Finite Group Databasebeta

← #10692804

Subgroups — #10692804

Order 768.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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#82956C211yesmaximal, characteristic
2#82939C211yesmaximal, characteristic
3C3×#1533C211yesmaximal, characteristic
4#70836C212yesmaximal, complemented
5#67748C212yesmaximal, complemented
6#9439—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#432C2212yes—
10#5229—21no—
11#5225—21no—
12#5220—21no—
13#5215—21no—
14#5056—22no—
15#5055C414yescomplemented
16#4951—22no—
17Dic3×C22⋊C4—22no—
18#4932C2212yes—
19C4×#208C2211yescharacteristic
20C3×#329—22no—
21#4467—22no—
22#4467—22no—
23#3140—31no—
24#2372—31no—
25#1533D311yescharacteristic
26#1506—32no—
27#1300—32no—
28C22×C3×Q8D411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×Dic3—21no—
31C3×#124—22no—
32C3×#124C2×C412yes—
33C3×#124D412yes—
34C2×C3×C22⋊C4—22no—
35C2×C3×C42C2311yescharacteristic
36C2×#217—22no—
37C2×#217—21no—
38#926—21no—
39C2×C4⋊Dic3—21no—
40#924—22no—
41C2×C4×Dic3—22no—
42C2×C4×Dic3—21no—
43C2×#208—21no—
44#870—21no—
45#869C2×C412yescomplemented
46C4×Dic6—22no—
47C3×C4⋊C8—22no—
48C3×C22⋊C8—22no—
49C3×#97—22no—
50#835—24no—
51#832—24no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95D412yes—
55C3⋊#95D412yes—
56#803C2×C412yescomplemented
57C4×C3⋊C8—21no—
58#533—31no—
59C2×#124C2×D311yescharacteristic
60#489—34no—
61#488—62no—
62#432C2×D312yes—
63#419—61no—
64#418—61no—
65C4×M16—31no—
66#391—61no—
67#388—62no—
68#386—61no—
69C4×C22⋊C4—62no—
70#376—32no—
71#329—22no—
72#328—62no—
73#328—62no—
74C24×C3—21no—
75C2×C3×Q8—21no—
76C2×C3×Q8—41no—
77C2×C3×Q8—41no—
78C2×C3×Q8C22⋊C412yes—
79C2×C3×D4—21no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4C22⋊C412yes—
83C22×C3×C4—21no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—44no—
89C22×Dic3—81no—
90C2×Dic6—22no—
91C2×C3×C8—42no—
92C3×C22⋊C4—24no—
93C3×C22⋊C4—24no—
94C3×C42C22×C411yescharacteristic
95C3×C42C2×D411yescharacteristic
96C3×C42#4112yes—
97#217—44no—
98#217—41no—
99#217—81no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102C4⋊Dic3—81no—
103Dic3⋊C4—42no—
104C4×Dic3—44no—
105C4×Dic3—41no—
106C4×Dic3—81no—
107#208—41no—
108#208—41no—
109C2×C3⋊C8—22no—
110C22×Q8D1211yescharacteristic
111C22×D4D1211yescharacteristic
112C23×C4—61no—
113C2×M16—61no—
114C4⋊Q8—32no—
115#125—61no—
116#124—22no—
117#124C4×D312yes—
118#124#6712yes—
119C22⋊Q8—62no—
120C4×Q8—62no—
121#117—61no—
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—22no—
130#103—64no—
131#102—64no—
132C22⋊C8—22no—
133#97—22no—
134#97—32no—
135C4×C8—61no—
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×D4—42no—
149C3×D4—81no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—22no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4C2×C22⋊C411yescharacteristic
157C2×C3×C4C4×D412yes—
158C2×C3×C4#12112yes—
159C2×C3×C4#12311yescharacteristic
160C2×Dic3—24no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—84no—
168Dic6—82no—
169C3×C8—82no—
170C3⋊C8—42no—
171C24—21no—
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—62no—
176C2×Q8(C2×D3)⋊C412yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4(C2×D3)⋊C412yes—
181C22×C4—21no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—124no—
185C22×C4—241no—
186C22×C4C2×D1211yescharacteristic
187C22×C4D3×D412yes—
188M16—121no—
189M16—121no—
190C2×C8—42no—
191C2×C8—62no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C4⋊C4—122no—
195C4⋊C4—241no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—124no—
199C22⋊C4—121no—
200C22⋊C4—241no—
201C42—124no—
202C42—121no—
203C42—241no—
204C42C2×C4×D311yescharacteristic
205C42#23712yes—
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—
221Dic3—44no—
222Dic3—82no—
223Dic3—82no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23#88211yescharacteristic
230Q8—42no—
231Q8—81no—
232Q8—242no—
233D4—42no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—64no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—244no—
249C2×C4#67⋊C412yes—
250C2×C4#88611yescharacteristic
251C2×C4C2×(C2×D3)⋊C411yescharacteristic
252C2×C4#93912yes—
253C8—82no—
254C8—122no—
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#474611yescharacteristic
267C22#474611yescharacteristic
268C22#495811yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—124no—
275C4—242no—
276C4—242no—
277C3#943911yescharacteristic, complemented, Sylow
278C2—21no—
279C2—21no—
280C2—82no—
281C2#7006911yescharacteristic
282C2#7039512yes—
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1533C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1533D3
hypocenterC3#9439
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
4. #70836C2semidirect
5. #67748C2semidirect
15. #5055C4semidirect
45. #869C2×C4semidirect
56. #803C2×C4semidirect
277. C3#9439semidirect

Maximal subgroups

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

subgroupindex
#829562
#829392
C3×#15332
#708362
#677482
#94393