Finite Group Databasebeta

← #24918490

Subgroups — #24918490

Order 10624.

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

normal subgroups only

This group has 386 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
1#19128235C211yesmaximal, characteristic
2#19128093C212yesmaximal, complemented
3#19128069C212yesmaximal, complemented
4#19127908C212yesmaximal, complemented
5#19127903C212yesmaximal, complemented
6#19127857C212yesmaximal, complemented
7#19127840C212yesmaximal
8#19127758C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
10C2×#13050580C2214yes—
11C2×#13050577C2214yes—
12#14581786—21no—
13#14581784C2211yescharacteristic
14#14581780C2211yescharacteristic
15#14581775—21no—
16#14581771C2211yescharacteristic
17#14581766C2211yescharacteristic
18C4×Dic166C2211yescharacteristic
19C4×D332C2211yescharacteristic
20C4×C22⋊D83—21no—
21#14581749—21no—
22C4⋊(C83⋊C8)C2212yes—
23#14581716C2214yes—
24#14581716C2214yes—
25C22⋊(C83⋊C8)C2212yes—
26#14581703C2212yes—
27C22×Q8×C83C2212yes—
28C4⋊Q8×C83—21no—
29C4⋊Q8×C83—21no—
30C4⋊Q8×C83C2212yes—
31C4⋊Q8×C83C2212yes—
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83—21no—
34C2×C42×C83C2211yescharacteristic
35C2×Dic166C2311yescharacteristic
36C2×D332C2311yescharacteristic
37C2×C22⋊D83—21no—
38C2×C4×D83—21no—
39#13050580—44no—
40#13050577—44no—
41C2×C83⋊C8C2314yes—
42C4⋊Dic83C2311yescharacteristic
43C4⋊Dic83C2311yescharacteristic
44Dic83⋊C4—21no—
45Dic83⋊C4—21no—
46Dic83⋊C4—21no—
47Dic83⋊C4—21no—
48C22⋊Dic83—21no—
49(C2×D83)⋊C4—21no—
50(C2×D83)⋊C4—21no—
51(C2×D83)⋊C4—21no—
52(C2×D83)⋊C4—21no—
53C4×Dic83—21no—
54C2×Q8×C83—22no—
55C2×Q8×C83—44no—
56C2×Q8×C83C2314yes—
57C22×C4×C83—22no—
58C22×C4×C83D411yescharacteristic
59C22×C4×C83D411yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C4⋊C4×C83—22no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C42×C83D411yescharacteristic
66C42×C83D411yescharacteristic
67C42×C83C2311yescharacteristic
68C42×C83C2311yescharacteristic
69C23×C83C2×D411yescharacteristic
70C22×D83—21no—
71Q8×C83—24no—
72Q8×C83—24no—
73Q8×C83—44no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—24no—
78C2×C4×C83—44no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2411yescharacteristic
85C22⋊D83—42no—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—21no—
89D332—22no—
90C4×D83—42no—
91Dic166—22no—
92C83⋊C8—24no—
93C22×C83—21no—
94C22×C83—21no—
95C22×C83#13712yes—
96C22×C83C22×D411yescharacteristic
97C2×D83—41no—
98C2×D83—41no—
99C2×D83—41no—
100C4×C83—21no—
101C4×C83—21no—
102C4×C83—21no—
103C4×C83—21no—
104C4×C83—44no—
105C4×C83#13712yes—
106C4×C83H(5,𝔽2)12yes—
107Dic83—41no—
108Dic83—41no—
109Dic83—41no—
110C2×C83—22no—
111C2×C83#53511yescharacteristic
112C2×C83C2×#13711yescharacteristic
113C2×C83C2×#13711yescharacteristic
114D83—81no—
115#3218—831nomaximal, Sylow
116C83#321811yescharacteristic, complemented, Sylow
117C2×C4⋊Q8D8311yescharacteristic, complemented
118#529—831no—
119#497—832no—
120#491—832no—
121#462—832no—
122#461—832no—
123#451—832no—
124#448—832no—
125#432—831no—
126C22×Q8C2×D8312yes—
127C2×Q16—834no—
128C2×SD16—834no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C2×D8312yes—
132C4⋊Q8C2×D8312yes—
133#126—1661no—
134#125—831no—
135#124—831no—
136#123—1661no—
137C22⋊Q8—831no—
138#121—831no—
139C4×Q8—831no—
140C4×D4—831no—
141C4×D4—1661no—
142#117—1661no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C42C2×D8311yescharacteristic
146C4⋊C8—832no—
147#103—834no—
148#103—834no—
149C22⋊C8—832no—
150#97—832no—
151C2×Q8—22no—
152C2×Q8—44no—
153C2×Q8—831no—
154C2×Q8C22×D8314yes—
155C2×D4—831no—
156C2×D4—1661no—
157C22×C4—22no—
158C22×C4—1661no—
159C22×C4C22⋊D8311yescharacteristic
160C22×C4C22⋊D8311yescharacteristic
161C22×C4C22×D8311yescharacteristic
162Q16—3324no—
163SD16—3324no—
164C2×C8—834no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—831no—
170C4⋊C4—831no—
171C4⋊C4—1661no—
172C4⋊C4—1661no—
173C4⋊C4—1661no—
174C4⋊C4—1661no—
175C22⋊C4—1661no—
176C22⋊C4—1661no—
177C22⋊C4—1661no—
178C22⋊C4—1661no—
179C22⋊C4—1661no—
180C42—1661no—
181C42C22⋊D8311yescharacteristic
182C42C22⋊D8311yescharacteristic
183C42C22×D8311yescharacteristic
184C42C22×D8311yescharacteristic
185C23—1661no—
186C23C2×C22⋊D8311yescharacteristic
187Q8—24no—
188Q8—24no—
189Q8—44no—
190Q8—1662no—
191D4—3322no—
192D4—1662no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—3322no—
199C2×C4—1661no—
200C2×C4—1661no—
201C2×C4—1661no—
202C2×C4C2×C22⋊D8311yescharacteristic
203C2×C4C2×C22⋊D8311yescharacteristic
204C2×C4C2×C22⋊D8311yescharacteristic
205C2×C4C2×C22⋊D8311yescharacteristic
206C2×C4C2×C22⋊D8311yescharacteristic
207C2×C4C23×D8311yescharacteristic
208C8—1664no—
209C22—21no—
210C22—21no—
211C22—3321no—
212C22—3321no—
213C22—3321no—
214C22#1458182312yes—
215C22C22×C22⋊D8311yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—44no—
221C4—3321no—
222C4—3321no—
223C4—3321no—
224C4#1458182312yes—
225C4#1458183712yes—
226C2—22no—
227C2—6641no—
228C2#1912826811yescharacteristic
229C2C2×#1458182311yescharacteristic
230C2C2×#1458182311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4⋊Q8×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenterC2×C4⋊Q8D83
hypocenterC83#3218
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19128093C2semidirect
3. #19128069C2semidirect
4. #19127908C2semidirect
5. #19127903C2semidirect
6. #19127857C2semidirect
8. #19127758C2semidirect
9. C2×C4⋊Q8×C83C2semidirect
116. C83#3218semidirect
117. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280932
#191280692
#191279082
#191279032
#191278572
#191278402
#191277582
C2×C4⋊Q8×C832
#321883