Finite Group Databasebeta

← #13136466

Subgroups — #13136466

Order 1728.

444 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C27×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11720195C212yesmaximal
3Q8×Dic27C212yesmaximal, complemented
4#11720146C212yesmaximal
5#11720141C212yesmaximal
6#11720139C212yesmaximal, complemented
7#11720136C212yesmaximal, complemented
8C42×D27C211yesmaximal, characteristic
9#11720118C211yesmaximal, characteristic, complemented
10#10589830—31nomaximal
11C2×Q8×C27C2212yes—
12C2×C4×D27C2211yescharacteristic
13C2×C4×D27C2211yescharacteristic
14C2×C4×D27C2211yescharacteristic
15C2×Dic54C2212yes—
16C4⋊C4×C27C2212yes—
17C4⋊C4×C27C2212yes—
18C42×C27C2211yescharacteristic
19(C2×D27)⋊C4C2214yes—
20C4⋊Dic27C2212yes—
21C4⋊Dic27C2214yes—
22C4⋊Dic27C2212yes—
23Dic27⋊C4C2214yes—
24C4×Dic27C2211yescharacteristic
25C4×Dic27C2214yes—
26C4×Dic27C2211yescharacteristic
27C4×Dic27C2211yescharacteristic
28C9×C4⋊Q8D311yescharacteristic
29#63508—32no—
30Q8×Dic9—32no—
31#63459—32no—
32#63454—32no—
33#63452—32no—
34#63449—32no—
35C42×D9—31no—
36#63431—31no—
37C22×D27C2311yescharacteristic
38Q8×C27—24no—
39C2×C4×C27C2314yes—
40C2×C4×C27C2311yescharacteristic
41C2×C4×C27C2311yescharacteristic
42C2×C4×C27C2311yescharacteristic
43C2×Dic27C2311yescharacteristic
44C2×Dic27C2311yescharacteristic
45C2×Dic27C2314yes—
46C2×Dic27C2311yescharacteristic
47C4×D27—22no—
48C4×D27—22no—
49C4×D27Q814yescomplemented
50Dic54—24no—
51#5728—91no—
52C2×Q8×C9C2×D312yes—
53C2×C4×D9—31no—
54C2×C4×D9—31no—
55C2×C4×D9—31no—
56C2×Dic18—32no—
57C9×C4⋊C4C2×D312yes—
58C9×C4⋊C4C2×D312yes—
59C42×C9C2×D311yescharacteristic
60(C2×D9)⋊C4—34no—
61C4⋊Dic9—32no—
62C4⋊Dic9—34no—
63C4⋊Dic9—32no—
64Dic9⋊C4—34no—
65C4×Dic9—31no—
66C4×Dic9—34no—
67C4×Dic9—31no—
68C4×Dic9—31no—
69C22×C27C2411yescharacteristic
70C2×D27—21no—
71C2×D27—21no—
72C2×D27C2×Q812yes—
73C4×C27—24no—
74C4×C27C2×Q812yes—
75C4×C27#4112yes—
76C4×C27#4112yes—
77Dic27—24no—
78Dic27—21no—
79Dic27—21no—
80Dic27C2×Q812yes—
81C3×C4⋊Q8D911yescharacteristic
82#946—92no—
83Q8×Dic3—92no—
84#897—92no—
85#892—92no—
86#890—92no—
87#887—92no—
88C42×D3—91no—
89#869—91no—
90C22×D9—31no—
91Q8×C9—24no—
92C2×C4×C9C22×D314yes—
93C2×C4×C9C22×D311yescharacteristic
94C2×C4×C9C22×D311yescharacteristic
95C2×C4×C9C22×D311yescharacteristic
96C2×Dic9—31no—
97C2×Dic9—31no—
98C2×Dic9—31no—
99C2×Dic9—34no—
100C4×D9—34no—
101C4×D9—62no—
102C4×D9—62no—
103Dic18—64no—
104#533—271noSylow
105C2×C27C22×Q811yescharacteristic
106C2×C27C2×#4111yescharacteristic
107C2×C27C2×#4111yescharacteristic
108D27—22no—
109C2×C3×Q8C2×D912yes—
110C2×C4×D3—91no—
111C2×C4×D3—91no—
112C2×C4×D3—91no—
113C2×Dic6—92no—
114C3×C4⋊C4C2×D912yes—
115C3×C4⋊C4C2×D912yes—
116C3×C42C2×D911yescharacteristic
117(C2×D3)⋊C4—94no—
118C4⋊Dic3—92no—
119C4⋊Dic3—92no—
120C4⋊Dic3—94no—
121Dic3⋊C4—94no—
122C4×Dic3—91no—
123C4×Dic3—91no—
124C4×Dic3—91no—
125C4×Dic3—94no—
126C22×C9C23×D311yescharacteristic
127C2×D9—32no—
128C2×D9—61no—
129C2×D9—61no—
130C4×C9—24no—
131C4×C9#23712yes—
132C4×C9D3×Q812yes—
133C4×C9#23912yes—
134Dic9—32no—
135Dic9—61no—
136Dic9—64no—
137Dic9—61no—
138C4⋊Q8—271no—
139C4⋊Q8D2711yescharacteristic, complemented
140#125—272no—
141C22⋊Q8—272no—
142C22⋊Q8—272no—
143C4×Q8—272no—
144C4×Q8—272no—
145#117—272no—
146C2×C42—271no—
147C27#53311yescharacteristic, complemented, Sylow
148C22×D3—91no—
149C3×Q8—24no—
150C2×C3×C4C22×D914yes—
151C2×C3×C4C22×D911yescharacteristic
152C2×C3×C4C22×D911yescharacteristic
153C2×C3×C4C22×D911yescharacteristic
154C2×Dic3—91no—
155C2×Dic3—91no—
156C2×Dic3—91no—
157C2×Dic3—94no—
158C4×D3—94no—
159C4×D3—182no—
160C4×D3—182no—
161Dic6—184no—
162C2×C9C2×#23711yescharacteristic
163C2×C9C2×D3×Q811yescharacteristic
164C2×C9C2×#23911yescharacteristic
165D9—62no—
166C2×Q8—272no—
167C2×Q8C2×D2712yes—
168C22×C4—271no—
169C22×C4—271no—
170C22×C4—271no—
171C4⋊C4—272no—
172C4⋊C4—274no—
173C4⋊C4—274no—
174C4⋊C4—272no—
175C4⋊C4C2×D2712yes—
176C4⋊C4C2×D2712yes—
177C22⋊C4—274no—
178C42—271no—
179C42—271no—
180C42—271no—
181C42—274no—
182C42C2×D2711yescharacteristic
183C22×C3C23×D911yescharacteristic
184C2×D3—92no—
185C2×D3—181no—
186C2×D3—181no—
187C3×C4—24no—
188C3×C4#369912yes—
189C3×C4Q8×D912yes—
190C3×C4#370112yes—
191Dic3—92no—
192Dic3—181no—
193Dic3—184no—
194Dic3—181no—
195C9#572811yescharacteristic
196C23—271no—
197Q8—24no—
198Q8—544no—
199C2×C4—271no—
200C2×C4—274no—
201C2×C4—271no—
202C2×C4—271no—
203C2×C4—274no—
204C2×C4—542no—
205C2×C4—542no—
206C2×C4C22×D2714yes—
207C2×C4C22×D2711yescharacteristic
208C2×C4C22×D2711yescharacteristic
209C2×C4C22×D2711yescharacteristic
210C2×C3C2×#369911yescharacteristic
211C2×C3C2×Q8×D911yescharacteristic
212C2×C3C2×#370111yescharacteristic
213D3—182no—
214C22—272no—
215C22—541no—
216C22—541no—
217C22C23×D2711yescharacteristic
218C4—24no—
219C4—272no—
220C4—541no—
221C4—541no—
222C4—544no—
223C4#8840112yes—
224C4Q8×D2712yes—
225C4#8840312yes—
226C3#1058983011yescharacteristic
227C2—542no—
228C2C2×#8840111yescharacteristic
229C2C2×Q8×D2711yescharacteristic
230C2C2×#8840311yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×C4⋊Q8C2
socleC22×C3C23×D9
prime radicalC22×D27C23
hypercenterC4⋊Q8D27
hypocenterC27#533
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C27×C4⋊Q8C2semidirect
3. Q8×Dic27C2semidirect
6. #11720139C2semidirect
7. #11720136C2semidirect
9. #11720118C2semidirect
49. C4×D27Q8semidirect
139. C4⋊Q8D27semidirect
147. C27#533semidirect

Maximal subgroups

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

subgroupindex
C27×C4⋊Q82
#117201952
Q8×Dic272
#117201462
#117201412
#117201392
#117201362
C42×D272
#117201182
#105898303