Finite Group Databasebeta

← #13136413

Subgroups — #13136413

Order 1728.

668 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#88403C211yesmaximal, characteristic, complemented
2C2×D4×D27C211yesmaximal, characteristic, complemented
3C27×#124C211yesmaximal, characteristic, complemented
4#11720196C211yesmaximal, characteristic, complemented
5#11720189C211yesmaximal, characteristic, complemented
6#11720133C214yesmaximal, complemented
7#11720131C214yesmaximal, complemented
8#11720123C211yesmaximal, characteristic, complemented
9#11720121C211yesmaximal, characteristic, complemented
10#10589776—31nomaximal
11C23×D27C2212yes—
12C2×Q8×C27C2211yescharacteristic, complemented
13C2×D4×C27C2211yescharacteristic
14C2×#6171C2212yes—
15C2×#6171C2214yescomplemented
16#88403—24no—
17D4×D27—24no—
18C2×D108C2211yescharacteristic
19C2×D108C2212yes—
20C2×D108C2211yescharacteristic, complemented
21C2×D108C2214yescomplemented
22C2×C4×D27C2211yescharacteristic
23C2×C4×D27C2212yescomplemented
24C22⋊C4×C27C2214yescomplemented
25C42×C27C2211yescharacteristic, complemented
26(C2×D27)⋊C4C2212yes—
27(C2×D27)⋊C4C2214yescomplemented
28Dic27⋊C4C2212yescomplemented
29C4×Dic27C2211yescharacteristic, complemented
30C2×#3701—31no—
31C2×D4×D9—31no—
32C9×#124D311yescharacteristic
33#63509—31no—
34#63502—31no—
35#63446—34no—
36#63444—34no—
37#63436—31no—
38#63434—31no—
39C23×C27C2312yes—
40C22×D27—24no—
41C22×D27—44no—
42C22×D27C2312yes—
43C22×D27C2312yes—
44C22×D27C2311yescharacteristic
45Q8×C27—22no—
46D4×C27—22no—
47C2×C4×C27C2312yes—
48C2×C4×C27C2312yes—
49C2×C4×C27C2311yescharacteristic
50#6171—24no—
51#6171—44no—
52C2×Dic27C2312yes—
53C2×Dic27C2311yescharacteristic
54D108—24no—
55D108—22no—
56D108—22no—
57D108—44no—
58C4×D27—24no—
59C4×D27D414yescomplemented
60#5674—91no—
61C23×D9—32no—
62C2×Q8×C9C2×D311yescharacteristic
63C2×D4×C9C2×D311yescharacteristic
64C2×#611—32no—
65C2×#611—34no—
66#3701—64no—
67D4×D9—64no—
68C2×D36—31no—
69C2×D36—31no—
70C2×D36—32no—
71C2×D36—34no—
72C2×C4×D9—31no—
73C2×C4×D9—32no—
74C9×C22⋊C4C2×D314yes—
75C42×C9C2×D311yescharacteristic
76(C2×D9)⋊C4—34no—
77(C2×D9)⋊C4—32no—
78Dic9⋊C4—32no—
79C4×Dic9—31no—
80C22×C27—22no—
81C22×C27—24no—
82C22×C27C2411yescharacteristic
83C2×D27—22no—
84C2×D27—22no—
85C2×D27—22no—
86C2×D27—24no—
87C2×D27—24no—
88C2×D27—48no—
89C2×D27C2×D412yes—
90C4×C27—22no—
91C4×C27—22no—
92C4×C27C2×D412yes—
93Dic27—22no—
94Dic27C2×D412yes—
95C2×#239—91no—
96C2×D3×D4—91no—
97C3×#124D911yescharacteristic
98#947—91no—
99#940—91no—
100#884—94no—
101#882—94no—
102#874—91no—
103#872—91no—
104C23×C9C22×D312yes—
105C22×D9—31no—
106C22×D9—32no—
107C22×D9—32no—
108C22×D9—64no—
109C22×D9—124no—
110Q8×C9—22no—
111D4×C9—22no—
112C2×C4×C9C22×D312yes—
113C2×C4×C9C22×D312yes—
114C2×C4×C9C22×D311yescharacteristic
115#611—64no—
116#611—124no—
117C2×Dic9—31no—
118C2×Dic9—32no—
119D36—64no—
120D36—62no—
121D36—62no—
122D36—124no—
123C4×D9—34no—
124C4×D9—64no—
125#535—271noSylow
126C2×C27—42no—
127C2×C27C22×D411yescharacteristic
128C2×C27H(5,𝔽2)12yes—
129D27—22no—
130D27—42no—
131D27—42no—
132C23×D3—92no—
133C2×C3×Q8C2×D911yescharacteristic
134C2×C3×D4C2×D911yescharacteristic
135C2×#67—92no—
136C2×#67—94no—
137#239—184no—
138D3×D4—184no—
139C2×D12—91no—
140C2×D12—91no—
141C2×D12—92no—
142C2×D12—94no—
143C2×C4×D3—91no—
144C2×C4×D3—92no—
145C3×C22⋊C4C2×D914yes—
146C3×C42C2×D911yescharacteristic
147(C2×D3)⋊C4—92no—
148(C2×D3)⋊C4—94no—
149Dic3⋊C4—92no—
150C4×Dic3—91no—
151C22×C9—22no—
152C22×C9—24no—
153C22×C9C23×D311yescharacteristic
154C2×D9—32no—
155C2×D9—62no—
156C2×D9—62no—
157C2×D9—62no—
158C2×D9—64no—
159C2×D9—64no—
160C2×D9—128no—
161C4×C9—22no—
162C4×C9—22no—
163C4×C9D3×D412yes—
164Dic9—32no—
165Dic9—62no—
166C2×#41—271no—
167C22×D4—271no—
168#127—271no—
169#127—271no—
170#124—271no—
171#124D2711yescharacteristic, complemented
172#121—274no—
173#120—274no—
174#117—271no—
175C27#53511yescharacteristic, complemented, Sylow
176C23×C3C22×D912yes—
177C22×D3—91no—
178C22×D3—92no—
179C22×D3—92no—
180C22×D3—184no—
181C22×D3—364no—
182C3×Q8—22no—
183C3×D4—22no—
184C2×C3×C4C22×D912yes—
185C2×C3×C4C22×D912yes—
186C2×C3×C4C22×D911yescharacteristic
187#67—184no—
188#67—364no—
189C2×Dic3—91no—
190C2×Dic3—92no—
191D12—184no—
192D12—182no—
193D12—182no—
194D12—364no—
195C4×D3—94no—
196C4×D3—184no—
197C2×C9—42no—
198C2×C9C2×D3×D411yescharacteristic
199C2×C9#100912yes—
200D9—62no—
201D9—122no—
202D9—122no—
203C24—272no—
204#41—544no—
205C2×Q8C2×D2711yescharacteristic, complemented
206C2×D4—272no—
207C2×D4—274no—
208C2×D4—272no—
209C2×D4—271no—
210C2×D4—271no—
211C2×D4—274no—
212C2×D4—544no—
213C2×D4C2×D2711yescharacteristic
214C22×C4—271no—
215C22×C4—272no—
216C4⋊C4—272no—
217C22⋊C4—272no—
218C22⋊C4—274no—
219C22⋊C4C2×D2714yescomplemented
220C42—271no—
221C42C2×D2711yescharacteristic, complemented
222C22×C3—22no—
223C22×C3—24no—
224C22×C3C23×D911yescharacteristic
225C2×D3—92no—
226C2×D3—182no—
227C2×D3—182no—
228C2×D3—182no—
229C2×D3—184no—
230C2×D3—184no—
231C2×D3—368no—
232C3×C4—22no—
233C3×C4—22no—
234C3×C4D4×D912yes—
235Dic3—92no—
236Dic3—182no—
237C9#567411yescharacteristic
238C23—271no—
239C23—272no—
240C23—272no—
241C23—544no—
242C23—1084no—
243C23C22×D2712yes—
244Q8—22no—
245D4—22no—
246D4—544no—
247D4—544no—
248D4—542no—
249D4—542no—
250D4—1084no—
251D4—1084no—
252C2×C4—271no—
253C2×C4—274no—
254C2×C4—272no—
255C2×C4—544no—
256C2×C4C22×D2712yes—
257C2×C4C22×D2712yes—
258C2×C4C22×D2711yescharacteristic
259C2×C3—42no—
260C2×C3C2×D4×D911yescharacteristic
261C2×C3#6371512yes—
262D3—182no—
263D3—362no—
264D3—362no—
265C22—22no—
266C22—24no—
267C22—272no—
268C22—542no—
269C22—542no—
270C22—542no—
271C22—544no—
272C22—544no—
273C22—1088no—
274C22C23×D2711yescharacteristic
275C4—22no—
276C4—22no—
277C4—272no—
278C4—542no—
279C4D4×D2712yes—
280C3#1058977611yescharacteristic
281C2—42no—
282C2—542no—
283C2—1082no—
284C2—1082no—
285C2C2×D4×D2711yescharacteristic
286C2#1172067512yes—
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×#124C2
socleC22×C3C23×D9
hypercenter#124D27
hypocenterC27#535
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#88403C2semidirect
2. C2×D4×D27C2semidirect
3. C27×#124C2semidirect
4. #11720196C2semidirect
5. #11720189C2semidirect
6. #11720133C2semidirect
7. #11720131C2semidirect
8. #11720123C2semidirect
9. #11720121C2semidirect
12. C2×Q8×C27C22semidirect
15. C2×#6171C22semidirect
20. C2×D108C22semidirect
21. C2×D108C22semidirect
23. C2×C4×D27C22semidirect
24. C22⋊C4×C27C22semidirect
25. C42×C27C22semidirect
27. (C2×D27)⋊C4C22semidirect
28. Dic27⋊C4C22semidirect
29. C4×Dic27C22semidirect
59. C4×D27D4semidirect
171. #124D27semidirect
175. C27#535semidirect
205. C2×Q8C2×D27semidirect
219. C22⋊C4C2×D27semidirect
221. C42C2×D27semidirect

Maximal subgroups

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

subgroupindex
C2×#884032
C2×D4×D272
C27×#1242
#117201962
#117201892
#117201332
#117201312
#117201232
#117201212
#105897763