Finite Group Databasebeta

← C3×#10592731 #13163070

Subgroups — C3×#10592731 #13163070

Order 1728.

1708 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1708 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64361C211yesmaximal, characteristic, complemented
2C33×#120C211yesmaximal, characteristic, complemented
3C3×#64162C212yesmaximal, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5C3×#63633C212yesmaximal, complemented
6#10592731C311yesmaximal, characteristic, direct product factor
7C3×#5032—34nomaximal
8C24×C33C2211yescharacteristic, complemented
9C2×C33×D4—21no—
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×#638—22no—
13C2×C3×#638C2212yes—
14C3×#3830—22no—
15C3×D4×C3⋊D3—22no—
16C3×#3828—21no—
17C33×C22⋊C4—21no—
18C33×C22⋊C4C2211yescharacteristic, complemented
19C3×#3757—22no—
20C3×#3757C2212yescomplemented
21#64361C2×C311yescharacteristic, complemented
22C3×#1004—34no—
23C32×#120—24no—
24C32×#120D314yescomplemented
25C32×#120C2×C311yescharacteristic, complemented
26#64162C2×C312yescomplemented
27C3×#953—38no—
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—34no—
31C3×#806—38no—
32C23×C33—21no—
33C23×C33—22no—
34C23×C33—42no—
35C23×C33D411yescharacteristic, complemented
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—21no—
38C22×C3×C3⋊D3D412yescomplemented
39C33×D4—22no—
40C33×D4—42no—
41C2×C33×C4—21no—
42C2×C33×C4D411yescharacteristic, complemented
43C3×#638—22no—
44C3×#638—22no—
45C3×#638—22no—
46C3×#638—44no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3D412yescomplemented
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4×C3⋊D3—22no—
52C3×#634—21no—
53C3×B(4,𝔽2)—91no—
54#5032—34no—
55C24×C32—24no—
56C24×C32C2×D314yescomplemented
57C24×C32C22×C311yescharacteristic, complemented
58C2×C32×D4—28no—
59C2×C32×D4—24no—
60C2×C32×D4—24no—
61C2×C32×D4—21no—
62C2×C32×D4C2×D314yes—
63C2×C32×D4C22×C311yescharacteristic
64C2×#638—21no—
65C2×#638—22no—
66C2×#638C22×C312yes—
67#3830—22no—
68D4×C3⋊D3—22no—
69#3828—21no—
70C2×C3×#67—38no—
71C2×C3×#67—64no—
72C2×C3×#67—68no—
73C3×#237—68no—
74C3×D3×D4—68no—
75C3×#235—64no—
76C32×C22⋊C4—28no—
77C32×C22⋊C4—24no—
78C32×C22⋊C4—24no—
79C32×C22⋊C4—21no—
80C32×C22⋊C4C2×D314yescomplemented
81C32×C22⋊C4C22×C311yescharacteristic, complemented
82#3757—22no—
83#3757C22×C312yescomplemented
84C3×#217—38no—
85C3×#217—68no—
86C22×C33—22no—
87C22×C33—24no—
88C22×C33—21no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33—42no—
92C22×C33—41no—
93C22×C33—41no—
94C22×C33C2×D412yes—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—42no—
98C2×C3×C3⋊D3—81no—
99C33×C4—21no—
100C33×C4—41no—
101C3×C3⋊Dic3—22no—
102C3×C3⋊Dic3—42no—
103C3×H(5,𝔽2)—91no—
104#1004—34no—
105C3×#120—24no—
106C3×#120—92no—
107C3×#120C3×D314yescomplemented
108C3×#120C3⋊D311yescharacteristic, complemented
109#953—38no—
110C3×#99—91no—
111C3×#99—92no—
112#834—34no—
113#806—38no—
114C23×C32—28no—
115C23×C32—24no—
116C23×C32—22no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32—24no—
120C23×C32—28no—
121C23×C32—48no—
122C23×C32—48no—
123C23×C32—42no—
124C23×C32—416no—
125C23×C32#6714yescomplemented
126C23×C32C3×D411yescharacteristic, complemented
127C23×C32C22×D314yes—
128C23×C32C23×C311yescharacteristic
129C22×C3⋊D3—21no—
130C22×C3⋊D3C3×D412yescomplemented
131C22×C3×D3—38no—
132C22×C3×D3—64no—
133C32×D4—22no—
134C32×D4—28no—
135C32×D4—48no—
136C32×D4—48no—
137C32×D4—42no—
138C32×D4—416no—
139C2×C32×C4—28no—
140C2×C32×C4—24no—
141C2×C32×C4—21no—
142C2×C32×C4—24no—
143C2×C32×C4#6714yescomplemented
144C2×C32×C4C3×D411yescharacteristic, complemented
145#638—22no—
146#638—22no—
147#638—22no—
148#638—44no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3C3×D412yescomplemented
152C4⋊(C3⋊D3)—21no—
153C4×C3⋊D3—22no—
154#634—21no—
155C3×#67—68no—
156C3×#67—68no—
157C3×#67—68no—
158C3×#67—1216no—
159C2×C3×Dic3—38no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—68no—
162C3×D12—64no—
163C3×C4×D3—68no—
164C3×Dic6—64no—
165B(4,𝔽2)—91noSylow
166C2×C33—21no—
167C2×C33—22no—
168C2×C33—42no—
169C2×C33—41no—
170C2×C33#12011yescharacteristic
171C3×C3⋊D3—42no—
172C24×C3—24no—
173C24×C3C2×C3×D314yescomplemented
174C24×C3C2×C3⋊D311yescharacteristic, complemented
175C3×#41—182no—
176C3×#41—181no—
177C2×C3×D4—28no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—24no—
181C2×C3×D4—92no—
182C2×C3×D4—182no—
183C2×C3×D4—181no—
184C2×C3×D4—182no—
185C2×C3×D4C2×C3×D314yes—
186C2×C3×D4C2×C3⋊D311yescharacteristic
187C2×#67—38no—
188C2×#67—64no—
189C2×#67—68no—
190#237—68no—
191D3×D4—68no—
192#235—64no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—182no—
199C3×C22⋊C4C2×C3×D314yescomplemented
200C3×C22⋊C4C2×C3⋊D311yescharacteristic, complemented
201#217—38no—
202#217—68no—
203C22×C32—24no—
204C22×C32—216no—
205C22×C32—28no—
206C22×C32—22no—
207C22×C32—21no—
208C22×C32—28no—
209C22×C32—24no—
210C22×C32—216no—
211C22×C32—24no—
212C22×C32—44no—
213C22×C32—416no—
214C22×C32—44no—
215C22×C32—41no—
216C22×C32—44no—
217C22×C32—41no—
218C22×C32—48no—
219C22×C32—48no—
220C22×C32—44no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32—48no—
224C22×C32—48no—
225C22×C32—48no—
226C22×C32—416no—
227C22×C32—44no—
228C22×C32—41no—
229C22×C32D3×D418yes—
230C22×C32C2×#6714yes—
231C22×C32C2×C3×D412yes—
232C22×C32C2×C3×D411yescharacteristic
233C2×C3⋊D3—22no—
234C2×C3⋊D3—42no—
235C2×C3⋊D3—81no—
236C2×C3×D3—68no—
237C2×C3×D3—128no—
238C2×C3×D3—244no—
239C32×C4—21no—
240C32×C4—24no—
241C32×C4—44no—
242C32×C4—48no—
243C32×C4—44no—
244C32×C4—41no—
245C3⋊Dic3—22no—
246C3⋊Dic3—42no—
247C3×Dic3—68no—
248C3×Dic3—128no—
249H(5,𝔽2)—91no—
250#120—92no—
251#120C3×C3⋊D311yescharacteristic, complemented
252#99—91no—
253#99—92no—
254C33B(4,𝔽2)11yescharacteristic, complemented, Sylow
255C23×C3—28no—
256C23×C3—28no—
257C23×C3—24no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—48no—
263C23×C3—48no—
264C23×C3—42no—
265C23×C3—416no—
266C23×C3—92no—
267C23×C3—181no—
268C23×C3C3×#6714yescomplemented
269C23×C3#63811yescharacteristic, complemented
270C23×C3C22×C3×D314yes—
271C23×C3C22×C3⋊D311yescharacteristic
272C22×D3—38no—
273C22×D3—64no—
274C3×Q8—181no—
275C3×D4—28no—
276C3×D4—22no—
277C3×D4—48no—
278C3×D4—48no—
279C3×D4—42no—
280C3×D4—416no—
281C3×D4—181no—
282C3×D4—182no—
283C3×D4—182no—
284C3×D4—182no—
285C3×D4—364no—
286C2×C3×C4—28no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—92no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×C3×C4—182no—
294C2×C3×C4C3×#6714yescomplemented
295C2×C3×C4#63811yescharacteristic, complemented
296#67—68no—
297#67—68no—
298#67—68no—
299#67—1216no—
300C2×Dic3—38no—
301C2×Dic3—64no—
302C2×Dic3—68no—
303D12—64no—
304C4×D3—68no—
305Dic6—64no—
306C2×C32—28no—
307C2×C32—24no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—24no—
311C2×C32—28no—
312C2×C32—48no—
313C2×C32—416no—
314C2×C32—48no—
315C2×C32—41no—
316C2×C32—42no—
317C2×C32—44no—
318C2×C32—84no—
319C2×C32#93714yes—
320C2×C32C3×#12011yescharacteristic
321C3⋊D3—42no—
322C3×D3—128no—
323C24C2×C3×C3⋊D311yescharacteristic, 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×C3⋊D311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—92no—
334C22⋊C4—182no—
335C22⋊C4C2×C3×C3⋊D311yescharacteristic, complemented
336C22×C3—216no—
337C22×C3—216no—
338C22×C3—28no—
339C22×C3—22no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—21no—
343C22×C3—28no—
344C22×C3—24no—
345C22×C3—44no—
346C22×C3—416no—
347C22×C3—44no—
348C22×C3—41no—
349C22×C3—44no—
350C22×C3—41no—
351C22×C3—48no—
352C22×C3—48no—
353C22×C3—48no—
354C22×C3—44no—
355C22×C3—41no—
356C22×C3—42no—
357C22×C3—48no—
358C22×C3—48no—
359C22×C3—416no—
360C22×C3—44no—
361C22×C3—41no—
362C22×C3—182no—
363C22×C3—362no—
364C22×C3—721no—
365C22×C3C3×D3×D418yes—
366C22×C3C2×C3×#6714yes—
367C22×C3D4×C3⋊D312yes—
368C22×C3C2×#63811yescharacteristic
369C2×D3—68no—
370C2×D3—128no—
371C2×D3—244no—
372C3×C4—24no—
373C3×C4—21no—
374C3×C4—44no—
375C3×C4—48no—
376C3×C4—44no—
377C3×C4—41no—
378C3×C4—182no—
379C3×C4—362no—
380Dic3—68no—
381Dic3—128no—
382C32—24no—
383C32#503214yescomplemented
384C32C3×B(4,𝔽2)11yescharacteristic, complemented
385C23—22no—
386C23—21no—
387C23—42no—
388C23—92no—
389C23—181no—
390C23C3×#63811yescharacteristic, complemented
391C23C22×C3×C3⋊D311yescharacteristic
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×#63811yescharacteristic, complemented
406C2×C3—28no—
407C2×C3—28no—
408C2×C3—24no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—24no—
412C2×C3—48no—
413C2×C3—416no—
414C2×C3—44no—
415C2×C3—48no—
416C2×C3—42no—
417C2×C3—41no—
418C2×C3—84no—
419C2×C3—362no—
420C2×C3C3×#93714yes—
421C2×C3#6414611yescharacteristic
422D3—128no—
423C22—22no—
424C22—24no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22—182no—
432C22—362no—
433C22—721no—
434C22C3×D4×C3⋊D312yes—
435C22C2×C3×#63811yescharacteristic
436C4—21no—
437C4—41no—
438C4—182no—
439C4—362no—
440C3—24no—
441C3C3×#503214yescomplemented
442C3#1059273111yescharacteristic, direct product factor
443C2—21no—
444C2—22no—
445C2—42no—
446C2—41no—
447C2—362no—
448C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC33×#120C2
socleC2×C33#120
hypercenterC3×#120C3⋊D3
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C3×#64361C2semidirect
2. C33×#120C2semidirect
3. C3×#64162C2semidirect
4. C3×#63661C2semidirect
5. C3×#63633C2semidirect
6. #10592731C3direct (N × H)
8. C24×C33C22semidirect
18. C33×C22⋊C4C22semidirect
20. C3×#3757C22semidirect
21. #64361C2×C3semidirect
24. C32×#120D3semidirect
25. C32×#120C2×C3semidirect
26. #64162C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
35. C23×C33D4semidirect
38. C22×C3×C3⋊D3D4semidirect
42. C2×C33×C4D4semidirect
49. C2×C3×C3⋊Dic3D4semidirect
56. C24×C32C2×D3semidirect
57. C24×C32C22×C3semidirect
80. C32×C22⋊C4C2×D3semidirect
81. C32×C22⋊C4C22×C3semidirect
83. #3757C22×C3semidirect
107. C3×#120C3×D3semidirect
108. C3×#120C3⋊D3semidirect
125. C23×C32#67semidirect
126. C23×C32C3×D4semidirect
130. C22×C3⋊D3C3×D4semidirect
143. C2×C32×C4#67semidirect
144. C2×C32×C4C3×D4semidirect
151. C2×C3⋊Dic3C3×D4semidirect
173. C24×C3C2×C3×D3semidirect
174. C24×C3C2×C3⋊D3semidirect
199. C3×C22⋊C4C2×C3×D3semidirect
200. C3×C22⋊C4C2×C3⋊D3semidirect
251. #120C3×C3⋊D3semidirect
254. C33B(4,𝔽2)semidirect
268. C23×C3C3×#67semidirect
269. C23×C3#638semidirect
294. C2×C3×C4C3×#67semidirect
295. C2×C3×C4#638semidirect
323. C24C2×C3×C3⋊D3semidirect
335. C22⋊C4C2×C3×C3⋊D3semidirect
383. C32#5032semidirect
384. C32C3×B(4,𝔽2)semidirect
390. C23C3×#638semidirect
405. C2×C4C3×#638semidirect
441. C3C3×#5032semidirect
442. C3#10592731direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C33×#1202
C3×#641622
C3×#636612
C3×#636332
#105927313
C3×#50323