Finite Group Databasebeta

← #24920465

Subgroups — #24920465

Order 10624.

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

normal subgroups only

This group has 348 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#19128429C211yesmaximal, characteristic
2#19128426C211yesmaximal, characteristic
3#19128080C211yesmaximal, characteristic
4#19128079C212yesmaximal
5#19127912C211yesmaximal, characteristic
6#19127910C212yesmaximal
7#19127909C212yesmaximal
8#19127825C211yesmaximal, characteristic
9C4×#13050580C211yesmaximal, characteristic
10#19127793C212yesmaximal
11#462×C83C211yesmaximal, characteristic
12C2×Dic332C2212yes—
13C2×#13050580C2211yescharacteristic
14C2×#13050580C2212yes—
15C2×#13050580C2212yes—
16#14581792C2211yescharacteristic
17#14581790C2212yes—
18#14581790C2212yes—
19#14581784—21no—
20#14581783—21no—
21#14581782—22no—
22#14581782—22no—
23Q8×Dic83C2212yes—
24C4×Dic166—21no—
25C4×Dic166C2211yescharacteristic
26C4×Dic166C2211yescharacteristic
27#14581757—21no—
28#14581757C2212yes—
29#14581725C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2212yes—
32#14581716C2211yescharacteristic
33#14581715C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C2×Q16×C83C2212yes—
37C4⋊Q8×C83C2211yescharacteristic
38C4×Q8×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166D411yescharacteristic
42C2×Dic166C2311yescharacteristic
43C2×Dic166C2312yes—
44Dic332—42no—
45#13050580—22no—
46#13050580—24no—
47#13050580—42no—
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2311yescharacteristic
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—21no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59Dic83⋊C4—22no—
60Dic83⋊C4—22no—
61Dic83⋊C4D412yes—
62C4×Dic83—21no—
63C4×Dic83—22no—
64C4×Dic83—22no—
65C4×Dic83C2312yes—
66C2×Q8×C83C2311yescharacteristic
67C2×Q8×C83C2312yes—
68Q16×C83—42no—
69C2×C8×C83C2312yes—
70C4⋊C4×C83—21no—
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83C2311yescharacteristic
74C42×C83—21no—
75C42×C83C2311yescharacteristic
76Q8×C83—21no—
77Q8×C83—22no—
78Q8×C83—21no—
79Q8×C83—22no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83C2×D412yes—
90C2×Dic83C2×D412yes—
91Dic166—21no—
92Dic166—22no—
93Dic166—22no—
94Dic166—21no—
95Dic166Q1614yescomplemented
96C8×C83—22no—
97C83⋊C8—21no—
98C83⋊C8#4112yes—
99C22×C83C22×D411yescharacteristic
100C4×C83—21no—
101C4×C83—41no—
102C4×C83—42no—
103C4×C83—41no—
104C4×C83C2×Q1612yes—
105C4×C83C2×#4111yescharacteristic
106C4×C83H(5,𝔽2)11yescharacteristic
107Dic83—22no—
108Dic83—42no—
109Dic83—41no—
110Dic83—41no—
111Dic83C2×Q1614yes—
112C2×C83#55011yescharacteristic
113C2×C83C22×Q1611yescharacteristic
114C2×C83#57811yescharacteristic
115#3395—831nomaximal, Sylow
116C83#339511yescharacteristic, complemented, Sylow
117#557—831no—
118#557—831no—
119#494—832no—
120#494—831no—
121#462—832no—
122#462—831no—
123#462—832no—
124#462D8311yescharacteristic
125C8×Q8—831no—
126C4×Q16—831no—
127C4×Q16—832no—
128C2×Q16—832no—
129C2×Q16—831no—
130C2×Q16—832no—
131C2×Q16—832no—
132C2×Q16C2×D8312yes—
133C4⋊Q8—832no—
134C4⋊Q8—832no—
135C4⋊Q8—831no—
136C4⋊Q8C2×D8311yescharacteristic
137#125—1661no—
138#125—1662no—
139#125—1662no—
140#125—1661no—
141C4×Q8—831no—
142C4×Q8—832no—
143C4×Q8—832no—
144C4×Q8—831no—
145C4×Q8—1661no—
146C4×Q8—1661no—
147C4×Q8C2×D8311yescharacteristic
148#107—831no—
149C4⋊C8—832no—
150C4⋊C8C2×D8311yescharacteristic
151#103—831no—
152#103—832no—
153#103—831no—
154#103C2×D8312yes—
155C4×C8—832no—
156C4×C8—831no—
157C2×Q8—831no—
158C2×Q8—832no—
159C2×Q8—831no—
160C2×Q8C22×D8311yescharacteristic
161C2×Q8C22×D8312yes—
162Q16—42no—
163Q16—3322no—
164Q16—3322no—
165Q16—1662no—
166Q16—1664no—
167C2×C8—831no—
168C2×C8—831no—
169C2×C8C22×D8312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—832no—
174C4⋊C4—831no—
175C4⋊C4—831no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C4⋊C4C22×D8311yescharacteristic
186C42—21no—
187C42—832no—
188C42—1662no—
189C42—1662no—
190C42—1661no—
191C42C22×D8311yescharacteristic
192Q8—21no—
193Q8—22no—
194Q8—21no—
195Q8—22no—
196Q8—834no—
197Q8—1662no—
198Q8—1661no—
199Q8—1661no—
200Q8—1662no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—832no—
205C2×C4—832no—
206C2×C4—1661no—
207C2×C4—1662no—
208C2×C4—1661no—
209C2×C4D4×D8311yescharacteristic
210C2×C4D4×D8311yescharacteristic
211C2×C4C23×D8311yescharacteristic
212C8—22no—
213C8—832no—
214C8—1661no—
215C22C2×D4×D8311yescharacteristic
216C4—21no—
217C4—41no—
218C4—42no—
219C4—41no—
220C4—3322no—
221C4—3321no—
222C4—3321no—
223C4—834no—
224C4—1662no—
225C4Q16×D8312yes—
226C4#41×D8311yescharacteristic
227C4#1458183711yescharacteristic
228C2#1912836911yescharacteristic
229C2C2×Q16×D8311yescharacteristic
230C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#462×C83C2
socle, prime radicalC22×C83C22×D4
hypercenter#462D83
hypocenterC83#3395
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
95. Dic166Q16semidirect
116. C83#3395semidirect

Maximal subgroups

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

subgroupindex
#191284292
#191284262
#191280802
#191280792
#191279122
#191279102
#191279092
#191278252
C4×#130505802
#191277932
#462×C832
#339583