Finite Group Databasebeta

← C3×#10591524 #13158551

Subgroups — C3×#10591524 #13158551

Order 1728.

850 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64298C211yesmaximal, characteristic, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×#932C212yesmaximal, complemented
4C3×#63568C212yesmaximal, complemented
5C3×#63561C211yesmaximal, characteristic, complemented
6C3×#5201—32nomaximal
7#10591524C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C32×#235C2212yes—
11C3×#3805—22no—
12C3×#3795—22no—
13C3×D3×Dic6—22no—
14C3×#3757—22no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×#214—22no—
17C32×#213—22no—
18C32×#208C2212yes—
19C3×#1010—32no—
20#64298C2×C311yescharacteristic, complemented
21C32×AGL(1,ℤ/8)—32no—
22#64145C2×C311yescharacteristic, complemented
23C3×#936—32no—
24C3×#936—61no—
25C3×#932—22no—
26C3×#932D312yescomplemented
27C3×#932C2×C312yescomplemented
28C3×#837—32no—
29C3×#825—32no—
30C3×#824—32no—
31#63568C2×C312yescomplemented
32#63561C2×C311yescharacteristic, complemented
33C23×C33—21no—
34C33×D4—22no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3D412yes—
38C3×#634—22no—
39C32×#67—22no—
40C32×D12D412yes—
41C32×C4×D3—22no—
42C32×Dic6D412yes—
43C3×#627—22no—
44C3×#625—22no—
45C3×D3×Dic3—24no—
46C32×C3⋊C8—22no—
47C3×#454—91no—
48#5201—32no—
49C32×#41—32no—
50C2×C32×D4—22no—
51C2×C32×D4—21no—
52C2×C32×D4—41no—
53C2×C32×D4C2×D312yes—
54C2×C32×D4C22×C311yescharacteristic
55C2×#634C22×C311yescharacteristic
56C3×D3×Q8—62no—
57C3×#237—62no—
58C3×#237—64no—
59C3×#235—22no—
60C3×#235—62no—
61C3×#235C2×D312yes—
62C3×#235C22×C312yes—
63C2×C3×Dic6—32no—
64C2×C3×Dic6—62no—
65C2×C3×Dic6—61no—
66#3805—22no—
67#3795—22no—
68D3×Dic6—22no—
69C32×SD16—62no—
70C32×D8—62no—
71C32×M16—32no—
72#3757—22no—
73C4×C3⋊Dic3C22×C311yescharacteristic
74C3×#217—64no—
75C3×#217—64no—
76C3×#214—24no—
77C3×#214—22no—
78C3×#214—22no—
79C3×#213—24no—
80C3×#213—22no—
81C3×#213—22no—
82C3×C4×Dic3—32no—
83C3×C4×Dic3—61no—
84C3×#208—22no—
85C3×#208C2×D312yes—
86C3×#208C22×C312yes—
87C22×C33—42no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—22no—
91C33×C4C2×D412yes—
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—22no—
95C3×2−1+4—91no—
96#1010—32no—
97C3×AGL(1,ℤ/8)—32no—
98C3×AGL(1,ℤ/8)—32no—
99C3×AGL(1,ℤ/8)—62no—
100C3×#124—91no—
101#936—32no—
102#936—61no—
103#932C3×D312yescomplemented
104C3×#104—92no—
105C3×#101—91no—
106#837—32no—
107#825—32no—
108#824—32no—
109C23×C32—21no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32—42no—
113C23×C32—42no—
114C32×Q8—32no—
115C32×D4—22no—
116C32×D4—24no—
117C32×D4—22no—
118C32×D4—22no—
119C32×D4—32no—
120C32×D4—42no—
121C32×D4—42no—
122C32×D4—44no—
123C32×D4—62no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—62no—
128C2×C32×C4C22×D312yes—
129C2×C32×C4C23×C311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3C3×D412yes—
132#634—22no—
133C3×#67—22no—
134C3×#67—22no—
135C3×#67—42no—
136C3×#67—64no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C3×D12—22no—
143C3×D12#6712yes—
144C3×D12C3×D412yes—
145C3×C4×D3—22no—
146C3×C4×D3—22no—
147C3×C4×D3—42no—
148C3×C4×D3—64no—
149C3×Dic6—22no—
150C3×Dic6—64no—
151C3×Dic6—62no—
152C3×Dic6—62no—
153C3×Dic6—122no—
154C3×Dic6#6712yes—
155C3×Dic6C3×D412yes—
156#627—22no—
157#625—22no—
158D3×Dic3—24no—
159C32×C8—62no—
160C3×C3⋊C8—24no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—22no—
163#454—91noSylow
164C2×C33—21no—
165C2×C33—81no—
166C2×C33#12011yescharacteristic
167C32×D3—42no—
168C3×#41—32no—
169C3×#41—32no—
170C3×#41—62no—
171C3×#41—182no—
172C3×#41—182no—
173C2×C3×Q8—91no—
174C2×C3×Q8—182no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C2×C3×D4D3211yescharacteristic
179C2×C3×D4C2×C3×D312yes—
180D3×Q8—62no—
181#237—62no—
182#237—64no—
183#235—62no—
184#235C2×C3×D312yes—
185C2×Dic6—32no—
186C2×Dic6—61no—
187C2×Dic6—62no—
188C3×SD16—62no—
189C3×SD16—62no—
190C3×SD16—64no—
191C3×D8—62no—
192C3×D8—62no—
193C3×D8—64no—
194C3×M16—32no—
195C3×M16—32no—
196C3×M16—62no—
197C3×C22⋊C4—182no—
198C3×C42—91no—
199#217—64no—
200#217—64no—
201#214—22no—
202#213—22no—
203C4×Dic3—32no—
204C4×Dic3—61no—
205#208C2×C3×D312yes—
206C22×C32—21no—
207C22×C32—22no—
208C22×C32—41no—
209C22×C32—44no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32—44no—
213C22×C32—42no—
214C22×C32—41no—
215C22×C32—44no—
216C22×C32—44no—
217C22×C32—62no—
218C22×C32—84no—
219C22×C32—84no—
220C22×C32D3×D412yes—
221C22×C32C2×C3×D411yescharacteristic
222C2×C3×D3—22no—
223C2×C3×D3—22no—
224C2×C3×D3—42no—
225C32×C4—22no—
226C32×C4—22no—
227C32×C4—22no—
228C32×C4—42no—
229C32×C4—62no—
230C32×C4D3×D412yes—
231C32×C4C2×#6712yes—
232C32×C4C2×C3×D412yes—
233C3⋊Dic3—22no—
234C3⋊Dic3—41no—
235C3×Dic3—22no—
236C3×Dic3—22no—
237C3×Dic3—42no—
238C3×Dic3—64no—
239C3×Dic3—122no—
240C3×Dic3—122no—
241C3×Dic3—122no—
2422−1+4—91no—
243AGL(1,ℤ/8)—32no—
244#124—91no—
245#104—92no—
246#101—91no—
247C33#45411yescharacteristic, complemented, Sylow
248C23×C3—22no—
249C23×C3—21no—
250C23×C3—22no—
251C23×C3—42no—
252C23×C3—42no—
253C3×Q8—32no—
254C3×Q8—32no—
255C3×Q8—62no—
256C3×Q8—182no—
257C3×Q8—182no—
258C3×D4—24no—
259C3×D4—22no—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—32no—
263C3×D4—32no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4—44no—
267C3×D4—62no—
268C3×D4—62no—
269C3×D4—62no—
270C3×D4—122no—
271C3×D4—182no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—92no—
278C2×C3×C4—122no—
279C2×C3×C4—184no—
280C2×C3×C4—181no—
281C2×C3×C4C2×D3211yescharacteristic
282C2×C3×C4C22×C3×D312yes—
283#67—22no—
284#67—64no—
285C2×Dic3—34no—
286C2×Dic3—62no—
287C2×Dic3—64no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290D12C3×#6712yes—
291C4×D3—22no—
292C4×D3—64no—
293Dic6—64no—
294Dic6—62no—
295Dic6—62no—
296Dic6—122no—
297Dic6C3×#6712yes—
298C3×C8—64no—
299C3×C8—62no—
300C3×C8—62no—
301C3⋊C8—22no—
302C2×C32—21no—
303C2×C32—22no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—42no—
310C2×C32—82no—
311C2×C32—84no—
312C2×C32—84no—
313C2×C32—82no—
314C2×C32—81no—
315C2×C32—122no—
316C2×C32#93712yes—
317C2×C32C3×#12011yescharacteristic
318C3×D3—42no—
319C3×D3—42no—
320C3×D3—82no—
321#41—32no—
322#41—182no—
323#41—182no—
324C2×Q8—91no—
325C2×Q8—182no—
326C2×D4C3×D3211yescharacteristic
327SD16—62no—
328D8—62no—
329M16—32no—
330C22⋊C4—182no—
331C42—91no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—44no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—44no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—62no—
344C22×C3—62no—
345C22×C3—84no—
346C22×C3—84no—
347C22×C3—122no—
348C22×C3#381111yescharacteristic
349C22×C3C3×D3×D412yes—
350C2×D3—22no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—42no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—122no—
358C3×C4—182no—
359C3×C4—361no—
360C3×C4D3×#6712yes—
361C3×C4C3×D3×D412yes—
362C3×C4C2×C3×#6712yes—
363Dic3—22no—
364Dic3—64no—
365Dic3—122no—
366Dic3—122no—
367Dic3—122no—
368C32—21no—
369C32—22no—
370C32—41no—
371C32#520112yescomplemented
372C32C3×#45411yescharacteristic, complemented
373C23—21no—
374Q8—32no—
375Q8—182no—
376Q8—182no—
377D4—22no—
378D4—32no—
379D4—62no—
380D4—182no—
381C2×C4—62no—
382C2×C4—92no—
383C2×C4—184no—
384C2×C4—181no—
385C2×C4C2×C3×D3211yescharacteristic
386C8—62no—
387C2×C3—22no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—42no—
393C2×C3—42no—
394C2×C3—41no—
395C2×C3—82no—
396C2×C3—84no—
397C2×C3—84no—
398C2×C3—81no—
399C2×C3—82no—
400C2×C3—122no—
401C2×C3—122no—
402C2×C3—242no—
403C2×C3#6398311yescharacteristic
404C2×C3C3×#93712yes—
405D3—42no—
406C22—42no—
407C22—41no—
408C22—62no—
409C22C3×#381111yescharacteristic
410C4—62no—
411C4—182no—
412C4—361no—
413C4C3×D3×#6712yes—
414C3—21no—
415C3—22no—
416C3—41no—
417C3#1059152411yescharacteristic, direct product factor
418C3C3×#520112yescomplemented
419C2—21no—
420C2—81no—
421C2—122no—
422C2C3×#6398311yescharacteristic
subgroupG/H
centerC2×C3#63983
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#120
hypercenterC2×C3×D4D32
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64298C2semidirect
2. C3×#64145C2semidirect
3. C32×#932C2semidirect
4. C3×#63568C2semidirect
5. C3×#63561C2semidirect
7. #10591524C3direct (N × H)
20. #64298C2×C3semidirect
22. #64145C2×C3semidirect
26. C3×#932D3semidirect
27. C3×#932C2×C3semidirect
31. #63568C2×C3semidirect
32. #63561C2×C3semidirect
103. #932C3×D3semidirect
247. C33#454semidirect
371. C32#5201semidirect
372. C32C3×#454semidirect
417. C3#10591524direct (N × H)
418. C3C3×#5201semidirect

Maximal subgroups

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

subgroupindex
C3×#642982
C3×#641452
C32×#9322
C3×#635682
C3×#635612
C3×#52013
#105915243