Finite Group Databasebeta

← #44558701

Subgroups — #44558701

Order 32128.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22548692C211yesmaximal, characteristic
2C22×C4×Dic251C211yesmaximal, characteristic
3#30631591C212yesmaximal, complemented
4#30631582C212yesmaximal, complemented
5#30631573C212yesmaximal, complemented
6#30631569C212yesmaximal, complemented
7#30631560C211yesmaximal, characteristic
8#30631548C211yesmaximal, characteristic
9#30631548C211yesmaximal, characteristic
10#30631545C211yesmaximal, characteristic
11#385×C251C211yesmaximal, characteristic
12C22×Dic502C2211yescharacteristic
13C23×Dic251C2211yescharacteristic
14C23×Dic251C2211yescharacteristic
15#22548692—24no—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×Dic251⋊C4C2212yes—
18C2×Dic251⋊C4C2212yes—
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×C22⋊Dic251C2211yescharacteristic
22C2×C22⋊Dic251C2211yescharacteristic
23C2×C4×Dic251—24no—
24C2×C4×Dic251C2212yes—
25C2×C4×Dic251C2212yes—
26C251⋊#95C2212yes—
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C23×C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2212yes—
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2211yescharacteristic
36#95×C251C2212yes—
37C2×Dic502—22no—
38C2×Dic502—24no—
39C22×Dic251—24no—
40C22×Dic251—22no—
41C22×Dic251—22no—
42C22×Dic251D414yes—
43C22×Dic251Q814yescomplemented
44C22×Dic251C2311yescharacteristic
45C22×Dic251C2312yes—
46C22×Dic251C2311yescharacteristic
47C22×Dic251C2311yescharacteristic
48C22×Dic251C2312yes—
49C22×Dic251C2311yescharacteristic
50C4⋊Dic251—22no—
51Dic251⋊C4—22no—
52Dic251⋊C4—22no—
53Dic251⋊C4—24no—
54Dic251⋊C4—24no—
55C22⋊Dic251—22no—
56C22⋊Dic251—22no—
57C4×Dic251—24no—
58C4×Dic251—24no—
59C24×C251C2311yescharacteristic
60C22×C4×C251—22no—
61C22×C4×C251—24no—
62C22×C4×C251C2311yescharacteristic
63C22×C4×C251C2312yes—
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2312yes—
66C4⋊C4×C251—24no—
67C22⋊C4×C251—22no—
68C22⋊C4×C251—22no—
69Dic502—44no—
70C2×Dic251—28no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—28no—
75C2×Dic251—21no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251C2×D412yes—
83C2×Dic251C2×D412yes—
84C2×Dic251C2×Q814yes—
85C2×Dic251#4114yes—
86C23×C251—22no—
87C23×C251—22no—
88C23×C251C2×D412yes—
89C23×C251C2×Q812yes—
90C23×C251#4112yes—
91C23×C251C2411yescharacteristic
92C2×C4×C251—28no—
93C2×C4×C251—22no—
94C2×C4×C251—22no—
95C2×C4×C251—21no—
96C2×C4×C251—21no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—22no—
101C2×C4×C251—22no—
102C2×C4×C251#4114yes—
103C22×C251—22no—
104C22×C251—22no—
105C22×C251—24no—
106C22×C251—24no—
107C22×C251C22⋊Q814yes—
108C22×C251C22×D411yescharacteristic
109C22×C251C22×Q811yescharacteristic
110C22×C251C2×#4112yes—
111C22×C251C2×#4112yes—
112C22×C251C2×#4111yescharacteristic
113C4×C251—24no—
114C4×C251—41no—
115C4×C251—41no—
116C4×C251—42no—
117Dic251—24no—
118Dic251—22no—
119Dic251—41no—
120Dic251—41no—
121Dic251—42no—
122Dic251#12314yes—
123C2×C251—24no—
124C2×C251C2×C22⋊Q811yescharacteristic
125C2×C251C2×#12311yescharacteristic
126C2×C251#52511yescharacteristic
127C2×C251#52912yes—
128C2×C251#53312yes—
129C251#285511yescharacteristic, complemented, Sylow
130#2855—2511nomaximal, Sylow
131C2×C22⋊Q8—2511no—
132C22×C42—2511no—
133#391—2512no—
134#389—2512no—
135#387—2512no—
136#387—2512no—
137#385—2511no—
138#385D25111yescharacteristic
139#381—2511no—
140#381—2511no—
141#380—2511no—
142C22×Q8—2511no—
143C23×C4—2511no—
144C23×C4—2511no—
145C23×C4C2×D25111yescharacteristic
146C22⋊Q8—5024no—
147C2×C4⋊C4—2511no—
148C2×C4⋊C4—2511no—
149C2×C4⋊C4—2511no—
150C2×C4⋊C4—2512no—
151C2×C4⋊C4—2512no—
152C2×C4⋊C4C2×D25112yes—
153C2×C22⋊C4—2511no—
154C2×C22⋊C4—2511no—
155C2×C22⋊C4C2×D25111yescharacteristic
156C2×C22⋊C4C2×D25111yescharacteristic
157C2×C42—5024no—
158C2×C42—2512no—
159C2×C42—2512no—
160#95—2512no—
161#95—2512no—
162#95—2512no—
163#95—2512no—
164#95—2512no—
165#95—2512no—
166#95C2×D25112yes—
167C24C22×D25111yescharacteristic
168C2×Q8—5022no—
169C2×Q8—5024no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—5022no—
173C22×C4—5022no—
174C22×C4—5024no—
175C22×C4—2512no—
176C22×C4—2511no—
177C22×C4—2511no—
178C22×C4—2514no—
179C22×C4—2514no—
180C22×C4—2512no—
181C22×C4—2511no—
182C22×C4—2511no—
183C22×C4C22×D25112yes—
184C22×C4C22×D25111yescharacteristic
185C22×C4C22×D25111yescharacteristic
186C22×C4C22×D25112yes—
187C4⋊C4—24no—
188C4⋊C4—5024no—
189C4⋊C4—5022no—
190C4⋊C4—5022no—
191C4⋊C4—5022no—
192C4⋊C4—5024no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—5022no—
196C22⋊C4—5022no—
197C42—5024no—
198C42—5024no—
199C23—22no—
200C23—22no—
201C23D4×D25112yes—
202C23Q8×D25112yes—
203C23#1686081012yes—
204C23C23×D25111yescharacteristic
205Q8—10044no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—5021no—
217C2×C4—5021no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—5022no—
221C2×C4—5022no—
222C2×C4—5022no—
223C2×C4—5022no—
224C2×C4—5022no—
225C2×C4—5022no—
226C2×C4—5028no—
227C2×C4—5028no—
228C2×C4—2512no—
229C2×C4—2514no—
230C2×C4—2512no—
231C2×C4—2514no—
232C2×C4#1686080914yes—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—22no—
237C22#2254869314yes—
238C22C2×D4×D25111yescharacteristic
239C22C2×Q8×D25111yescharacteristic
240C22#41×D25112yes—
241C22C2×#1686080912yes—
242C22C2×#1686081011yescharacteristic
243C4—24no—
244C4—41no—
245C4—42no—
246C4—41no—
247C4—10041no—
248C4—10042no—
249C4—10041no—
250C4—5024no—
251C4—5022no—
252C4—2514no—
253C2—24no—
254C2C2×#2254869311yescharacteristic
255C2#123×D25111yescharacteristic
256C2#3063219311yescharacteristic
257C2#3063221712yes—
258C2#3063223912yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#385×C251C2
prime radicalC24×C251C23
hypercenter#385D251
hypocenterC251#2855
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #30631591C2semidirect
4. #30631582C2semidirect
5. #30631573C2semidirect
6. #30631569C2semidirect
43. C22×Dic251Q8semidirect
129. C251#2855semidirect

Maximal subgroups

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

subgroupindex
C2×#225486922
C22×C4×Dic2512
#306315912
#306315822
#306315732
#306315692
#306315602
#306315482
#306315452
#385×C2512
#2855251