Finite Group Databasebeta

← #13147691

Subgroups — #13147691

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721706C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×C4×D36C211yesmaximal, characteristic, complemented
4#11721042C214yesmaximal, complemented
5#11721037C212yesmaximal, complemented
6#11721036C212yesmaximal, complemented
7#11720907C212yesmaximal, complemented
8#11720906C212yesmaximal, complemented
9#10593988—31nomaximal
10#10589665—31nomaximal
11C2×#6226C2212yes—
12C2×C9×D12C2211yescharacteristic
13C2×C4×D3×C9C2212yes—
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9C2212yes—
16C2×#6194C2214yes—
17C3×C42×C9C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2212yes—
19C4⋊(C3⋊Dic9)C2212yes—
20C9×(C2×D3)⋊C4C2212yes—
21C3×(C2×D9)⋊C4C2212yes—
22C9×C4⋊Dic3C2211yescharacteristic
23C3×C4⋊Dic9C2211yescharacteristic
24(C2×D3)⋊Dic9C2214yes—
25(C2×D9)⋊Dic3C2214yes—
26Dic3×Dic9C2214yes—
27#64078—31no—
28C3×C4×D12—31no—
29C3×C4×D12D311yescharacteristic
30#63913—34no—
31#63908—32no—
32#63908—32no—
33#63841—32no—
34#63841—32no—
35C4×D4×C9—31no—
36#63500—32no—
37#63492—32no—
38#63488—32no—
39#63454—32no—
40#63447—34no—
41C4×D36D311yescharacteristic, complemented
42#63431—31no—
43#63431—61no—
44C22×D3×C9C2312yes—
45C22×C3×D9C2312yes—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3⋊Dic9C2314yes—
50#6226—42no—
51C2×C9×Dic3C2312yes—
52C2×C3×Dic9C2312yes—
53C9×D12—22no—
54C4×D3×C9—24no—
55C3×D36—22no—
56C3×C4×D9—24no—
57#6194—44no—
58#5563—91no—
59#5563—91no—
60C2×#634—32no—
61C2×C3×D12—31no—
62C2×C3×D12C2×D311yescharacteristic
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3C2×D312yes—
65C2×#625—34no—
66C32×C42C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)—32no—
68C4⋊(C3⋊Dic3)—32no—
69C3×(C2×D3)⋊C4—32no—
70C3×(C2×D3)⋊C4C2×D312yes—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3C2×D311yescharacteristic
73(C2×D3)⋊Dic3—34no—
74(C2×D3)⋊Dic3—34no—
75Dic32—34no—
76C2×D4×C9—31no—
77C22×C4×C9—32no—
78C2×#611—34no—
79C2×D36C2×D311yescharacteristic
80C2×C4×D9C2×D312yes—
81C2×Dic18—32no—
82C2×Dic18—62no—
83C9×C4⋊C4—31no—
84C9×C22⋊C4—32no—
85C42×C9—21no—
86C42×C9C2×D311yescharacteristic
87#3676—34no—
88(C2×D9)⋊C4—34no—
89(C2×D9)⋊C4C2×D312yes—
90C4⋊Dic9—32no—
91C4⋊Dic9—32no—
92C4⋊Dic9—62no—
93C4⋊Dic9—62no—
94C4⋊Dic9C2×D311yescharacteristic
95C4×Dic9—34no—
96C22×C3×C9C2411yescharacteristic
97C2×D3×C9—22no—
98C2×D3×C9—22no—
99C2×D3×C9—22no—
100C2×C3×D9—22no—
101C2×C3×D9—22no—
102C2×C3×D9—22no—
103C3×C4×C9—21no—
104C3×C4×C9#4112yes—
105C3×C4×C9#4112yes—
106C3⋊Dic9—24no—
107C9×Dic3—22no—
108C3×Dic9—22no—
109C3×C4×D4—31no—
110C3×C4×D4—91no—
111#938—92no—
112#938—92no—
113#930—92no—
114#930—92no—
115#926—92no—
116#926—92no—
117#892—92no—
118#892—92no—
119#885—94no—
120#885—94no—
121C4×D12—31no—
122C4×D12D911yescharacteristic, complemented
123#869—91no—
124#869—91no—
125#869—181no—
126C22×C3×D3—32no—
127C22×C3×D3C22×D312yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—34no—
132#634—122no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3C22×D312yes—
135C3×D12—22no—
136C3×D12—62no—
137C3×C4×D3—24no—
138C3×C4×D3—64no—
139#625—124no—
140C23×C9—32no—
141C22×D9C22×D312yes—
142D4×C9—62no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—32no—
147C2×C4×C9—64no—
148C2×C4×C9C22×D311yescharacteristic
149C2×C4×C9C22×D311yescharacteristic
150C2×C4×C9C22×D311yescharacteristic
151#611—124no—
152C2×Dic9—34no—
153C2×Dic9—64no—
154C2×Dic9C22×D312yes—
155D36—22no—
156C4×D9—24no—
157Dic18—122no—
158Dic18—124no—
159#555—271noSylow
160C2×C3×C9C2×#4111yescharacteristic
161C2×C3×C9C2×#4111yescharacteristic
162C2×C3×C9H(5,𝔽2)11yescharacteristic
163D3×C9—42no—
164C3×D9—42no—
165C2×C3×D4—31no—
166C2×C3×D4—91no—
167C22×C3×C4—32no—
168C22×C3×C4—92no—
169C2×#67—94no—
170C2×#67—94no—
171C2×D12—31no—
172C2×D12C2×D911yescharacteristic
173C2×C4×D3—32no—
174C2×C4×D3C2×D912yes—
175C2×Dic6—92no—
176C2×Dic6—92no—
177C2×Dic6—182no—
178C3×C4⋊C4—31no—
179C3×C4⋊C4—91no—
180C3×C22⋊C4—32no—
181C3×C22⋊C4—92no—
182C3×C42—21no—
183C3×C42C2×D911yescharacteristic
184C3×C42D3211yescharacteristic
185#217—94no—
186#217—94no—
187(C2×D3)⋊C4—32no—
188(C2×D3)⋊C4—94no—
189(C2×D3)⋊C4—94no—
190(C2×D3)⋊C4C2×D912yes—
191C4⋊Dic3—31no—
192C4⋊Dic3—92no—
193C4⋊Dic3—92no—
194C4⋊Dic3—92no—
195C4⋊Dic3—92no—
196C4⋊Dic3—182no—
197C4⋊Dic3—182no—
198C4⋊Dic3C2×D911yescharacteristic
199C4×Dic3—94no—
200C4×Dic3—94no—
201C22×C32C23×D311yescharacteristic
202C2×C3×D3—22no—
203C2×C3×D3—22no—
204C2×C3×D3—22no—
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—62no—
208C32×C4—21no—
209C32×C4#23512yes—
210C32×C4#23712yes—
211C3⋊Dic3—64no—
212C3×Dic3—22no—
213C3×Dic3—62no—
214C22×C9—21no—
215C22×C9—62no—
216C22×C9—62no—
217C22×C9—62no—
218C22×C9C23×D311yescharacteristic
219C2×D9—22no—
220C2×D9—22no—
221C2×D9—22no—
222C4×C9—22no—
223C4×C9—21no—
224C4×C9—22no—
225C4×C9—22no—
226C4×C9—62no—
227C4×C9#23512yes—
228C4×C9#23712yes—
229Dic9—22no—
230Dic9—64no—
231Dic9—124no—
232C4⋊Q8—271no—
233#124—274no—
234#121—272no—
235#121—272no—
236C4×D4—31no—
237C4×D4—91no—
238#117—272no—
239#117—272no—
240C3×C9#55511yescharacteristic, complemented, Sylow
241C23×C3—32no—
242C23×C3—92no—
243C22×D3—32no—
244C22×D3C22×D912yes—
245C3×D4—62no—
246C3×D4—182no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—32no—
251C2×C3×C4—64no—
252C2×C3×C4—92no—
253C2×C3×C4—184no—
254C2×C3×C4C22×D911yescharacteristic
255C2×C3×C4C22×D911yescharacteristic
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4C2×D3211yescharacteristic
258C2×C3×C4C2×D3211yescharacteristic
259C2×C3×C4C2×D3211yescharacteristic
260#67—364no—
261#67—364no—
262C2×Dic3—32no—
263C2×Dic3—94no—
264C2×Dic3—94no—
265C2×Dic3—184no—
266C2×Dic3C22×D912yes—
267D12—22no—
268D12—62no—
269C4×D3—24no—
270C4×D3—64no—
271Dic6—362no—
272Dic6—362no—
273Dic6—364no—
274C2×C32C2×#23511yescharacteristic
275C2×C32C2×#23711yescharacteristic
276C2×C32#100911yescharacteristic
277C3×D3—42no—
278C3×D3—122no—
279C2×C9—21no—
280C2×C9—21no—
281C2×C9—21no—
282C2×C9—122no—
283C2×C9C2×#23511yescharacteristic
284C2×C9C2×#23711yescharacteristic
285C2×C9#100911yescharacteristic
286D9—42no—
287C2×Q8—272no—
288C2×D4—31no—
289C2×D4—91no—
290C2×D4—274no—
291C22×C4—32no—
292C22×C4—92no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—272no—
296C4⋊C4—272no—
297C22⋊C4—32no—
298C22⋊C4—92no—
299C22⋊C4—274no—
300C22⋊C4—274no—
301C42—274no—
302C42D3×D911yescharacteristic
303C22×C3—21no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—62no—
307C22×C3—182no—
308C22×C3—182no—
309C22×C3—182no—
310C22×C3C23×D911yescharacteristic
311C22×C3C22×D3211yescharacteristic
312C2×D3—22no—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—62no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—62no—
324C3×C4—182no—
325C3×C4#369712yes—
326C3×C4#369912yes—
327C3×C4#379512yes—
328C3×C4#379512yes—
329Dic3—22no—
330Dic3—62no—
331Dic3—184no—
332Dic3—184no—
333Dic3—364no—
334C32#556311yescharacteristic
335C9—21no—
336C9#556311yescharacteristic, complemented
337C23—32no—
338C23—92no—
339Q8—1082no—
340D4—62no—
341D4—182no—
342D4—1084no—
343C2×C4—32no—
344C2×C4—64no—
345C2×C4—92no—
346C2×C4—184no—
347C2×C4—274no—
348C2×C4C2×D3×D911yescharacteristic
349C2×C4C2×D3×D911yescharacteristic
350C2×C4C2×D3×D911yescharacteristic
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—122no—
355C2×C3—362no—
356C2×C3C2×#369711yescharacteristic
357C2×C3C2×#369911yescharacteristic
358C2×C3#6371511yescharacteristic
359C2×C3C2×#379511yescharacteristic
360C2×C3C2×#379511yescharacteristic
361C2×C3#6430811yescharacteristic
362D3—42no—
363D3—122no—
364C22—62no—
365C22—62no—
366C22—62no—
367C22—182no—
368C22—182no—
369C22—182no—
370C22C22×D3×D911yescharacteristic
371C4—21no—
372C4—62no—
373C4—182no—
374C4—544no—
375C4#8863812yes—
376C4#8864112yes—
377C3—21no—
378C3#1058966511yescharacteristic, complemented
379C3#1059398811yescharacteristic
380C2—122no—
381C2—362no—
382C2C2×#8863811yescharacteristic
383C2C2×#8864111yescharacteristic
384C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#555
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721706C2semidirect
2. C4×C9×D12C2semidirect
3. C3×C4×D36C2semidirect
4. #11721042C2semidirect
5. #11721037C2semidirect
6. #11721036C2semidirect
7. #11720907C2semidirect
8. #11720906C2semidirect
41. C4×D36D3semidirect
122. C4×D12D9semidirect
240. C3×C9#555semidirect
336. C9#5563semidirect
378. C3#10589665semidirect

Maximal subgroups

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

subgroupindex
#117217062
C4×C9×D122
C3×C4×D362
#117210422
#117210372
#117210362
#117209072
#117209062
#105939883
#105896653