Finite Group Databasebeta

← #10697683

Subgroups — #10697683

Order 768.

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

normal subgroups only

This group has 526 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
1#82296C212yesmaximal
2#79037C211yesmaximal, characteristic, complemented
3C3×#1954C211yesmaximal, characteristic
4#69738C212yesmaximal, complemented
5#67705C211yesmaximal, characteristic, complemented
6#10258—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8#5470C2212yes—
9C2×C3×C4⋊C8C2211yescharacteristic
10C2×C3×C22⋊C8—22no—
11C2×C3×C4×C8C2211yescharacteristic
12#5215—22no—
13#5116—22no—
14#5009—24no—
15#5008—22no—
16#5007—22no—
17#4964—22no—
18#4924C2212yes—
19#4850—22no—
20#4849—22no—
21#4456—24no—
22#3249—31no—
23#3077—32no—
24#1954D311yescharacteristic
25#1493—32no—
26#1292—31no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4—21no—
29C22×C3×C8—21no—
30C22×C3×C8D412yes—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C22⋊C4—21no—
35C2×C3×C42C2311yescharacteristic
36#926—22no—
37#924—22no—
38#924—22no—
39C2×#208—22no—
40C2×Dic12—22no—
41#870—22no—
42#869D412yes—
43#869D412yes—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C22⋊C8—44no—
48C3×C4×C8—22no—
49#832—24no—
50#832—24no—
51#818—22no—
52#816—24no—
53#808—24no—
54#807—24no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)D414yes—
57#533—32no—
58C2×C4×D4C2×D311yescharacteristic
59#478—64no—
60#477—62no—
61#476—62no—
62#474—62no—
63#467—62no—
64#462—62no—
65#428—62no—
66#423—32no—
67C2×C4⋊C8C2×D311yescharacteristic
68#418—62no—
69C2×C22⋊C8—22no—
70C2×C4×C8C2×D311yescharacteristic
71Q8⋊C8—64no—
72C24×C3—21no—
73C2×C3×D4—22no—
74C2×C3×D4—21no—
75C2×C3×D4—22no—
76C2×C3×D4Q1612yescomplemented
77C22×C3×C4—21no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4—41no—
81C22×C3×C4—41no—
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4#4111yescharacteristic
85C2×Dic6—22no—
86C2×Dic6—22no—
87C2×C3×C8—22no—
88C2×C3×C8—22no—
89C2×C3×C8—41no—
90C2×C3×C8—41no—
91C2×C3×C8—42no—
92C3×C4⋊C4—22no—
93C3×C22⋊C4—42no—
94C3×C42C2×D412yes—
95C3×C42C2×D412yes—
96#217—42no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101Dic3⋊C4—42no—
102Dic3⋊C4—42no—
103C4×Dic3—42no—
104#208—42no—
105#208—42no—
106C2×C3⋊C8—24no—
107Dic12—82no—
108C22×D4D1211yescharacteristic
109C23×C4—21no—
110C2×Q16—62no—
111C2×M16—62no—
112C22×C8—21no—
113C22×C8#6712yes—
114C4⋊Q8—32no—
115C4⋊Q8—32no—
116#125—62no—
117C22⋊Q8—62no—
118C22⋊Q8—62no—
119C4×Q8—62no—
120C4×Q8—62no—
121C4×D4—22no—
122C4×D4—22no—
123#117—62no—
124C2×C4⋊C4D1211yescharacteristic
125C2×C22⋊C4—21no—
126C2×C42C22×D311yescharacteristic
127#107—64no—
128#107—62no—
129#106—64no—
130C4⋊C8—22no—
131C4⋊C8—34no—
132C4⋊C8—62no—
133#103—64no—
134#102—64no—
135#102—64no—
136C22⋊C8—44no—
137C4×C8—22no—
138C23×C3—22no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3#12111yescharacteristic
144C3×D4—24no—
145C3×D4—42no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—41no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—82no—
152C2×C3×C4—81no—
153C2×C3×C4—81no—
154C2×C3×C4#12012yes—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C2×C3×C4#12711yescharacteristic
158C2×C3×C4C2×Q1611yescharacteristic
159C2×C3×C4#13511yescharacteristic
160C2×C3×C4#13712yes—
161C2×Dic3—42no—
162C2×Dic3—42no—
163Dic6—44no—
164Dic6—42no—
165Dic6—42no—
166C3×C8—42no—
167C3×C8—42no—
168C3⋊C8—44no—
169C24—21no—
170C2×Q8—62no—
171C2×Q8—62no—
172C2×D4—22no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4Dic1212yescomplemented
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4C2×D1211yescharacteristic
182C22×C4#23511yescharacteristic
183C22×C4C2×#6711yescharacteristic
184Q16—242no—
185M16—122no—
186M16—122no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—64no—
193C4⋊C4—22no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C4⋊C4—62no—
198C4⋊C4—122no—
199C4⋊C4—122no—
200C22⋊C4—42no—
201C22⋊C4—122no—
202C42—122no—
203C42D3×D412yes—
204C42D3×D412yes—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—44no—
208C22×C3—42no—
209C22×C3—82no—
210C22×C3—81no—
211C22×C3—81no—
212C22×C3#39211yescharacteristic
213C22×C3#46711yescharacteristic
214C22×C3#47011yescharacteristic
215C3×C4—21no—
216C3×C4—22no—
217C3×C4—81no—
218C3×C4#45112yes—
219C3×C4#45212yes—
220C3×C4#45311yescharacteristic
221C3×C4#45411yescharacteristic
222Dic3—82no—
223Dic3—82no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23#93011yescharacteristic
230Q8—124no—
231Q8—122no—
232Q8—122no—
233D4—24no—
234D4—42no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4#88212yes—
246C2×C4#88412yes—
247C2×C4#90411yescharacteristic
248C2×C4C2×Dic1211yescharacteristic
249C2×C4#90912yes—
250C2×C4#93711yescharacteristic
251C2×C4#94011yescharacteristic
252C8—42no—
253C8—42no—
254C8—124no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—82no—
258C2×C3#200111yescharacteristic
259C2×C3#200611yescharacteristic
260C2×C3#201711yescharacteristic
261C22—21no—
262C22—21no—
263C22—44no—
264C22—42no—
265C22—82no—
266C22—81no—
267C22—81no—
268C22#496011yescharacteristic
269C22#511611yescharacteristic
270C22#513711yescharacteristic
271C4—21no—
272C4—22no—
273C4—81no—
274C4—242no—
275C4—242no—
276C4#473312yes—
277C4#473912yes—
278C4#506211yescharacteristic
279C4#507711yescharacteristic
280C3#1025811yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—82no—
284C2#6966411yescharacteristic
285C2#6996411yescharacteristic
286C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1954C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1954D3
hypocenterC3#10258
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. #79037C2semidirect
4. #69738C2semidirect
5. #67705C2semidirect
76. C2×C3×D4Q16semidirect
175. C2×D4Dic12semidirect
280. C3#10258semidirect

Maximal subgroups

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

subgroupindex
#822962
#790372
C3×#19542
#697382
#677052
#102583