Finite Group Databasebeta

← #11713133

Subgroups — #11713133

Order 768.

505 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 63 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×#5398C211yesmaximal, characteristic
2#85302C212yesmaximal
3A4×#106C211yesmaximal, characteristic
4#73108C211yesmaximal, characteristic
5#73052C212yesmaximal
6#23028—31nomaximal, Sylow
7C2×A4⋊Q8C2212yes—
8C2×C8×A4C2211yescharacteristic
9A4×C4⋊C4C2212yes—
10SL(2,ℤ/4)⋊C4—22no—
11C4×SL(2,ℤ/4)—22no—
12C4×SL(2,ℤ/4)C2211yescharacteristic
13C2×A4⋊C8C2211yescharacteristic
14#5398C414yescomplemented
15#4853—41nomaximal
16C2×#467—31no—
17C2×#441—31no—
18C22×#106D311yescharacteristic
19#2360—32no—
20#1971—31no—
21#1943—31no—
22#1927—32no—
23#1872—32no—
24#1865—32no—
25#1776—31no—
26#1752—31no—
27C2×C4×A4—22no—
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)—22no—
30C2×SL(2,ℤ/4)D412yes—
31A4⋊Q8—22no—
32A4⋊Q8C2×C414yes—
33C2×Dic12—41no—
34#887—42no—
35C8×A4C2×C412yes—
36A4⋊C8—21no—
37C3×#106—41no—
38#815—41no—
39#810—42no—
40C22×Q16—31no—
41C23×C8C2×D311yescharacteristic
42C2×C22⋊Q8—32no—
43C2×C4×Q8—32no—
44C22×C4⋊C4C2×D312yes—
45#467—34no—
46#467—34no—
47#441—62no—
48#441—61no—
49#441—61no—
50C2×#107—31no—
51C2×#106—31no—
52C2×#106—32no—
53C2×#106—31no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#103—32no—
57C2×C22⋊C8—31no—
58C2×#97—31no—
59C2×#97—31no—
60#391—32no—
61#389—32no—
62#389—62no—
63#387—62no—
64#385—62no—
65C4×C4⋊C4—32no—
66C4×C22⋊C4—31no—
67C4×C22⋊C4—62no—
68#340—32no—
69#340—32no—
70#336—31no—
71#336—31no—
72C22×A4C2×D411yescharacteristic
73C2×Dic6—42no—
74C4×A4—42no—
75C4×A4C22×C411yescharacteristic
76C4×A4#4111yescharacteristic
77SL(2,ℤ/4)—24no—
78SL(2,ℤ/4)—22no—
79C2×C3×C8—41no—
80C3×C4⋊C4—42no—
81Dic3⋊C4—82no—
82C4×Dic3—41no—
83C4×Dic3—82no—
84C2×C3⋊C8—41no—
85Dic12—44no—
86C22×Q8—32no—
87C23×C4—22no—
88C23×C4C22×D311yescharacteristic
89C2×Q16—34no—
90C2×Q16—62no—
91C2×Q16—61no—
92C2×Q16—61no—
93C22×C8—31no—
94C22×C8—31no—
95C22×C8—32no—
96C22×C8—31no—
97C22×C8—31no—
98C22×C8C4×D312yes—
99C22⋊Q8—34no—
100C22⋊Q8—34no—
101C22⋊Q8—62no—
102C22⋊Q8—62no—
103C4×Q8—62no—
104C4×Q8—62no—
105C4×Q8—122no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—62no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—62no—
115C2×C42—31no—
116C2×C42—31no—
117C2×C42—62no—
118C2×C42—62no—
119#107—34no—
120#106—31no—
121#106—32no—
122#106—61no—
123#106S411yescharacteristic
124#103—38no—
125#103—62no—
126#103—62no—
127#103—122no—
128C22⋊C8—61no—
129C22⋊C8—61no—
130#97—62no—
131#97—61no—
132#97—61no—
133#95—32no—
134#95—62no—
135#95—62no—
136#95—62no—
137#95—62no—
138C2×A4C4×D411yescharacteristic
139C2×A4#13712yes—
140C2×C3×C4—41no—
141C2×C3×C4—82no—
142C2×Dic3—42no—
143C2×Dic3—82no—
144Dic6—44no—
145Dic6—82no—
146C3×C8—42no—
147C3⋊C8—81no—
148C24D3×D411yescharacteristic
149C2×Q8—34no—
150C2×Q8—62no—
151C2×Q8—62no—
152C2×Q8—62no—
153C2×Q8—62no—
154C2×Q8—62no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—32no—
160C22×C4—32no—
161C22×C4—42no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—122no—
167C22×C4C2×C4×D311yescharacteristic
168C22×C4#23911yescharacteristic
169Q16—64no—
170Q16—64no—
171C2×C8—31no—
172C2×C8—32no—
173C2×C8—32no—
174C2×C8—32no—
175C2×C8—61no—
176C2×C8—61no—
177C2×C8—61no—
178C2×C8—61no—
179C2×C8—61no—
180C2×C8—61no—
181C2×C8—61no—
182C2×C8C2×S411yescharacteristic
183C4⋊C4—34no—
184C4⋊C4—32no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—64no—
189C4⋊C4—64no—
190C4⋊C4—122no—
191C4⋊C4—122no—
192C4⋊C4—242no—
193C4⋊C4C2×S412yes—
194C22⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—62no—
198C42—61no—
199C42—61no—
200C42—61no—
201C42—61no—
202C42—122no—
203C42—122no—
204C42—242no—
205C22×C3—41no—
206A4#44111yescharacteristic, complemented
207C3×C4—41no—
208C3×C4—41no—
209C3×C4—162no—
210Dic3—84no—
211Dic3—82no—
212C23—31no—
213C23—31no—
214C23—32no—
215C23#89311yescharacteristic
216C23#91512yes—
217Q8—64no—
218Q8—64no—
219Q8—122no—
220Q8—122no—
221C2×C4—22no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—61no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—64no—
233C2×C4—64no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4C22×S411yescharacteristic
248C8—32no—
249C8—121no—
250C8—121no—
251C8C4×S412yes—
252C2×C3—41no—
253C2×C3—42no—
254C22—31no—
255C22—32no—
256C22—31no—
257C22—31no—
258C22—32no—
259C22—32no—
260C22—61no—
261C22#485311yescharacteristic, complemented
262C22D4×S411yescharacteristic
263C4—31no—
264C4—31no—
265C4—42no—
266C4—122no—
267C4—124no—
268C4—124no—
269C4—122no—
270C4—122no—
271C4C2×C4×S411yescharacteristic
272C4#591911yescharacteristic
273C3—41noSylow
274C2—31no—
275C2—32no—
276C2—31no—
277C2#8530811yescharacteristic
278C2#8546112yes—
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#106D3
socleC24D3×D4
prime radicalC22×A4C2×D4
cosocleC2×C4×A4C23
hypercenter#106S4
hypocenterA4#441
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
14. #5398C4semidirect
206. A4#441semidirect
261. C22#4853semidirect

Maximal subgroups

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

subgroupindex
C2×#53982
#853022
A4×#1062
#731082
#730522
#230283
#48534