Finite Group Databasebeta

← C3×#10588712 #13142812

Subgroups — C3×#10588712 #13142812

Order 1728.

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

normal subgroups only

This group has 724 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×C9×#120C211yesmaximal, characteristic, complemented
3C3×#63515C212yesmaximal, complemented
4C3×#63393C211yesmaximal, characteristic, complemented
5C3×#63365C212yesmaximal, complemented
6C3×#5032—31nomaximal
7#10588712C311yesmaximal, characteristic, direct product factor
8C24×C3×C9C2211yescharacteristic, complemented
9C2×C3×D4×C9—21no—
10C2×C3×D4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C2×C3×#611—22no—
13C2×C3×#611C2212yes—
14C3×#3699—22no—
15C3×D4×D9—22no—
16C3×#3697—21no—
17C3×C9×C22⋊C4—21no—
18C3×C9×C22⋊C4C2211yescharacteristic, complemented
19C3×#3676—22no—
20C3×#3676C2212yescomplemented
21C3×#1004—31no—
22C32×#120D311yescharacteristic
23C3×#953—32no—
24#63710C2×C311yescharacteristic, complemented
25C3×#834—31no—
26C3×#806—32no—
27C9×#120—21no—
28C9×#120C2×C311yescharacteristic, complemented
29#63515C2×C312yescomplemented
30#63393C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×C3×C9—21no—
33C23×C3×C9—22no—
34C23×C3×C9—42no—
35C23×C3×C9D411yescharacteristic, complemented
36C23×C3×C9C2311yescharacteristic
37C22×C3×D9—21no—
38C22×C3×D9D412yescomplemented
39C3×D4×C9—22no—
40C3×D4×C9—42no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic, complemented
43C3×#611—22no—
44C3×#611—22no—
45C3×#611—22no—
46C3×#611—44no—
47C2×C3×Dic9—21no—
48C2×C3×Dic9—22no—
49C2×C3×Dic9D412yescomplemented
50C3×D36—21no—
51C3×C4×D9—22no—
52C3×Dic18—21no—
53C3×B(4,𝔽2)—91no—
54#5032—31no—
55C24×C32C2×D311yescharacteristic
56C2×C32×D4—21no—
57C2×C32×D4C2×D311yescharacteristic
58C2×C3×#67—32no—
59C2×C3×#67—61no—
60C2×C3×#67—62no—
61C3×#237—62no—
62C3×D3×D4—62no—
63C3×#235—61no—
64C24×C9—21no—
65C24×C9C22×C311yescharacteristic, complemented
66C32×C22⋊C4—21no—
67C32×C22⋊C4C2×D311yescharacteristic
68C3×#217—32no—
69C3×#217—62no—
70C2×D4×C9—22no—
71C2×D4×C9—21no—
72C2×D4×C9—21no—
73C2×D4×C9C22×C311yescharacteristic
74C2×#611—21no—
75C2×#611—22no—
76C2×#611C22×C312yes—
77#3699—22no—
78D4×D9—22no—
79#3697—21no—
80C9×C22⋊C4—22no—
81C9×C22⋊C4—21no—
82C9×C22⋊C4—21no—
83C9×C22⋊C4C22×C311yescharacteristic, complemented
84#3676—22no—
85#3676C22×C312yescomplemented
86C22×C3×C9—22no—
87C22×C3×C9—24no—
88C22×C3×C9—21no—
89C22×C3×C9—41no—
90C22×C3×C9—41no—
91C22×C3×C9—42no—
92C22×C3×C9—41no—
93C22×C3×C9—41no—
94C22×C3×C9C2×D412yes—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3×D9—22no—
97C2×C3×D9—42no—
98C2×C3×D9—81no—
99C3×C4×C9—21no—
100C3×C4×C9—41no—
101C3×Dic9—22no—
102C3×Dic9—42no—
103C3×H(5,𝔽2)—91no—
104#1004—31no—
105C3×#120—21no—
106C3×#120—92no—
107C3×#120D911yescharacteristic, complemented
108C3×#120C3×D311yescharacteristic
109#953—32no—
110C3×#99—91no—
111C3×#99—92no—
112#834—31no—
113#806—32no—
114C23×C32—21no—
115C23×C32—22no—
116C23×C32—42no—
117C23×C32#6711yescharacteristic
118C23×C32C22×D311yescharacteristic
119C22×C3×D3—32no—
120C22×C3×D3—61no—
121C32×D4—22no—
122C32×D4—42no—
123C2×C32×C4—21no—
124C2×C32×C4#6711yescharacteristic
125C3×#67—62no—
126C3×#67—62no—
127C3×#67—62no—
128C3×#67—124no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—62no—
132C3×D12—61no—
133C3×C4×D3—62no—
134C3×Dic6—61no—
135C23×C9—22no—
136C23×C9—22no—
137C23×C9—21no—
138C23×C9—21no—
139C23×C9—21no—
140C23×C9—42no—
141C23×C9—42no—
142C23×C9—44no—
143C23×C9C3×D411yescharacteristic, complemented
144C23×C9C23×C311yescharacteristic
145C22×D9—21no—
146C22×D9C3×D412yescomplemented
147D4×C9—22no—
148D4×C9—42no—
149D4×C9—42no—
150D4×C9—44no—
151C2×C4×C9—22no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9C3×D411yescharacteristic, complemented
155#611—22no—
156#611—22no—
157#611—22no—
158#611—44no—
159C2×Dic9—21no—
160C2×Dic9—22no—
161C2×Dic9C3×D412yescomplemented
162D36—21no—
163C4×D9—22no—
164Dic18—21no—
165B(4,𝔽2)—91noSylow
166C2×C3×C9—21no—
167C2×C3×C9—22no—
168C2×C3×C9—42no—
169C2×C3×C9—41no—
170C2×C3×C9#12011yescharacteristic
171C3×D9—42no—
172C24×C3—21no—
173C24×C3C2×D911yescharacteristic, complemented
174C24×C3C2×C3×D311yescharacteristic
175C3×#41—182no—
176C3×#41—181no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—92no—
182C2×C3×D4—182no—
183C2×C3×D4—181no—
184C2×C3×D4—182no—
185C2×C3×D4C2×D911yescharacteristic
186C2×C3×D4C2×C3×D311yescharacteristic
187C2×#67—32no—
188C2×#67—61no—
189C2×#67—62no—
190#237—62no—
191D3×D4—62no—
192#235—61no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—21no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—182no—
199C3×C22⋊C4C2×D911yescharacteristic, complemented
200C3×C22⋊C4C2×C3×D311yescharacteristic
201#217—32no—
202#217—62no—
203C22×C32—22no—
204C22×C32—24no—
205C22×C32—21no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32D3×D412yes—
212C22×C32C2×#6711yescharacteristic
213C2×C3×D3—62no—
214C2×C3×D3—122no—
215C2×C3×D3—241no—
216C32×C4—21no—
217C32×C4—41no—
218C3×Dic3—62no—
219C3×Dic3—122no—
220C22×C9—24no—
221C22×C9—24no—
222C22×C9—22no—
223C22×C9—21no—
224C22×C9—22no—
225C22×C9—21no—
226C22×C9—41no—
227C22×C9—44no—
228C22×C9—41no—
229C22×C9—41no—
230C22×C9—42no—
231C22×C9—42no—
232C22×C9—42no—
233C22×C9—41no—
234C22×C9—42no—
235C22×C9—42no—
236C22×C9—44no—
237C22×C9—41no—
238C22×C9C2×C3×D412yes—
239C22×C9C2×C3×D411yescharacteristic
240C2×D9—22no—
241C2×D9—42no—
242C2×D9—81no—
243C4×C9—21no—
244C4×C9—41no—
245C4×C9—42no—
246C4×C9—41no—
247Dic9—22no—
248Dic9—42no—
249H(5,𝔽2)—91no—
250#120—92no—
251#120C3×D911yescharacteristic, complemented
252#99—91no—
253#99—92no—
254C3×C9B(4,𝔽2)11yescharacteristic, complemented, Sylow
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—44no—
266C23×C3—92no—
267C23×C3—181no—
268C23×C3#61111yescharacteristic, complemented
269C23×C3C22×D911yescharacteristic
270C23×C3C3×#6711yescharacteristic
271C23×C3C22×C3×D311yescharacteristic
272C22×D3—32no—
273C22×D3—61no—
274C3×Q8—181no—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—44no—
281C3×D4—181no—
282C3×D4—182no—
283C3×D4—182no—
284C3×D4—182no—
285C3×D4—364no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—92no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×C3×C4—182no—
294C2×C3×C4#61111yescharacteristic, complemented
295C2×C3×C4C3×#6711yescharacteristic
296#67—62no—
297#67—62no—
298#67—62no—
299#67—124no—
300C2×Dic3—32no—
301C2×Dic3—61no—
302C2×Dic3—62no—
303D12—61no—
304C4×D3—62no—
305Dic6—61no—
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—42no—
309C2×C32—41no—
310C2×C32#93711yescharacteristic
311C3×D3—122no—
312C2×C9—22no—
313C2×C9—22no—
314C2×C9—21no—
315C2×C9—21no—
316C2×C9—42no—
317C2×C9—44no—
318C2×C9—41no—
319C2×C9—42no—
320C2×C9—81no—
321C2×C9C3×#12011yescharacteristic
322D9—42no—
323C24C2×C3×D911yescharacteristic, complemented
324#41—182no—
325#41—181no—
326C2×D4—21no—
327C2×D4—92no—
328C2×D4—182no—
329C2×D4—181no—
330C2×D4—182no—
331C2×D4C2×C3×D911yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—92no—
334C22⋊C4—182no—
335C22⋊C4C2×C3×D911yescharacteristic, complemented
336C22×C3—24no—
337C22×C3—22no—
338C22×C3—24no—
339C22×C3—22no—
340C22×C3—24no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—44no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—41no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—44no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—182no—
363C22×C3—362no—
364C22×C3—721no—
365C22×C3D4×D912yes—
366C22×C3C2×#61111yescharacteristic
367C22×C3C3×D3×D412yes—
368C22×C3C2×C3×#6711yescharacteristic
369C2×D3—62no—
370C2×D3—122no—
371C2×D3—241no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—42no—
377C3×C4—41no—
378C3×C4—182no—
379C3×C4—362no—
380Dic3—62no—
381Dic3—122no—
382C32#503211yescharacteristic
383C9—21no—
384C9C3×B(4,𝔽2)11yescharacteristic, complemented
385C23—22no—
386C23—21no—
387C23—42no—
388C23—92no—
389C23—181no—
390C23C3×#61111yescharacteristic, complemented
391C23C22×C3×D911yescharacteristic
392Q8—181no—
393D4—22no—
394D4—42no—
395D4—181no—
396D4—182no—
397D4—182no—
398D4—182no—
399D4—364no—
400C2×C4—21no—
401C2×C4—92no—
402C2×C4—182no—
403C2×C4—181no—
404C2×C4—182no—
405C2×C4C3×#61111yescharacteristic, complemented
406C2×C3—22no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—42no—
413C2×C3—44no—
414C2×C3—42no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—42no—
418C2×C3—81no—
419C2×C3—362no—
420C2×C3#6349911yescharacteristic
421C2×C3C3×#93711yescharacteristic
422D3—122no—
423C22—22no—
424C22—24no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22—182no—
432C22—362no—
433C22—721no—
434C22C3×D4×D912yes—
435C22C2×C3×#61111yescharacteristic
436C4—21no—
437C4—41no—
438C4—182no—
439C4—362no—
440C3—21no—
441C3#1058871211yescharacteristic, direct product factor
442C3C3×#503211yescharacteristic
443C2—21no—
444C2—22no—
445C2—42no—
446C2—41no—
447C2—362no—
448C2C3×#6349911yescharacteristic
subgroupG/H
centerC2×C3#63499
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#120C2
socleC2×C32#937
hypercenterC3×#120D9
hypocenterC9C3×B(4,𝔽2)
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C3×#63710C2semidirect
2. C3×C9×#120C2semidirect
3. C3×#63515C2semidirect
4. C3×#63393C2semidirect
5. C3×#63365C2semidirect
7. #10588712C3direct (N × H)
8. C24×C3×C9C22semidirect
18. C3×C9×C22⋊C4C22semidirect
20. C3×#3676C22semidirect
24. #63710C2×C3semidirect
28. C9×#120C2×C3semidirect
29. #63515C2×C3semidirect
30. #63393C2×C3semidirect
31. #63365C2×C3semidirect
35. C23×C3×C9D4semidirect
38. C22×C3×D9D4semidirect
42. C2×C3×C4×C9D4semidirect
49. C2×C3×Dic9D4semidirect
65. C24×C9C22×C3semidirect
83. C9×C22⋊C4C22×C3semidirect
85. #3676C22×C3semidirect
107. C3×#120D9semidirect
143. C23×C9C3×D4semidirect
146. C22×D9C3×D4semidirect
154. C2×C4×C9C3×D4semidirect
161. C2×Dic9C3×D4semidirect
173. C24×C3C2×D9semidirect
199. C3×C22⋊C4C2×D9semidirect
251. #120C3×D9semidirect
254. C3×C9B(4,𝔽2)semidirect
268. C23×C3#611semidirect
294. C2×C3×C4#611semidirect
323. C24C2×C3×D9semidirect
335. C22⋊C4C2×C3×D9semidirect
384. C9C3×B(4,𝔽2)semidirect
390. C23C3×#611semidirect
405. C2×C4C3×#611semidirect
441. C3#10588712direct (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×C9×#1202
C3×#635152
C3×#633932
C3×#633652
C3×#50323
#105887123