Finite Group Databasebeta

← #13150517

Subgroups — #13150517

Order 1728.

974 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
4#11721290C214yesmaximal, complemented
5#11721037C212yesmaximal, complemented
6#11721036C212yesmaximal, complemented
7#11720960C212yesmaximal, complemented
8#11720944C212yesmaximal, complemented
9#10595044—31nomaximal
10#10589935—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2212yes—
13C2×C9×#67C2212yes—
14C2×C3×#611C2212yes—
15C9×#235—24no—
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2212yes—
18C3×#3697—24no—
19C2×C3×D36C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C2×C9×Dic6C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×#6194C2214yes—
24C4⋊(C3⋊Dic9)C2212yes—
25C4⋊(C3⋊Dic9)C2212yes—
26(C2×D3)⋊Dic9C2214yes—
27(C2×D9)⋊Dic3C2214yes—
28#88442C2214yes—
29C2×C3×#235—31no—
30C2×C3×#235D311yescharacteristic
31C2×C4⋊(C3⋊Dic3)—31no—
32#63980—34no—
33#63908—32no—
34#63908—32no—
35#63856—32no—
36#63856—32no—
37C2×C9×#41—31no—
38C2×#3697D311yescharacteristic, complemented
39#63508—32no—
40#63501—34no—
41#63500—32no—
42#63492—32no—
43C2×C4⋊Dic9—31no—
44C2×C4⋊Dic9—61no—
45#63486—32no—
46C23×C3×C9C2311yescharacteristic
47C22×D3×C9C2312yes—
48C22×C3×D9C2312yes—
49C2×C3×C4×C9D414yescomplemented
50C2×C3×C4×C9C2311yescharacteristic
51C2×C3×C4×C9C2311yescharacteristic
52C2×C3⋊Dic9—42no—
53C2×C3⋊Dic9C2314yes—
54C9×#67—24no—
55C3×#611—24no—
56C2×C9×Dic3C2312yes—
57C2×C3×Dic9C2312yes—
58C9×D12—22no—
59C4×D3×C9—24no—
60C3×D36—22no—
61C3×C4×D9—24no—
62C9×Dic6—22no—
63C3×Dic18—22no—
64#6194—44no—
65#5833—91no—
66#5833—91no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3—32no—
69C2×C3×#67—32no—
70C2×C3×#67C2×D312yes—
71C3×#235—24no—
72C3×#235—64no—
73C2×C3×D12—31no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3C2×D312yes—
77C2×C3×Dic6—31no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×#625—34no—
80C4⋊(C3⋊Dic3)—32no—
81C4⋊(C3⋊Dic3)—32no—
82#3724—34no—
83(C2×D3)⋊Dic3—34no—
84(C2×D3)⋊Dic3—34no—
85C9×#41—64no—
86C2×Q8×C9—31no—
87C2×D4×C9—32no—
88C2×D4×C9—31no—
89C22×C4×C9—21no—
90C22×C4×C9—32no—
91C22×C4×C9C2×D311yescharacteristic
92C2×#611—34no—
93C2×#611C2×D312yes—
94C22×Dic9—32no—
95C22×Dic9—62no—
96#3697—24no—
97C2×D36C2×D311yescharacteristic
98C2×C4×D9C2×D312yes—
99C2×Dic18C2×D311yescharacteristic
100#3676—34no—
101(C2×D9)⋊C4—34no—
102C4⋊Dic9—32no—
103C4⋊Dic9—32no—
104C4⋊Dic9—62no—
105C4⋊Dic9—62no—
106Dic9⋊C4—34no—
107C22×C3×C9—21no—
108C22×C3×C9—21no—
109C22×C3×C9C2×D412yes—
110C22×C3×C9C2411yescharacteristic
111C2×D3×C9—22no—
112C2×D3×C9—22no—
113C2×D3×C9—22no—
114C2×C3×D9—22no—
115C2×C3×D9—22no—
116C2×C3×D9—22no—
117C3×C4×C9C2×D412yes—
118C3×C4×C9C2×D412yes—
119C3⋊Dic9—24no—
120C9×Dic3—22no—
121C3×Dic9—22no—
122C2×C3×#41—31no—
123C2×C3×#41—91no—
124C2×#235—31no—
125C2×#235D911yescharacteristic, complemented
126#946—92no—
127#946—92no—
128#939—94no—
129#939—94no—
130#938—92no—
131#938—92no—
132#930—92no—
133#930—92no—
134C2×C4⋊Dic3—91no—
135C2×C4⋊Dic3—91no—
136C2×C4⋊Dic3—181no—
137#924—92no—
138#924—92no—
139C23×C32C22×D311yescharacteristic
140C22×C3×D3—32no—
141C22×C3×D3C22×D312yes—
142C2×C32×C4#6714yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—122no—
147C3×#67—24no—
148C3×#67—64no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3C22×D312yes—
151C3×D12—22no—
152C3×D12—62no—
153C3×C4×D3—24no—
154C3×C4×D3—64no—
155C3×Dic6—22no—
156C3×Dic6—62no—
157#625—124no—
158C23×C9—21no—
159C23×C9—32no—
160C23×C9C22×D311yescharacteristic
161C22×D9C22×D312yes—
162Q8×C9—62no—
163D4×C9—64no—
164D4×C9—62no—
165C2×C4×C9—24no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—32no—
169C2×C4×C9—64no—
170C2×C4×C9#6714yescomplemented
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173#611—24no—
174#611—124no—
175C2×Dic9—34no—
176C2×Dic9—64no—
177C2×Dic9—122no—
178C2×Dic9—124no—
179C2×Dic9C22×D312yes—
180D36—22no—
181C4×D9—24no—
182Dic18—22no—
183#537—271noSylow
184C2×C3×C9—22no—
185C2×C3×C9C22×D411yescharacteristic
186C2×C3×C9H(5,𝔽2)11yescharacteristic
187C2×C3×C92−1+411yescharacteristic
188D3×C9—42no—
189C3×D9—42no—
190C3×#41—64no—
191C3×#41—184no—
192C2×C3×Q8—31no—
193C2×C3×Q8—91no—
194C2×C3×D4—32no—
195C2×C3×D4—31no—
196C2×C3×D4—92no—
197C2×C3×D4—91no—
198C22×C3×C4—21no—
199C22×C3×C4—32no—
200C22×C3×C4—92no—
201C22×C3×C4C2×D911yescharacteristic
202C22×C3×C4D3211yescharacteristic
203C2×#67—32no—
204C2×#67—94no—
205C2×#67—94no—
206C2×#67C2×D912yes—
207C22×Dic3—92no—
208C22×Dic3—92no—
209C22×Dic3—182no—
210#235—24no—
211#235—64no—
212C2×D12—31no—
213C2×D12C2×D911yescharacteristic
214C2×C4×D3—32no—
215C2×C4×D3C2×D912yes—
216C2×Dic6—31no—
217C2×Dic6C2×D911yescharacteristic
218#217—94no—
219#217—94no—
220(C2×D3)⋊C4—94no—
221(C2×D3)⋊C4—94no—
222C4⋊Dic3—92no—
223C4⋊Dic3—92no—
224C4⋊Dic3—92no—
225C4⋊Dic3—92no—
226C4⋊Dic3—182no—
227C4⋊Dic3—182no—
228Dic3⋊C4—94no—
229Dic3⋊C4—94no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32C2×#6712yes—
233C22×C32C23×D311yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—22no—
236C2×C3×D3—22no—
237C2×C3×D3—62no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C32×C4C2×#6712yes—
241C32×C4C2×#6712yes—
242C3⋊Dic3—64no—
243C3×Dic3—22no—
244C3×Dic3—62no—
245C22×C9—22no—
246C22×C9—22no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—22no—
251C22×C9—62no—
252C22×C9—62no—
253C22×C9—62no—
254C22×C9C2×#6712yes—
255C22×C9C23×D311yescharacteristic
256C2×D9—22no—
257C2×D9—22no—
258C2×D9—22no—
259C4×C9—22no—
260C4×C9—22no—
261C4×C9—62no—
262C4×C9C2×#6712yes—
263C4×C9C2×#6712yes—
264Dic9—22no—
265Dic9—64no—
266Dic9—124no—
267C2×#41—31no—
268C2×#41—91no—
269C22⋊Q8—272no—
270C22⋊Q8—272no—
271#121—272no—
272#121—272no—
273#121—274no—
274C2×C4⋊C4—271no—
275C3×C9#53711yescharacteristic, complemented, Sylow
276C23×C3—21no—
277C23×C3—32no—
278C23×C3—92no—
279C23×C3C22×D911yescharacteristic
280C23×C3C2×D3211yescharacteristic
281C22×D3—32no—
282C22×D3C22×D912yes—
283C3×Q8—62no—
284C3×Q8—182no—
285C3×D4—64no—
286C3×D4—62no—
287C3×D4—184no—
288C3×D4—182no—
289C2×C3×C4—24no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—32no—
293C2×C3×C4—64no—
294C2×C3×C4—92no—
295C2×C3×C4—184no—
296C2×C3×C4#61114yescomplemented
297C2×C3×C4C22×D911yescharacteristic
298C2×C3×C4C22×D911yescharacteristic
299C2×C3×C4#62514yes—
300C2×C3×C4C2×D3211yescharacteristic
301C2×C3×C4C2×D3211yescharacteristic
302#67—24no—
303#67—64no—
304#67—364no—
305#67—364no—
306C2×Dic3—32no—
307C2×Dic3—94no—
308C2×Dic3—94no—
309C2×Dic3—184no—
310C2×Dic3—362no—
311C2×Dic3—362no—
312C2×Dic3—364no—
313C2×Dic3C22×D912yes—
314D12—22no—
315D12—62no—
316C4×D3—24no—
317C4×D3—64no—
318Dic6—22no—
319Dic6—62no—
320C2×C32—22no—
321C2×C32#100911yescharacteristic
322C2×C32#101011yescharacteristic
323C2×C32C22×#6711yescharacteristic
324C3×D3—42no—
325C3×D3—122no—
326C2×C9—24no—
327C2×C9—22no—
328C2×C9—21no—
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—122no—
332C2×C9#100911yescharacteristic
333C2×C9#101011yescharacteristic
334C2×C9C22×#6711yescharacteristic
335D9—42no—
336#41—64no—
337#41—184no—
338C2×Q8—31no—
339C2×Q8—91no—
340C2×D4—32no—
341C2×D4—31no—
342C2×D4—92no—
343C2×D4—91no—
344C2×D4—274no—
345C22×C4—32no—
346C22×C4—92no—
347C22×C4—272no—
348C22×C4D3×D911yescharacteristic
349C4⋊C4—274no—
350C4⋊C4—272no—
351C4⋊C4—272no—
352C22⋊C4—274no—
353C22⋊C4—274no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—62no—
365C22×C3—182no—
366C22×C3—182no—
367C22×C3—182no—
368C22×C3C2×#61112yes—
369C22×C3C23×D911yescharacteristic
370C22×C3C2×#62512yes—
371C22×C3C22×D3211yescharacteristic
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—62no—
376C2×D3—62no—
377C2×D3—62no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—62no—
381C3×C4—182no—
382C3×C4C2×#61112yes—
383C3×C4C2×#61112yes—
384C3×C4C2×#62512yes—
385C3×C4C2×#62512yes—
386Dic3—22no—
387Dic3—62no—
388Dic3—184no—
389Dic3—184no—
390Dic3—364no—
391C32#583311yescharacteristic
392C9—21no—
393C9#583311yescharacteristic, complemented
394C23—32no—
395C23—92no—
396C23C2×D3×D911yescharacteristic
397Q8—62no—
398Q8—182no—
399D4—64no—
400D4—62no—
401D4—184no—
402D4—182no—
403D4—1084no—
404C2×C4—32no—
405C2×C4—64no—
406C2×C4—92no—
407C2×C4—184no—
408C2×C4—274no—
409C2×C4—1082no—
410C2×C4#619414yescomplemented
411C2×C4C2×D3×D911yescharacteristic
412C2×C4C2×D3×D911yescharacteristic
413C2×C3—24no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—122no—
420C2×C3—362no—
421C2×C3#6371511yescharacteristic
422C2×C3#6371611yescharacteristic
423C2×C3C22×#61111yescharacteristic
424C2×C3#6429811yescharacteristic
425C2×C3#6430811yescharacteristic
426C2×C3C22×#62511yescharacteristic
427D3—42no—
428D3—122no—
429C22—21no—
430C22—21no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—182no—
435C22—182no—
436C22—182no—
437C22C2×#619412yes—
438C22C22×D3×D911yescharacteristic
439C4—62no—
440C4—182no—
441C4—544no—
442C4C2×#619412yes—
443C4C2×#619412yes—
444C3—21no—
445C3#1058993511yescharacteristic, complemented
446C3#1059504411yescharacteristic
447C2—22no—
448C2—122no—
449C2—362no—
450C2#1172246511yescharacteristic
451C2#1172247811yescharacteristic
452C2C22×#619411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#537
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#235C2semidirect
2. C2×C3×#3697C2semidirect
3. C2×C4⋊(C3⋊Dic9)C2semidirect
4. #11721290C2semidirect
5. #11721037C2semidirect
6. #11721036C2semidirect
7. #11720960C2semidirect
8. #11720944C2semidirect
38. C2×#3697D3semidirect
49. C2×C3×C4×C9D4semidirect
125. C2×#235D9semidirect
170. C2×C4×C9#67semidirect
275. C3×C9#537semidirect
296. C2×C3×C4#611semidirect
393. C9#5833semidirect
410. C2×C4#6194semidirect
445. C3#10589935semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×C3×#36972
C2×C4⋊(C3⋊Dic9)2
#117212902
#117210372
#117210362
#117209602
#117209442
#105950443
#105899353