Finite Group Databasebeta

← #24908499

Subgroups — #24908499

Order 10624.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#14581823C211yesmaximal, characteristic
2#19128275C211yesmaximal, characteristic
3C2×#14581720C211yesmaximal, characteristic
4#19127710C211yesmaximal, characteristic, complemented
5C2×#14581711C211yesmaximal, characteristic, complemented
6#19127461C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9#14581823—22no—
10#14581823—22no—
11C2×#13050580—21no—
12C2×#13050577—21no—
13C2×#13050574C2211yescharacteristic
14C2×#13050574C2211yescharacteristic
15#14581790—21no—
16#14581781—21no—
17#14581775—22no—
18#14581768—22no—
19#14581751C2211yescharacteristic
20#14581720—22no—
21#14581716—22no—
22#14581711—22no—
23C22×Q8×C83C2211yescharacteristic
24C22×Q8×C83C2211yescharacteristic
25C4⋊Q8×C83—22no—
26C4⋊Q8×C83—22no—
27C2×C4⋊C4×C83—22no—
28C2×C42×C83C2211yescharacteristic
29#13050590—21no—
30#13050590—21no—
31#13050590—21no—
32#13050590—21no—
33C2×Dic166D411yescharacteristic
34C2×D332D411yescharacteristic
35C2×C22⋊D83—21no—
36C2×C4×D83—21no—
37#13050580—24no—
38#13050577—24no—
39#13050574—21no—
40#13050574—21no—
41#13050574—21no—
42#13050574D412yes—
43C2×C83⋊C8—21no—
44C2×C83⋊C8—21no—
45C4⋊Dic83D412yes—
46Dic83⋊C4—24no—
47C22⋊Dic83—21no—
48(C2×D83)⋊C4—24no—
49C4×Dic83—21no—
50C2×Q8×C83—21no—
51C2×Q8×C83—21no—
52C2×Q8×C83—21no—
53C2×Q8×C83—21no—
54C2×Q8×C83—21no—
55C2×Q8×C83—21no—
56C2×Q8×C83—21no—
57C2×Q8×C83—21no—
58C2×Q8×C83—21no—
59C2×Q8×C83—21no—
60C2×Q8×C83—21no—
61C2×Q8×C83D412yes—
62C22×C4×C83—21no—
63C22×C4×C83—21no—
64C22×C4×C83D412yes—
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—28no—
67C42×C83—21no—
68C42×C83D412yes—
69C23×C83C2×D411yescharacteristic
70C22×D83—21no—
71Q8×C83—22no—
72Q8×C83—22no—
73Q8×C83—41no—
74Q8×C83—41no—
75Q8×C83—41no—
76Q8×C83—41no—
77Q8×C83—41no—
78Q8×C83—41no—
79C2×C4×C83—24no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—41no—
86C2×C4×C83—41no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83#4111yescharacteristic
93C22⋊D83—41no—
94C22⋊D83—41no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97D332—21no—
98D332—21no—
99C4×D83—41no—
100C4×D83—41no—
101Dic166—21no—
102Dic166—21no—
103C83⋊C8—22no—
104C83⋊C8—41no—
105C22×C83—22no—
106C22×C83#12011yescharacteristic
107C22×C83#12011yescharacteristic
108C22×C83#12111yescharacteristic
109C2×D83—41no—
110C2×D83—42no—
111C4×C83—24no—
112C4×C83—42no—
113C4×C83—42no—
114C4×C83#12011yescharacteristic
115C4×C83#12111yescharacteristic
116C4×C83#12111yescharacteristic
117C4×C83#12711yescharacteristic
118Dic83—41no—
119Dic83—42no—
120C2×C83—21no—
121C2×C83—21no—
122C2×C83#39211yescharacteristic
123C2×C83#45612yes—
124D83—81no—
125#2005—831nomaximal, Sylow
126C83#200511yescharacteristic, complemented, Sylow
127C2×#137—831no—
128#536—831no—
129C2×C4⋊Q8D8311yescharacteristic, complemented
130C2×#104—831no—
131#419—831no—
132C2×#101—831no—
133#339—831no—
134C2×#41—831no—
135C22×Q8C2×D8311yescharacteristic
136C22×Q8C2×D8311yescharacteristic
137#137—1662no—
138#137—1662no—
139C2×Q16—1661no—
140C2×SD16—1661no—
141C2×M16—831no—
142C2×M16—831no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—1661no—
146#124—1661no—
147#123—1662no—
148C22⋊Q8—1662no—
149#117—831no—
150C2×C4⋊C4—22no—
151C2×C42C2×D8311yescharacteristic
152#104—1662no—
153#103—1662no—
154#101—1662no—
155#41—1661no—
156#41—1661no—
157#41—1661no—
158#41—1661no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—831no—
171C2×Q8C22⋊D8312yes—
172C2×D4—831no—
173C2×D4—1661no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1661no—
177C22×C4C22⋊D8312yes—
178C22×C4C22×D8311yescharacteristic
179Q16—1664no—
180SD16—1664no—
181M16—832no—
182M16—1661no—
183M16—1661no—
184M16—1661no—
185C2×C8—1661no—
186C2×C8—1661no—
187C4⋊C4—28no—
188C4⋊C4—832no—
189C4⋊C4—1664no—
190C22⋊C4—1664no—
191C22⋊C4—1661no—
192C42—21no—
193C42—1661no—
194C42C22⋊D8312yes—
195C23—1661no—
196C23C2×C22⋊D8311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—1661no—
206Q8—1661no—
207D4—3321no—
208D4—3321no—
209D4—1661no—
210D4—1661no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3321no—
220C2×C4—3321no—
221C2×C4—1662no—
222C2×C4—1661no—
223C2×C4D4×D8311yescharacteristic
224C2×C4D4×D8311yescharacteristic
225C2×C4D4×D8311yescharacteristic
226C2×C4C2×C22⋊D8311yescharacteristic
227C2×C4C2×C22⋊D8311yescharacteristic
228C2×C4#1305058911yescharacteristic
229C8—3321no—
230C8—1662no—
231C22—22no—
232C22—3321no—
233C22—3322no—
234C22#1458176011yescharacteristic
235C22#1458176111yescharacteristic
236C22#1458176411yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—3321no—
241C4—3322no—
242C4#1458176011yescharacteristic
243C4#1458176311yescharacteristic
244C4#1458176411yescharacteristic
245C4#1458178811yescharacteristic
246C2—21no—
247C2—21no—
248C2—6641no—
249C2#1912760911yescharacteristic
250C2#1912788312yes—
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
FittingC2×C4⋊Q8×C83C2
socleC22×C83#120
prime radicalC2×#13050590C22
hypercenterC2×C4⋊Q8D83
hypocenterC83#2005
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #19127710C2semidirect
5. C2×#14581711C2semidirect
6. #19127461C2semidirect
7. C2×C4⋊Q8×C83C2semidirect
126. C83#2005semidirect
129. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
#191282752
C2×#145817202
#191277102
C2×#145817112
#191274612
C2×C4⋊Q8×C832
#200583