Finite Group Databasebeta

← #44562083

Subgroups — #44562083

Order 32128.

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

normal subgroups only

This group has 484 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#30632342C211yesmaximal, characteristic, complemented
3#30632066C211yesmaximal, characteristic, complemented
4#30631926C212yesmaximal, complemented
5#30631892C211yesmaximal, characteristic, complemented
6#30631889C211yesmaximal, characteristic, complemented
7#30631849C212yesmaximal, complemented
8#30631847C212yesmaximal, complemented
9C4×#16860795C211yesmaximal, characteristic, complemented
10D4×C251⋊C8C211yesmaximal, characteristic, complemented
11#547×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2212yes—
13#22548735—28no—
14C2×#16860800C2212yes—
15C2×#16860800C2212yes—
16C2×#16860798C2212yes—
17C2×#16860797C2212yes—
18C2×#16860795C2211yescharacteristic
19C22×C251⋊C8C2212yes—
20#22548715C2211yescharacteristic
21#22548697—22no—
22#22548694C2212yes—
23#22548686—21no—
24C4×D1004C2211yescharacteristic
25C4⋊C4×D251—21no—
26#22548648C2211yescharacteristic
27C4⋊(C251⋊C8)C2211yescharacteristic
28#22548639C2212yes—
29#22548638C2212yes—
30#22548636C2211yescharacteristic
31#22548635C2211yescharacteristic
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×Dic502C2312yes—
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—22no—
47C2×C4×D251—22no—
48C2×C4×D251—21no—
49#16860800—24no—
50#16860800—42no—
51#16860798—24no—
52#16860797—24no—
53#16860795D414yescomplemented
54C2×C251⋊C8—24no—
55C2×C251⋊C8C2311yescharacteristic
56C2×C251⋊C8C2312yes—
57C2×C251⋊C8C2311yescharacteristic
58C4⋊Dic251—21no—
59C4⋊Dic251—21no—
60C4⋊Dic251C2311yescharacteristic
61Dic251⋊C4—22no—
62C22⋊Dic251—22no—
63(C2×D251)⋊C4—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—21no—
83Dic502—22no—
84Dic502—22no—
85D1004—21no—
86D1004C2×D412yes—
87C22⋊D251—44no—
88C2×Dic251—21no—
89C2×Dic251—22no—
90C4×D251—44no—
91C4×D251—42no—
92C251⋊C8—22no—
93C251⋊C8—21no—
94C251⋊C8C2×D412yes—
95C23×C251—21no—
96C23×C251C2×D412yes—
97Q8×C251—22no—
98Q8×C251—22no—
99D4×C251—22no—
100D4×C251—21no—
101D4×C251—44no—
102D4×C251C2×D412yes—
103C2×C4×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251—44no—
108C2×C4×C251—42no—
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D412yes—
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C22×C251—22no—
114C22×C251—22no—
115C22×C251—22no—
116C22×C251—22no—
117C22×C251—41no—
118C22×C251—41no—
119C22×C251C22×D411yescharacteristic
120C2×D251—22no—
121C2×D251—41no—
122C2×D251—41no—
123C4×C251—22no—
124C4×C251—21no—
125C4×C251—42no—
126C4×C251—41no—
127C4×C251#13512yescomplemented
128C4×C251C22×D411yescharacteristic
129C4×C251H(5,𝔽2)11yescharacteristic
130Dic251—42no—
131Dic251—41no—
132C2×C251—42no—
133C2×C251—42no—
134C2×C251D4211yescharacteristic
135C2×C251C2×#13511yescharacteristic
136C2×C251#57811yescharacteristic
137D251—42no—
138C251#328311yescharacteristic, complemented, Sylow
139#3283—2511nomaximal, Sylow
140C2×#135—2512no—
141#547—2511no—
142#547D25111yescharacteristic, complemented
143#494—2511no—
144#467—2512no—
145#461—2511no—
146#461—2511no—
147#452—2512no—
148#452—2512no—
149C4×D8—2511no—
150C8×D4—2511no—
151C2×#41—2512no—
152C2×#41C2×D25112yes—
153#135—5028no—
154C2×Q16—2512no—
155C2×Q16—2512no—
156C2×SD16—2512no—
157C2×SD16—2512no—
158C2×D8—2511no—
159C22×C8—2512no—
160C4⋊Q8—2511no—
161C4⋊Q8C2×D25111yescharacteristic
162#123—22no—
163#123—5022no—
164C22⋊Q8—2512no—
165C22⋊Q8C2×D25112yes—
166#121—21no—
167#121—5021no—
168C4×D4—2511no—
169C4×D4C2×D25111yescharacteristic
170C4×D4C2×D25111yescharacteristic
171C2×C4⋊C4—21no—
172C2×C4⋊C4—5021no—
173#107—2511no—
174C4⋊C8—2511no—
175#103—2512no—
176#103—2512no—
177#102—2511no—
178#102—2511no—
179C22⋊C8—2512no—
180C4×C8—2511no—
181#41—24no—
182#41—42no—
183#41—10042no—
184#41—5024no—
185C2×Q8—2512no—
186C2×Q8C22×D25112yes—
187C2×D4—22no—
188C2×D4—5022no—
189C2×D4—2511no—
190C2×D4C22⋊D25111yescharacteristic
191C2×D4C22×D25111yescharacteristic
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—5021no—
195C22×C4—5022no—
196C22×C4C22×D25112yes—
197Q16—10042no—
198Q16—5024no—
199SD16—5024no—
200SD16—5024no—
201D8—2514no—
202C2×C8—5024no—
203C2×C8—2511no—
204C2×C8—2512no—
205C2×C8—2511no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—5022no—
210C4⋊C4—5021no—
211C4⋊C4—5021no—
212C4⋊C4—2511no—
213C4⋊C4C22⋊D25111yescharacteristic
214C4⋊C4C22×D25111yescharacteristic
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—5022no—
218C22⋊C4—5021no—
219C22⋊C4C22⋊D25112yes—
220C42C22×D25111yescharacteristic
221C23—21no—
222C23—5021no—
223C23C2×C22⋊D25112yes—
224Q8—22no—
225Q8—22no—
226Q8—5022no—
227Q8—5022no—
228D4—21no—
229D4—22no—
230D4—44no—
231D4—10044no—
232D4—5021no—
233D4—2512no—
234D4D4×D25112yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—10044no—
242C2×C4—10042no—
243C2×C4—5021no—
244C2×C4—5022no—
245C2×C4C2×C22⋊D25111yescharacteristic
246C2×C4C2×C22⋊D25112yes—
247C2×C4C2×C22⋊D25111yescharacteristic
248C2×C4C23×D25111yescharacteristic
249C8—5022no—
250C8—5021no—
251C8—2512no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—10041no—
259C22—10041no—
260C22—5022no—
261C22C22×C22⋊D25111yescharacteristic
262C4—21no—
263C4—22no—
264C4—41no—
265C4—42no—
266C4—10041no—
267C4—10042no—
268C4#2254873512yescomplemented
269C4C2×D4×D25111yescharacteristic
270C4#2254876011yescharacteristic
271C2—42no—
272C2—42no—
273C2—10042no—
274C2D4×C22⋊D25111yescharacteristic
275C2C2×#2254873511yescharacteristic
276C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#547×C251C2
socleC22×C251C22×D4
hypercenter#547D251
hypocenterC251#3283
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. #30632342C2semidirect
3. #30632066C2semidirect
4. #30631926C2semidirect
5. #30631892C2semidirect
6. #30631889C2semidirect
7. #30631849C2semidirect
8. #30631847C2semidirect
9. C4×#16860795C2semidirect
10. D4×C251⋊C8C2semidirect
11. #547×C251C2semidirect
53. #16860795D4semidirect
127. C4×C251#135semidirect
138. C251#3283semidirect
142. #547D251semidirect
268. C4#22548735semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
#306323422
#306320662
#306319262
#306318922
#306318892
#306318492
#306318472
C4×#168607952
D4×C251⋊C82
#547×C2512
#3283251