Finite Group Databasebeta

← #24917993

Subgroups — #24917993

Order 10624.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#19128332C211yesmaximal, characteristic
2#19128329C211yesmaximal, characteristic
3#19128109C212yesmaximal, complemented
4#19128108C212yesmaximal, complemented
5#19128102C212yesmaximal, complemented
6#19128098C212yesmaximal, complemented
7#19127743C211yesmaximal, characteristic
8#19127737C211yesmaximal, characteristic
9C2×#14581723C211yesmaximal, characteristic
10#19127659C211yesmaximal, characteristic
11#428×C83C211yesmaximal, characteristic
12C22×C83⋊C8C2211yescharacteristic
13#14581792—21no—
14#14581791—21no—
15#14581791C2212yes—
16#14581791C2212yes—
17#14581784—21no—
18#14581783—21no—
19#14581783C2212yes—
20#14581783C2212yes—
21#14581771—22no—
22#14581769—22no—
23#14581751C2211yescharacteristic
24#14581751C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26#14581726C2212yes—
27#14581725C2211yescharacteristic
28#14581725C2211yescharacteristic
29#14581724C2212yes—
30#14581723C2211yescharacteristic
31#14581723C2212yes—
32#14581723C2211yescharacteristic
33#14581723C2212yes—
34#14581702C2212yes—
35C8×Dic83C2212yes—
36C2×M16×C83C2211yescharacteristic
37#117×C83C2211yescharacteristic
38C2×C4⋊C4×C83C2211yescharacteristic
39#107×C83C2212yes—
40#106×C83C2212yes—
41C2×Dic166—22no—
42C2×Dic166—22no—
43C22×Dic83—21no—
44C2×C83⋊C8Q814yescomplemented
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C4⋊Dic83—21no—
48C4⋊Dic83—22no—
49C4⋊Dic83—22no—
50C4⋊Dic83D411yescharacteristic
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2312yes—
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59C22⋊Dic83—21no—
60C22⋊Dic83D412yes—
61C4×Dic83—21no—
62C4×Dic83C2312yes—
63C22×C4×C83—21no—
64C22×C4×C83C2311yescharacteristic
65M16×C83Q814yescomplemented
66C2×C8×C83C2312yes—
67C4⋊C4×C83—21no—
68C4⋊C4×C83C2311yescharacteristic
69C4⋊C4×C83C2312yes—
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C42×C83—21no—
73C23×C83C2×D411yescharacteristic
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—81no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×Q812yes—
80C2×C4×C83C2×Q812yes—
81C2×C4×C83C2411yescharacteristic
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83—22no—
85C2×Dic83—81no—
86C2×Dic83C2×D412yes—
87C2×Dic83C2×D412yes—
88Dic166—82no—
89Dic166—82no—
90C8×C83C2×Q814yes—
91C83⋊C8C2×Q812yes—
92C83⋊C8C2×Q812yes—
93C22×C83—21no—
94C22×C83—21no—
95C22×C83C4⋊Q812yes—
96C22×C83C22×D411yescharacteristic
97C4×C83—41no—
98C4×C83—42no—
99C4×C83—41no—
100C4×C83C4⋊Q812yes—
101C4×C83C22×Q811yescharacteristic
102C4×C83C22×Q811yescharacteristic
103Dic83—22no—
104Dic83—22no—
105Dic83—41no—
106Dic83—42no—
107Dic83—41no—
108C2×C83—22no—
109C2×C83C2×C4⋊Q811yescharacteristic
110C2×C83#57711yescharacteristic
111C2×C83#57711yescharacteristic
112#3163—831nomaximal, Sylow
113C83#316311yescharacteristic, complemented, Sylow
114#544—831no—
115#544—831no—
116#501—832no—
117#501—832no—
118#499—832no—
119#498—832no—
120#428—831no—
121#428D8311yescharacteristic
122#427—831no—
123C2×#106—831no—
124#405—831no—
125C2×M16C2×D8311yescharacteristic
126C22×C8—831no—
127C4⋊Q8—832no—
128C4⋊Q8—832no—
129C4⋊Q8—1661no—
130C4⋊Q8—1661no—
131#125—832no—
132#125—832no—
133#125—1661no—
134#125—1661no—
135C22⋊Q8—1662no—
136C22⋊Q8—1662no—
137#117—831no—
138#117—831no—
139#117C2×D8311yescharacteristic
140C2×C4⋊C4—831no—
141C2×C4⋊C4C2×D8311yescharacteristic
142#107—832no—
143#107—831no—
144#107—831no—
145#107C2×D8312yes—
146#106—832no—
147#106—832no—
148#106—832no—
149#106—831no—
150#106—831no—
151#106C2×D8312yes—
152#97—832no—
153C4×C8—832no—
154C2×Q8—1662no—
155C2×Q8—1662no—
156C22×C4—21no—
157C22×C4—1661no—
158C22×C4C22×D8311yescharacteristic
159M16Dic16614yescomplemented
160C2×C8—834no—
161C2×C8—831no—
162C2×C8—831no—
163C2×C8C22×D8312yes—
164C4⋊C4—21no—
165C4⋊C4—831no—
166C4⋊C4—831no—
167C4⋊C4—832no—
168C4⋊C4—831no—
169C4⋊C4—831no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1661no—
177C4⋊C4C22×D8312yes—
178C4⋊C4C22×D8311yescharacteristic
179C4⋊C4C22×D8311yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—832no—
182C22⋊C4—1661no—
183C42—21no—
184C42—832no—
185C42—1661no—
186C23D4×D8311yescharacteristic
187Q8—6642no—
188Q8—6642no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—832no—
194C2×C4—832no—
195C2×C4—6641no—
196C2×C4—1661no—
197C2×C4—1661no—
198C2×C4—1662no—
199C2×C4D4×D8311yescharacteristic
200C2×C4Q8×D8312yes—
201C2×C4C2×Dic16612yes—
202C2×C4C23×D8311yescharacteristic
203C8—832no—
204C8—832no—
205C8C2×Dic16614yes—
206C22—21no—
207C22—21no—
208C22#1458179112yes—
209C22C2×D4×D8311yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—3322no—
214C4—3321no—
215C4—3321no—
216C4—1662no—
217C4—1662no—
218C4#1458179112yes—
219C4C2×Q8×D8311yescharacteristic
220C4C22×Dic16611yescharacteristic
221C2—22no—
222C2C2×#1458179111yescharacteristic
223C2#1912851011yescharacteristic
224C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#428×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenter#428D83
hypocenterC83#3163
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #19128109C2semidirect
4. #19128108C2semidirect
5. #19128102C2semidirect
6. #19128098C2semidirect
44. C2×C83⋊C8Q8semidirect
65. M16×C83Q8semidirect
113. C83#3163semidirect
159. M16Dic166semidirect

Maximal subgroups

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

subgroupindex
#191283322
#191283292
#191281092
#191281082
#191281022
#191280982
#191277432
#191277372
C2×#145817232
#191276592
#428×C832
#316383