Finite Group Databasebeta

← #13136408

Subgroups — #13136408

Order 1728.

472 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 135 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×#124C211yesmaximal, characteristic, complemented
2#11720196C211yesmaximal, characteristic, complemented
3#11720193C211yesmaximal, characteristic, complemented
4#11720188C212yesmaximal, complemented
5#11720134C212yesmaximal, complemented
6#11720133C212yesmaximal, complemented
7#6171⋊C4C212yesmaximal, complemented
8#11720126C212yesmaximal, complemented
9#11720121C212yesmaximal, complemented
10#10589772—31nomaximal
11C2×Q8×C27C2211yescharacteristic
12C2×D4×C27C2211yescharacteristic
13C2×#6171C2212yes—
14C2×#6171C2212yes—
15C22×Dic27C2212yes—
16C2×D108C2211yescharacteristic
17C2×C4×D27C2212yes—
18C2×Dic54C2211yescharacteristic
19C22⋊C4×C27C2212yes—
20C22⋊C4×C27C2212yes—
21C42×C27C2211yescharacteristic
22#88372C2212yes—
23(C2×D27)⋊C4C2212yes—
24(C2×D27)⋊C4C2212yes—
25(C2×D27)⋊C4C2212yes—
26Dic27⋊C4C2212yes—
27Dic27⋊C4C2212yes—
28Dic27⋊C4C2212yes—
29C4×Dic27C2211yescharacteristic
30C4×Dic27C2211yescharacteristic
31C4×Dic27C2212yes—
32C9×#124D311yescharacteristic
33#63509—31no—
34#63506—31no—
35#63501—32no—
36#63447—32no—
37#63446—32no—
38#611⋊C4—32no—
39#63439—32no—
40#63434—32no—
41C23×C27C2312yes—
42C22×D27C2312yes—
43Q8×C27—41no—
44D4×C27—41no—
45C2×C4×C27C2311yescharacteristic
46C2×C4×C27C2312yes—
47C2×C4×C27C2311yescharacteristic
48C2×C4×C27C2311yescharacteristic
49#6171—24no—
50#6171—42no—
51C2×Dic27—24no—
52C2×Dic27C2312yes—
53C2×Dic27C2312yes—
54C2×Dic27C2312yes—
55D108—41no—
56C4×D27—24no—
57Dic54—41no—
58#5670—91no—
59C2×Q8×C9C2×D311yescharacteristic
60C2×D4×C9C2×D311yescharacteristic
61C2×#611—32no—
62C2×#611—32no—
63C22×Dic9—32no—
64C2×D36—31no—
65C2×C4×D9—32no—
66C2×Dic18—31no—
67C9×C22⋊C4C2×D312yes—
68C9×C22⋊C4C2×D312yes—
69C42×C9C2×D311yescharacteristic
70#3676—32no—
71(C2×D9)⋊C4—32no—
72(C2×D9)⋊C4—32no—
73(C2×D9)⋊C4—32no—
74Dic9⋊C4—32no—
75Dic9⋊C4—32no—
76Dic9⋊C4—32no—
77C4×Dic9—31no—
78C4×Dic9—31no—
79C4×Dic9—32no—
80C22×C27—22no—
81C22×C27—22no—
82C22×C27—22no—
83C22×C27C2411yescharacteristic
84C2×D27—22no—
85C2×D27—22no—
86C2×D27—22no—
87C4×C27—22no—
88C4×C27—21no—
89C4×C27—21no—
90C4×C27—21no—
91Dic27—22no—
92Dic27—22no—
93Dic27#4114yes—
94C3×#124D911yescharacteristic
95#947—91no—
96#944—91no—
97#939—92no—
98#885—92no—
99#884—92no—
100#67⋊C4—92no—
101#877—92no—
102#872—92no—
103C23×C9C22×D312yes—
104C22×D9—32no—
105Q8×C9—41no—
106D4×C9—41no—
107C2×C4×C9C22×D311yescharacteristic
108C2×C4×C9C22×D312yes—
109C2×C4×C9C22×D311yescharacteristic
110C2×C4×C9C22×D311yescharacteristic
111#611—64no—
112#611—122no—
113C2×Dic9—32no—
114C2×Dic9—32no—
115C2×Dic9—32no—
116C2×Dic9—64no—
117D36—121no—
118C4×D9—64no—
119Dic18—121no—
120#555—271noSylow
121C2×C27—42no—
122C2×C27C2×#4112yes—
123C2×C27H(5,𝔽2)11yescharacteristic
124D27—42no—
125C2×C3×Q8C2×D911yescharacteristic
126C2×C3×D4C2×D911yescharacteristic
127C2×#67—92no—
128C2×#67—92no—
129C22×Dic3—92no—
130C2×D12—91no—
131C2×C4×D3—92no—
132C2×Dic6—91no—
133C3×C22⋊C4C2×D912yes—
134C3×C22⋊C4C2×D912yes—
135C3×C42C2×D911yescharacteristic
136#217—92no—
137(C2×D3)⋊C4—92no—
138(C2×D3)⋊C4—92no—
139(C2×D3)⋊C4—92no—
140Dic3⋊C4—92no—
141Dic3⋊C4—92no—
142Dic3⋊C4—92no—
143C4×Dic3—92no—
144C4×Dic3—91no—
145C4×Dic3—91no—
146C22×C9—22no—
147C22×C9—22no—
148C22×C9—22no—
149C22×C9C23×D311yescharacteristic
150C2×D9—62no—
151C2×D9—62no—
152C2×D9—62no—
153C4×C9—21no—
154C4×C9—21no—
155C4×C9—21no—
156C4×C9—22no—
157Dic9—34no—
158Dic9—62no—
159Dic9—62no—
160C4⋊Q8—271no—
161#124—271no—
162#124—272no—
163#124D2711yescharacteristic, complemented
164#121—272no—
165#121—272no—
166C4×D4—272no—
167#117—272no—
168#117—272no—
169C27#55511yescharacteristic, complemented, Sylow
170C23×C3C22×D912yes—
171C22×D3—92no—
172C3×Q8—41no—
173C3×D4—41no—
174C2×C3×C4C22×D911yescharacteristic
175C2×C3×C4C22×D912yes—
176C2×C3×C4C22×D911yescharacteristic
177C2×C3×C4C22×D911yescharacteristic
178#67—184no—
179#67—362no—
180C2×Dic3—92no—
181C2×Dic3—92no—
182C2×Dic3—92no—
183C2×Dic3—184no—
184D12—361no—
185C4×D3—184no—
186Dic6—361no—
187C2×C9—42no—
188C2×C9#100411yescharacteristic
189C2×C9D3×#4112yes—
190D9—122no—
191C2×Q8—271no—
192C2×Q8C2×D2711yescharacteristic
193C2×D4—272no—
194C2×D4—272no—
195C2×D4—271no—
196C2×D4C2×D2711yescharacteristic
197C22×C4—272no—
198C22×C4—272no—
199C4⋊C4—272no—
200C4⋊C4—272no—
201C4⋊C4—272no—
202C22⋊C4—272no—
203C22⋊C4—272no—
204C22⋊C4—272no—
205C22⋊C4—272no—
206C22⋊C4C2×D2712yes—
207C22⋊C4C2×D2712yes—
208C42—271no—
209C42—272no—
210C42—271no—
211C42C2×D2711yescharacteristic
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3C23×D911yescharacteristic
216C2×D3—182no—
217C2×D3—182no—
218C2×D3—182no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—22no—
223Dic3—94no—
224Dic3—182no—
225Dic3—182no—
226C9#567011yescharacteristic
227C23—272no—
228C23C22×D2712yes—
229Q8—41no—
230Q8—1081no—
231D4—41no—
232D4—544no—
233D4—1081no—
234D4—1082no—
235C2×C4—272no—
236C2×C4—272no—
237C2×C4—272no—
238C2×C4—544no—
239C2×C4—544no—
240C2×C4C22×D2712yes—
241C2×C4C22×D2711yescharacteristic
242C2×C4C22×D2711yescharacteristic
243C2×C4C22×D2711yescharacteristic
244C2×C3—42no—
245C2×C3#6371011yescharacteristic
246C2×C3#41×D912yes—
247D3—362no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—542no—
252C22—542no—
253C22—542no—
254C22C23×D2711yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—22no—
259C4—274no—
260C4—542no—
261C4—542no—
262C3#1058977211yescharacteristic
263C2—42no—
264C2—1082no—
265C2#1172067011yescharacteristic
266C2#41×D2712yes—
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#124C2
socleC22×C3C23×D9
hypercenter#124D27
hypocenterC27#555
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C27×#124C2semidirect
2. #11720196C2semidirect
3. #11720193C2semidirect
4. #11720188C2semidirect
5. #11720134C2semidirect
6. #11720133C2semidirect
7. #6171⋊C4C2semidirect
8. #11720126C2semidirect
9. #11720121C2semidirect
163. #124D27semidirect
169. C27#555semidirect

Maximal subgroups

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

subgroupindex
C27×#1242
#117201962
#117201932
#117201882
#117201342
#117201332
#6171⋊C42
#117201262
#117201212
#105897723