Finite Group Databasebeta

← #13147584

Subgroups — #13147584

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic, complemented
3C9×#869C211yesmaximal, characteristic, complemented
4#11721007C212yesmaximal, complemented
5#11721004C212yesmaximal, complemented
6#11721003C214yesmaximal, complemented
7#11720922C212yesmaximal, complemented
8#11720907C212yesmaximal, complemented
9#10593987—31nomaximal
10#10589836—31nomaximal
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2212yes—
13C2×C3×D36C2211yescharacteristic
14C2×C3×C4×D9C2212yes—
15C2×C9×Dic6C2212yes—
16C2×#6192C2214yes—
17C3×C42×C9C2211yescharacteristic
18#88536C2212yes—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C3×(C2×D9)⋊C4C2212yes—
21C9×C4⋊Dic3C2212yes—
22C9×C4⋊Dic3C2212yes—
23C3×C4⋊Dic9C2211yescharacteristic
24#88445C2214yes—
25(C2×D9)⋊Dic3C2214yes—
26Dic3×Dic9C2214yes—
27C4×C4⋊(C3⋊D3)—31no—
28C3×C4×D12—31no—
29C3×#869D311yescharacteristic
30#63916—32no—
31#63909—32no—
32#63907—34no—
33#63849—32no—
34#63841—32no—
35C9×C4⋊Q8—31no—
36#63509—34no—
37#63457—32no—
38#63457—32no—
39#63454—32no—
40#63454—32no—
41C4×D36—31no—
42C4×D36—61no—
43C4×D36D311yescharacteristic, complemented
44C22×C3⋊D9C2312yes—
45C22×C3×D9C2312yes—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3⋊Dic9C2312yes—
50C4⋊(C3⋊D9)—22no—
51C4×C3⋊D9—24no—
52C2×C9×Dic3C2314yes—
53C2×C3×Dic9C2312yes—
54C3×D36—22no—
55C3×C4×D9—24no—
56C9×Dic6—42no—
57#6192—44no—
58#5734—91no—
59#5563—91no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4×C3⋊D3—32no—
62C2×C3×D12—31no—
63C2×C3×C4×D3—32no—
64C2×C3×Dic6C2×D312yes—
65C2×#626—34no—
66C32×C42C2×D311yescharacteristic
67#3752—32no—
68C4⋊(C3⋊Dic3)—31no—
69C3×(C2×D3)⋊C4—32no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3C2×D312yes—
72C3×C4⋊Dic3C2×D312yes—
73#3722—34no—
74(C2×D3)⋊Dic3—34no—
75Dic32—34no—
76C2×Q8×C9—32no—
77C2×D36—31no—
78C2×D36—34no—
79C2×D36—61no—
80C2×D36C2×D311yescharacteristic
81C2×C4×D9—32no—
82C2×C4×D9—62no—
83C2×C4×D9C2×D312yes—
84C9×C4⋊C4—32no—
85C9×C4⋊C4—32no—
86C42×C9—21no—
87C42×C9C2×D311yescharacteristic
88(C2×D9)⋊C4—34no—
89(C2×D9)⋊C4—32no—
90(C2×D9)⋊C4—34no—
91(C2×D9)⋊C4—62no—
92(C2×D9)⋊C4C2×D312yes—
93C4⋊Dic9—31no—
94C4⋊Dic9—61no—
95C4⋊Dic9C2×D311yescharacteristic
96C4×Dic9—34no—
97C22×C3×C9C2411yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—22no—
100C2×C3⋊D9—22no—
101C2×C3×D9—22no—
102C2×C3×D9—22no—
103C2×C3×D9—22no—
104C3×C4×C9—21no—
105C3×C4×C9#4112yes—
106C3×C4×C9#4112yes—
107C3⋊Dic9—22no—
108C9×Dic3—24no—
109C3×Dic9—22no—
110C3×C4⋊Q8—31no—
111C3×C4×D4—91no—
112#947—94no—
113#938—92no—
114#930—92no—
115#926—92no—
116#895—92no—
117#895—92no—
118#892—92no—
119#892—92no—
120#892—92no—
121#885—94no—
122C4×D12—31no—
123C4×D12—91no—
124C4×D12—91no—
125C4×D12—181no—
126#869D911yescharacteristic, complemented
127C22×C3⋊D3—32no—
128C22×C3×D3—32no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—32no—
133C4⋊(C3⋊D3)—62no—
134C4×C3⋊D3—64no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3C22×D314yes—
137C3×D12—62no—
138C3×C4×D3—64no—
139C3×Dic6—42no—
140#626—124no—
141C22×D9—32no—
142C22×D9—62no—
143C22×D9C22×D312yes—
144Q8×C9—122no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—34no—
149C2×C4×C9C22×D311yescharacteristic
150C2×C4×C9C22×D311yescharacteristic
151C2×C4×C9C22×D311yescharacteristic
152C2×Dic9—32no—
153C2×Dic9—62no—
154C2×Dic9C22×D312yes—
155D36—22no—
156D36—64no—
157D36—62no—
158D36—124no—
159C4×D9—24no—
160C4×D9—64no—
161C4×D9—124no—
162#555—271noSylow
163C2×C3×C9C2×#4111yescharacteristic
164C2×C3×C9C2×#4111yescharacteristic
165C2×C3×C9H(5,𝔽2)11yescharacteristic
166C3⋊D9—42no—
167C3×D9—42no—
168C2×C3×Q8—32no—
169C2×C3×D4—91no—
170C22×C3×C4—92no—
171C2×#67—94no—
172C2×D12—31no—
173C2×D12—91no—
174C2×D12—91no—
175C2×D12—94no—
176C2×D12—181no—
177C2×C4×D3—32no—
178C2×C4×D3—92no—
179C2×C4×D3—92no—
180C2×C4×D3—182no—
181C2×Dic6C2×D912yes—
182C3×C4⋊C4—32no—
183C3×C4⋊C4—32no—
184C3×C4⋊C4—91no—
185C3×C22⋊C4—92no—
186C3×C42—21no—
187C3×C42C2×D911yescharacteristic
188C3×C42D3211yescharacteristic
189#217—94no—
190(C2×D3)⋊C4—32no—
191(C2×D3)⋊C4—94no—
192(C2×D3)⋊C4—94no—
193(C2×D3)⋊C4—92no—
194(C2×D3)⋊C4—92no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—182no—
197C4⋊Dic3—31no—
198C4⋊Dic3—91no—
199C4⋊Dic3—91no—
200C4⋊Dic3—181no—
201C4⋊Dic3C2×D912yes—
202C4⋊Dic3C2×D912yes—
203C4×Dic3—94no—
204C4×Dic3—94no—
205C22×C32C23×D311yescharacteristic
206C2×C3⋊D3—62no—
207C2×C3⋊D3—62no—
208C2×C3⋊D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—62no—
212C32×C4—21no—
213C32×C4#23912yes—
214C32×C4#23912yes—
215C3⋊Dic3—62no—
216C3×Dic3—24no—
217C3×Dic3—62no—
218C22×C9—21no—
219C22×C9C23×D311yescharacteristic
220C2×D9—22no—
221C2×D9—22no—
222C2×D9—22no—
223C2×D9—64no—
224C2×D9—62no—
225C2×D9—62no—
226C2×D9—62no—
227C2×D9—122no—
228C2×D9—122no—
229C4×C9—21no—
230C4×C9—22no—
231C4×C9—22no—
232C4×C9—41no—
233C4×C9—64no—
234C4×C9#23512yes—
235C4×C9#23712yes—
236Dic9—22no—
237Dic9—62no—
238Dic9—122no—
239C4⋊Q8—31no—
240#124—274no—
241#121—272no—
242#121—272no—
243C4×D4—91no—
244C4×D4—271no—
245#117—272no—
246#117—272no—
247C3×C9#55511yescharacteristic, complemented, Sylow
248C23×C3—92no—
249C22×D3—32no—
250C22×D3—92no—
251C22×D3—92no—
252C22×D3—182no—
253C3×Q8—122no—
254C3×D4—182no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—34no—
259C2×C3×C4—92no—
260C2×C3×C4—184no—
261C2×C3×C4C22×D911yescharacteristic
262C2×C3×C4C22×D911yescharacteristic
263C2×C3×C4C22×D911yescharacteristic
264C2×C3×C4C2×D3211yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266C2×C3×C4C2×D3211yescharacteristic
267#67—364no—
268C2×Dic3—32no—
269C2×Dic3—92no—
270C2×Dic3—92no—
271C2×Dic3—182no—
272C2×Dic3C22×D914yes—
273D12—62no—
274D12—184no—
275D12—182no—
276D12—182no—
277D12—364no—
278C4×D3—64no—
279C4×D3—184no—
280C4×D3—184no—
281C4×D3—364no—
282Dic6—42no—
283C2×C32#100411yescharacteristic
284C2×C32C2×#23911yescharacteristic
285C2×C32C2×#23911yescharacteristic
286C3⋊D3—122no—
287C3×D3—122no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—21no—
291C2×C9C2×#23511yescharacteristic
292C2×C9C2×#23711yescharacteristic
293C2×C9#100911yescharacteristic
294D9—42no—
295D9—124no—
296D9—122no—
297C2×Q8—32no—
298C2×D4—91no—
299C2×D4—274no—
300C2×D4—271no—
301C22×C4—92no—
302C22×C4—272no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—91no—
306C4⋊C4—271no—
307C22⋊C4—92no—
308C22⋊C4—274no—
309C22⋊C4—274no—
310C22⋊C4—272no—
311C42—274no—
312C42D3×D911yescharacteristic
313C22×C3—21no—
314C22×C3—182no—
315C22×C3—182no—
316C22×C3—182no—
317C22×C3C23×D911yescharacteristic
318C22×C3C22×D3211yescharacteristic
319C2×D3—62no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—184no—
323C2×D3—182no—
324C2×D3—182no—
325C2×D3—182no—
326C2×D3—182no—
327C2×D3—182no—
328C2×D3—182no—
329C2×D3—362no—
330C2×D3—362no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—41no—
336C3×C4—64no—
337C3×C4—182no—
338C3×C4#370112yes—
339C3×C4#370112yes—
340C3×C4#379612yes—
341C3×C4#379912yes—
342Dic3—24no—
343Dic3—62no—
344Dic3—182no—
345Dic3—182no—
346Dic3—362no—
347C32#573411yescharacteristic
348C9—21no—
349C9#556311yescharacteristic, complemented
350C23—92no—
351C23—272no—
352Q8—122no—
353D4—182no—
354D4—542no—
355D4—1084no—
356C2×C4—34no—
357C2×C4—92no—
358C2×C4—184no—
359C2×C4—272no—
360C2×C4—544no—
361C2×C4C2×D3×D911yescharacteristic
362C2×C4C2×D3×D911yescharacteristic
363C2×C4C2×D3×D911yescharacteristic
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—362no—
368C2×C3#6371011yescharacteristic
369C2×C3C2×#370111yescharacteristic
370C2×C3C2×#370111yescharacteristic
371C2×C3C2×#379611yescharacteristic
372C2×C3C2×#379911yescharacteristic
373C2×C3#6430711yescharacteristic
374D3—122no—
375D3—364no—
376D3—362no—
377D3—362no—
378C22—182no—
379C22—182no—
380C22—182no—
381C22—542no—
382C22—542no—
383C22—542no—
384C22C22×D3×D911yescharacteristic
385C4—21no—
386C4—64no—
387C4—182no—
388C4—542no—
389C4#8863412yes—
390C4#8863512yes—
391C3—21no—
392C3#1058983611yescharacteristic, complemented
393C3#1059398711yescharacteristic
394C2—362no—
395C2—1082no—
396C2C2×#8863411yescharacteristic
397C2C2×#8863511yescharacteristic
398C2#1172247611yescharacteristic
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. C4×C4⋊(C3⋊D9)C2semidirect
2. C3×C4×D36C2semidirect
3. C9×#869C2semidirect
4. #11721007C2semidirect
5. #11721004C2semidirect
6. #11721003C2semidirect
7. #11720922C2semidirect
8. #11720907C2semidirect
43. C4×D36D3semidirect
126. #869D9semidirect
247. C3×C9#555semidirect
349. C9#5563semidirect
392. C3#10589836semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D9)2
C3×C4×D362
C9×#8692
#117210072
#117210042
#117210032
#117209222
#117209072
#105939873
#105898363