63C22×C4×C251C2311yescharacteristic
64Q16×C251—28no—
65C2×C8×C251—22no—
66C2×C8×C251C2312yes—
67C22×D251—21no—
68Dic502—21no—
69Dic502—21no—
70D1004—21no—
71D1004—21no—
72C22⋊D251—42no—
73C2×Dic251—22no—
74C2×Dic251—21no—
75C2×Dic251C2×D414yes—
76C4×D251—42no—
77C251⋊C8—22no—
78C23×C251C2×D411yescharacteristic
79Q8×C251—24no—
80Q8×C251—22no—
81Q8×C251—22no—
82Q8×C251—22no—
83Q8×C251—42no—
84Q8×C251C2×D414yes—
85C2×C4×C251—24no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—42no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D412yes—
91C2×C4×C251#4112yes—
92C2×C4×C251C2411yescharacteristic
93C8×C251—22no—
94C22×C251—22no—
95C22×C251#12112yes—
96C22×C251C22×D411yescharacteristic
97C2×D251—41no—
98C2×D251—42no—
99C4×C251—24no—
100C4×C251—42no—
101C4×C251#12112yes—
102C4×C251C22×D411yescharacteristic
103C4×C251C2×#4111yescharacteristic
104Dic251—24no—
105Dic251—41no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251C2×#12111yescharacteristic
109C2×C251#57812yes—
110D251—81no—
111C251#304411yescharacteristic, complemented, Sylow
112#3044—2511nomaximal, Sylow
113C2×#137—2511no—
114C22×Q16D25111yescharacteristic, complemented
115#536—2511no—
116#519—2511no—
117#464—2514no—
118#462—2514no—
119#424—2511no—
120#419—2511no—
121#416—2511no—
122C2×#41—2511no—
123C22×Q8C2×D25111yescharacteristic
124C22×Q8C2×D25111yescharacteristic
125#137—5024no—
126C2×Q16—24no—
127C2×Q16—2512no—
128C2×Q16C2×D25114yes—
129C2×SD16—2512no—
130C2×M16—2511no—
131C22×C8C2×D25111yescharacteristic
132C4⋊Q8—2512no—
133#124—2512no—
134#123—5022no—
135C22⋊Q8—5022no—
136C4×Q8—5022no—
137C4×Q8—2514no—
138#117—5022no—
139#117—2511no—
140#117—2511no—
141C4⋊C8—2514no—
142#103—2514no—
143#103—2512no—
144#102—2512no—
145#41—5022no—
146#41—5022no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—2511no—
153C2×Q8C22⋊D25114yes—
154C2×Q8C22×D25112yes—
155C2×Q8C22×D25112yes—
156C2×D4—5021no—
157C2×D4—2511no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—5021no—
161C22×C4C22×D25111yescharacteristic
162Q16—28no—
163Q16—5024no—
164SD16—5024no—
165M16—5022no—
166C2×C8—22no—
167C2×C8—2512no—
168C2×C8C22×D25112yes—
169C4⋊C4—5024no—
170C4⋊C4—5024no—
171C4⋊C4—2512no—
172C4⋊C4—2512no—
173C22⋊C4—5024no—
174C22⋊C4—5021no—
175C22⋊C4—2512no—
176C42—5024no—
177C42—5021no—
178C42—2512no—
179C23—5021no—
180C23D4×D25111yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—5021no—
187Q8—5021no—
188Q8C2×C22⋊D25114yes—
189D4—10042no—
190D4—5021no—
191D4—5021no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—10042no—
197C2×C4—5022no—
198C2×C4—5021no—
199C2×C4—2514no—
200C2×C4D4×D25111yescharacteristic
201C2×C4C2×C22⋊D25112yes—
202C2×C4#1686080912yes—
203C2×C4C23×D25111yescharacteristic
204C8—22no—
205C8—5022no—
206C22—22no—
207C22—10042no—
208C22—10041no—
209C22#2254868712yes—
210C22C2×D4×D25111yescharacteristic
211C4—24no—
212C4—42no—
213C4—10042no—
214C4—10041no—
215C4—5024no—
216C4#2254868712yes—
217C4C22×C22⋊D25111yescharacteristic
218C4C2×#1686080911yescharacteristic
219C2—22no—
220C2—20081no—
221C2C2×#2254868711yescharacteristic
222C2#3063250012yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×Q16×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860810C22
hypercenterC22×Q16D251
hypocenterC251#3044
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #30632165C2semidirect
4. #30631911C2semidirect
5. #30631895C2semidirect
6. #30631713C2semidirect
7. #30631696C2semidirect
9. C22×Q16×C251C2semidirect
111. C251#3044semidirect
114. C22×Q16D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
#306322612
#306321652
#306319112
#306318952
#306317132
#306316962
#306316842
C22×Q16×C2512
#3044251