Finite Group Databasebeta

← #13150302

Subgroups — #13150302

Order 1728.

1046 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic
2C9×#926C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
4#11721241C214yesmaximal, complemented
5#11721000C212yesmaximal, complemented
6#11720993C212yesmaximal, complemented
7#11720922C212yesmaximal, complemented
8#11720912C212yesmaximal, complemented
9#10594990—31nomaximal
10#10589625—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C2×#6233C2212yes—
13#88736C418yescomplemented
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×#6226C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×(C3⋊D9)⋊C4C2214yes—
19C9×#217C2212yes—
20C9×C4⋊Dic3C2211yescharacteristic
21C9×C4⋊Dic3C2211yescharacteristic
22C4×C9×Dic3C2212yes—
23C3×C4⋊Dic9C2212yes—
24C3×C4⋊Dic9C2212yes—
25#88445C2214yes—
26#88442C2214yes—
27Dic3×Dic9C2214yes—
28C2×#3828—31no—
29C3×#926D311yescharacteristic
30C2×C3×C4⋊Dic3—31no—
31#63952—34no—
32#63900—32no—
33#63894—32no—
34#63849—32no—
35#63844—32no—
36C2×#3697—31no—
37C2×#3697—61no—
38C9×#117—31no—
39C2×C4⋊Dic9D311yescharacteristic, complemented
40#63454—32no—
41C4⋊C4×D9—32no—
42#611⋊C4—34no—
43C4×D36—32no—
44C4×Dic18—32no—
45C23×C3×C9C2311yescharacteristic
46C22×C3⋊D9C2312yes—
47C2×C3×C4×C9C2×C414yescomplemented
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3×C4×C9C2311yescharacteristic
50#6233C2×C418yescomplemented
51C2×C3⋊Dic9C2312yes—
52C4⋊(C3⋊D9)C2×C414yescomplemented
53C4×C3⋊D9C2×C418yescomplemented
54#6226C2×C414yescomplemented
55C2×C9×Dic3C2312yes—
56C2×C9×Dic3C2312yes—
57C2×C3×Dic9—42no—
58C2×C3×Dic9C2314yes—
59(C3⋊D9)⋊C4—44no—
60#5523—91no—
61#5504—91no—
62C22×C32×C4C2×D311yescharacteristic
63C2×#638—32no—
64#3828—38no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4×C3⋊D3—32no—
67C2×#634—31no—
68C22×C3×Dic3—32no—
69C2×#624—34no—
70C3×#217C2×D312yes—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—32no—
73C3×C4⋊Dic3C2×D311yescharacteristic
74C3×C4⋊Dic3C2×D311yescharacteristic
75C3×C4×Dic3C2×D312yes—
76#3724—34no—
77#3722—34no—
78Dic32—34no—
79C22×C4×C9—21no—
80C22×C4×C9C2×D311yescharacteristic
81C2×#611—32no—
82C2×#611—62no—
83C22×Dic9C2×D312yes—
84#3697—38no—
85#3697—68no—
86C2×D36—31no—
87C2×D36—61no—
88C2×C4×D9—32no—
89C2×C4×D9—34no—
90C2×C4×D9—62no—
91C2×Dic18—31no—
92C2×Dic18—61no—
93C9×C4⋊C4—31no—
94C9×C4⋊C4—31no—
95C9×C22⋊C4—32no—
96C42×C9—32no—
97(C2×D9)⋊C4—34no—
98C4⋊Dic9C2×D312yes—
99C4⋊Dic9C2×D312yes—
100Dic9⋊C4—34no—
101C4×Dic9—34no—
102C22×C3×C9—21no—
103C22×C3×C9—21no—
104C22×C3×C9C22×C412yes—
105C22×C3×C9C2411yescharacteristic
106C2×C3⋊D9—22no—
107C2×C3⋊D9—22no—
108C2×C3⋊D9C22×C414yes—
109C3×C4×C9C22×C412yes—
110C3×C4×C9C22×C412yes—
111C3⋊Dic9C22×C414yes—
112C9×Dic3—22no—
113C9×Dic3—22no—
114C3×Dic9—24no—
115C2×#235—91no—
116C2×#235—91no—
117C2×#235—181no—
118C3×#117—31no—
119C2×C3×C4⋊C4—91no—
120C4×#67—94no—
121#926D911yescharacteristic, complemented
122C2×C4⋊Dic3—31no—
123#893—92no—
124#892—92no—
125#892—92no—
126D3×C4⋊C4—92no—
127D3×C4⋊C4—92no—
128#887—92no—
129#67⋊C4—94no—
130C4×D12—92no—
131C4×Dic6—92no—
132C23×C32C22×D311yescharacteristic
133C22×C3⋊D3—32no—
134C2×C32×C4C4×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137#638—38no—
138C2×C3⋊Dic3—32no—
139C4⋊(C3⋊D3)—34no—
140C4×C3⋊D3—38no—
141#634—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—122no—
144C2×C3×Dic3C22×D312yes—
145C2×C3×Dic3C22×D312yes—
146#624—124no—
147C23×C9—21no—
148C23×C9C22×D311yescharacteristic
149C22×D9—32no—
150C22×D9—62no—
151C2×C4×C9—24no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—32no—
155C2×C4×C9—32no—
156C2×C4×C9C4×D314yescomplemented
157C2×C4×C9C22×D311yescharacteristic
158C2×C4×C9C22×D311yescharacteristic
159#611—38no—
160#611—68no—
161C2×Dic9—32no—
162C2×Dic9—42no—
163C2×Dic9—62no—
164C2×Dic9C22×D314yes—
165D36—34no—
166D36—64no—
167C4×D9—38no—
168C4×D9—68no—
169C4×D9—124no—
170Dic18—34no—
171Dic18—64no—
172#520—271noSylow
173C2×C3×C9—22no—
174C2×C3×C9C23×C411yescharacteristic
175C2×C3×C9H(5,𝔽2)11yescharacteristic
176C2×C3×C92−1+411yescharacteristic
177C3⋊D9—24no—
178C22×C3×C4—21no—
179C22×C3×C4—92no—
180C22×C3×C4C2×D911yescharacteristic
181C22×C3×C4D3211yescharacteristic
182C2×#67—92no—
183C2×#67—92no—
184C2×#67—182no—
185C22×Dic3—32no—
186#235—98no—
187#235—98no—
188#235—188no—
189C2×D12—91no—
190C2×D12—91no—
191C2×D12—181no—
192C2×C4×D3—92no—
193C2×C4×D3—92no—
194C2×C4×D3—94no—
195C2×C4×D3—94no—
196C2×C4×D3—182no—
197C2×Dic6—91no—
198C2×Dic6—91no—
199C2×Dic6—181no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4—92no—
204C3×C22⋊C4—32no—
205C3×C42—32no—
206#217C2×D912yes—
207(C2×D3)⋊C4—94no—
208(C2×D3)⋊C4—94no—
209C4⋊Dic3—32no—
210C4⋊Dic3—32no—
211C4⋊Dic3C2×D911yescharacteristic
212C4⋊Dic3C2×D911yescharacteristic
213Dic3⋊C4—94no—
214Dic3⋊C4—94no—
215C4×Dic3—94no—
216C4×Dic3—94no—
217C4×Dic3C2×D912yes—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32C2×C4×D312yes—
221C22×C32C23×D311yescharacteristic
222C2×C3⋊D3—34no—
223C2×C3⋊D3—62no—
224C2×C3⋊D3—62no—
225C32×C4C2×C4×D312yes—
226C32×C4C2×C4×D312yes—
227C3⋊Dic3—34no—
228C3×Dic3—22no—
229C3×Dic3—22no—
230C3×Dic3—64no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—21no—
234C22×C9—22no—
235C22×C9—41no—
236C22×C9—41no—
237C22×C9C2×C4×D312yes—
238C22×C9C23×D311yescharacteristic
239C2×D9—34no—
240C2×D9—64no—
241C2×D9—62no—
242C2×D9—62no—
243C2×D9—122no—
244C2×D9—122no—
245C4×C9—22no—
246C4×C9—22no—
247C4×C9—62no—
248C4×C9—62no—
249C4×C9C2×C4×D312yes—
250C4×C9C2×C4×D312yes—
251Dic9—24no—
252Dic9—34no—
253Dic9—64no—
254C2×#41—271no—
255C4×Q8—272no—
256C4×D4—272no—
257C4×D4—274no—
258#117—31no—
259#117—272no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—272no—
262C3×C9#52011yescharacteristic, complemented, Sylow
263C23×C3—21no—
264C23×C3C22×D911yescharacteristic
265C23×C3C2×D3211yescharacteristic
266C22×D3—92no—
267C22×D3—92no—
268C22×D3—182no—
269C2×C3×C4—24no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—32no—
273C2×C3×C4—32no—
274C2×C3×C4—94no—
275C2×C3×C4—362no—
276C2×C3×C4C4×D914yescomplemented
277C2×C3×C4C22×D911yescharacteristic
278C2×C3×C4C22×D911yescharacteristic
279C2×C3×C4#62414yes—
280C2×C3×C4C2×D3211yescharacteristic
281C2×C3×C4C2×D3211yescharacteristic
282#67—98no—
283#67—98no—
284#67—188no—
285C2×Dic3—34no—
286C2×Dic3—92no—
287C2×Dic3—92no—
288C2×Dic3—122no—
289C2×Dic3—182no—
290C2×Dic3C22×D912yes—
291C2×Dic3C22×D912yes—
292D12—94no—
293D12—94no—
294D12—184no—
295C4×D3—98no—
296C4×D3—98no—
297C4×D3—188no—
298C4×D3—364no—
299C4×D3—364no—
300Dic6—94no—
301Dic6—94no—
302Dic6—184no—
303C2×C32—22no—
304C2×C32C22×C4×D311yescharacteristic
305C2×C32#100911yescharacteristic
306C2×C32#101011yescharacteristic
307C3⋊D3—64no—
308C2×C9—22no—
309C2×C9—21no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9—42no—
313C2×C9C22×C4×D311yescharacteristic
314C2×C9#100411yescharacteristic
315C2×C9#100711yescharacteristic
316D9—64no—
317D9—124no—
318#41—278no—
319C2×Q8—271no—
320C2×D4—272no—
321C2×D4—271no—
322C22×C4—92no—
323C22×C4—274no—
324C22×C4—272no—
325C22×C4D3×D911yescharacteristic
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—92no—
329C4⋊C4—92no—
330C4⋊C4—274no—
331C22⋊C4—32no—
332C22⋊C4—274no—
333C42—32no—
334C42—274no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3C2×C4×D912yes—
344C22×C3C23×D911yescharacteristic
345C22×C3C2×#62412yes—
346C22×C3C22×D3211yescharacteristic
347C2×D3—94no—
348C2×D3—94no—
349C2×D3—184no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—182no—
353C2×D3—182no—
354C2×D3—362no—
355C2×D3—362no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—184no—
361C3×C4C2×C4×D912yes—
362C3×C4C2×C4×D912yes—
363C3×C4C2×#62412yes—
364C3×C4C2×#62412yes—
365Dic3—22no—
366Dic3—22no—
367Dic3—64no—
368Dic3—94no—
369Dic3—94no—
370Dic3—184no—
371C32#552311yescharacteristic
372C9—21no—
373C9#550411yescharacteristic, complemented
374C23—272no—
375C23C2×D3×D911yescharacteristic
376Q8—274no—
377D4—278no—
378D4—274no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—94no—
382C2×C4—272no—
383C2×C4—278no—
384C2×C4—362no—
385C2×C4—1084no—
386C2×C4(C3⋊D9)⋊C414yescomplemented
387C2×C4C2×D3×D911yescharacteristic
388C2×C4C2×D3×D911yescharacteristic
389C2×C3—22no—
390C2×C3—22no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—42no—
395C2×C3C22×C4×D911yescharacteristic
396C2×C3#6371511yescharacteristic
397C2×C3#6371611yescharacteristic
398C2×C3#6429711yescharacteristic
399C2×C3#6430711yescharacteristic
400C2×C3C22×#62411yescharacteristic
401D3—184no—
402D3—184no—
403D3—364no—
404C22—21no—
405C22—21no—
406C22—274no—
407C22—542no—
408C22—542no—
409C22C2×(C3⋊D9)⋊C412yes—
410C22C22×D3×D911yescharacteristic
411C4—62no—
412C4—62no—
413C4—184no—
414C4—274no—
415C4C2×(C3⋊D9)⋊C412yes—
416C4C2×(C3⋊D9)⋊C412yes—
417C3—21no—
418C3#1058962511yescharacteristic, complemented
419C3#1059499011yescharacteristic
420C2—22no—
421C2—544no—
422C2#1172246411yescharacteristic
423C2#1172247711yescharacteristic
424C2C22×(C3⋊D9)⋊C411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#88736C2
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9#520
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#926C2semidirect
3. C2×C3×C4⋊Dic9C2semidirect
4. #11721241C2semidirect
5. #11721000C2semidirect
6. #11720993C2semidirect
7. #11720922C2semidirect
8. #11720912C2semidirect
13. #88736C4semidirect
39. C2×C4⋊Dic9D3semidirect
47. C2×C3×C4×C9C2×C4semidirect
50. #6233C2×C4semidirect
52. C4⋊(C3⋊D9)C2×C4semidirect
53. C4×C3⋊D9C2×C4semidirect
54. #6226C2×C4semidirect
121. #926D9semidirect
156. C2×C4×C9C4×D3semidirect
262. C3×C9#520semidirect
276. C2×C3×C4C4×D9semidirect
373. C9#5504semidirect
386. C2×C4(C3⋊D9)⋊C4semidirect
418. C3#10589625semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C9×#9262
C2×C3×C4⋊Dic92
#117212412
#117210002
#117209932
#117209222
#117209122
#105949903
#105896253