Finite Group Databasebeta

← #24920179

Subgroups — #24920179

Order 10624.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128421C211yesmaximal, characteristic, complemented
2D4×Dic166C211yesmaximal, characteristic, complemented
3#19128059C212yesmaximal, complemented
4#19128052C211yesmaximal, characteristic, complemented
5#19127995C212yesmaximal, complemented
6#19127981C211yesmaximal, characteristic, complemented
7#19127914C212yesmaximal
8#19127906C211yesmaximal, characteristic, complemented
9#19127831C211yesmaximal, characteristic, complemented
10#19127798C212yesmaximal, complemented
11C4×D8×C83C211yesmaximal, characteristic
12C22×Dic166—21no—
13C2×#13050578C2212yes—
14#14581792C2211yescharacteristic
15#14581792C2211yescharacteristic
16#14581791C2212yes—
17#14581785—22no—
18#14581779C2212yes—
19#14581771—21no—
20#14581771—21no—
21#14581769—22no—
22C4×Dic166C2211yescharacteristic
23#14581757C2212yes—
24D4×Dic83C2212yes—
25#14581751—21no—
26#14581725C2212yes—
27C4⋊(C83⋊C8)C2211yescharacteristic
28Dic83⋊C8C2212yes—
29#14581717C2212yes—
30#14581717C2212yes—
31#14581715C2212yes—
32#14581713C2212yes—
33#14581702C2212yes—
34C2×D8×C83C2211yescharacteristic
35C4×D4×C83C2211yescharacteristic
36C4×D4×C83C2211yescharacteristic
37#107×C83C2211yescharacteristic
38#102×C83C2211yescharacteristic
39#102×C83C2211yescharacteristic
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166—44no—
44C2×Dic166D411yescharacteristic
45C2×Dic166C2312yes—
46C22×Dic83—22no—
47#13050578—24no—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83—21no—
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83D411yescharacteristic
53C4⋊Dic83C2312yes—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56Dic83⋊C4D412yes—
57C22⋊Dic83—22no—
58C22⋊Dic83—22no—
59C22⋊Dic83—22no—
60C4×Dic83—22no—
61C4×Dic83C2312yes—
62C2×D4×C83C2311yescharacteristic
63C2×D4×C83C2311yescharacteristic
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66D8×C83—22no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2311yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C42×C83C2311yescharacteristic
74C23×C83—21no—
75C23×C83—21no—
76D4×C83—21no—
77D4×C83—21no—
78D4×C83—21no—
79D4×C83#4112yes—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—42no—
83C2×C4×C83—42no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—44no—
90C2×Dic83C2×D412yes—
91C2×Dic83C2×D412yes—
92Dic166—22no—
93Dic166—22no—
94Dic166—22no—
95Dic166—44no—
96Dic166—42no—
97C8×C83—21no—
98C8×C83—21no—
99C83⋊C8—22no—
100C22×C83—22no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83C22×D411yescharacteristic
107C4×C83—21no—
108C4×C83—41no—
109C4×C83—41no—
110C4×C83AGL(1,ℤ/8)12yes—
111C4×C83C2×#4111yescharacteristic
112C4×C832−1+411yescharacteristic
113Dic83—22no—
114Dic83—22no—
115Dic83—42no—
116Dic83—42no—
117C2×C83—42no—
118C2×C83—81no—
119C2×C83#54811yescharacteristic
120C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C83#57811yescharacteristic
122#3367—831nomaximal, Sylow
123C83#336711yescharacteristic, complemented, Sylow
124#556—831no—
125D4×Q8—831no—
126#489—832no—
127#487—831no—
128#477—832no—
129#474—831no—
130#463—832no—
131#461—831no—
132#446—831no—
133#440—832no—
134C4×D8D8311yescharacteristic
135C22×Q8—1661no—
136C2×SD16—832no—
137C2×D8C2×D8311yescharacteristic
138C4⋊Q8—832no—
139C4⋊Q8—831no—
140C4⋊Q8—831no—
141#126—1662no—
142#124—832no—
143C22⋊Q8—1662no—
144C22⋊Q8—1661no—
145C22⋊Q8—1661no—
146C4×Q8—831no—
147C4×Q8—832no—
148C4×D4—832no—
149C4×D4C2×D8311yescharacteristic
150C4×D4C2×D8311yescharacteristic
151#117—1661no—
152#107—832no—
153#107C2×D8311yescharacteristic
154C4⋊C8—832no—
155C4⋊C8—831no—
156#103—832no—
157#103—832no—
158#103—832no—
159#102—832no—
160#102C2×D8311yescharacteristic
161#102C2×D8311yescharacteristic
162#97—832no—
163C4×C8C2×D8311yescharacteristic
164C2×Q8—3324no—
165C2×Q8—831no—
166C2×Q8—832no—
167C2×Q8—1662no—
168C2×Q8—1661no—
169C2×D4C22×D8311yescharacteristic
170C2×D4C22×D8311yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—1662no—
174SD16—1664no—
175D8—22no—
176C2×C8—832no—
177C2×C8C22×D8311yescharacteristic
178C2×C8C22×D8311yescharacteristic
179C4⋊C4—831no—
180C4⋊C4—832no—
181C4⋊C4—832no—
182C4⋊C4—1662no—
183C4⋊C4—1662no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4C22×D8311yescharacteristic
188C4⋊C4C22×D8311yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—1662no—
192C22⋊C4—1662no—
193C22⋊C4—1662no—
194C42—832no—
195C42—1662no—
196C42C22×D8311yescharacteristic
197C23—21no—
198C23—21no—
199Q8—3324no—
200Q8—3322no—
201Q8—1662no—
202Q8—1662no—
203Q8—1662no—
204D4—21no—
205D4—21no—
206D4—21no—
207D4#1305059012yes—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—3324no—
213C2×C4—832no—
214C2×C4—832no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4D4×D8311yescharacteristic
218C2×C4D4×D8311yescharacteristic
219C2×C4C23×D8311yescharacteristic
220C8—21no—
221C8—21no—
222C8—1662no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22C2×D4×D8311yescharacteristic
230C4—21no—
231C4—41no—
232C4—41no—
233C4—3322no—
234C4—3322no—
235C4—1662no—
236C4—1662no—
237C4#1458181612yes—
238C4C2×#1305059011yescharacteristic
239C4#1458183911yescharacteristic
240C2—42no—
241C2—81no—
242C2#1912836011yescharacteristic
243C2C2×#1458181611yescharacteristic
244C2#1912851511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
prime radicalC2×D8×C83C22
hypercenterC4×D8D83
hypocenterC83#3367
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
2. D4×Dic166C2semidirect
3. #19128059C2semidirect
4. #19128052C2semidirect
5. #19127995C2semidirect
6. #19127981C2semidirect
8. #19127906C2semidirect
9. #19127831C2semidirect
10. #19127798C2semidirect
123. C83#3367semidirect

Maximal subgroups

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

subgroupindex
#191284212
D4×Dic1662
#191280592
#191280522
#191279952
#191279812
#191279142
#191279062
#191278312
#191277982
C4×D8×C832
#336783