Finite Group Databasebeta

← #44560536

Subgroups — #44560536

Order 32128.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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#30632221C211yesmaximal, characteristic
2#30632022C211yesmaximal, characteristic, complemented
3#30632005C211yesmaximal, characteristic
4#30631994C212yesmaximal, complemented
5#30631909C212yesmaximal, complemented
6#30631843C211yesmaximal, characteristic, complemented
7#30631826C211yesmaximal, characteristic
8#30631792C212yesmaximal, complemented
9#30631785C212yesmaximal, complemented
10#30631744C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C251C211yesmaximal, characteristic, complemented
12C2×#16860800C2212yes—
13C2×#16860797C2212yes—
14#22548709—21no—
15#22548707C2211yescharacteristic
16#22548703C2211yescharacteristic
17#22548698—21no—
18#22548694C2211yescharacteristic
19#22548689C2211yescharacteristic
20C4×Dic502C2211yescharacteristic
21C4×D1004C2211yescharacteristic
22C4×C22⋊D251—21no—
23#22548672—21no—
24#22548648C2212yes—
25#22548646C2212yes—
26C4⋊(C251⋊C8)C2212yes—
27#22548639C2212yes—
28#22548639C2212yes—
29#22548638C2212yes—
30#22548635C2212yes—
31C22⋊(C251⋊C8)C2211yescharacteristic
32C22⋊(C251⋊C8)C2211yescharacteristic
33#22548626C2212yes—
34C22×Q8×C251C2211yescharacteristic
35C4×Q8×C251—21no—
36C4×Q8×C251—21no—
37C4×Q8×C251C2212yes—
38C4×Q8×C251C2212yes—
39C2×C4⋊C4×C251—21no—
40C2×C4⋊C4×C251C2211yescharacteristic
41C2×C42×C251—21no—
42C2×C42×C251C2211yescharacteristic
43C2×Dic502C2311yescharacteristic
44C2×D1004C2311yescharacteristic
45C2×C22⋊D251—21no—
46C2×C4×D251—21no—
47#16860800—24no—
48#16860797—24no—
49C2×C251⋊C8C2312yes—
50C2×C251⋊C8C2312yes—
51C4⋊Dic251C2311yescharacteristic
52C4⋊Dic251C2311yescharacteristic
53Dic251⋊C4—21no—
54Dic251⋊C4—21no—
55Dic251⋊C4—21no—
56Dic251⋊C4—21no—
57C22⋊Dic251—21no—
58(C2×D251)⋊C4—21no—
59(C2×D251)⋊C4—21no—
60(C2×D251)⋊C4—21no—
61(C2×D251)⋊C4—21no—
62C4×Dic251—21no—
63C2×Q8×C251—24no—
64C2×Q8×C251—21no—
65C2×Q8×C251C2312yes—
66C22×C4×C251—21no—
67C22×C4×C251—21no—
68C22×C4×C251D411yescharacteristic
69C22×C4×C251D411yescharacteristic
70C22×C4×C251C2311yescharacteristic
71C4⋊C4×C251—22no—
72C4⋊C4×C251—22no—
73C4⋊C4×C251—21no—
74C4⋊C4×C251C2312yes—
75C42×C251—22no—
76C42×C251—22no—
77C42×C251D411yescharacteristic
78C42×C251D411yescharacteristic
79C42×C251C2311yescharacteristic
80C42×C251C2311yescharacteristic
81C22×D251—21no—
82Dic502—22no—
83D1004—22no—
84C22⋊D251—42no—
85C2×Dic251—21no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C4×D251—42no—
89C251⋊C8—22no—
90C251⋊C8—22no—
91C23×C251C2×D411yescharacteristic
92Q8×C251—24no—
93Q8×C251—22no—
94Q8×C251#4114yes—
95C2×C4×C251—24no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—41no—
101C2×C4×C251—42no—
102C2×C4×C251C2×D411yescharacteristic
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251C2×D411yescharacteristic
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2411yescharacteristic
108C22×C251—21no—
109C22×C251—21no—
110C22×C251#13712yes—
111C22×C251C22×D411yescharacteristic
112C2×D251—41no—
113C2×D251—41no—
114C2×D251—41no—
115C4×C251—24no—
116C4×C251—21no—
117C4×C251—21no—
118C4×C251—21no—
119C4×C251—21no—
120C4×C251—21no—
121C4×C251—42no—
122C4×C251C2×#4112yes—
123Dic251—41no—
124Dic251—41no—
125Dic251—41no—
126C2×C251—22no—
127C2×C251#52511yescharacteristic
128C2×C251C2×#13711yescharacteristic
129C2×C251#57511yescharacteristic
130D251—81no—
131C251#311711yescharacteristic, complemented, Sylow
132#3117—2511nomaximal, Sylow
133#529—2511no—
134C2×C4×Q8D25111yescharacteristic, complemented
135#484—2511no—
136#481—2511no—
137#479—2512no—
138#464—2512no—
139#451—2511no—
140#448—2511no—
141#441—2512no—
142#440—2512no—
143#432—2511no—
144C22×Q8C2×D25111yescharacteristic
145C2×Q16—2512no—
146C2×SD16—2512no—
147#126—5021no—
148#125—2511no—
149#124—2511no—
150#123—5021no—
151C22⋊Q8—2511no—
152#121—2511no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—2511no—
156C4×Q8C2×D25112yes—
157C4×Q8C2×D25112yes—
158C4×D4—5021no—
159C4×D4—2511no—
160#117—5021no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D25111yescharacteristic
163C2×C42—21no—
164C2×C42C2×D25111yescharacteristic
165#107—2512no—
166#106—2512no—
167C4⋊C8—2512no—
168#103—2512no—
169#103—2512no—
170#103—2512no—
171#102—2512no—
172C22⋊C8—2511no—
173C22⋊C8—2511no—
174#97—2512no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—2511no—
178C2×Q8C22×D25112yes—
179C2×D4—5021no—
180C2×D4—2511no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—5021no—
184C22×C4C22⋊D25111yescharacteristic
185C22×C4C22⋊D25111yescharacteristic
186C22×C4C22×D25111yescharacteristic
187Q16—5024no—
188SD16—5024no—
189C2×C8—2512no—
190C2×C8—2512no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—5021no—
195C4⋊C4—5021no—
196C4⋊C4—5021no—
197C4⋊C4—5021no—
198C4⋊C4—2511no—
199C4⋊C4—2511no—
200C4⋊C4C22×D25112yes—
201C22⋊C4—5021no—
202C22⋊C4—5021no—
203C22⋊C4—5021no—
204C22⋊C4—5021no—
205C22⋊C4—5021no—
206C42—22no—
207C42—22no—
208C42—5021no—
209C42C22⋊D25111yescharacteristic
210C42C22⋊D25111yescharacteristic
211C42C22×D25111yescharacteristic
212C42C22×D25111yescharacteristic
213C23—5021no—
214C23C2×C22⋊D25111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—5022no—
218Q8#1686081014yes—
219D4—10042no—
220D4—5022no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—10042no—
229C2×C4—5021no—
230C2×C4—5021no—
231C2×C4—5021no—
232C2×C4C2×C22⋊D25111yescharacteristic
233C2×C4C2×C22⋊D25111yescharacteristic
234C2×C4C2×C22⋊D25111yescharacteristic
235C2×C4C2×C22⋊D25111yescharacteristic
236C2×C4C2×C22⋊D25111yescharacteristic
237C2×C4C23×D25111yescharacteristic
238C8—5022no—
239C8—5022no—
240C22—21no—
241C22—21no—
242C22—10041no—
243C22—10041no—
244C22—10041no—
245C22#2254874612yes—
246C22C22×C22⋊D25111yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—10041no—
255C4—10041no—
256C4—10041no—
257C4C2×#1686081012yes—
258C2—22no—
259C2—20081no—
260C2#3063219811yescharacteristic
261C2C2×#2254874611yescharacteristic
262C2#3063248511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C4×Q8×C251C2
socleC22×C251C22×D4
prime radical#22548689C22
hypercenterC2×C4×Q8D251
hypocenterC251#3117
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #30632022C2semidirect
4. #30631994C2semidirect
5. #30631909C2semidirect
6. #30631843C2semidirect
8. #30631792C2semidirect
9. #30631785C2semidirect
10. #30631744C2semidirect
11. C2×C4×Q8×C251C2semidirect
131. C251#3117semidirect
134. C2×C4×Q8D251semidirect

Maximal subgroups

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

subgroupindex
#306322212
#306320222
#306320052
#306319942
#306319092
#306318432
#306318262
#306317922
#306317852
#306317442
C2×C4×Q8×C2512
#3117251