Finite Group Databasebeta

← #10692807

Subgroups — #10692807

Order 768.

490 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3140C211yesmaximal, characteristic
2#82956C211yesmaximal, characteristic
3#71836C212yesmaximal, complemented
4#71719C211yesmaximal, characteristic
5#67665C212yesmaximal, complemented
6#9439—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C3×#124C2211yescharacteristic
9C3×#489C414yescomplemented
10C3×#488—22no—
11C3×#419—21no—
12C3×#418—21no—
13C3×C4×M16C2211yescharacteristic
14#5229—21no—
15#5220—21no—
16#4951—22no—
17Dic3×C22⋊C4—22no—
18#4932C2212yes—
19#4926C2212yes—
20#4540—22no—
21#4538—22no—
22#4538—22no—
23#3140D311yescharacteristic
24#2372—31no—
25#1533—31no—
26#1506—32no—
27#1300—32no—
28C22×C3×Q8D411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×Dic3—21no—
31C2×C3×M16—21no—
32C3×C4⋊Q8C2×C412yescomplemented
33C3×#125—21no—
34C3×#124—22no—
35C3×#124C2×C412yes—
36C3×#124D412yes—
37C3×C22⋊Q8—22no—
38C3×C4×Q8—22no—
39C3×#117—21no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C42C2311yescharacteristic
42C2×#217—22no—
43C2×#217—21no—
44C2×C4⋊Dic3—21no—
45C2×C4×Dic3—22no—
46C2×C4×Dic3—21no—
47C3×#103—24no—
48C3×#102—24no—
49C3×#97C2×C412yescomplemented
50C3×C4×C8—21no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95D412yes—
54C3⋊#95D412yes—
55#830—22no—
56C4⋊(C3⋊C8)—22no—
57#803—22no—
58#533C2×D311yescharacteristic
59C2×#124C2×D311yescharacteristic
60#489Dic314yescomplemented
61#488—22no—
62#432—32no—
63#419—21no—
64#418—21no—
65C4×M16C2×D311yescharacteristic
66#391—61no—
67#388—62no—
68#386—61no—
69C4×C22⋊C4—62no—
70#376—32no—
71#329—62no—
72#328—62no—
73#328—62no—
74C24×C3—21no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8—41no—
79C2×C3×Q8C22⋊C412yes—
80C2×C3×D4—21no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×C3×D4C22⋊C412yes—
84C22×C3×C4—21no—
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—44no—
90C22×Dic3—81no—
91C3×M16—41no—
92C3×M16—41no—
93C2×C3×C8—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—42no—
97C3×C22⋊C4—24no—
98C3×C22⋊C4—24no—
99C3×C22⋊C4—41no—
100C3×C42—41no—
101C3×C42C22×C411yescharacteristic
102C3×C42C2×D411yescharacteristic
103C3×C42#4112yes—
104#217—44no—
105#217—81no—
106C4⋊Dic3—81no—
107C4×Dic3—44no—
108C4×Dic3—81no—
109C2×C3⋊C8—42no—
110C22×Q8#6711yescharacteristic
111C22×D4#6711yescharacteristic
112C23×C4—61no—
113C2×M16—21no—
114C4⋊Q8C2×Dic312yescomplemented
115#125—21no—
116#124—22no—
117#124C2×Dic312yes—
118#124#6712yes—
119C22⋊Q8—22no—
120C4×Q8—22no—
121#117—21no—
122C2×C4⋊C4—61no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—61no—
126C2×C42—62no—
127C2×C42—61no—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—62no—
130#103—24no—
131#102—24no—
132C22⋊C8—62no—
133#97—62no—
134#97C2×Dic312yescomplemented
135C4×C8—21no—
136#95—32no—
137#95—32no—
138#95—62no—
139#95—62no—
140C23×C3—22no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—41no—
144C23×C3—41no—
145C23×C3#12011yescharacteristic
146C3×Q8—42no—
147C3×Q8—81no—
148C3×Q8—82no—
149C3×D4—42no—
150C3×D4—81no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4C2×C22⋊C411yescharacteristic
159C2×C3×C4C4×D412yes—
160C2×C3×C4#12112yes—
161C2×C3×C4#12311yescharacteristic
162C2×Dic3—24no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—84no—
169C3×C8—42no—
170C3⋊C8—82no—
171C24—21no—
172C2×Q8—21no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8#21712yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#21712yes—
181C22×C4—21no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—124no—
185C22×C4—241no—
186C22×C4D3×D412yes—
187C22×C4C2×#6711yescharacteristic
188M16—41no—
189M16—41no—
190C2×C8—22no—
191C2×C8—122no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—241no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—41no—
199C22⋊C4—124no—
200C22⋊C4—241no—
201C42—41no—
202C42—124no—
203C42—241no—
204C42#23712yes—
205C42C22×Dic311yescharacteristic
206C42C2×#6711yescharacteristic
207C22×C3—22no—
208C22×C3—42no—
209C22×C3—44no—
210C22×C3—81no—
211C22×C3—81no—
212C22×C3—82no—
213C22×C3#38611yescharacteristic
214C22×C3#45511yescharacteristic
215C22×C3#45511yescharacteristic
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—42no—
221C3×C4—82no—
222Dic3—44no—
223Dic3—82no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23#93711yescharacteristic
230Q8—42no—
231Q8—81no—
232Q8—82no—
233D4—42no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—64no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—244no—
249C2×C4D4×Dic312yes—
250C2×C4#93511yescharacteristic
251C2×C4#93912yes—
252C2×C4C2×#21711yescharacteristic
253C8—42no—
254C8—242no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—82no—
258C2×C3#188511yescharacteristic
259C2×C3#190612yes—
260C22—22no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—81no—
265C22—82no—
266C22#506111yescharacteristic
267C22#506111yescharacteristic
268C22#522011yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—82no—
275C4—124no—
276C4—242no—
277C3#943911yescharacteristic, complemented, Sylow
278C2—21no—
279C2—21no—
280C2—82no—
281C2#7172311yescharacteristic
282C2#7211412yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3140C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3140D3
hypocenterC3#9439
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
3. #71836C2semidirect
5. #67665C2semidirect
9. C3×#489C4semidirect
32. C3×C4⋊Q8C2×C4semidirect
49. C3×#97C2×C4semidirect
60. #489Dic3semidirect
114. C4⋊Q8C2×Dic3semidirect
134. #97C2×Dic3semidirect
277. C3#9439semidirect

Maximal subgroups

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

subgroupindex
C3×#31402
#829562
#718362
#717192
#676652
#94393