Finite Group Databasebeta

← #44563027

Subgroups — #44563027

Order 32128.

362 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 123 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#30632422C211yesmaximal, characteristic, complemented
2#30632397C211yesmaximal, characteristic, complemented
3#30632042C212yesmaximal, complemented
4#30632034C211yesmaximal, characteristic, complemented
5#30631988C212yesmaximal, complemented
6#30631983C211yesmaximal, characteristic, complemented
7#30631903C212yesmaximal, complemented
8#30631883C211yesmaximal, characteristic
9C4×#16860799C211yesmaximal, characteristic, complemented
10C4×D2008C212yesmaximal, complemented
11C8×Q8×C251C211yesmaximal, characteristic, complemented
12C2×#16860799C2211yescharacteristic
13C2×D2008C2212yes—
14#22548715C2211yescharacteristic
15#22548712C2211yescharacteristic
16#22548707C2212yes—
17#22548704—21no—
18#22548703C2212yes—
19#22548700—22no—
20#22548693—22no—
21#22548691—21no—
22C4×Dic502C2211yescharacteristic
23C4×D1004C2212yes—
24C4×D1004C2212yes—
25C4×D1004C2211yescharacteristic
26#22548677—21no—
27C4⋊C4×D251—21no—
28#22548649C2212yes—
29#22548647C2211yescharacteristic
30#22548640C2212yes—
31#22548640C2211yescharacteristic
32#22548637C2212yes—
33#22548637C2212yes—
34#22548637C2211yescharacteristic
35#22548637C2212yes—
36C4×Q8×C251C2211yescharacteristic
37C4⋊C8×C251C2211yescharacteristic
38C4⋊C8×C251C2212yes—
39C4×C8×C251C2212yes—
40C4×C8×C251C2211yescharacteristic
41C2×Dic502—21no—
42C2×Dic502C2311yescharacteristic
43C2×D1004—21no—
44C2×D1004—21no—
45C2×D1004C2312yes—
46C2×D1004C2311yescharacteristic
47C2×C4×D251—21no—
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50#16860799—22no—
51D2008—24no—
52C4⋊Dic251—21no—
53C4⋊Dic251C2311yescharacteristic
54C4⋊Dic251C2312yes—
55C4⋊Dic251C2311yescharacteristic
56Dic251⋊C4—22no—
57Dic251⋊C4—21no—
58Dic251⋊C4—22no—
59(C2×D251)⋊C4—22no—
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4—21no—
62(C2×D251)⋊C4—22no—
63(C2×D251)⋊C4—22no—
64C4×Dic251—21no—
65C2×Q8×C251D411yescharacteristic
66C2×C8×C251C2311yescharacteristic
67C2×C8×C251C2311yescharacteristic
68C2×C8×C251C2312yes—
69C4⋊C4×C251D411yescharacteristic
70C4⋊C4×C251D412yes—
71C42×C251C2312yes—
72C42×C251C2311yescharacteristic
73C22×D251—21no—
74C22×D251—22no—
75Dic502—21no—
76Dic502—21no—
77Dic502—81no—
78D1004—22no—
79D1004—21no—
80D1004—22no—
81D1004—42no—
82D1004—81no—
83D1004#4112yes—
84C2×Dic251—21no—
85C2×Dic251—21no—
86C2×Dic251—21no—
87C2×Dic251—22no—
88C4×D251—42no—
89C4×D251—82no—
90C4×D251—82no—
91Q8×C251—22no—
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×D411yescharacteristic
94C2×C4×C251C2×D412yes—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2411yescharacteristic
97C8×C251—22no—
98C8×C251—21no—
99C8×C251—21no—
100C22×C251C22×D411yescharacteristic
101C2×D251—22no—
102C2×D251—41no—
103C2×D251—41no—
104C2×D251—42no—
105C2×D251—42no—
106C2×D251—42no—
107C4×C251—22no—
108C4×C251—22no—
109C4×C251—21no—
110C4×C251#13512yes—
111C4×C251C2×#4111yescharacteristic
112C4×C2512−1+411yescharacteristic
113Dic251—41no—
114Dic251—41no—
115Dic251—41no—
116Dic251—42no—
117C2×C251#54811yescharacteristic
118C2×C251C2×#13511yescharacteristic
119C2×C251#57611yescharacteristic
120D251—42no—
121D251—82no—
122C251#336111yescharacteristic, complemented, Sylow
123#3361—2511nomaximal, Sylow
124#559—2511no—
125#554—2511no—
126#488—2512no—
127#486—2511no—
128#478—2512no—
129#477—2511no—
130#463—2512no—
131#459—2511no—
132C8×Q8D25111yescharacteristic, complemented
133C4×SD16—2511no—
134C4×D8—2512no—
135C2×SD16—2511no—
136C2×D8—2512no—
137C4⋊Q8—2511no—
138#127—2511no—
139#125—2512no—
140#124—5021no—
141#124—2512no—
142#123—5022no—
143C22⋊Q8—5021no—
144C22⋊Q8—5022no—
145C4×Q8—2511no—
146C4×Q8C2×D25111yescharacteristic
147C4×D4—5021no—
148C4×D4—2512no—
149C4×D4—2511no—
150C4×D4—2512no—
151C2×C4⋊C4—5021no—
152#107—2512no—
153#106—2511no—
154C4⋊C8C2×D25111yescharacteristic
155C4⋊C8C2×D25112yes—
156#103—2511no—
157#103—2512no—
158#102—2511no—
159#102—2512no—
160#102—2512no—
161#102—2512no—
162C4×C8C2×D25112yes—
163C4×C8C2×D25111yescharacteristic
164C2×Q8—5021no—
165C2×Q8—2511no—
166C2×Q8D100411yescharacteristic
167C2×D4—5021no—
168C2×D4—5021no—
169C2×D4—2512no—
170C2×D4—2511no—
171C22×C4—5022no—
172C22×C4—5022no—
173C22×C4—5021no—
174SD16—5022no—
175D8—5024no—
176C2×C8C22×D25111yescharacteristic
177C2×C8C22×D25111yescharacteristic
178C2×C8C22×D25112yes—
179C4⋊C4—5022no—
180C4⋊C4—5021no—
181C4⋊C4—5021no—
182C4⋊C4—5022no—
183C4⋊C4—2511no—
184C4⋊C4—2512no—
185C4⋊C4—2511no—
186C4⋊C4D100411yescharacteristic
187C4⋊C4D100412yes—
188C22⋊C4—5022no—
189C22⋊C4—5021no—
190C22⋊C4—5022no—
191C22⋊C4—5022no—
192C22⋊C4—5022no—
193C42—5021no—
194C42C22×D25112yes—
195C42C22×D25111yescharacteristic
196C23—5022no—
197C23—5021no—
198Q8—22no—
199Q8—20081no—
200Q8—5021no—
201Q8—5021no—
202D4—20081no—
203D4—10042no—
204D4—5022no—
205D4—5022no—
206D4—5021no—
207D4—2512no—
208C2×C4—20082no—
209C2×C4—20082no—
210C2×C4—10042no—
211C2×C4—5021no—
212C2×C4—5021no—
213C2×C4—5021no—
214C2×C4—5022no—
215C2×C4C2×D100412yes—
216C2×C4C2×D100412yes—
217C2×C4C2×D100411yescharacteristic
218C2×C4C2×D100411yescharacteristic
219C2×C4C23×D25111yescharacteristic
220C8—22no—
221C8—21no—
222C8—21no—
223C22—10042no—
224C22—10042no—
225C22—10042no—
226C22—10041no—
227C22—10041no—
228C22—5022no—
229C22C22×D100411yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—10041no—
234C4—10041no—
235C4—10042no—
236C4—10041no—
237C4#2254873712yes—
238C4#41×D25111yescharacteristic
239C4#2254876311yescharacteristic
240C2—20082no—
241C2—10042no—
242C2#3063234911yescharacteristic
243C2C2×#2254873711yescharacteristic
244C2#3063249211yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×Q8×C251C2
socleC22×C251C22×D4
prime radical#30631883C2
hypercenterC8×Q8D251
hypocenterC251#3361
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
1. #30632422C2semidirect
2. #30632397C2semidirect
3. #30632042C2semidirect
4. #30632034C2semidirect
5. #30631988C2semidirect
6. #30631983C2semidirect
7. #30631903C2semidirect
9. C4×#16860799C2semidirect
10. C4×D2008C2semidirect
11. C8×Q8×C251C2semidirect
122. C251#3361semidirect
132. C8×Q8D251semidirect

Maximal subgroups

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

subgroupindex
#306324222
#306323972
#306320422
#306320342
#306319882
#306319832
#306319032
#306318832
C4×#168607992
C4×D20082
C8×Q8×C2512
#3361251