Finite Group Databasebeta

← C3×#10589826 #13153294

Subgroups — C3×#10589826 #13153294

Order 1728.

878 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3701C211yesmaximal, characteristic
2C2×C3×Q8×D9C211yesmaximal, characteristic, complemented
3C3×C9×C4⋊Q8C211yesmaximal, characteristic, complemented
4C3×#63509C211yesmaximal, characteristic, complemented
5C3×#63506C211yesmaximal, characteristic, complemented
6C3×#63458C214yesmaximal, complemented
7C3×#63456C214yesmaximal, complemented
8C3×#63437C211yesmaximal, characteristic, complemented
9C3×#63434C211yesmaximal, characteristic, complemented
10C3×#5724—31nomaximal
11#10589826C311yesmaximal, characteristic, direct product factor
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×Q8×C9C2211yescharacteristic
14C3×#3701—24no—
15C3×Q8×D9—24no—
16C2×C3×D36C2211yescharacteristic
17C2×C3×D36C2212yes—
18C2×C3×C4×D9C2212yes—
19C2×C3×C4×D9C2212yes—
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic
22C2×C3×Dic18C2212yes—
23C3×C9×C4⋊C4C2214yes—
24C3×C42×C9C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2214yes—
27C3×(C2×D9)⋊C4C2214yes—
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2214yes—
30C3×C4×Dic9C2211yescharacteristic
31C2×C3×#239—31no—
32C2×C3×D3×Q8—31no—
33C32×C4⋊Q8D311yescharacteristic
34C3×#947—31no—
35C3×#944—31no—
36C3×#896—34no—
37C3×#894—34no—
38C3×#875—31no—
39C3×#872—31no—
40C2×#3701C2×C311yescharacteristic
41C2×Q8×D9C2×C311yescharacteristic, complemented
42C9×C4⋊Q8—21no—
43C9×C4⋊Q8C2×C311yescharacteristic, complemented
44#63509C2×C311yescharacteristic, complemented
45#63506C2×C311yescharacteristic, complemented
46#63458C2×C314yescomplemented
47#63456C2×C314yescomplemented
48#63437C2×C311yescharacteristic, complemented
49#63434C2×C311yescharacteristic, complemented
50C22×C3×D9C2312yes—
51C22×C3×D9C2311yescharacteristic
52C3×Q8×C9—22no—
53C3×Q8×C9—22no—
54C2×C3×C4×C9C2312yes—
55C2×C3×C4×C9C2312yes—
56C2×C3×C4×C9C2312yes—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×Dic9C2312yes—
59C2×C3×Dic9C2312yes—
60C2×C3×Dic9C2311yescharacteristic
61C3×D36—24no—
62C3×D36—22no—
63C3×C4×D9—24no—
64C3×C4×D9—24no—
65C3×C4×D9D414yes—
66C3×Dic18—22no—
67C3×Dic18—24no—
68C3×#536—91no—
69#5724—31no—
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×Q8C2×D311yescharacteristic
72C3×#239—64no—
73C3×D3×Q8—64no—
74C2×C3×D12—32no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—31no—
81C32×C4⋊C4C2×D314yes—
82C32×C42C2×D311yescharacteristic
83C3×(C2×D3)⋊C4—32no—
84C3×(C2×D3)⋊C4—34no—
85C3×(C2×D3)⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—32no—
88C3×C4×Dic3—31no—
89C2×Q8×C9—21no—
90C2×Q8×C9—21no—
91C2×Q8×C9C22×C311yescharacteristic
92C2×Q8×C9C22×C311yescharacteristic
93#3701—24no—
94Q8×D9—24no—
95C2×D36C22×C311yescharacteristic
96C2×D36C22×C312yes—
97C2×C4×D9C22×C312yes—
98C2×C4×D9C22×C312yes—
99C2×C4×D9C22×C311yescharacteristic
100C2×Dic18C22×C311yescharacteristic
101C2×Dic18C22×C312yes—
102C9×C4⋊C4—24no—
103C9×C4⋊C4C22×C314yes—
104C42×C9—21no—
105C42×C9C22×C311yescharacteristic
106(C2×D9)⋊C4C22×C314yes—
107(C2×D9)⋊C4C22×C312yes—
108(C2×D9)⋊C4C22×C314yes—
109Dic9⋊C4C22×C314yes—
110Dic9⋊C4C22×C312yes—
111C4×Dic9C22×C311yescharacteristic
112C22×C3×C9C2411yescharacteristic
113C2×C3×D9—22no—
114C2×C3×D9—22no—
115C2×C3×D9—24no—
116C2×C3×D9C2×D412yes—
117C3×C4×C9—22no—
118C3×C4×C9—22no—
119C3×C4×C9—22no—
120C3×C4×C9C2×D412yes—
121C3×Dic9—22no—
122C3×Dic9—22no—
123C3×Dic9C2×D412yes—
124C2×C3×#41—91no—
125C22×C3×Q8—91no—
126C2×#239—31no—
127C2×D3×Q8—31no—
128C3×C4⋊Q8—21no—
129C3×C4⋊Q8—91no—
130C3×C4⋊Q8D911yescharacteristic, complemented
131C3×C4⋊Q8C3×D311yescharacteristic
132C3×#124—91no—
133C3×#124—91no—
134C3×#123—94no—
135C3×C22⋊Q8—94no—
136C3×#117—91no—
137#947—31no—
138#944—31no—
139#896—34no—
140#894—34no—
141#875—31no—
142#872—31no—
143C22×C3×D3—31no—
144C22×C3×D3—32no—
145C32×Q8—22no—
146C32×Q8—22no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3×Dic3—31no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C3×D12—64no—
155C3×D12—62no—
156C3×C4×D3—34no—
157C3×C4×D3—64no—
158C3×C4×D3—64no—
159C3×Dic6—64no—
160C3×Dic6—62no—
161C22×D9C23×C312yes—
162C22×D9C23×C311yescharacteristic
163Q8×C9—22no—
164Q8×C9—22no—
165Q8×C9—42no—
166Q8×C9—42no—
167C2×C4×C9—22no—
168C2×C4×C9—21no—
169C2×C4×C9—22no—
170C2×C4×C9—22no—
171C2×C4×C9C23×C312yes—
172C2×C4×C9C23×C312yes—
173C2×C4×C9C23×C312yes—
174C2×C4×C9C23×C311yescharacteristic
175C2×Dic9C23×C312yes—
176C2×Dic9C23×C312yes—
177C2×Dic9C23×C311yescharacteristic
178D36—24no—
179D36—22no—
180C4×D9—24no—
181C4×D9—24no—
182C4×D9C3×D414yes—
183Dic18—22no—
184Dic18—24no—
185#536—91noSylow
186C2×C3×C9C22×D411yescharacteristic
187C2×C3×C92−1+412yes—
188C3×D9—22no—
189C3×D9—42no—
190C3×#41—184no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—92no—
194C2×C3×Q8—91no—
195C2×C3×Q8—184no—
196C2×C3×Q8C2×D911yescharacteristic
197C2×C3×Q8C2×D911yescharacteristic
198C2×C3×Q8C2×C3×D311yescharacteristic
199C2×C3×Q8C2×C3×D311yescharacteristic
200C2×C3×D4—92no—
201C2×C3×D4—91no—
202C22×C3×C4—91no—
203C22×C3×C4—92no—
204C22×C3×C4—92no—
205#239—64no—
206D3×Q8—64no—
207C2×D12—31no—
208C2×D12—32no—
209C2×C4×D3—31no—
210C2×C4×D3—32no—
211C2×C4×D3—32no—
212C2×Dic6—31no—
213C2×Dic6—32no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—94no—
217C3×C4⋊C4C2×D914yes—
218C3×C4⋊C4C2×C3×D314yes—
219C3×C22⋊C4—92no—
220C3×C22⋊C4—94no—
221C3×C22⋊C4—94no—
222C3×C42—21no—
223C3×C42—91no—
224C3×C42C2×D911yescharacteristic
225C3×C42C2×C3×D311yescharacteristic
226(C2×D3)⋊C4—32no—
227(C2×D3)⋊C4—34no—
228(C2×D3)⋊C4—34no—
229Dic3⋊C4—32no—
230Dic3⋊C4—34no—
231C4×Dic3—31no—
232C22×C32C23×D311yescharacteristic
233C2×C3×D3—32no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—64no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4D3×D412yes—
241C3×Dic3—32no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C22×C9—21no—
245C22×C9C24×C311yescharacteristic
246C2×D9—22no—
247C2×D9—22no—
248C2×D9—24no—
249C2×D9C2×C3×D412yes—
250C4×C9—24no—
251C4×C9—22no—
252C4×C9—22no—
253C4×C9—22no—
254C4×C9—22no—
255C4×C9—42no—
256C4×C9—42no—
257C4×C9C2×C3×D412yes—
258Dic9—22no—
259Dic9—22no—
260Dic9C2×C3×D412yes—
261C2×#41—91no—
262C22×Q8—91no—
263C4⋊Q8—91no—
264C4⋊Q8C3×D911yescharacteristic, complemented
265#124—91no—
266#124—91no—
267#123—94no—
268C22⋊Q8—94no—
269#117—91no—
270C3×C9#53611yescharacteristic, complemented, Sylow
271C23×C3—91no—
272C23×C3—92no—
273C22×D3—31no—
274C22×D3—32no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—42no—
280C3×Q8—42no—
281C3×Q8—184no—
282C3×Q8—182no—
283C3×D4—184no—
284C3×D4—182no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—91no—
290C2×C3×C4—94no—
291C2×C3×C4—92no—
292C2×C3×C4—92no—
293C2×C3×C4—184no—
294C2×C3×C4—184no—
295C2×C3×C4C22×D912yes—
296C2×C3×C4C22×D912yes—
297C2×C3×C4C22×D912yes—
298C2×C3×C4C22×D911yescharacteristic
299C2×C3×C4C22×C3×D312yes—
300C2×C3×C4C22×C3×D312yes—
301C2×C3×C4C22×C3×D312yes—
302C2×C3×C4C22×C3×D311yescharacteristic
303C2×Dic3—31no—
304C2×Dic3—32no—
305C2×Dic3—32no—
306D12—64no—
307D12—62no—
308C4×D3—34no—
309C4×D3—64no—
310C4×D3—64no—
311Dic6—64no—
312Dic6—62no—
313C2×C32C2×D3×D411yescharacteristic
314C2×C32#100712yes—
315C3×D3—62no—
316C3×D3—122no—
317C2×C9—21no—
318C2×C9—22no—
319C2×C9C22×C3×D411yescharacteristic
320C2×C9C3×2−1+412yes—
321D9—22no—
322D9—42no—
323#41—184no—
324C2×Q8—92no—
325C2×Q8—91no—
326C2×Q8—184no—
327C2×Q8C2×C3×D911yescharacteristic
328C2×Q8C2×C3×D911yescharacteristic
329C2×D4—92no—
330C2×D4—91no—
331C22×C4—91no—
332C22×C4—92no—
333C22×C4—92no—
334C4⋊C4—92no—
335C4⋊C4—94no—
336C4⋊C4C2×C3×D914yes—
337C22⋊C4—92no—
338C22⋊C4—94no—
339C22⋊C4—94no—
340C42—91no—
341C42C2×C3×D911yescharacteristic
342C22×C3—21no—
343C22×C3—92no—
344C22×C3—182no—
345C22×C3—182no—
346C22×C3—184no—
347C22×C3C23×D911yescharacteristic
348C22×C3C23×C3×D311yescharacteristic
349C2×D3—32no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—64no—
353C3×C4—24no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4—92no—
364C3×C4—182no—
365C3×C4—182no—
366C3×C4D4×D912yes—
367C3×C4C3×D3×D412yes—
368Dic3—32no—
369Dic3—62no—
370Dic3—62no—
371C32#572411yescharacteristic
372C9—21no—
373C9C3×#53611yescharacteristic, complemented
374C23—91no—
375C23—92no—
376Q8—22no—
377Q8—22no—
378Q8—184no—
379Q8—182no—
380D4—184no—
381D4—182no—
382C2×C4—91no—
383C2×C4—94no—
384C2×C4—92no—
385C2×C4—92no—
386C2×C4—184no—
387C2×C4—184no—
388C2×C4C22×C3×D912yes—
389C2×C4C22×C3×D912yes—
390C2×C4C22×C3×D912yes—
391C2×C4C22×C3×D911yescharacteristic
392C2×C3—21no—
393C2×C3—22no—
394C2×C3—182no—
395C2×C3—362no—
396C2×C3C2×D4×D911yescharacteristic
397C2×C3#6371312yes—
398C2×C3C2×C3×D3×D411yescharacteristic
399C2×C3C3×#100712yes—
400D3—62no—
401D3—122no—
402C22—92no—
403C22—182no—
404C22—182no—
405C22—184no—
406C22C23×C3×D911yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—92no—
411C4—182no—
412C4—182no—
413C4C3×D4×D912yes—
414C3—21no—
415C3#1058982611yescharacteristic, direct product factor
416C3C3×#572411yescharacteristic
417C2—182no—
418C2—362no—
419C2C2×C3×D4×D911yescharacteristic
420C2C3×#6371312yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C4⋊Q8C2
socleC22×C32C23×D3
prime radicalC2×C3×#3701C2
hypercenterC3×C4⋊Q8D9
hypocenterC9C3×#536
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×D9C2semidirect
3. C3×C9×C4⋊Q8C2semidirect
4. C3×#63509C2semidirect
5. C3×#63506C2semidirect
6. C3×#63458C2semidirect
7. C3×#63456C2semidirect
8. C3×#63437C2semidirect
9. C3×#63434C2semidirect
11. #10589826C3direct (N × H)
41. C2×Q8×D9C2×C3semidirect
43. C9×C4⋊Q8C2×C3semidirect
44. #63509C2×C3semidirect
45. #63506C2×C3semidirect
46. #63458C2×C3semidirect
47. #63456C2×C3semidirect
48. #63437C2×C3semidirect
49. #63434C2×C3semidirect
130. C3×C4⋊Q8D9semidirect
264. C4⋊Q8C3×D9semidirect
270. C3×C9#536semidirect
373. C9C3×#536semidirect
415. C3#10589826direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×C3×Q8×D92
C3×C9×C4⋊Q82
C3×#635092
C3×#635062
C3×#634582
C3×#634562
C3×#634372
C3×#634342
C3×#57243
#105898263