Finite Group Databasebeta

← C3×#10594813 #13174647

Subgroups — C3×#10594813 #13174647

Order 1728.

1118 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C211yesmaximal, characteristic
2C32×#924C212yesmaximal
3C3×#63982C212yesmaximal
4C3×#63957C212yesmaximal
5C3×#63847C214yesmaximal, complemented
6C3×#63839C214yesmaximal, complemented
7C3×#5623—32nomaximal
8#10594813C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C2×C3×#627C2212yes—
14C3×#3750C2214yes—
15C32×#217C2214yes—
16C32×C4⋊Dic3C2212yes—
17C32×Dic3⋊C4C2214yes—
18C3×#3724C2212yes—
19C3×#3724C2212yes—
20C3×#3724C2212yes—
21C3×Dic3⋊Dic3C2214yes—
22C3×Dic32C2214yes—
23C32×C22⋊Q8—32no—
24C2×#3750C2×C311yescharacteristic
25C3×#944—34no—
26C3×#924—22no—
27C3×#924D312yes—
28C3×#924C2×C312yes—
29C2×C3×Dic3⋊C4—32no—
30C2×C3×Dic3⋊C4—61no—
31C3×#889—34no—
32C3×#889—34no—
33C3×#888—34no—
34C3×#878—34no—
35C3×#877—34no—
36#63982C2×C312yes—
37#63957C2×C312yes—
38#63847C2×C314yescomplemented
39#63839C2×C314yescomplemented
40C23×C33C2311yescharacteristic
41C2×C33×C4—41no—
42C2×C33×C4C2312yes—
43C2×C3×C3⋊Dic3—41no—
44C2×C3×C3⋊Dic3Q814yescomplemented
45C2×C3×C3⋊Dic3C2312yes—
46C2×C3×C3⋊Dic3C2312yes—
47C2×C32×Dic3C2314yes—
48C2×C32×Dic3C2314yes—
49C32×Dic6—42no—
50C3×#627—42no—
51C3×#544—91no—
52#5623—32no—
53C2×C32×Q8—32no—
54C22×C32×C4—21no—
55C22×C32×C4—22no—
56C22×C32×C4—41no—
57C22×C32×C4C2×D312yes—
58C22×C32×C4C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C311yescharacteristic
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—61no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—34no—
67C2×C3×Dic6C2×D312yes—
68C2×C3×Dic6C22×C312yes—
69C2×#627C22×C312yes—
70C32×C4⋊C4—34no—
71C32×C4⋊C4—32no—
72C32×C22⋊C4—34no—
73#3750C22×C314yes—
74C3×#217—24no—
75C3×#217C2×D314yes—
76C3×#217C22×C314yes—
77C3×C4⋊Dic3—22no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3C2×D312yes—
80C3×C4⋊Dic3C22×C312yes—
81C3×Dic3⋊C4—24no—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4—34no—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—34no—
88C3×Dic3⋊C4—64no—
89C3×Dic3⋊C4C2×D314yes—
90C3×Dic3⋊C4C22×C314yes—
91C3×C4×Dic3—34no—
92C3×C4×Dic3—34no—
93#3724C22×C312yes—
94#3724C22×C312yes—
95#3724C22×C312yes—
96Dic3⋊Dic3C22×C314yes—
97Dic32C22×C314yes—
98C22×C33—21no—
99C22×C33—21no—
100C22×C33C2×Q812yes—
101C22×C33C2411yescharacteristic
102C33×C4—22no—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3C2×Q814yes—
105C32×Dic3—24no—
106C32×Dic3—24no—
107C3×C4⋊Q8—94no—
108C3×#125—94no—
109C3×C22⋊Q8—32no—
110C3×C22⋊Q8—32no—
111C3×C22⋊Q8—62no—
112C3×C22⋊Q8—92no—
113C3×#117—92no—
114C2×C3×C4⋊C4—91no—
115#944—34no—
116#924C3×D312yes—
117C2×Dic3⋊C4—32no—
118C2×Dic3⋊C4—61no—
119#889—34no—
120#889—34no—
121#888—34no—
122#878—34no—
123#877—34no—
124C23×C32—21no—
125C23×C32—22no—
126C23×C32—41no—
127C23×C32C22×D312yes—
128C23×C32C23×C311yescharacteristic
129C32×Q8—122no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—34no—
134C2×C32×C4—34no—
135C2×C32×C4—42no—
136C2×C32×C4—42no—
137C2×C32×C4—41no—
138C2×C32×C4—44no—
139C2×C32×C4—42no—
140C2×C32×C4—44no—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C23×C312yes—
144C2×C3⋊Dic3—41no—
145C2×C3⋊Dic3C3×Q814yescomplemented
146C2×C3⋊Dic3C23×C312yes—
147C2×C3⋊Dic3C23×C312yes—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—38no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—122no—
157C2×C3×Dic3—122no—
158C2×C3×Dic3C22×D314yes—
159C2×C3×Dic3C22×D314yes—
160C2×C3×Dic3C23×C314yes—
161C2×C3×Dic3C23×C314yes—
162C3×Dic6—42no—
163C3×Dic6—42no—
164C3×Dic6—44no—
165C3×Dic6—124no—
166#627—42no—
167#544—91noSylow
168C2×C33—22no—
169C2×C33C22×Q811yescharacteristic
170C2×C33H(5,𝔽2)11yescharacteristic
171C2×C332−1+411yescharacteristic
172C2×C3×Q8—32no—
173C2×C3×Q8—32no—
174C2×C3×Q8—62no—
175C2×C3×Q8—92no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—41no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4D3211yescharacteristic
182C22×C3×C4C2×C3×D312yes—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—61no—
186C22×Dic3—61no—
187C2×Dic6—34no—
188C2×Dic6C2×C3×D312yes—
189C3×C4⋊C4—34no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—34no—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—64no—
194C3×C4⋊C4—62no—
195C3×C4⋊C4—92no—
196C3×C4⋊C4—92no—
197C3×C4⋊C4—92no—
198C3×C4⋊C4—94no—
199C3×C4⋊C4—94no—
200C3×C22⋊C4—34no—
201C3×C22⋊C4—34no—
202C3×C22⋊C4—64no—
203C3×C42—94no—
204#217C2×C3×D314yes—
205C4⋊Dic3—34no—
206C4⋊Dic3C2×C3×D312yes—
207Dic3⋊C4—34no—
208Dic3⋊C4—34no—
209Dic3⋊C4—34no—
210Dic3⋊C4—34no—
211Dic3⋊C4—34no—
212Dic3⋊C4—34no—
213Dic3⋊C4—64no—
214Dic3⋊C4C2×C3×D314yes—
215C4×Dic3—34no—
216C4×Dic3—34no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—21no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—24no—
227C22×C32—42no—
228C22×C32—41no—
229C22×C32—42no—
230C22×C32—42no—
231C22×C32—42no—
232C22×C32—42no—
233C22×C32D3×Q814yes—
234C22×C32C2×C3×Q812yes—
235C22×C32C23×D312yes—
236C22×C32C24×C311yescharacteristic
237C32×C4—22no—
238C32×C4—24no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—42no—
242C32×C4—42no—
243C32×C4—44no—
244C32×C4—64no—
245C32×C4—64no—
246C3⋊Dic3—22no—
247C3⋊Dic3C2×C3×Q814yes—
248C3×Dic3—24no—
249C3×Dic3—24no—
250C3×Dic3—24no—
251C3×Dic3—24no—
252C3×Dic3—38no—
253C3×Dic3—44no—
254C3×Dic3—44no—
255C3×Dic3—64no—
256C3×Dic3—64no—
257C3×Dic3—122no—
258C4⋊Q8—94no—
259#125—94no—
260C22⋊Q8—32no—
261C22⋊Q8—92no—
262#117—92no—
263C2×C4⋊C4—91no—
264C33#54411yescharacteristic, complemented, Sylow
265C23×C3—21no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3C2×D3211yescharacteristic
269C23×C3C22×C3×D312yes—
270C3×Q8—122no—
271C3×Q8—122no—
272C3×Q8—124no—
273C3×Q8—362no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—34no—
278C2×C3×C4—34no—
279C2×C3×C4—34no—
280C2×C3×C4—34no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—44no—
286C2×C3×C4—42no—
287C2×C3×C4—64no—
288C2×C3×C4—64no—
289C2×C3×C4—92no—
290C2×C3×C4—94no—
291C2×C3×C4—92no—
292C2×C3×C4—361no—
293C2×C3×C4C2×D3212yes—
294C2×C3×C4C22×C3×D312yes—
295C2×C3×C4C22×C3×D312yes—
296C2×Dic3—34no—
297C2×Dic3—38no—
298C2×Dic3—34no—
299C2×Dic3—64no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C2×Dic3—122no—
303C2×Dic3—122no—
304C2×Dic3C22×C3×D314yes—
305C2×Dic3C22×C3×D314yes—
306Dic6—42no—
307Dic6—124no—
308C2×C32—22no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—44no—
321C2×C32—44no—
322C2×C32#100412yes—
323C2×C32C2×D3×Q812yes—
324C2×C32#101012yes—
325C2×C32C22×C3×Q811yescharacteristic
326C2×C32C3×H(5,𝔽2)11yescharacteristic
327C2×C32C3×2−1+411yescharacteristic
328C2×Q8—32no—
329C2×Q8—92no—
330C22×C4—91no—
331C22×C4—91no—
332C22×C4C3×D3211yescharacteristic
333C4⋊C4—32no—
334C4⋊C4—34no—
335C4⋊C4—92no—
336C4⋊C4—92no—
337C4⋊C4—92no—
338C4⋊C4—94no—
339C4⋊C4—94no—
340C22⋊C4—34no—
341C42—94no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—24no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3#379712yes—
359C22×C3C3×D3×Q814yes—
360C22×C3C22×D3211yescharacteristic
361C22×C3C23×C3×D312yes—
362C3×C4—24no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—44no—
369C3×C4—64no—
370C3×C4—64no—
371C3×C4—64no—
372C3×C4—64no—
373C3×C4—94no—
374C3×C4—124no—
375C3×C4—124no—
376C3×C4—182no—
377Dic3—24no—
378Dic3—24no—
379Dic3—38no—
380Dic3—64no—
381Dic3—64no—
382Dic3—122no—
383C32—22no—
384C32—21no—
385C32—41no—
386C32#562312yescomplemented
387C32C3×#54411yescharacteristic, complemented
388C23C2×C3×D3211yescharacteristic
389Q8—122no—
390Q8—362no—
391C2×C4—34no—
392C2×C4—34no—
393C2×C4—41no—
394C2×C4—92no—
395C2×C4—94no—
396C2×C4—92no—
397C2×C4—361no—
398C2×C4C2×C3×D3212yes—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—22no—
408C2×C3—44no—
409C2×C3—44no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3#6429811yescharacteristic
414C2×C3C2×#379711yescharacteristic
415C2×C3#6433011yescharacteristic
416C2×C3C3×#100412yes—
417C2×C3C2×C3×D3×Q812yes—
418C2×C3C3×#101012yes—
419C22—21no—
420C22—21no—
421C22C3×#379712yes—
422C22C22×C3×D3211yescharacteristic
423C4—22no—
424C4—64no—
425C4—64no—
426C4—94no—
427C4—182no—
428C3—21no—
429C3—22no—
430C3—41no—
431C3#1059481311yescharacteristic, direct product factor
432C3C3×#562312yescomplemented
433C2—22no—
434C2C3×#6429811yescharacteristic
435C2C2×C3×#379711yescharacteristic
436C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. C3×#63847C2semidirect
6. C3×#63839C2semidirect
8. #10594813C3direct (N × H)
38. #63847C2×C3semidirect
39. #63839C2×C3semidirect
44. C2×C3×C3⋊Dic3Q8semidirect
145. C2×C3⋊Dic3C3×Q8semidirect
264. C33#544semidirect
386. C32#5623semidirect
387. C32C3×#544semidirect
431. C3#10594813direct (N × H)
432. C3C3×#5623semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C32×#9242
C3×#639822
C3×#639572
C3×#638472
C3×#638392
C3×#56233
#105948133