Finite Group Databasebeta

← #13136473

Subgroups — #13136473

Order 1728.

392 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 131 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#11720196C211yesmaximal, characteristic
3#11720195C212yesmaximal, complemented
4#11720193C211yesmaximal, characteristic, complemented
5#11720147C212yesmaximal, complemented
6#11720145C212yesmaximal, complemented
7#11720143C212yesmaximal
8#11720138C212yesmaximal, complemented
9#11720125C212yesmaximal, complemented
10#10589837—31nomaximal
11C2×Q8×C27C2211yescharacteristic
12C2×Q8×C27C2211yescharacteristic
13C2×D108C2211yescharacteristic
14C2×C4×D27C2212yes—
15C2×Dic54C2211yescharacteristic
16C4⋊C4×C27C2212yes—
17C4⋊C4×C27C2212yes—
18C42×C27C2211yescharacteristic
19(C2×D27)⋊C4C2212yes—
20(C2×D27)⋊C4C2212yes—
21(C2×D27)⋊C4C2212yes—
22(C2×D27)⋊C4C2212yes—
23(C2×D27)⋊C4C2212yes—
24C4⋊Dic27C2212yes—
25Dic27⋊C4C2212yes—
26Dic27⋊C4C2212yes—
27Dic27⋊C4C2212yes—
28Dic27⋊C4C2212yes—
29Dic27⋊C4C2212yes—
30C4×Dic27C2211yescharacteristic
31C4×Dic27C2211yescharacteristic
32C9×C4⋊Q8D311yescharacteristic
33#63509—31no—
34#63508—32no—
35#63506—31no—
36#63460—32no—
37#63458—32no—
38#63456—32no—
39#63451—32no—
40#63438—32no—
41C22×D27C2312yes—
42Q8×C27—41no—
43Q8×C27—41no—
44C2×C4×C27C2311yescharacteristic
45C2×C4×C27C2311yescharacteristic
46C2×C4×C27C2312yes—
47C2×C4×C27C2311yescharacteristic
48C2×C4×C27C2312yes—
49C2×Dic27C2312yes—
50C2×Dic27C2312yes—
51C2×Dic27C2312yes—
52D108—41no—
53C4×D27—42no—
54Dic54—41no—
55#5735—91no—
56C2×Q8×C9C2×D311yescharacteristic
57C2×Q8×C9C2×D311yescharacteristic
58C2×D36—31no—
59C2×C4×D9—32no—
60C2×Dic18—31no—
61C9×C4⋊C4C2×D312yes—
62C9×C4⋊C4C2×D312yes—
63C42×C9C2×D311yescharacteristic
64(C2×D9)⋊C4—32no—
65(C2×D9)⋊C4—32no—
66(C2×D9)⋊C4—32no—
67(C2×D9)⋊C4—32no—
68(C2×D9)⋊C4—32no—
69C4⋊Dic9—32no—
70Dic9⋊C4—32no—
71Dic9⋊C4—32no—
72Dic9⋊C4—32no—
73Dic9⋊C4—32no—
74Dic9⋊C4—32no—
75C4×Dic9—31no—
76C4×Dic9—31no—
77C22×C27C2411yescharacteristic
78C2×D27—22no—
79C2×D27—24no—
80C4×C27—21no—
81C4×C27—22no—
82C4×C27—21no—
83C4×C27—21no—
84C4×C27—22no—
85Dic27—22no—
86Dic27—22no—
87Dic27—22no—
88C3×C4⋊Q8D911yescharacteristic
89#947—91no—
90#946—92no—
91#944—91no—
92#898—92no—
93#896—92no—
94#894—92no—
95#889—92no—
96#876—92no—
97C22×D9—32no—
98Q8×C9—41no—
99Q8×C9—41no—
100C2×C4×C9C22×D311yescharacteristic
101C2×C4×C9C22×D311yescharacteristic
102C2×C4×C9C22×D312yes—
103C2×C4×C9C22×D311yescharacteristic
104C2×C4×C9C22×D312yes—
105C2×Dic9—32no—
106C2×Dic9—32no—
107C2×Dic9—32no—
108D36—121no—
109C4×D9—122no—
110Dic18—121no—
111#563—271noSylow
112C2×C27H(5,𝔽2)11yescharacteristic
113C2×C272−1+412yes—
114D27—42no—
115C2×C3×Q8C2×D911yescharacteristic
116C2×C3×Q8C2×D911yescharacteristic
117C2×D12—91no—
118C2×C4×D3—92no—
119C2×Dic6—91no—
120C3×C4⋊C4C2×D912yes—
121C3×C4⋊C4C2×D912yes—
122C3×C42C2×D911yescharacteristic
123(C2×D3)⋊C4—92no—
124(C2×D3)⋊C4—92no—
125(C2×D3)⋊C4—92no—
126(C2×D3)⋊C4—92no—
127(C2×D3)⋊C4—92no—
128C4⋊Dic3—92no—
129Dic3⋊C4—92no—
130Dic3⋊C4—92no—
131Dic3⋊C4—92no—
132Dic3⋊C4—92no—
133Dic3⋊C4—92no—
134C4×Dic3—91no—
135C4×Dic3—91no—
136C22×C9C23×D311yescharacteristic
137C2×D9—62no—
138C2×D9—64no—
139C4×C9—21no—
140C4×C9—22no—
141C4×C9—21no—
142C4×C9—21no—
143C4×C9—22no—
144Dic9—62no—
145Dic9—62no—
146Dic9—62no—
147C4⋊Q8—271no—
148C4⋊Q8D2711yescharacteristic, complemented
149#126—272no—
150#126—272no—
151#125—272no—
152#124—271no—
153#123—272no—
154C22⋊Q8—272no—
155C22⋊Q8—272no—
156C27#56311yescharacteristic, complemented, Sylow
157C22×D3—92no—
158C3×Q8—41no—
159C3×Q8—41no—
160C2×C3×C4C22×D911yescharacteristic
161C2×C3×C4C22×D911yescharacteristic
162C2×C3×C4C22×D912yes—
163C2×C3×C4C22×D911yescharacteristic
164C2×C3×C4C22×D912yes—
165C2×Dic3—92no—
166C2×Dic3—92no—
167C2×Dic3—92no—
168D12—361no—
169C4×D3—362no—
170Dic6—361no—
171C2×C9#100411yescharacteristic
172C2×C9#100712yes—
173D9—122no—
174C2×Q8—271no—
175C2×Q8C2×D2711yescharacteristic
176C2×Q8C2×D2711yescharacteristic
177C2×D4—271no—
178C22×C4—272no—
179C4⋊C4—272no—
180C4⋊C4—272no—
181C4⋊C4—272no—
182C4⋊C4—272no—
183C4⋊C4—272no—
184C4⋊C4—272no—
185C4⋊C4C2×D2712yes—
186C4⋊C4C2×D2712yes—
187C22⋊C4—272no—
188C22⋊C4—272no—
189C22⋊C4—272no—
190C22⋊C4—272no—
191C22⋊C4—272no—
192C42—271no—
193C42—271no—
194C42C2×D2711yescharacteristic
195C22×C3C23×D911yescharacteristic
196C2×D3—182no—
197C2×D3—184no—
198C3×C4—21no—
199C3×C4—22no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—22no—
203Dic3—182no—
204Dic3—182no—
205Dic3—182no—
206C9#573511yescharacteristic
207C23—272no—
208Q8—41no—
209Q8—41no—
210Q8—1081no—
211D4—1081no—
212C2×C4—272no—
213C2×C4—272no—
214C2×C4—272no—
215C2×C4—1082no—
216C2×C4C22×D2711yescharacteristic
217C2×C4C22×D2711yescharacteristic
218C2×C4C22×D2712yes—
219C2×C4C22×D2711yescharacteristic
220C2×C4C22×D2712yes—
221C2×C3#6371011yescharacteristic
222C2×C3#6371312yes—
223D3—362no—
224C22—542no—
225C22—544no—
226C22C23×D2711yescharacteristic
227C4—22no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—22no—
232C4—542no—
233C4—542no—
234C4—542no—
235C3#1058983711yescharacteristic
236C2—1082no—
237C2#1172067011yescharacteristic
238C2#1172067312yes—
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×C4⋊Q8C2
socleC22×C3C23×D9
prime radicalC2×D108C22
hypercenterC4⋊Q8D27
hypocenterC27#563
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. #11720195C2semidirect
4. #11720193C2semidirect
5. #11720147C2semidirect
6. #11720145C2semidirect
8. #11720138C2semidirect
9. #11720125C2semidirect
148. C4⋊Q8D27semidirect
156. C27#563semidirect

Maximal subgroups

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

subgroupindex
C27×C4⋊Q82
#117201962
#117201952
#117201932
#117201472
#117201452
#117201432
#117201382
#117201252
#105898373