Finite Group Databasebeta

← #44562091

Subgroups — #44562091

Order 32128.

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

normal subgroups only

This group has 494 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×#22548735C212yesmaximal, complemented
2#30632358C211yesmaximal, characteristic, complemented
3#30632061C211yesmaximal, characteristic, complemented
4#30631918C212yesmaximal, complemented
5#30631894C211yesmaximal, characteristic, complemented
6#30631882C211yesmaximal, characteristic, complemented
7#30631847C212yesmaximal, complemented
8#30631832C212yesmaximal, complemented
9C4×#16860798C211yesmaximal, characteristic, complemented
10D4×C251⋊C8C211yesmaximal, characteristic, complemented
11#547×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2212yes—
13#22548735—28no—
14C2×#16860800C2212yes—
15C2×#16860798C2211yescharacteristic
16C2×#16860797C2212yes—
17C2×#16860797C2212yes—
18C2×#16860795C2212yes—
19C22×C251⋊C8C2212yes—
20#22548712C2211yescharacteristic
21#22548711—21no—
22#22548689C2212yes—
23#22548688—22no—
24C4×Dic502C2211yescharacteristic
25#22548677—21no—
26#22548646C2211yescharacteristic
27C4⋊(C251⋊C8)C2211yescharacteristic
28#22548639C2212yes—
29#22548638C2211yescharacteristic
30#22548636C2211yescharacteristic
31#22548635C2212yes—
32C22⋊(C251⋊C8)C2212yes—
33C4×C251⋊C8C2211yescharacteristic
34C2×#41×C251C2212yes—
35C4⋊Q8×C251C2211yescharacteristic
36#123×C251—22no—
37C22⋊Q8×C251C2212yes—
38#121×C251—21no—
39C4×D4×C251C2211yescharacteristic
40C4×D4×C251C2211yescharacteristic
41C2×C4⋊C4×C251—21no—
42#16860810—24no—
43#16860810—42no—
44C2×Dic502C2311yescharacteristic
45C2×D1004—21no—
46C2×D1004—21no—
47C2×D1004C2312yes—
48C2×C22⋊D251—22no—
49C2×C22⋊D251—22no—
50C2×C4×D251—22no—
51#16860800—24no—
52#16860798D414yescomplemented
53#16860797—24no—
54#16860797—42no—
55#16860795—24no—
56C2×C251⋊C8—24no—
57C2×C251⋊C8C2311yescharacteristic
58C2×C251⋊C8C2312yes—
59C2×C251⋊C8C2311yescharacteristic
60C4⋊Dic251C2311yescharacteristic
61Dic251⋊C4—21no—
62(C2×D251)⋊C4—22no—
63C4×Dic251—21no—
64#41×C251—24no—
65#41×C251—42no—
66C2×Q8×C251C2312yes—
67C2×D4×C251—22no—
68C2×D4×C251D411yescharacteristic
69C2×D4×C251C2311yescharacteristic
70C22×C4×C251—21no—
71C22×C4×C251—22no—
72C22×C4×C251C2312yes—
73C4⋊C4×C251—21no—
74C4⋊C4×C251—22no—
75C4⋊C4×C251—21no—
76C4⋊C4×C251D411yescharacteristic
77C4⋊C4×C251C2311yescharacteristic
78C22⋊C4×C251—22no—
79C22⋊C4×C251—21no—
80C22⋊C4×C251D412yes—
81C42×C251C2311yescharacteristic
82C22×D251—22no—
83Dic502—21no—
84Dic502C2×D412yes—
85D1004—22no—
86D1004—22no—
87D1004—42no—
88D1004—81no—
89C22⋊D251—44no—
90C22⋊D251—82no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C4×D251—44no—
94C251⋊C8—22no—
95C251⋊C8—21no—
96C251⋊C8C2×D412yes—
97C23×C251—21no—
98C23×C251C2×D412yes—
99Q8×C251—22no—
100Q8×C251—22no—
101D4×C251—22no—
102D4×C251—21no—
103D4×C251—44no—
104D4×C251C2×D412yes—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—21no—
108C2×C4×C251—22no—
109C2×C4×C251—44no—
110C2×C4×C251—42no—
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2×D412yes—
113C2×C4×C251C2×D411yescharacteristic
114C2×C4×C251C2411yescharacteristic
115C22×C251—22no—
116C22×C251—22no—
117C22×C251—22no—
118C22×C251—22no—
119C22×C251—41no—
120C22×C251—41no—
121C22×C251C22×D411yescharacteristic
122C2×D251—42no—
123C2×D251—42no—
124C2×D251—42no—
125C4×C251—22no—
126C4×C251—21no—
127C4×C251—42no—
128C4×C251—41no—
129C4×C251#13512yescomplemented
130C4×C251C22×D411yescharacteristic
131C4×C251H(5,𝔽2)11yescharacteristic
132Dic251—22no—
133Dic251—41no—
134C2×C251—42no—
135C2×C251—42no—
136C2×C251D4211yescharacteristic
137C2×C251C2×#13511yescharacteristic
138C2×C251#57711yescharacteristic
139D251—82no—
140C251#328411yescharacteristic, complemented, Sylow
141#3284—2511nomaximal, Sylow
142C2×#135—2512no—
143#550—2511no—
144#547D25111yescharacteristic, complemented
145#492—2511no—
146#465—2512no—
147#462—2511no—
148#459—2511no—
149#452—2512no—
150#449—2512no—
151C4×SD16—2511no—
152C8×D4—2511no—
153C2×#41—2512no—
154C2×#41C2×D25112yes—
155#135—5028no—
156C2×Q16—2512no—
157C2×SD16—2512no—
158C2×SD16—2512no—
159C2×SD16—2511no—
160C2×D8—2512no—
161C22×C8—2512no—
162C4⋊Q8C2×D25111yescharacteristic
163#127—5021no—
164#127—2511no—
165#123—22no—
166C22⋊Q8C2×D25112yes—
167#121—21no—
168#121—5022no—
169#121—2512no—
170C4×Q8—2511no—
171C4×D4—5021no—
172C4×D4C2×D25111yescharacteristic
173C4×D4C2×D25111yescharacteristic
174C2×C4⋊C4—21no—
175#106—2511no—
176C4⋊C8—2511no—
177#103—2512no—
178#103—2511no—
179#102—2511no—
180#102—2512no—
181C22⋊C8—2512no—
182C4×C8—2511no—
183#41—24no—
184#41—42no—
185#41—10042no—
186#41—5024no—
187C2×Q8—2511no—
188C2×Q8C22×D25112yes—
189C2×D4—22no—
190C2×D4—5022no—
191C2×D4—5022no—
192C2×D4—5021no—
193C2×D4—5021no—
194C2×D4—2512no—
195C2×D4C22⋊D25111yescharacteristic
196C2×D4C22×D25111yescharacteristic
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—5022no—
200C22×C4C22×D25112yes—
201Q16—5024no—
202SD16—10042no—
203SD16—5024no—
204SD16—2514no—
205D8—5024no—
206C2×C8—5024no—
207C2×C8—2511no—
208C2×C8—2512no—
209C2×C8—2511no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—5021no—
214C4⋊C4—2511no—
215C4⋊C4C22⋊D25111yescharacteristic
216C4⋊C4C22×D25111yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—5022no—
220C22⋊C4C22⋊D25112yes—
221C42—5021no—
222C42C22×D25111yescharacteristic
223C23—21no—
224C23—5022no—
225C23C2×C22⋊D25112yes—
226Q8—22no—
227Q8—22no—
228Q8—5021no—
229Q8—2512no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—20081no—
234D4—20082no—
235D4—10044no—
236D4—10042no—
237D4—5022no—
238D4—5022no—
239D4D4×D25112yes—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—10044no—
247C2×C4—5021no—
248C2×C4—5021no—
249C2×C4C2×C22⋊D25111yescharacteristic
250C2×C4C2×C22⋊D25112yes—
251C2×C4C2×C22⋊D25111yescharacteristic
252C2×C4C23×D25111yescharacteristic
253C8—5022no—
254C8—5021no—
255C8—2512no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—22no—
260C22—41no—
261C22—41no—
262C22—10042no—
263C22—10042no—
264C22—10042no—
265C22C22×C22⋊D25111yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—10041no—
271C4—5022no—
272C4#2254873512yescomplemented
273C4C2×D4×D25111yescharacteristic
274C4#2254876011yescharacteristic
275C2—42no—
276C2—42no—
277C2—20082no—
278C2D4×C22⋊D25111yescharacteristic
279C2C2×#2254873511yescharacteristic
280C2#3063249511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#547×C251C2
socleC22×C251C22×D4
hypercenter#547D251
hypocenterC251#3284
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#22548735C2semidirect
2. #30632358C2semidirect
3. #30632061C2semidirect
4. #30631918C2semidirect
5. #30631894C2semidirect
6. #30631882C2semidirect
7. #30631847C2semidirect
8. #30631832C2semidirect
9. C4×#16860798C2semidirect
10. D4×C251⋊C8C2semidirect
11. #547×C251C2semidirect
52. #16860798D4semidirect
129. C4×C251#135semidirect
140. C251#3284semidirect
144. #547D251semidirect
272. C4#22548735semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
#306323582
#306320612
#306319182
#306318942
#306318822
#306318472
#306318322
C4×#168607982
D4×C251⋊C82
#547×C2512
#3284251