Finite Group Databasebeta

← #44563038

Subgroups — #44563038

Order 32128.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic502C211yesmaximal, characteristic
2#30632412C211yesmaximal, characteristic
3#30632037C211yesmaximal, characteristic
4#30632036C212yesmaximal
5#30631983C211yesmaximal, characteristic
6#30631980C212yesmaximal
7#30631895C212yesmaximal
8#30631811C211yesmaximal, characteristic
9Q16×Dic251C212yesmaximal
10C4×#16860800C211yesmaximal, characteristic
11#462×C251C211yesmaximal, characteristic
12C2×#16860800C2211yescharacteristic
13#22548715—21no—
14#22548715C2211yescharacteristic
15#22548714—22no—
16#22548714C2212yes—
17#22548713C2212yes—
18#22548706—21no—
19#22548705—22no—
20Q8×Dic251C2212yes—
21Q8×Dic251C2212yes—
22C4×Dic502—21no—
23C4×Dic502C2211yescharacteristic
24C4×Dic502C2211yescharacteristic
25#22548680—21no—
26#22548649C2212yes—
27#22548648C2211yescharacteristic
28Dic251⋊C8C2212yes—
29#22548640C2212yes—
30#22548639C2212yes—
31#22548639C2211yescharacteristic
32#22548639C2212yes—
33#22548638C2211yescharacteristic
34C4×C251⋊C8C2211yescharacteristic
35C8×Dic251C2212yes—
36C2×Q16×C251C2212yes—
37C4⋊Q8×C251C2211yescharacteristic
38C4×Q8×C251C2211yescharacteristic
39C4⋊C8×C251C2211yescharacteristic
40#103×C251C2212yes—
41C2×Dic502—22no—
42C2×Dic502—21no—
43C2×Dic502D411yescharacteristic
44C2×Dic502C2311yescharacteristic
45#16860800—22no—
46C2×C251⋊C8C2311yescharacteristic
47C2×C251⋊C8C2311yescharacteristic
48C4⋊Dic251—22no—
49C4⋊Dic251—22no—
50C4⋊Dic251—21no—
51C4⋊Dic251D411yescharacteristic
52C4⋊Dic251C2312yes—
53C4⋊Dic251C2311yescharacteristic
54Dic251⋊C4—22no—
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57Dic251⋊C4—21no—
58Dic251⋊C4D412yes—
59C4×Dic251—22no—
60C4×Dic251—22no—
61C4×Dic251—21no—
62C4×Dic251C2312yes—
63C2×Q8×C251C2312yes—
64C2×Q8×C251C2311yescharacteristic
65Q16×C251—24no—
66C2×C8×C251C2312yes—
67C4⋊C4×C251—21no—
68C4⋊C4×C251—21no—
69C4⋊C4×C251—21no—
70C4⋊C4×C251C2311yescharacteristic
71C42×C251—21no—
72C42×C251C2311yescharacteristic
73Dic502—21no—
74Dic502—21no—
75Dic502—82no—
76Dic502—81no—
77Dic502Q1614yescomplemented
78C2×Dic251—21no—
79C2×Dic251—21no—
80C2×Dic251—22no—
81C2×Dic251C2×D412yes—
82C2×Dic251C2×D412yes—
83C251⋊C8—21no—
84C251⋊C8—21no—
85Q8×C251—21no—
86Q8×C251—22no—
87Q8×C251—22no—
88Q8×C251#4112yes—
89C2×C4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251—22no—
92C2×C4×C251C2×D411yescharacteristic
93C2×C4×C251C2×D411yescharacteristic
94C2×C4×C251C2411yescharacteristic
95C8×C251—22no—
96C22×C251C22×D411yescharacteristic
97C4×C251—22no—
98C4×C251—21no—
99C4×C251—42no—
100C4×C251—41no—
101C4×C251C2×Q1612yes—
102C4×C251C2×#4111yescharacteristic
103C4×C2512−1+411yescharacteristic
104Dic251—22no—
105Dic251—41no—
106Dic251—41no—
107Dic251—42no—
108Dic251C2×Q1614yes—
109C2×C251#54811yescharacteristic
110C2×C251C22×Q1611yescharacteristic
111C2×C251#57711yescharacteristic
112C251#336311yescharacteristic, complemented, Sylow
113#3363—2511nomaximal, Sylow
114Q82—2511no—
115#557—2511no—
116#487—2511no—
117#487—2512no—
118#477—2511no—
119#477—2512no—
120#462—2512no—
121#462D25111yescharacteristic
122C8×Q8—2511no—
123C4×Q16—2512no—
124C4×Q16—2511no—
125C2×Q16—2511no—
126C2×Q16C2×D25112yes—
127C4⋊Q8—5021no—
128C4⋊Q8—5022no—
129C4⋊Q8—2512no—
130C4⋊Q8—2512no—
131C4⋊Q8—2511no—
132C4⋊Q8C2×D25111yescharacteristic
133#125—5022no—
134#125—5021no—
135C4×Q8—5021no—
136C4×Q8—5021no—
137C4×Q8—2512no—
138C4×Q8—2512no—
139C4×Q8—2511no—
140C4×Q8—2511no—
141C4×Q8C2×D25111yescharacteristic
142#107—2512no—
143#107—2511no—
144C4⋊C8—2512no—
145C4⋊C8C2×D25111yescharacteristic
146#103—2511no—
147#103—2512no—
148#103—2512no—
149#103—2512no—
150#103—2511no—
151#103C2×D25112yes—
152C4×C8—2512no—
153C4×C8—2511no—
154C2×Q8—5022no—
155C2×Q8—5021no—
156C2×Q8—2511no—
157C2×Q8—2511no—
158C2×Q8C22×D25112yes—
159C2×Q8C22×D25111yescharacteristic
160Q16—24no—
161Q16—5022no—
162C2×C8—2511no—
163C2×C8—2511no—
164C2×C8C22×D25112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—5022no—
169C4⋊C4—5022no—
170C4⋊C4—5022no—
171C4⋊C4—5022no—
172C4⋊C4—5022no—
173C4⋊C4—5021no—
174C4⋊C4—5021no—
175C4⋊C4—2512no—
176C4⋊C4—2511no—
177C4⋊C4—2512no—
178C4⋊C4—2511no—
179C4⋊C4C22×D25111yescharacteristic
180C42—21no—
181C42—5022no—
182C42—5022no—
183C42—5021no—
184C42—2512no—
185C42C22×D25111yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—20081no—
190Q8—20082no—
191Q8—5021no—
192Q8—5021no—
193Q8—2514no—
194Q8#1686080912yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—5021no—
199C2×C4—5021no—
200C2×C4—5022no—
201C2×C4—2512no—
202C2×C4—2512no—
203C2×C4D4×D25111yescharacteristic
204C2×C4D4×D25111yescharacteristic
205C2×C4C23×D25111yescharacteristic
206C8—22no—
207C8—5021no—
208C8—5021no—
209C22C2×D4×D25111yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—42no—
214C4—10042no—
215C4—10041no—
216C4—10041no—
217C4—5022no—
218C4—2514no—
219C4Q16×D25112yes—
220C4C2×#1686080911yescharacteristic
221C4#2254876211yescharacteristic
222C2#3063234311yescharacteristic
223C2C2×Q16×D25111yescharacteristic
224C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#462×C251C2
socle, prime radicalC22×C251C22×D4
hypercenter#462D251
hypocenterC251#3363
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
77. Dic502Q16semidirect
112. C251#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic5022
#306324122
#306320372
#306320362
#306319832
#306319802
#306318952
#306318112
Q16×Dic2512
C4×#168608002
#462×C2512
#3363251