Finite Group Databasebeta

← #24917268

Subgroups — #24917268

Order 10624.

376 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 153 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#19128258C211yesmaximal, characteristic
2#19128008C214yesmaximal, complemented
3#19127996C212yesmaximal, complemented
4#19127986C212yesmaximal, complemented
5#19127741C212yesmaximal
6#19127725C211yesmaximal, characteristic
7#19127710C212yesmaximal
8C2×C4×Q8×C83C211yesmaximal, characteristic
9C2×#13050574C2212yes—
10#14581792C2211yescharacteristic
11#14581792C2211yescharacteristic
12#14581784C2212yes—
13#14581771—21no—
14#14581771—21no—
15C4×Dic166—21no—
16C4×Dic166—21no—
17#14581751C2212yes—
18#14581725C2214yes—
19#14581723C2214yes—
20C4⋊(C83⋊C8)C2212yes—
21C4⋊(C83⋊C8)C2212yes—
22#14581716C2214yes—
23#14581716C2214yes—
24C22×Q8×C83C2211yescharacteristic
25C4×Q8×C83—21no—
26C4×Q8×C83—21no—
27C4×Q8×C83C2212yes—
28C4×Q8×C83C2212yes—
29C2×C4⋊C4×C83—21no—
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C42×C83—21no—
32C2×C42×C83C2211yescharacteristic
33C2×Dic166—21no—
34C2×Dic166—21no—
35#13050574—24no—
36C2×C83⋊C8C2314yes—
37C4⋊Dic83—21no—
38C4⋊Dic83—21no—
39C4⋊Dic83C2312yes—
40C4⋊Dic83C2312yes—
41Dic83⋊C4—22no—
42Dic83⋊C4—22no—
43C22⋊Dic83—22no—
44C4×Dic83—22no—
45C2×Q8×C83—21no—
46C2×Q8×C83—22no—
47C2×Q8×C83Q814yescomplemented
48C2×Q8×C83C2312yes—
49C22×C4×C83—21no—
50C22×C4×C83—21no—
51C22×C4×C83D411yescharacteristic
52C22×C4×C83D411yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—22no—
55C4⋊C4×C83—22no—
56C4⋊C4×C83—21no—
57C4⋊C4×C83C2312yes—
58C42×C83—22no—
59C42×C83—22no—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83C2×D411yescharacteristic
65Q8×C83—24no—
66Q8×C83—22no—
67Q8×C83C2×Q814yes—
68C2×C4×C83—24no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—41no—
73C2×C4×C83—42no—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×Q812yes—
80C2×C4×C83#4112yes—
81C2×C4×C83C2411yescharacteristic
82C2×Dic83—22no—
83C2×Dic83—22no—
84Dic166—42no—
85Dic166—42no—
86C83⋊C8—24no—
87C22×C83—21no—
88C22×C83—21no—
89C22×C83C22⋊Q812yes—
90C22×C83C22×D411yescharacteristic
91C4×C83—24no—
92C4×C83—21no—
93C4×C83—21no—
94C4×C83—21no—
95C4×C83—42no—
96C4×C83C22⋊Q812yes—
97C4×C83#13712yes—
98C4×C83C22×Q811yescharacteristic
99C4×C83C2×#4111yescharacteristic
100Dic83—42no—
101Dic83—42no—
102C2×C83—22no—
103C2×C83C2×C22⋊Q811yescharacteristic
104C2×C83C2×#13711yescharacteristic
105C2×C83#57511yescharacteristic
106#3078—831nomaximal, Sylow
107C83#307811yescharacteristic, complemented, Sylow
108#533—831no—
109C2×C4×Q8D8311yescharacteristic
110#479—834no—
111#477—832no—
112#475—832no—
113#428—832no—
114#423—831no—
115#419—832no—
116C22×Q8C2×D8311yescharacteristic
117C2×M16—832no—
118C4⋊Q8—831no—
119C4⋊Q8—831no—
120#125—832no—
121C22⋊Q8—1661no—
122C22⋊Q8—1661no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—1661no—
126C4×Q8—1661no—
127C4×Q8C2×D8312yes—
128C4×Q8C2×D8312yes—
129#117—832no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C2×D8311yescharacteristic
132C2×C42—21no—
133C2×C42C2×D8311yescharacteristic
134#107—834no—
135#106—834no—
136C4⋊C8—832no—
137C4⋊C8—832no—
138#103—834no—
139#103—834no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—1661no—
143C2×Q8—1661no—
144C2×Q8Dic16614yescomplemented
145C2×Q8C22×D8312yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22⋊D8311yescharacteristic
149C22×C4C22⋊D8311yescharacteristic
150C22×C4C22×D8311yescharacteristic
151M16—1664no—
152C2×C8—834no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—832no—
157C4⋊C4—832no—
158C4⋊C4—1662no—
159C4⋊C4—1662no—
160C4⋊C4—1661no—
161C4⋊C4—1661no—
162C4⋊C4C22×D8312yes—
163C22⋊C4—1662no—
164C42—22no—
165C42—22no—
166C42—1662no—
167C42C22⋊D8311yescharacteristic
168C42C22⋊D8311yescharacteristic
169C42C22×D8311yescharacteristic
170C42C22×D8311yescharacteristic
171C23C2×C22⋊D8311yescharacteristic
172Q8—24no—
173Q8—22no—
174Q8—3322no—
175Q8—3322no—
176Q8C2×Dic16614yes—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4C2×C22⋊D8311yescharacteristic
186C2×C4C2×C22⋊D8311yescharacteristic
187C2×C4C2×C22⋊D8311yescharacteristic
188C2×C4C2×C22⋊D8311yescharacteristic
189C2×C4C2×C22⋊D8311yescharacteristic
190C2×C4C2×Dic16612yes—
191C2×C4#1305059012yes—
192C2×C4C23×D8311yescharacteristic
193C8—1664no—
194C22—21no—
195C22—21no—
196C22#1458177112yes—
197C22C22×C22⋊D8311yescharacteristic
198C4—24no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—3322no—
204C4—3322no—
205C4#1458177112yes—
206C4#1458182312yes—
207C4C22×Dic16611yescharacteristic
208C4C2×#1305059011yescharacteristic
209C2—22no—
210C2C2×#1458177111yescharacteristic
211C2C2×#1458182311yescharacteristic
212C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4×Q8×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenterC2×C4×Q8D83
hypocenterC83#3078
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. #19128008C2semidirect
3. #19127996C2semidirect
4. #19127986C2semidirect
47. C2×Q8×C83Q8semidirect
107. C83#3078semidirect
144. C2×Q8Dic166semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191280082
#191279962
#191279862
#191277412
#191277252
#191277102
C2×C4×Q8×C832
#307883