Finite Group Databasebeta

← #72215118

Subgroups — #72215118

Order 1536.

405 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 36 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#11710976C212yesmaximal
2C2×#1842⋊C3C211yesmaximal, characteristic
3#120306—31nomaximal, Sylow
4C2×C4×C42⋊C3C2211yescharacteristic
5#73105—41nomaximal
6#1842⋊C3C412yes—
7#67995—22no—
8#21969—31no—
9C2×#1842D311yescharacteristic
10#8082—32no—
11#7988—31no—
12#7867—32no—
13C2×C8×A4—41no—
14C22×C42⋊C3Q811yescharacteristic
15#5409—42no—
16C4×C42⋊C3C2×C411yescharacteristic
17C4×C42⋊C3D411yescharacteristic
18#2920—32no—
19#2920—32no—
20C22×C4⋊C8—31no—
21C22×C4⋊C8—31no—
22C2×C43C2×D311yescharacteristic
23#1950—62no—
24#1950—62no—
25#1842—32no—
26#1842Dic312yes—
27#1810—61no—
28#1810—61no—
29#1301—62no—
30#1301—62no—
31#1290—62no—
32#1276—62no—
33#1276—62no—
34C2×C4×A4—41no—
35C8×A4—42no—
36C2×C42⋊C3—21no—
37C2×C42⋊C3C4⋊C411yescharacteristic
38A4⋊C8—82no—
39#819—161no—
40C23×C8—31no—
41C23×C8S411yescharacteristic
42C22×C42—31no—
43C22×C42—31no—
44C22×C42Dic611yescharacteristic
45#432—34no—
46#432—62no—
47#432—62no—
48#432—62no—
49#431—62no—
50#431—62no—
51#431—62no—
52#431—62no—
53C2×#108—61no—
54C2×#108—61no—
55C2×#108—61no—
56C2×#108—61no—
57#423—32no—
58#423—32no—
59#423—62no—
60C2×C4⋊C8—32no—
61C2×C4⋊C8—32no—
62C2×C4⋊C8—32no—
63C2×C4⋊C8—62no—
64C2×C4⋊C8—62no—
65C2×C4⋊C8—62no—
66C2×C4⋊C8—62no—
67C2×C4⋊C8—61no—
68#407—62no—
69#407—62no—
70C4×M16—62no—
71C4×M16—62no—
72C43—31no—
73C43—31no—
74C43D1211yescharacteristic
75C43C2×Dic311yescharacteristic
76#341—62no—
77#341—62no—
78C22×A4—41no—
79C4×A4—41no—
80C4×A4—41no—
81C2×C3×C8—161no—
82#208—162no—
83C42⋊C3#10811yescharacteristic, complemented
84C23×C4—31no—
85C23×C4—31no—
86C23×C4C2×S411yescharacteristic
87C2×M16—62no—
88C2×M16—62no—
89C2×M16—122no—
90C22×C8—32no—
91C22×C8—61no—
92C22×C8—61no—
93C22×C8—61no—
94C22×C8—61no—
95C22×C8—121no—
96C22×C8—121no—
97C22×C8SL(2,ℤ/4)12yes—
98C2×C42—21no—
99C2×C42—32no—
100C2×C42—32no—
101C2×C42—32no—
102C2×C42—32no—
103C2×C42—31no—
104C2×C42—31no—
105C2×C42—31no—
106C2×C42—31no—
107C2×C42—31no—
108C2×C42—31no—
109C2×C42—31no—
110C2×C42—61no—
111C2×C42—61no—
112C2×C42—122no—
113C2×C42C4⋊Dic311yescharacteristic
114#108—241no—
115#108—241no—
116#108—241no—
117#108—241no—
118C4⋊C8—64no—
119C4⋊C8—64no—
120C4⋊C8—64no—
121C4⋊C8—62no—
122C4⋊C8—62no—
123C4⋊C8—62no—
124C4⋊C8—62no—
125C4⋊C8—64no—
126C4⋊C8—122no—
127C4⋊C8—122no—
128C22⋊C8—64no—
129C22⋊C8—122no—
130C22⋊C8—122no—
131C22⋊C8—122no—
132#97—64no—
133#97—122no—
134C4×C8—122no—
135C4×C8—122no—
136C2×A4—41no—
137C2×A4—81no—
138C2×C3×C4—161no—
139C3×C8—162no—
140C3⋊C8—322no—
141C24A4⋊Q811yescharacteristic
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—32no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—61no—
155C22×C4—61no—
156C22×C4—62no—
157C22×C4—122no—
158C22×C4C4⋊S411yescharacteristic
159C22×C4C2×SL(2,ℤ/4)11yescharacteristic
160M16—122no—
161M16—122no—
162M16—122no—
163M16—122no—
164M16—244no—
165C2×C8—41no—
166C2×C8—62no—
167C2×C8—62no—
168C2×C8—121no—
169C2×C8—121no—
170C2×C8—121no—
171C2×C8—122no—
172C2×C8—124no—
173C2×C8—122no—
174C2×C8—122no—
175C2×C8—241no—
176C42—31no—
177C42—32no—
178C42—32no—
179C42—32no—
180C42—32no—
181C42—32no—
182C42—32no—
183C42—32no—
184C42—32no—
185C42—32no—
186C42—61no—
187C42—62no—
188C42—62no—
189C42—122no—
190C42#81911yescharacteristic, complemented
191C22×C3—161no—
192A4—41no—
193C3×C4—161no—
194C3×C4—161no—
195C23—21no—
196C23—31no—
197C23—31no—
198C23—61no—
199C23C4⋊SL(2,ℤ/4)11yescharacteristic
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—61no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—64no—
221C2×C4—61no—
222C2×C4—121no—
223C2×C4—121no—
224C2×C4—122no—
225C2×C4#857⋊C211yescharacteristic
226C8—42no—
227C8—122no—
228C8—242no—
229C8—244no—
230C8—242no—
231C2×C3—161no—
232C2×C3—321no—
233C22—31no—
234C22—31no—
235C22—32no—
236C22—31no—
237C22—61no—
238C22—61no—
239C22—61no—
240C22#7299211yescharacteristic
241C22#7310511yescharacteristic
242C4—32no—
243C4—32no—
244C4—31no—
245C4—31no—
246C4—62no—
247C4—62no—
248C4—62no—
249C4#7299411yescharacteristic
250C4#7308311yescharacteristic
251C3—161noSylow
252C2—21no—
253C2—31no—
254C2—31no—
255C2—61no—
256C2#1171133611yescharacteristic
subgroupG/H
centerC4#72994
commutatorC4×C42⋊C3C2×C4
FrattiniC23×C4C2×S4
FittingC2×#1842D3
socleC23C4⋊SL(2,ℤ/4)
prime radicalC22×C42⋊C3Q8
cosocleC2×C4×C42⋊C3C22
hypercenterC2×C4#857⋊C2
hypocenterC42⋊C3#108
LCSG ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
83. C42⋊C3#108semidirect
190. C42#819semidirect

Maximal subgroups

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

subgroupindex
#117109762
C2×#1842⋊C32
#1203063
#731054