Finite Group Databasebeta

← #44558376

Subgroups — #44558376

Order 32128.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548698C211yesmaximal, characteristic
2#30631620C212yesmaximal, complemented
3#30631618C211yesmaximal, characteristic, complemented
4#30631615C212yesmaximal, complemented
5#30631613C211yesmaximal, characteristic, complemented
6#30631609C212yesmaximal, complemented
7#30631606C211yesmaximal, characteristic, complemented
8#30631602C212yesmaximal
9#30631598C211yesmaximal, characteristic, complemented
10#30631547C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C251C211yesmaximal, characteristic, complemented
12C22×C22⋊D251C2211yescharacteristic
13#22548698—24no—
14C2×C4⋊Dic251C2212yes—
15C2×Dic251⋊C4C2211yescharacteristic
16C2×Dic251⋊C4C2211yescharacteristic
17C2×Dic251⋊C4C2212yes—
18C2×Dic251⋊C4C2212yes—
19C2×C22⋊Dic251C2211yescharacteristic
20C2×C22⋊Dic251C2212yes—
21C2×(C2×D251)⋊C4C2211yescharacteristic
22C2×(C2×D251)⋊C4C2211yescharacteristic
23C2×(C2×D251)⋊C4C2212yes—
24C2×(C2×D251)⋊C4C2212yes—
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2211yescharacteristic
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C22×Q8×C251C2211yescharacteristic
32C23×C4×C251C2211yescharacteristic
33C22⋊Q8×C251—24no—
34C2×C4⋊C4×C251C2212yes—
35C2×C4⋊C4×C251C2211yescharacteristic
36C2×C22⋊C4×C251C2212yes—
37C23×D251C2311yescharacteristic
38C2×C22⋊D251—22no—
39C2×C22⋊D251—42no—
40C22×Dic251C2311yescharacteristic
41C22×Dic251C2311yescharacteristic
42C22×Dic251C2312yes—
43C22×Dic251C2312yes—
44C22×Dic251C2311yescharacteristic
45C4⋊Dic251—42no—
46Dic251⋊C4—22no—
47Dic251⋊C4—22no—
48Dic251⋊C4—42no—
49Dic251⋊C4—42no—
50C22⋊Dic251—22no—
51C22⋊Dic251—42no—
52(C2×D251)⋊C4—22no—
53(C2×D251)⋊C4—22no—
54(C2×D251)⋊C4—42no—
55(C2×D251)⋊C4—42no—
56C24×C251C2311yescharacteristic
57C2×Q8×C251—22no—
58C2×Q8×C251—42no—
59C22×C4×C251—42no—
60C22×C4×C251D414yes—
61C22×C4×C251C2312yes—
62C22×C4×C251C2312yes—
63C22×C4×C251C2311yescharacteristic
64C22×C4×C251C2311yescharacteristic
65C4⋊C4×C251—22no—
66C4⋊C4×C251—24no—
67C22⋊C4×C251—24no—
68C22×D251—21no—
69C22×D251—21no—
70C22×D251—22no—
71C22×D251—21no—
72C22×D251—22no—
73C22⋊D251—44no—
74C2×Dic251—21no—
75C2×Dic251—21no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—21no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—21no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C23×C251—22no—
88C23×C251—21no—
89C23×C251—21no—
90C23×C251—22no—
91C23×C251C2×D412yes—
92C23×C251C2411yescharacteristic
93Q8×C251—44no—
94C2×C4×C251—22no—
95C2×C4×C251—22no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—22no—
101C2×C4×C251—22no—
102C2×C4×C251—44no—
103C2×C4×C251C2×D412yes—
104C2×C4×C251C2×D412yes—
105C22×C251—22no—
106C22×C251—22no—
107C22×C251—22no—
108C22×C251—42no—
109C22×C251—42no—
110C22×C251C22×D411yescharacteristic
111C22×C251H(5,𝔽2)12yes—
112C22×C251H(5,𝔽2)11yescharacteristic
113C22×C2512−1+411yescharacteristic
114C22×C2512−1+412yes—
115C2×D251—41no—
116C2×D251—41no—
117C2×D251—41no—
118C2×D251—42no—
119C2×D251—42no—
120C4×C251—22no—
121C4×C251—22no—
122C4×C251—42no—
123C4×C251—42no—
124Dic251—41no—
125Dic251—41no—
126Dic251—41no—
127Dic251—42no—
128Dic251—42no—
129C2×C251—42no—
130C2×C251#53411yescharacteristic
131C2×C251#53611yescharacteristic
132C2×C251#53711yescharacteristic
133C2×C251#56212yes—
134C2×C251#56312yes—
135D251—81no—
136C251#281411yescharacteristic, complemented, Sylow
137#2814—2511nomaximal, Sylow
138C2×#123—2511no—
139C2×C22⋊Q8D25111yescharacteristic, complemented
140#398—2511no—
141#398—2512no—
142#397—2511no—
143#397—2512no—
144#396—2512no—
145#395—2511no—
146#394—2512no—
147#393—2511no—
148#380—2511no—
149C22×Q8C2×D25111yescharacteristic
150C22×D4—2511no—
151C23×C4C2×D25111yescharacteristic
152#123—5024no—
153C22⋊Q8—24no—
154C2×C4⋊C4—2512no—
155C2×C4⋊C4—2512no—
156C2×C4⋊C4—2512no—
157C2×C4⋊C4—2511no—
158C2×C4⋊C4—2511no—
159C2×C4⋊C4C2×D25112yes—
160C2×C4⋊C4C2×D25111yescharacteristic
161C2×C22⋊C4—2512no—
162C2×C22⋊C4—2512no—
163C2×C22⋊C4—2512no—
164C2×C22⋊C4—2511no—
165C2×C22⋊C4—2511no—
166C2×C22⋊C4—2511no—
167C2×C22⋊C4C2×D25112yes—
168#95—2511no—
169#95—2511no—
170#95—2512no—
171#95—2512no—
172#95—2512no—
173#95—2512no—
174C24—2511no—
175C24C22×D25111yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—10042no—
179C2×D4—5022no—
180C22×C4—42no—
181C22×C4—2511no—
182C22×C4—2512no—
183C22×C4—2512no—
184C22×C4—2511no—
185C22×C4—2511no—
186C22×C4C22⋊D25114yes—
187C22×C4C22×D25112yes—
188C22×C4C22×D25112yes—
189C22×C4C22×D25111yescharacteristic
190C22×C4C22×D25111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—10042no—
194C4⋊C4—10042no—
195C4⋊C4—10042no—
196C4⋊C4—5022no—
197C4⋊C4—5022no—
198C22⋊C4—24no—
199C22⋊C4—10042no—
200C22⋊C4—10042no—
201C22⋊C4—10042no—
202C22⋊C4—5022no—
203C22⋊C4—5022no—
204C22⋊C4—5022no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—5021no—
210C23—5022no—
211C23—5021no—
212C23—5021no—
213C23—5022no—
214C23C2×C22⋊D25112yes—
215C23C23×D25111yescharacteristic
216Q8—44no—
217D4—10044no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—5021no—
228C2×C4—5021no—
229C2×C4—5021no—
230C2×C4—5022no—
231C2×C4—5022no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4—5021no—
236C2×C4—5021no—
237C2×C4—5022no—
238C2×C4—5022no—
239C2×C4—5022no—
240C2×C4C2×C22⋊D25112yes—
241C2×C4C2×C22⋊D25112yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—10041no—
248C22—10041no—
249C22—10041no—
250C22—10042no—
251C22—10042no—
252C22C22×C22⋊D25111yescharacteristic
253C22#2254876012yes—
254C22#2254876111yescharacteristic
255C22#2254876211yescharacteristic
256C22#2254876312yes—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—10041no—
262C4—10041no—
263C4—10041no—
264C4—10042no—
265C4—10042no—
266C2—42no—
267C2—20081no—
268C2#3063224611yescharacteristic
269C2#3063226111yescharacteristic
270C2#3063226511yescharacteristic
271C2#3063243312yes—
272C2#3063243812yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×C22⋊Q8×C251C2
prime radicalC22×C22⋊D251C22
hypercenterC2×C22⋊Q8D251
hypocenterC251#2814
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #30631620C2semidirect
3. #30631618C2semidirect
4. #30631615C2semidirect
5. #30631613C2semidirect
6. #30631609C2semidirect
7. #30631606C2semidirect
9. #30631598C2semidirect
10. #30631547C2semidirect
11. C2×C22⋊Q8×C251C2semidirect
136. C251#2814semidirect
139. C2×C22⋊Q8D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
#306316202
#306316182
#306316152
#306316132
#306316092
#306316062
#306316022
#306315982
#306315472
C2×C22⋊Q8×C2512
#2814251