Finite Group Databasebeta

← C3×#10588421 #13141645

Subgroups — C3×#10588421 #13141645

Order 1728.

646 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63710C211yesmaximal, characteristic, complemented
2C3×#63499C211yesmaximal, characteristic, complemented
3C3×#63444C212yesmaximal, complemented
4C3×C9×#99C211yesmaximal, characteristic, complemented
5C3×#63365C212yesmaximal, complemented
6C3×#4741—31nomaximal
7#10588421C311yesmaximal, characteristic, direct product factor
8C23×C3×D9C2211yescharacteristic, complemented
9C2×C3×D4×C9C2211yescharacteristic
10C2×C3×#611—21no—
11C2×C3×#611—21no—
12C2×C3×#611C2212yes—
13C3×#3699—22no—
14C3×D4×D9—22no—
15C3×#3697—21no—
16C2×C3×D36—22no—
17C3×C9×C22⋊C4C2212yescomplemented
18C3×#3676C2211yescharacteristic, complemented
19C3×(C2×D9)⋊C4—21no—
20C3×(C2×D9)⋊C4—22no—
21C3×#1004—31no—
22C3×#937—31no—
23C3×#882—32no—
24#63710C2×C311yescharacteristic, complemented
25C32×#99D311yescharacteristic
26C3×#806—32no—
27#63499C2×C311yescharacteristic, complemented
28#63444C2×C312yescomplemented
29C9×#99—21no—
30C9×#99C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×C3×C9D411yescharacteristic, complemented
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9—22no—
35C22×C3×D9—21no—
36C22×C3×D9—21no—
37C22×C3×D9—42no—
38C22×C3×D9D412yescomplemented
39C3×D4×C9—22no—
40C2×C3×C4×C9—22no—
41C2×C3×C4×C9D411yescharacteristic, complemented
42C3×#611—22no—
43C3×#611—22no—
44C3×#611—22no—
45C3×#611—42no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9D412yescomplemented
49C3×D36—21no—
50C3×D36—44no—
51C3×C4×D9—22no—
52C3×Dic18—21no—
53C3×B(4,𝔽2)—91no—
54#4741—31no—
55C23×C3×D3—31no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C3×#67—32no—
58C2×C3×#67—61no—
59C2×C3×#67—61no—
60C3×#237—62no—
61C3×D3×D4—62no—
62C3×#235—61no—
63C2×C3×D12—62no—
64C23×D9C22×C311yescharacteristic, complemented
65C32×C22⋊C4C2×D312yes—
66C3×#217—31no—
67C3×(C2×D3)⋊C4—62no—
68C3×(C2×D3)⋊C4—61no—
69C2×D4×C9—21no—
70C2×D4×C9C22×C311yescharacteristic
71C2×#611—21no—
72C2×#611—21no—
73C2×#611C22×C312yes—
74#3699—22no—
75D4×D9—22no—
76#3697—21no—
77C2×D36—22no—
78C9×C22⋊C4—22no—
79C9×C22⋊C4C22×C312yescomplemented
80#3676C22×C311yescharacteristic, complemented
81(C2×D9)⋊C4—21no—
82(C2×D9)⋊C4—22no—
83C22×C3×C9—21no—
84C22×C3×C9—41no—
85C22×C3×C9—41no—
86C22×C3×C9C2×D411yescharacteristic
87C22×C3×C9C2×D412yes—
88C2×C3×D9—24no—
89C2×C3×D9—22no—
90C2×C3×D9—22no—
91C2×C3×D9—41no—
92C2×C3×D9—42no—
93C2×C3×D9—41no—
94C2×C3×D9—42no—
95C2×C3×D9—81no—
96C3×C4×C9—21no—
97C3×C4×C9—42no—
98C3×Dic9—22no—
99C3×Dic9—41no—
100C3×H(5,𝔽2)—91no—
101#1004—31no—
102C3×#120—91no—
103C3×#120—92no—
104#937—31no—
105#882—32no—
106C3×#99—21no—
107C3×#99—92no—
108C3×#99D911yescharacteristic, complemented
109C3×#99C3×D311yescharacteristic
110#806—32no—
111C23×C32D1211yescharacteristic
112C23×C32C22×D311yescharacteristic
113C22×C3×D3—32no—
114C22×C3×D3—61no—
115C22×C3×D3—62no—
116C22×C3×D3—61no—
117C22×C3×D3—122no—
118C32×D4—22no—
119C2×C32×C4—22no—
120C2×C32×C4D1211yescharacteristic
121C3×#67—62no—
122C3×#67—62no—
123C3×#67—62no—
124C3×#67—122no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—61no—
129C3×D12—124no—
130C3×C4×D3—62no—
131C3×Dic6—61no—
132C23×C9—21no—
133C23×C9—21no—
134C23×C9C3×D411yescharacteristic, complemented
135C23×C9C23×C311yescharacteristic
136C22×D9—22no—
137C22×D9—21no—
138C22×D9—21no—
139C22×D9—42no—
140C22×D9C3×D412yescomplemented
141D4×C9—22no—
142D4×C9—42no—
143C2×C4×C9—22no—
144C2×C4×C9—21no—
145C2×C4×C9—42no—
146C2×C4×C9C3×D411yescharacteristic, complemented
147#611—22no—
148#611—22no—
149#611—22no—
150#611—42no—
151C2×Dic9—21no—
152C2×Dic9—21no—
153C2×Dic9C3×D412yescomplemented
154D36—21no—
155D36—44no—
156C4×D9—22no—
157Dic18—21no—
158B(4,𝔽2)—91noSylow
159C2×C3×C9—21no—
160C2×C3×C9—22no—
161C2×C3×C9—41no—
162C2×C3×C9#12011yescharacteristic
163C3×D9—42no—
164C3×D9—42no—
165C24×C3—91no—
166C23×D3—31no—
167C3×#41—182no—
168C3×#41—181no—
169C2×C3×D4—21no—
170C2×C3×D4—92no—
171C2×C3×D4—182no—
172C2×C3×D4—182no—
173C2×C3×D4—181no—
174C2×C3×D4—181no—
175C2×C3×D4C2×D911yescharacteristic
176C2×C3×D4C2×C3×D311yescharacteristic
177C2×#67—32no—
178C2×#67—61no—
179C2×#67—61no—
180#237—62no—
181D3×D4—62no—
182#235—61no—
183C2×D12—62no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—91no—
186C3×C22⋊C4—182no—
187C3×C22⋊C4—181no—
188C3×C22⋊C4C2×D912yescomplemented
189C3×C22⋊C4C2×C3×D312yes—
190#217—31no—
191(C2×D3)⋊C4—62no—
192(C2×D3)⋊C4—61no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32C2×D1211yescharacteristic
197C22×C32D3×D412yes—
198C2×C3×D3—64no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—121no—
202C2×C3×D3—122no—
203C2×C3×D3—121no—
204C2×C3×D3—122no—
205C2×C3×D3—241no—
206C32×C4—21no—
207C32×C4—42no—
208C3×Dic3—62no—
209C3×Dic3—121no—
210C22×C9—21no—
211C22×C9—21no—
212C22×C9—22no—
213C22×C9—41no—
214C22×C9—41no—
215C22×C9—41no—
216C22×C9—81no—
217C22×C9—81no—
218C22×C9C2×C3×D411yescharacteristic
219C22×C9C2×C3×D412yes—
220C2×D9—24no—
221C2×D9—22no—
222C2×D9—22no—
223C2×D9—41no—
224C2×D9—42no—
225C2×D9—41no—
226C2×D9—42no—
227C2×D9—81no—
228C4×C9—21no—
229C4×C9—41no—
230C4×C9—42no—
231C4×C9—82no—
232Dic9—22no—
233Dic9—41no—
234H(5,𝔽2)—91no—
235#120—91no—
236#120—92no—
237#99—92no—
238#99C3×D911yescharacteristic, complemented
239C3×C9B(4,𝔽2)11yescharacteristic, complemented, Sylow
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—92no—
243C23×C3—181no—
244C23×C3—181no—
245C23×C3—182no—
246C23×C3—362no—
247C23×C3D3611yescharacteristic, complemented
248C23×C3C22×D911yescharacteristic
249C23×C3C3×D1211yescharacteristic
250C23×C3C22×C3×D311yescharacteristic
251C22×D3—32no—
252C22×D3—61no—
253C22×D3—62no—
254C22×D3—61no—
255C22×D3—122no—
256C3×Q8—181no—
257C3×D4—22no—
258C3×D4—22no—
259C3×D4—42no—
260C3×D4—181no—
261C3×D4—182no—
262C3×D4—182no—
263C3×D4—182no—
264C3×D4—364no—
265C3×D4—362no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—42no—
270C2×C3×C4—92no—
271C2×C3×C4—182no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4D3611yescharacteristic, complemented
275C2×C3×C4C3×D1211yescharacteristic
276#67—62no—
277#67—62no—
278#67—62no—
279#67—122no—
280C2×Dic3—32no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283D12—61no—
284D12—124no—
285C4×D3—62no—
286Dic6—61no—
287C2×C32—21no—
288C2×C32—22no—
289C2×C32—41no—
290C2×C32#88211yescharacteristic
291C3×D3—122no—
292C3×D3—122no—
293C2×C9—21no—
294C2×C9—22no—
295C2×C9—21no—
296C2×C9—42no—
297C2×C9—41no—
298C2×C9—41no—
299C2×C9—81no—
300C2×C9C3×#12011yescharacteristic
301D9—42no—
302D9—42no—
303C24—91no—
304#41—182no—
305#41—181no—
306C2×D4—92no—
307C2×D4—182no—
308C2×D4—182no—
309C2×D4—181no—
310C2×D4—181no—
311C2×D4C2×C3×D911yescharacteristic
312C22⋊C4—91no—
313C22⋊C4—182no—
314C22⋊C4—181no—
315C22⋊C4C2×C3×D912yescomplemented
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—184no—
328C22×C3—182no—
329C22×C3—182no—
330C22×C3—362no—
331C22×C3—362no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—721no—
335C22×C3C2×D3611yescharacteristic
336C22×C3D4×D912yes—
337C22×C3C2×C3×D1211yescharacteristic
338C22×C3C3×D3×D412yes—
339C2×D3—64no—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—121no—
343C2×D3—122no—
344C2×D3—121no—
345C2×D3—122no—
346C2×D3—241no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—42no—
351C3×C4—42no—
352C3×C4—82no—
353C3×C4—182no—
354C3×C4—361no—
355Dic3—62no—
356Dic3—121no—
357C32#474111yescharacteristic
358C9—21no—
359C9C3×B(4,𝔽2)11yescharacteristic, complemented
360C23—92no—
361C23—181no—
362C23—181no—
363C23—182no—
364C23—362no—
365C23C3×D3611yescharacteristic, complemented
366C23C22×C3×D911yescharacteristic
367Q8—181no—
368D4—22no—
369D4—181no—
370D4—182no—
371D4—182no—
372D4—182no—
373D4—364no—
374D4—362no—
375C2×C4—22no—
376C2×C4—92no—
377C2×C4—182no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4C3×D3611yescharacteristic, complemented
381C2×C3—22no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—42no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—81no—
391C2×C3—362no—
392C2×C3—362no—
393C2×C3#6344411yescharacteristic
394C2×C3C3×#88211yescharacteristic
395D3—122no—
396D3—122no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—184no—
401C22—182no—
402C22—182no—
403C22—362no—
404C22—362no—
405C22—361no—
406C22—361no—
407C22—721no—
408C22C2×C3×D3611yescharacteristic
409C22C3×D4×D912yes—
410C4—21no—
411C4—42no—
412C4—182no—
413C4—361no—
414C3—21no—
415C3#1058842111yescharacteristic, direct product factor
416C3C3×#474111yescharacteristic
417C2—22no—
418C2—21no—
419C2—41no—
420C2—362no—
421C2—362no—
422C2C3×#6344411yescharacteristic
subgroupG/H
centerC2×C3#63444
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#99C2
socleC2×C32#882
hypercenterC3×#99D9
hypocenterC9C3×B(4,𝔽2)
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
1. C3×#63710C2semidirect
2. C3×#63499C2semidirect
3. C3×#63444C2semidirect
4. C3×C9×#99C2semidirect
5. C3×#63365C2semidirect
7. #10588421C3direct (N × H)
8. C23×C3×D9C22semidirect
17. C3×C9×C22⋊C4C22semidirect
18. C3×#3676C22semidirect
24. #63710C2×C3semidirect
27. #63499C2×C3semidirect
28. #63444C2×C3semidirect
30. C9×#99C2×C3semidirect
31. #63365C2×C3semidirect
32. C23×C3×C9D4semidirect
38. C22×C3×D9D4semidirect
41. C2×C3×C4×C9D4semidirect
48. C2×C3×Dic9D4semidirect
64. C23×D9C22×C3semidirect
79. C9×C22⋊C4C22×C3semidirect
80. #3676C22×C3semidirect
108. C3×#99D9semidirect
134. C23×C9C3×D4semidirect
140. C22×D9C3×D4semidirect
146. C2×C4×C9C3×D4semidirect
153. C2×Dic9C3×D4semidirect
188. C3×C22⋊C4C2×D9semidirect
238. #99C3×D9semidirect
239. C3×C9B(4,𝔽2)semidirect
247. C23×C3D36semidirect
274. C2×C3×C4D36semidirect
315. C22⋊C4C2×C3×D9semidirect
359. C9C3×B(4,𝔽2)semidirect
365. C23C3×D36semidirect
380. C2×C4C3×D36semidirect
415. C3#10588421direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637102
C3×#634992
C3×#634442
C3×C9×#992
C3×#633652
C3×#47413
#105884213