Finite Group Databasebeta

← #44550721

Subgroups — #44550721

Order 32128.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#16860799C211yesmaximal, characteristic
2C2×#22548680C211yesmaximal, characteristic, complemented
3C2×#22548638C211yesmaximal, characteristic, complemented
4C2×#22548635C211yesmaximal, characteristic
5#30631588C211yesmaximal, characteristic
6#30631441C211yesmaximal, characteristic, complemented
7#340×C251C211yesmaximal, characteristic, complemented
8C22×Dic502C2211yescharacteristic
9C22×D1004C2211yescharacteristic
10C2×#16860799—22no—
11C2×#16860799—22no—
12C2×#16860799C414yescomplemented
13C22×C251⋊C8C2211yescharacteristic
14#22548680—22no—
15#22548680—22no—
16C2×Dic251⋊C4—21no—
17C2×(C2×D251)⋊C4—21no—
18C2×(C2×D251)⋊C4—21no—
19C2×C4×Dic251—21no—
20C2×C4×Dic251C2211yescharacteristic
21#22548638—22no—
22#22548635—22no—
23C22×C8×C251C2211yescharacteristic
24C2×C4⋊C4×C251C2211yescharacteristic
25C2×C4⋊C4×C251C2211yescharacteristic
26C23×D251—21no—
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502—21no—
30C2×Dic502C2×C412yes—
31C2×Dic502D412yes—
32C2×Dic502D412yes—
33C2×D1004—21no—
34C2×D1004—21no—
35C2×D1004—21no—
36C2×D1004—21no—
37C2×D1004—21no—
38C2×D1004C2×C412yes—
39C22×Dic251—21no—
40C22×Dic251D411yescharacteristic
41C22×Dic251D411yescharacteristic
42#16860799—28no—
43C2×C251⋊C8—21no—
44C2×C251⋊C8—21no—
45C2×C251⋊C8—21no—
46Dic251⋊C4—24no—
47(C2×D251)⋊C4—42no—
48(C2×D251)⋊C4—42no—
49C4×Dic251—24no—
50C4×Dic251—21no—
51C4×Dic251—21no—
52C22×C4×C251—21no—
53C22×C4×C251—21no—
54C22×C4×C251C2311yescharacteristic
55C2×C8×C251—21no—
56C2×C8×C251—21no—
57C2×C8×C251C2×C412yes—
58C4⋊C4×C251—21no—
59C4⋊C4×C251—21no—
60C4⋊C4×C251—21no—
61C4⋊C4×C251D412yes—
62C22×D251—22no—
63C22×D251—41no—
64C22×D251—41no—
65C22×D251—41no—
66C22×D251—41no—
67C22×D251—41no—
68C22×D251—41no—
69Dic502—22no—
70Dic502—22no—
71Dic502—22no—
72Dic502C22⋊C414yescomplemented
73D1004—22no—
74D1004—22no—
75D1004—41no—
76D1004—41no—
77C2×Dic251—22no—
78C2×Dic251—21no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—21no—
82C2×Dic251—21no—
83C2×Dic251—21no—
84C2×Dic251—21no—
85C2×Dic251SD1612yescomplemented
86C251⋊C8—41no—
87C23×C251C2×D411yescharacteristic
88C2×C4×C251—22no—
89C2×C4×C251—41no—
90C2×C4×C251—41no—
91C2×C4×C251—41no—
92C2×C4×C251—41no—
93C2×C4×C251—41no—
94C2×C4×C251C22×C411yescharacteristic
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2×D411yescharacteristic
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251#4111yescharacteristic
99C2×C4×C251#4111yescharacteristic
100C8×C251—22no—
101C22×C251C4×D411yescharacteristic
102C22×C251#12011yescharacteristic
103C22×C251#12111yescharacteristic
104C22×C251C2×SD1611yescharacteristic
105C22×C251#13511yescharacteristic
106C22×C251AGL(1,ℤ/8)11yescharacteristic
107C22×C251#13711yescharacteristic
108C2×D251—42no—
109C2×D251—42no—
110C2×D251—42no—
111C2×D251—81no—
112C2×D251—81no—
113C2×D251—81no—
114C2×D251—81no—
115C4×C251—42no—
116C4×C251—81no—
117C4×C251C2×C22⋊C411yescharacteristic
118C4×C251C4×D411yescharacteristic
119C4×C251#12111yescharacteristic
120C4×C251#12311yescharacteristic
121Dic251—24no—
122Dic251—22no—
123Dic251—41no—
124C2×C251#38611yescharacteristic
125C2×C251C4×SD1611yescharacteristic
126C2×C251#44011yescharacteristic
127C2×C251#44811yescharacteristic
128C2×C251#44911yescharacteristic
129C2×C251#46011yescharacteristic
130C2×C251#46411yescharacteristic
131D251—82no—
132C251#188011yescharacteristic, complemented, Sylow
133#1880—2511nomaximal, Sylow
134C22×SD16—2511no—
135C2×C4×Q8—2511no—
136C2×#103—2511no—
137C2×#102—2511no—
138#390—2511no—
139#340D25111yescharacteristic, complemented
140#336—2511no—
141C22×Q8—2511no—
142C22×D4—2511no—
143C2×SD16—5022no—
144C2×SD16—5022no—
145C2×SD16—2514no—
146C22×C8—2511no—
147C22×C8C2×D25111yescharacteristic
148C4×Q8—5022no—
149C4×Q8—5022no—
150C2×C4⋊C4—5021no—
151C2×C4⋊C4C2×D25111yescharacteristic
152C2×C4⋊C4C2×D25111yescharacteristic
153C2×C22⋊C4—5021no—
154C2×C22⋊C4—5021no—
155C2×C42—5021no—
156C2×C42—2511no—
157#103—5022no—
158#102—5022no—
159C24—5021no—
160C2×Q8—5021no—
161C2×Q8—5021no—
162C2×Q8—5021no—
163C2×Q8—2512no—
164C2×Q8—2512no—
165C2×Q8—2512no—
166C2×D4—5021no—
167C2×D4—5021no—
168C2×D4—5021no—
169C2×D4—5021no—
170C2×D4—5021no—
171C2×D4—2512no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—5021no—
175C22×C4—2511no—
176C22×C4—2511no—
177C22×C4C22×D25111yescharacteristic
178SD16—5028no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—5021no—
182C2×C8—5021no—
183C2×C8—5021no—
184C2×C8C4×D25112yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—5024no—
189C4⋊C4D100412yes—
190C22⋊C4—10042no—
191C22⋊C4—10042no—
192C42—5024no—
193C42—5021no—
194C42—5021no—
195C23—10041no—
196C23—10041no—
197C23—10041no—
198C23—10041no—
199C23—10041no—
200C23—10041no—
201C23—5022no—
202C23D4×D25111yescharacteristic
203Q8—5022no—
204Q8—5022no—
205Q8—5022no—
206Q8—2514no—
207D4—10041no—
208D4—10041no—
209D4—5022no—
210D4—5022no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—5021no—
218C2×C4—5021no—
219C2×C4—5021no—
220C2×C4—5021no—
221C2×C4—5022no—
222C2×C4—5021no—
223C2×C4—5022no—
224C2×C4—5022no—
225C2×C4—2512no—
226C2×C4C2×C4×D25111yescharacteristic
227C2×C4D4×D25111yescharacteristic
228C2×C4D4×D25111yescharacteristic
229C2×C4C2×D100411yescharacteristic
230C2×C4#1686081011yescharacteristic
231C2×C4#1686081111yescharacteristic
232C8—22no—
233C8—10041no—
234C22—20081no—
235C22—20081no—
236C22—20081no—
237C22—20081no—
238C22—10042no—
239C22—10042no—
240C22—10042no—
241C22#2254867711yescharacteristic
242C22#2254868511yescharacteristic
243C22#2254868811yescharacteristic
244C22SD16×D25111yescharacteristic
245C22#2254873611yescharacteristic
246C22#2254873911yescharacteristic
247C22#2254874711yescharacteristic
248C4—42no—
249C4—81no—
250C4—10041no—
251C4—5024no—
252C4—5022no—
253C4C22⋊C4×D25111yescharacteristic
254C4C4×D100411yescharacteristic
255C4#2254869011yescharacteristic
256C4#2254870011yescharacteristic
257C2—20082no—
258C2#3063156511yescharacteristic
259C2#3063177411yescharacteristic
260C2#3063178611yescharacteristic
261C2#3063182711yescharacteristic
262C2#3063183311yescharacteristic
263C2#3063188611yescharacteristic
264C2#3063191211yescharacteristic
subgroupG/H
centerC23D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#340×C251C2
socleC23×C251C2×D4
prime radicalC22×D1004C22
cosocleC22×C4×C251C23
hypercenter#340D251
hypocenterC251#1880
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#22548680C2semidirect
3. C2×#22548638C2semidirect
6. #30631441C2semidirect
7. #340×C251C2semidirect
12. C2×#16860799C4semidirect
72. Dic502C22⋊C4semidirect
85. C2×Dic251SD16semidirect
132. C251#1880semidirect
139. #340D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607992
C2×#225486802
C2×#225486382
C2×#225486352
#306315882
#306314412
#340×C2512
#1880251