Finite Group Databasebeta

← #44549357

Subgroups — #44549357

Order 32128.

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

normal subgroups only

This group has 228 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#30632312C211yesmaximal, characteristic
2#30632018C211yesmaximal, characteristic, complemented
3#30632015C211yesmaximal, characteristic, complemented
4#30631614C211yesmaximal, characteristic
5#30631423C211yesmaximal, characteristic, complemented
6#30631410C211yesmaximal, characteristic
7#328×C251C211yesmaximal, characteristic, complemented
8#22548695—21no—
9#22548694C2211yescharacteristic
10#22548693—21no—
11#22548693—21no—
12#22548692—21no—
13#22548691—21no—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×(C2×D251)⋊C4—21no—
16C2×(C2×D251)⋊C4C2211yescharacteristic
17C22⋊C4×D251—21no—
18#22548647—21no—
19#22548646—21no—
20#22548640—21no—
21#22548639—21no—
22C22⋊(C251⋊C8)C2211yescharacteristic
23C251⋊#95—21no—
24C22⋊Q8×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C23×D251D411yescharacteristic
28C2×Dic502—21no—
29C2×Dic502D411yescharacteristic
30C22×Dic251—21no—
31C22×Dic251D411yescharacteristic
32C2×C4×D251—21no—
33C2×C4×D251—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251—21no—
38C4⋊Dic251D411yescharacteristic
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41Dic251⋊C4—21no—
42C22⋊Dic251—21no—
43(C2×D251)⋊C4—21no—
44(C2×D251)⋊C4—21no—
45(C2×D251)⋊C4—21no—
46(C2×D251)⋊C4—21no—
47(C2×D251)⋊C4—42no—
48C2×Q8×C251D411yescharacteristic
49C22×C4×C251—21no—
50C22×C4×C251C2311yescharacteristic
51C2×C8×C251—21no—
52C4⋊C4×C251—21no—
53C4⋊C4×C251D411yescharacteristic
54C22⋊C4×C251—21no—
55C22×D251—21no—
56C22×D251—21no—
57C22×D251—41no—
58C22×D251—41no—
59C22×D251SD1612yes—
60Dic502—41no—
61Dic502—81no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—41no—
66C2×Dic251—41no—
67C2×Dic251—41no—
68C2×Dic251—41no—
69C4×D251—81no—
70C4×D251—81no—
71C251⋊C8—41no—
72C23×C251C2×D411yescharacteristic
73Q8×C251—41no—
74C2×C4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251—41no—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C8×C251—41no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251#12011yescharacteristic
86C22×C251C2×SD1611yescharacteristic
87C22×C251#13711yescharacteristic
88C2×D251—22no—
89C2×D251—41no—
90C2×D251—41no—
91C2×D251—42no—
92C2×D251—81no—
93C4×C251—21no—
94C4×C251—21no—
95C4×C251—41no—
96C4×C251—41no—
97C4×C251—81no—
98Dic251—41no—
99Dic251—41no—
100Dic251—41no—
101Dic251—81no—
102C2×C251—21no—
103C2×C251—21no—
104C2×C251#44811yescharacteristic
105C2×C251#45511yescharacteristic
106C2×C251#45811yescharacteristic
107D251—42no—
108C251#160811yescharacteristic, complemented, Sylow
109#1608—2511nomaximal, Sylow
110#543—2511no—
111#483—2511no—
112#483—2511no—
113#397—2511no—
114#328—2511no—
115#328D25111yescharacteristic, complemented
116#323—2511no—
117C22⋊Q8—5021no—
118C22⋊Q8—5021no—
119C22⋊Q8—5021no—
120C22⋊Q8—5021no—
121C22⋊Q8—5021no—
122C22⋊Q8—2511no—
123C22⋊Q8C2×D25111yescharacteristic
124C2×C4⋊C4—2511no—
125C2×C22⋊C4—5021no—
126C2×C22⋊C4—5021no—
127C2×C22⋊C4—2511no—
128#106—5021no—
129#106—5021no—
130#103—5021no—
131#103—5021no—
132C22⋊C8—2511no—
133C22⋊C8C2×D25111yescharacteristic
134#95—5021no—
135#95C2×D25111yescharacteristic
136C24—2511no—
137C2×Q8—5021no—
138C2×Q8—2511no—
139C2×Q8D100411yescharacteristic
140C22×C4—21no—
141C22×C4—5021no—
142C22×C4—5021no—
143C22×C4—5021no—
144C22×C4—2511no—
145C22×C4C22×D25111yescharacteristic
146C2×C8—21no—
147C2×C8—5021no—
148C4⋊C4—21no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C4⋊C4—5021no—
152C4⋊C4—5021no—
153C4⋊C4—5021no—
154C4⋊C4—5021no—
155C4⋊C4—2511no—
156C4⋊C4D100411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—10042no—
159C22⋊C4—5021no—
160C22⋊C4—5021no—
161C22⋊C4—5021no—
162C22⋊C4—5021no—
163C22⋊C4—5021no—
164C23—10041no—
165C23—10041no—
166C23—5021no—
167C23—5021no—
168C23—2512no—
169C23D4×D25111yescharacteristic
170Q8—41no—
171Q8—20081no—
172Q8—10041no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—20081no—
180C2×C4—20081no—
181C2×C4—10041no—
182C2×C4—10041no—
183C2×C4—10041no—
184C2×C4—10041no—
185C2×C4—5021no—
186C2×C4—5021no—
187C2×C4—5021no—
188C2×C4D4×D25111yescharacteristic
189C2×C4C2×D100411yescharacteristic
190C8—41no—
191C8—10041no—
192C22—21no—
193C22—21no—
194C22—20081no—
195C22—10042no—
196C22—10041no—
197C22—10041no—
198C22—5022no—
199C22#2254868511yescharacteristic
200C22SD16×D25111yescharacteristic
201C22#2254874711yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—20081no—
208C4—10041no—
209C4—10041no—
210C4—10041no—
211C2—21no—
212C2—21no—
213C2—10042no—
214C2#3063182711yescharacteristic
215C2#3063186611yescharacteristic
216C2#3063187811yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#328×C251C2
socleC22×C251#120
prime radicalC23×D251D4
hypercenter#328D251
hypocenterC251#1608
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #30632018C2semidirect
3. #30632015C2semidirect
5. #30631423C2semidirect
7. #328×C251C2semidirect
108. C251#1608semidirect
115. #328D251semidirect

Maximal subgroups

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

subgroupindex
#306323122
#306320182
#306320152
#306316142
#306314232
#306314102
#328×C2512
#1608251