Finite Group Databasebeta

← #13151142

Subgroups — #13151142

Order 1728.

886 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×C9×#237C211yesmaximal, characteristic
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3#11721836C211yesmaximal, characteristic, complemented
4C3×#63498C211yesmaximal, characteristic, complemented
5#11721280C214yesmaximal, complemented
6#11721249C214yesmaximal, complemented
7#11720959C211yesmaximal, characteristic, complemented
8#11720930C211yesmaximal, characteristic, complemented
9#11720924C211yesmaximal, characteristic, complemented
10#10595147—31nomaximal
11#10589927—31nomaximal
12C2×C3×D4×C9C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C9×#67C2212yes—
15C22×C9×Dic3C2212yes—
16C9×#237—24no—
17C2×C4×D3×C9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×D3×Dic9C2212yes—
21C2×#6189C2212yes—
22D3×Dic18—24no—
23#88551C2214yes—
24C4×C3⋊Dic9C2211yescharacteristic
25C3×#3676C2214yes—
26C3×C4×Dic9C2211yescharacteristic
27(C2×D3)⋊Dic9C2212yes—
28Dic3⋊Dic9C2212yes—
29#88442C2214yes—
30Dic9⋊Dic3C2214yes—
31C2×C3×#237D311yescharacteristic
32C2×D3×Dic6—31no—
33#64145—31no—
34C3×#936—31no—
35#63969—34no—
36#63964—34no—
37#63873—31no—
38#63866—31no—
39#63855—31no—
40C2×C9×#41—31no—
41C22×Dic18—31no—
42#63506—31no—
43#63506—31no—
44#63498—31no—
45#63498—61no—
46#63498D311yescharacteristic, complemented
47#63497—34no—
48#63488—31no—
49#63486—34no—
50C23×C3×C9C2312yes—
51C22×D3×C9C2311yescharacteristic
52C3×D4×C9—22no—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9C2312yes—
55C2×C3⋊Dic9C2312yes—
56#6226—22no—
57C9×#67—24no—
58C2×C9×Dic3—24no—
59C2×C9×Dic3C2312yes—
60C2×C9×Dic3C2311yescharacteristic
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2312yes—
63C4×D3×C9D414yes—
64C9×Dic6—22no—
65C3×Dic18—22no—
66D3×Dic9—24no—
67#6189—24no—
68#5825—91no—
69#5675—91no—
70C2×C32×D4C2×D311yescharacteristic
71C2×#634—31no—
72C2×C3×#67C2×D312yes—
73C22×C3×Dic3C2×D312yes—
74C3×#237—24no—
75C2×C3×C4×D3C2×D311yescharacteristic
76C2×C3×Dic6—31no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×#627—32no—
79C2×D3×Dic3—32no—
80D3×Dic6—64no—
81#3757—34no—
82C4×C3⋊Dic3—31no—
83C3×#217—34no—
84C3×C4×Dic3—31no—
85#3724—34no—
86Dic3⋊Dic3—32no—
87Dic3⋊Dic3—34no—
88(C2×D3)⋊Dic3—32no—
89C9×#41—64no—
90C2×Q8×C9—31no—
91C2×D4×C9—21no—
92C2×D4×C9—32no—
93C2×D4×C9C2×D311yescharacteristic
94C22×C4×C9—31no—
95C22×C4×C9—32no—
96C22×Dic9—32no—
97C2×Dic18—31no—
98C2×Dic18—32no—
99C2×Dic18—64no—
100C2×Dic18—61no—
101C2×Dic18C2×D311yescharacteristic
102#3676—34no—
103#3676—32no—
104#3676—64no—
105#3676C2×D314yes—
106C4⋊Dic9—32no—
107Dic9⋊C4—34no—
108Dic9⋊C4—34no—
109C4×Dic9—31no—
110C4×Dic9—61no—
111C4×Dic9C2×D311yescharacteristic
112C22×C3×C9—22no—
113C22×C3×C9—24no—
114C22×C3×C9C2411yescharacteristic
115C2×D3×C9—22no—
116C2×D3×C9C2×D412yes—
117C3×C4×C9C2×D412yes—
118C3⋊Dic9—22no—
119C3⋊Dic9—22no—
120C9×Dic3—22no—
121C9×Dic3C2×D412yes—
122C3×Dic9—22no—
123C3×Dic9—22no—
124C2×C3×#41—31no—
125C2×D3×Q8—91no—
126C2×#237D911yescharacteristic
127C22×Dic6—91no—
128C3×#124—91no—
129#944—91no—
130#944—91no—
131#944—91no—
132#936—31no—
133#936—91no—
134#936—91no—
135#936—181no—
136#935—94no—
137#926—91no—
138#924—94no—
139#883—94no—
140#878—94no—
141#872—91no—
142#869—91no—
143C23×C32C22×D312yes—
144C22×C3×D3C22×D311yescharacteristic
145C32×D4—22no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—32no—
149#634—62no—
150C3×#67—24no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3C22×D312yes—
155C2×C3×Dic3C22×D311yescharacteristic
156C3×C4×D3#6714yes—
157C3×Dic6—22no—
158C3×Dic6—62no—
159#627—64no—
160D3×Dic3—64no—
161C23×C9—22no—
162C23×C9—31no—
163C23×C9C22×D312yes—
164Q8×C9—62no—
165D4×C9—22no—
166D4×C9—42no—
167D4×C9—64no—
168C2×C4×C9—21no—
169C2×C4×C9—31no—
170C2×C4×C9—34no—
171C2×C4×C9—32no—
172C2×C4×C9—64no—
173C2×C4×C9C22×D311yescharacteristic
174C2×Dic9—32no—
175C2×Dic9—32no—
176C2×Dic9—62no—
177C2×Dic9—62no—
178C2×Dic9—64no—
179C2×Dic9C22×D312yes—
180C2×Dic9C22×D312yes—
181Dic18—22no—
182Dic18—64no—
183Dic18—62no—
184Dic18—122no—
185#536—271noSylow
186C2×C3×C9—42no—
187C2×C3×C9C22×D411yescharacteristic
188C2×C3×C92−1+412yes—
189D3×C9—22no—
190C3×#41—64no—
191C2×C3×Q8—31no—
192C2×C3×Q8—91no—
193C2×C3×D4—21no—
194C2×C3×D4—32no—
195C2×C3×D4C2×D911yescharacteristic
196C2×C3×D4D3211yescharacteristic
197C22×C3×C4—31no—
198C22×C3×C4—32no—
199C2×#67C2×D912yes—
200C22×Dic3—92no—
201C22×Dic3C2×D912yes—
202D3×Q8—184no—
203#237—24no—
204C2×C4×D3—92no—
205C2×C4×D3C2×D911yescharacteristic
206C2×Dic6—31no—
207C2×Dic6—92no—
208C2×Dic6—91no—
209C2×Dic6—91no—
210C2×Dic6—92no—
211C2×Dic6—181no—
212C2×Dic6—184no—
213C2×Dic6C2×D911yescharacteristic
214C3×C22⋊C4—94no—
215C3×C42—91no—
216#217—34no—
217#217—92no—
218#217—94no—
219#217—94no—
220#217—184no—
221(C2×D3)⋊C4—92no—
222C4⋊Dic3—92no—
223C4⋊Dic3—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—94no—
226Dic3⋊C4—92no—
227Dic3⋊C4—94no—
228C4×Dic3—31no—
229C4×Dic3—91no—
230C4×Dic3—91no—
231C4×Dic3—181no—
232C22×C32—24no—
233C22×C32—22no—
234C22×C32C23×D311yescharacteristic
235C2×C3×D3—22no—
236C2×C3×D3C2×#6712yes—
237C32×C4C2×#6712yes—
238C3⋊Dic3—62no—
239C3⋊Dic3—62no—
240C3×Dic3—22no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C3×Dic3C2×#6712yes—
244C22×C9—22no—
245C22×C9—24no—
246C22×C9—21no—
247C22×C9—32no—
248C22×C9—42no—
249C22×C9—44no—
250C22×C9—62no—
251C22×C9C23×D311yescharacteristic
252C4×C9—22no—
253C4×C9—32no—
254C4×C9—62no—
255C4×C9D3×D412yes—
256Dic9—22no—
257Dic9—22no—
258Dic9—62no—
259Dic9—62no—
260Dic9—122no—
261Dic9—122no—
262C2×#41—31no—
263C22×Q8—271no—
264C4⋊Q8—271no—
265C4⋊Q8—271no—
266#124—91no—
267#124—271no—
268#123—274no—
269C22⋊Q8—274no—
270#117—271no—
271C3×C9#53611yescharacteristic, complemented, Sylow
272C23×C3—22no—
273C23×C3—31no—
274C23×C3C22×D912yes—
275C23×C3C2×D3212yes—
276C22×D3C22×D911yescharacteristic
277C3×Q8—62no—
278C3×Q8—182no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—42no—
282C3×D4—64no—
283C2×C3×C4—21no—
284C2×C3×C4—31no—
285C2×C3×C4—34no—
286C2×C3×C4—32no—
287C2×C3×C4—64no—
288C2×C3×C4—92no—
289C2×C3×C4—92no—
290C2×C3×C4C22×D911yescharacteristic
291C2×C3×C4C2×D3211yescharacteristic
292#67—24no—
293C2×Dic3—24no—
294C2×Dic3—32no—
295C2×Dic3—32no—
296C2×Dic3—92no—
297C2×Dic3—92no—
298C2×Dic3—92no—
299C2×Dic3—92no—
300C2×Dic3—182no—
301C2×Dic3—182no—
302C2×Dic3—184no—
303C2×Dic3C22×D912yes—
304C2×Dic3C22×D911yescharacteristic
305C4×D3—184no—
306C4×D3#61114yes—
307Dic6—22no—
308Dic6—62no—
309Dic6—184no—
310Dic6—182no—
311Dic6—182no—
312Dic6—184no—
313Dic6—362no—
314C2×C32—42no—
315C2×C32#101012yes—
316C2×C32C22×#6711yescharacteristic
317C3×D3—22no—
318C2×C9—22no—
319C2×C9—21no—
320C2×C9—42no—
321C2×C9—62no—
322C2×C9—82no—
323C2×C9C2×D3×D411yescharacteristic
324C2×C9#101012yes—
325#41—64no—
326C2×Q8—31no—
327C2×Q8—91no—
328C2×Q8—272no—
329C2×Q8—271no—
330C2×Q8—544no—
331C2×D4—32no—
332C2×D4D3×D911yescharacteristic
333C22×C4—31no—
334C22×C4—32no—
335C22×C4—272no—
336C4⋊C4—272no—
337C4⋊C4—274no—
338C4⋊C4—274no—
339C22⋊C4—94no—
340C22⋊C4—272no—
341C22⋊C4—274no—
342C42—91no—
343C42—271no—
344C22×C3—24no—
345C22×C3—22no—
346C22×C3—24no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—32no—
350C22×C3—42no—
351C22×C3—44no—
352C22×C3—62no—
353C22×C3C23×D911yescharacteristic
354C22×C3C22×D3211yescharacteristic
355C2×D3—22no—
356C2×D3C2×#61112yes—
357C3×C4—22no—
358C3×C4—32no—
359C3×C4—62no—
360C3×C4—182no—
361C3×C4—182no—
362C3×C4C2×#61112yes—
363C3×C4D3×#6712yes—
364Dic3—22no—
365Dic3—62no—
366Dic3—62no—
367Dic3—182no—
368Dic3—182no—
369Dic3—182no—
370Dic3—182no—
371Dic3—362no—
372Dic3—362no—
373Dic3C2×#61112yes—
374C32#582511yescharacteristic
375C9—21no—
376C9#567511yescharacteristic, complemented
377C23—31no—
378C23C2×D3×D912yes—
379Q8—62no—
380Q8—182no—
381Q8—544no—
382Q8—542no—
383D4—22no—
384D4—64no—
385C2×C4—31no—
386C2×C4—34no—
387C2×C4—32no—
388C2×C4—64no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—272no—
392C2×C4—272no—
393C2×C4—544no—
394C2×C4C2×D3×D911yescharacteristic
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—42no—
398C2×C3—42no—
399C2×C3—62no—
400C2×C3—82no—
401C2×C3#6371612yes—
402C2×C3C22×#61111yescharacteristic
403C2×C3#6430912yes—
404C2×C3C2×D3×#6711yescharacteristic
405D3—22no—
406C22—22no—
407C22—24no—
408C22—32no—
409C22—62no—
410C22C22×D3×D911yescharacteristic
411C4—32no—
412C4—62no—
413C4—182no—
414C4—182no—
415C4—542no—
416C4—542no—
417C4D3×#61112yes—
418C3—21no—
419C3#1058992711yescharacteristic, complemented
420C3#1059514711yescharacteristic
421C2—42no—
422C2—62no—
423C2#1172249412yes—
424C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#237D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×D3×Dic18C2semidirect
3. #11721836C2semidirect
4. C3×#63498C2semidirect
5. #11721280C2semidirect
6. #11721249C2semidirect
7. #11720959C2semidirect
8. #11720930C2semidirect
9. #11720924C2semidirect
46. #63498D3semidirect
271. C3×C9#536semidirect
376. C9#5675semidirect
419. C3#10589927semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C2×D3×Dic182
#117218362
C3×#634982
#117212802
#117212492
#117209592
#117209302
#117209242
#105951473
#105899273