Finite Group Databasebeta

← #10698700

Subgroups — #10698700

Order 768.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×#2537C211yesmaximal, characteristic
2#71604C212yesmaximal, complemented
3C2×D4⋊(C3⋊C8)C212yesmaximal, complemented
4#69576C211yesmaximal, characteristic, complemented
5#67634C211yesmaximal, characteristic, complemented
6#10432—31nomaximal, Sylow
7C2×C3×#121—22no—
8C2×C3×C4×D4C2212yes—
9C3×#394—22no—
10C3×#384C2211yescharacteristic
11C3×#380—22no—
12C2×#832—22no—
13C2×#808—22no—
14#4935C2211yescharacteristic
15C2×C4⋊(C3⋊C8)C2212yes—
16C2×C4×C3⋊C8C2211yescharacteristic
17D4⋊(C3⋊C8)—24no—
18#4529—22no—
19#2537D311yescharacteristic
20#1844—31no—
21#1802—32no—
22C2×D4⋊C8—32no—
23#1296—31no—
24C22×C3×D4—22no—
25C22×C3×D4D412yes—
26C23×C3×C4—22no—
27C3×#121—44no—
28C3×#121—44no—
29C3×C4×D4—22no—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C22⋊C4—24no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C42C2311yescharacteristic
39C2×C4⋊Dic3—22no—
40C2×C4⋊Dic3—21no—
41C22×C3⋊C8—21no—
42C22×C3⋊C8—22no—
43C3×#95—22no—
44#832—44no—
45#808—44no—
46C4⋊(C3⋊C8)—22no—
47C4⋊(C3⋊C8)—22no—
48C4×C3⋊C8—21no—
49C4×C3⋊C8D412yescomplemented
50C2×#121—22no—
51C2×C4×D4C2×D312yes—
52C2×#106—62no—
53C2×C4⋊C8—32no—
54C2×#102—62no—
55C2×C4×C8—31no—
56#394—22no—
57#384—31no—
58#384C2×D311yescharacteristic
59#380—22no—
60#340—62no—
61D4⋊C8—64no—
62C24×C3—22no—
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4—22no—
66C2×C3×D4—22no—
67C2×C3×D4—22no—
68C2×C3×D4—48no—
69C2×C3×D4—82no—
70C2×C3×D4SD1614yes—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—42no—
74C22×C3×C4—42no—
75C22×C3×C4—42no—
76C22×C3×C4—42no—
77C22×C3×C4—42no—
78C22×C3×C4—42no—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D412yes—
81C22×Dic3—41no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—42no—
87C3×C22⋊C4—48no—
88C3×C22⋊C4—42no—
89C3×C22⋊C4—42no—
90C3×C42C2×D411yescharacteristic
91C3×C42#4111yescharacteristic
92C3×C42#4112yes—
93C4⋊Dic3—42no—
94C4⋊Dic3—42no—
95C4⋊Dic3—42no—
96C2×C3⋊C8—22no—
97C2×C3⋊C8—22no—
98C2×C3⋊C8—22no—
99C2×C3⋊C8—42no—
100C2×C3⋊C8—42no—
101C2×C3⋊C8—42no—
102C22×D4—22no—
103C22×D4#6712yes—
104C23×C4—22no—
105C22×C8—61no—
106C22×C8—62no—
107#121—44no—
108#121—44no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4#6712yes—
117C2×C4⋊C4#6712yes—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—22no—
120C2×C42C22×D311yescharacteristic
121#106—124no—
122C4⋊C8—62no—
123C4⋊C8—62no—
124#102—124no—
125C4×C8—32no—
126C4×C8—61no—
127#95—22no—
128C23×C3—24no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3—42no—
134C23×C3—42no—
135C23×C3#12011yescharacteristic
136C3×D4—24no—
137C3×D4—24no—
138C3×D4—42no—
139C3×D4—42no—
140C3×D4—88no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—44no—
147C2×C3×C4—42no—
148C2×C3×C4—82no—
149C2×C3×C4—82no—
150C2×C3×C4—82no—
151C2×C3×C4—82no—
152C2×C3×C4#12111yescharacteristic
153C2×C3×C4#12112yes—
154C2×C3×C4#12312yes—
155C2×C3×C4#12411yescharacteristic
156C2×C3×C4C2×SD1612yes—
157C2×C3×C4#13512yes—
158C2×C3×C4AGL(1,ℤ/8)12yes—
159C2×C3×C4AGL(1,ℤ/8)12yes—
160C2×Dic3—82no—
161C2×Dic3—81no—
162C3⋊C8—24no—
163C3⋊C8—82no—
164C24—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—48no—
171C2×D4—82no—
172C2×D4#21414yes—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—121no—
182C22×C4C2×#6711yescharacteristic
183C22×C4C2×#6712yes—
184C2×C8—62no—
185C2×C8—62no—
186C2×C8—62no—
187C2×C8—122no—
188C2×C8—122no—
189C2×C8—122no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—122no—
196C4⋊C4—122no—
197C4⋊C4—122no—
198C22⋊C4—48no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C42D3×D411yescharacteristic
202C42#23711yescharacteristic
203C42#23712yes—
204C22×C3—44no—
205C22×C3—44no—
206C22×C3—44no—
207C22×C3—82no—
208C22×C3—82no—
209C22×C3—82no—
210C22×C3—82no—
211C22×C3#39411yescharacteristic
212C22×C3#44912yes—
213C22×C3#45012yes—
214C22×C3#45311yescharacteristic
215C22×C3#45611yescharacteristic
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—82no—
219C3×C4—82no—
220C3×C4#46512yes—
221C3×C4#46912yes—
222C3×C4#48112yes—
223C3×C4#48212yes—
224Dic3—161no—
225C23—24no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23#95311yescharacteristic
233D4—24no—
234D4—24no—
235D4—42no—
236D4—42no—
237D4—88no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—242no—
250C2×C4—241no—
251C2×C4#93212yes—
252C2×C4C2×#21412yes—
253C2×C4#93512yes—
254C2×C4#93611yescharacteristic
255C2×C4#93811yescharacteristic
256C2×C4#93912yes—
257C2×C4#94912yes—
258C2×C4#95012yes—
259C8—64no—
260C8—242no—
261C2×C3—84no—
262C2×C3#162411yescharacteristic
263C2×C3#162911yescharacteristic
264C2×C3#165412yes—
265C2×C3#203412yes—
266C2×C3#203911yescharacteristic
267C22—44no—
268C22—44no—
269C22—44no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22#521812yes—
275C22#522211yescharacteristic
276C22#523812yes—
277C22#524111yescharacteristic
278C22#524611yescharacteristic
279C4—21no—
280C4—21no—
281C4—82no—
282C4—82no—
283C4—481no—
284C4#503412yes—
285C4#503512yes—
286C4#503912yes—
287C4#504112yes—
288C3#1043211yescharacteristic, complemented, Sylow
289C2—84no—
290C2#7168012yes—
291C2#7187611yescharacteristic
292C2#7190711yescharacteristic
293C2#7191312yes—
294C2#7192711yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3#120
hypercenter#2537D3
hypocenterC3#10432
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #71604C2semidirect
3. C2×D4⋊(C3⋊C8)C2semidirect
4. #69576C2semidirect
5. #67634C2semidirect
49. C4×C3⋊C8D4semidirect
288. C3#10432semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
#716042
C2×D4⋊(C3⋊C8)2
#695762
#676342
#104323