Finite Group Databasebeta

← #44551274

Subgroups — #44551274

Order 32128.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#16860795C211yesmaximal, characteristic
2C2×#22548650C211yesmaximal, characteristic, complemented
3#30631691C211yesmaximal, characteristic, complemented
4C2×#22548632C211yesmaximal, characteristic, complemented
5C2×#22548631C211yesmaximal, characteristic, complemented
6#30631643C211yesmaximal, characteristic, complemented
7#418×C251C211yesmaximal, characteristic, complemented
8C22×D1004C2211yescharacteristic
9C2×#16860795—21no—
10C2×#16860795—21no—
11C2×#16860795—22no—
12C2×#16860795C414yescomplemented
13C2×#16860794C2211yescharacteristic
14C2×#16860794C2211yescharacteristic
15C22×C251⋊C8C2211yescharacteristic
16#22548650—22no—
17#22548635—22no—
18#22548632—22no—
19#22548631—22no—
20#22548626—21no—
21C4×C251⋊C8—21no—
22C22×D4×C251C2211yescharacteristic
23C2×M16×C251C2211yescharacteristic
24#117×C251C2211yescharacteristic
25#102×C251—22no—
26C23×D251—21no—
27C2×D1004—21no—
28C2×D1004—21no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004—21no—
32C2×D1004C2×C412yes—
33#16860795—24no—
34#16860795—24no—
35#16860794—21no—
36#16860794—21no—
37#16860794—21no—
38#16860794D412yes—
39C2×C251⋊C8—21no—
40C2×C251⋊C8—21no—
41C2×C251⋊C8C2×C412yes—
42C2×C251⋊C8D411yescharacteristic
43C2×C251⋊C8D411yescharacteristic
44C2×C251⋊C8D412yes—
45C24×C251—21no—
46C2×D4×C251—21no—
47C2×D4×C251—21no—
48C2×D4×C251—21no—
49C2×D4×C251—21no—
50C2×D4×C251—21no—
51C2×D4×C251C2×C412yes—
52C22×C4×C251C2311yescharacteristic
53M16×C251—21no—
54M16×C251—21no—
55C2×C8×C251—21no—
56C4⋊C4×C251D412yes—
57C22⋊C4×C251—21no—
58C42×C251—21no—
59C22×D251—22no—
60C22×D251—41no—
61C22×D251—41no—
62C22×D251—82no—
63D1004—22no—
64D1004—22no—
65D1004—41no—
66D1004—41no—
67C251⋊C8—22no—
68C251⋊C8—41no—
69C251⋊C8C22⋊C412yes—
70C251⋊C8C22⋊C412yes—
71C23×C251—22no—
72C23×C251—41no—
73C23×C251—41no—
74C23×C251—82no—
75C23×C251#4111yescharacteristic
76D4×C251—22no—
77D4×C251—22no—
78D4×C251—41no—
79D4×C251—41no—
80C2×C4×C251—22no—
81C2×C4×C251C22×C411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251#4111yescharacteristic
87C8×C251—41no—
88C22×C251—22no—
89C22×C251—42no—
90C22×C251—82no—
91C22×C251—82no—
92C22×C251—82no—
93C22×C251C4×D411yescharacteristic
94C22×C251#12111yescharacteristic
95C22×C251#12411yescharacteristic
96C2×D251—42no—
97C2×D251—82no—
98C2×D251—82no—
99C2×D251—82no—
100C4×C251—42no—
101C4×C251C2×C22⋊C411yescharacteristic
102C4×C251C4×D411yescharacteristic
103C4×C251#12111yescharacteristic
104C4×C251#12711yescharacteristic
105C2×C251—21no—
106C2×C251—21no—
107C2×C251—82no—
108C2×C251#39011yescharacteristic
109C2×C251#47212yes—
110D251—82no—
111C251#197811yescharacteristic, complemented, Sylow
112#1978—2511nomaximal, Sylow
113C22×D8—2511no—
114C2×#108—2511no—
115#418—2511no—
116#418D25111yescharacteristic, complemented
117C2×#100—2511no—
118C2×#100—2511no—
119#405—2511no—
120C22×D4—2511no—
121C22×D4C2×D25111yescharacteristic
122C2×D8—5021no—
123C2×D8—5021no—
124C2×D8—5022no—
125C2×D8—2514no—
126C2×M16—2511no—
127C2×M16—2511no—
128C2×M16C2×D25111yescharacteristic
129C22×C8—2511no—
130#117C2×D25111yescharacteristic
131#108—5022no—
132#102—22no—
133#102—5022no—
134#100—5022no—
135#100—5022no—
136#97—5021no—
137C4×C8—5021no—
138C24—21no—
139C24—5021no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—5021no—
146C2×D4—5021no—
147C2×D4—5021no—
148C2×D4—5021no—
149C2×D4—5021no—
150C2×D4—2512no—
151C2×D4C4×D25112yes—
152C22×C4C22×D25111yescharacteristic
153D8—5024no—
154D8—5024no—
155M16—21no—
156M16—21no—
157M16—5021no—
158M16—5021no—
159M16—5021no—
160M16—2512no—
161C2×C8—21no—
162C2×C8—5021no—
163C2×C8—5021no—
164C2×C8—2512no—
165C2×C8—2511no—
166C2×C8—2511no—
167C2×C8—2512no—
168C4⋊C4C22⋊D25112yes—
169C22⋊C4—21no—
170C42—21no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—82no—
175C23—20082no—
176C23—10041no—
177C23—10041no—
178C23—5022no—
179C23#1686080911yescharacteristic
180D4—22no—
181D4—22no—
182D4—41no—
183D4—41no—
184D4—10041no—
185D4—10041no—
186D4—5022no—
187D4—5022no—
188C2×C4—22no—
189C2×C4C2×C4×D25111yescharacteristic
190C2×C4D4×D25111yescharacteristic
191C2×C4D4×D25111yescharacteristic
192C2×C4D4×D25111yescharacteristic
193C2×C4C2×C22⋊D25111yescharacteristic
194C2×C4#1686081011yescharacteristic
195C8—41no—
196C8—10041no—
197C8—5022no—
198C8—2512no—
199C8—2512no—
200C22—22no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—20082no—
206C22—20082no—
207C22—20082no—
208C22—10042no—
209C22(C22⋊D251)⋊C411yescharacteristic
210C22#2254868711yescharacteristic
211C22#2254870111yescharacteristic
212C4—42no—
213C4C22⋊C4×D25111yescharacteristic
214C4C4×C22⋊D25111yescharacteristic
215C4#2254868811yescharacteristic
216C4#2254871111yescharacteristic
217C2—21no—
218C2—21no—
219C2—82no—
220C2—20082no—
221C2#3063158711yescharacteristic
222C2#3063195512yes—
subgroupG/H
centerC22#22548687
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#418×C251C2
socleC22×C251#121
prime radicalC22×#16860795C2
cosocleC22×C4×C251C23
hypercenter#418D251
hypocenterC251#1978
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#22548650C2semidirect
3. #30631691C2semidirect
4. C2×#22548632C2semidirect
5. C2×#22548631C2semidirect
6. #30631643C2semidirect
7. #418×C251C2semidirect
12. C2×#16860795C4semidirect
111. C251#1978semidirect
116. #418D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607952
C2×#225486502
#306316912
C2×#225486322
C2×#225486312
#306316432
#418×C2512
#1978251