Finite Group Databasebeta

← C3×#10591522 #13158549

Subgroups — C3×#10591522 #13158549

Order 1728.

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

normal subgroups only

This group has 970 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×#64308C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C32×#932C212yesmaximal, complemented
4C3×#63568C212yesmaximal, complemented
5C3×#63559C211yesmaximal, characteristic, complemented
6C3×#5198—32nomaximal
7#10591522C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic, complemented
9C2×C3×#638—22no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C32×#235C2212yes—
12C3×#3811—22no—
13C3×D3×D12—22no—
14C3×#3799—22no—
15C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
16C32×#214—22no—
17C32×#213—22no—
18C32×#208C2212yescomplemented
19C3×#1009—32no—
20#64308C2×C311yescharacteristic, complemented
21C32×AGL(1,ℤ/8)—32no—
22#64149C2×C311yescharacteristic, complemented
23C3×#940—32no—
24C3×#940—61no—
25C3×#932—22no—
26C3×#932D312yescomplemented
27C3×#932C2×C312yescomplemented
28C3×#837—32no—
29C3×#825—32no—
30C3×#823—32no—
31#63568C2×C312yescomplemented
32#63559C2×C311yescharacteristic, complemented
33C23×C33—21no—
34C22×C3×C3⋊D3D412yes—
35C2×C3×D32—22no—
36C33×D4—22no—
37C2×C33×C4C2311yescharacteristic
38C3×#638—44no—
39C2×C3×C3⋊Dic3—21no—
40C3×C4⋊(C3⋊D3)—22no—
41C32×#67—22no—
42C32×D12D412yescomplemented
43C32×C4×D3—22no—
44C32×Dic6D412yescomplemented
45C3×#626—24no—
46C3×#624—22no—
47C32×C3⋊C8—22no—
48C3×#453—91no—
49#5198—32no—
50C32×#41—32no—
51C2×C32×D4—22no—
52C2×C32×D4—21no—
53C2×C32×D4—41no—
54C2×C32×D4C2×D312yescomplemented
55C2×C32×D4C22×C311yescharacteristic, complemented
56C2×#638—22no—
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C3×#67—64no—
59C2×C3×#67—64no—
60C3×#239—62no—
61C3×D3×D4—62no—
62C3×D3×D4—64no—
63C3×#235—22no—
64C3×#235—62no—
65C3×#235C2×D312yes—
66C3×#235C22×C312yes—
67C2×C3×D12—32no—
68C2×C3×D12—62no—
69C2×C3×D12—61no—
70#3811—22no—
71D3×D12—22no—
72#3799—22no—
73C32×SD16—62no—
74C32×D8—62no—
75C32×M16—32no—
76C4×C3⋊Dic3C22×C311yescharacteristic, complemented
77C3×#214—24no—
78C3×#214—22no—
79C3×#214—22no—
80C3×#213—24no—
81C3×#213—22no—
82C3×#213—22no—
83C3×C4×Dic3—32no—
84C3×C4×Dic3—61no—
85C3×#208—22no—
86C3×#208C2×D312yescomplemented
87C3×#208C22×C312yescomplemented
88C22×C33—42no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—42no—
93C2×C32×D3—22no—
94C3×D32—82no—
95C33×C4C2×D412yes—
96C3×C3⋊Dic3—22no—
97C32×Dic3—22no—
98C3×H(5,𝔽2)—91no—
99#1009—32no—
100C3×AGL(1,ℤ/8)—32no—
101C3×AGL(1,ℤ/8)—32no—
102C3×AGL(1,ℤ/8)—62no—
103C3×#127—91no—
104#940—32no—
105#940—61no—
106#932C3×D312yescomplemented
107C3×#104—92no—
108C3×#100—91no—
109#837—32no—
110#825—32no—
111#823—32no—
112C23×C32—21no—
113C23×C32—22no—
114C23×C32—22no—
115C23×C32—42no—
116C23×C32—42no—
117C22×C3⋊D3C3×D412yes—
118C22×C3×D3—34no—
119C22×C3×D3—62no—
120C22×C3×D3—64no—
121C2×D32—22no—
122C32×Q8—32no—
123C32×D4—22no—
124C32×D4—24no—
125C32×D4—22no—
126C32×D4—22no—
127C32×D4—32no—
128C32×D4—42no—
129C32×D4—42no—
130C32×D4—44no—
131C32×D4—62no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—41no—
135C2×C32×C4—62no—
136C2×C32×C4C22×D312yes—
137C2×C32×C4C23×C311yescharacteristic
138#638—44no—
139C2×C3⋊Dic3—21no—
140C4⋊(C3⋊D3)—22no—
141C3×#67—22no—
142C3×#67—22no—
143C3×#67—42no—
144C3×#67—64no—
145C3×#67—128no—
146C3×#67—128no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C3×D12—22no—
150C3×D12—64no—
151C3×D12—62no—
152C3×D12—62no—
153C3×D12—122no—
154C3×D12#6712yescomplemented
155C3×D12C3×D412yescomplemented
156C3×C4×D3—22no—
157C3×C4×D3—22no—
158C3×C4×D3—42no—
159C3×C4×D3—64no—
160C3×Dic6—22no—
161C3×Dic6#6712yescomplemented
162C3×Dic6C3×D412yescomplemented
163#626—24no—
164#624—22no—
165C32×C8—62no—
166C3×C3⋊C8—24no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8—22no—
169#453—91noSylow
170C2×C33—21no—
171C2×C33—81no—
172C2×C33#12011yescharacteristic
173C3×C3⋊D3—42no—
174C32×D3—42no—
175C3×#41—32no—
176C3×#41—32no—
177C3×#41—62no—
178C3×#41—182no—
179C2×C3×D4—22no—
180C2×C3×D4—21no—
181C2×C3×D4—41no—
182C2×C3×D4—91no—
183C2×C3×D4—182no—
184C2×C3×D4—182no—
185C2×C3×D4—182no—
186C2×C3×D4D3211yescharacteristic, complemented
187C2×C3×D4C2×C3×D312yescomplemented
188C2×#67—64no—
189C2×#67—64no—
190#239—62no—
191D3×D4—62no—
192D3×D4—64no—
193#235—62no—
194#235C2×C3×D312yes—
195C2×D12—32no—
196C2×D12—61no—
197C2×D12—62no—
198C3×SD16—62no—
199C3×SD16—62no—
200C3×SD16—64no—
201C3×D8—62no—
202C3×D8—62no—
203C3×D8—64no—
204C3×M16—32no—
205C3×M16—32no—
206C3×M16—62no—
207C3×C42—91no—
208#214—22no—
209#213—22no—
210C4×Dic3—32no—
211C4×Dic3—61no—
212#208C2×C3×D312yescomplemented
213C22×C32—21no—
214C22×C32—22no—
215C22×C32—41no—
216C22×C32—42no—
217C22×C32—44no—
218C22×C32—42no—
219C22×C32—44no—
220C22×C32—42no—
221C22×C32—41no—
222C22×C32—44no—
223C22×C32—44no—
224C22×C32—62no—
225C22×C32—84no—
226C22×C32—84no—
227C22×C32D3×D412yes—
228C22×C32C2×C3×D411yescharacteristic
229C2×C3⋊D3—22no—
230C2×C3⋊D3—42no—
231C2×C3×D3—22no—
232C2×C3×D3—22no—
233C2×C3×D3—42no—
234C2×C3×D3—64no—
235C2×C3×D3—122no—
236C2×C3×D3—124no—
237C2×C3×D3—124no—
238C2×C3×D3—244no—
239D32—82no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—22no—
243C32×C4—42no—
244C32×C4—62no—
245C32×C4D3×D412yes—
246C32×C4C2×#6712yes—
247C32×C4C2×C3×D412yes—
248C3⋊Dic3—22no—
249C3×Dic3—22no—
250C3×Dic3—22no—
251C3×Dic3—42no—
252C3×Dic3—64no—
253C3×Dic3—64no—
254H(5,𝔽2)—91no—
255AGL(1,ℤ/8)—32no—
256#127—91no—
257#104—92no—
258#100—91no—
259C33#45311yescharacteristic, complemented, Sylow
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—92no—
266C23×C3—182no—
267C22×D3—34no—
268C22×D3—62no—
269C22×D3—64no—
270C3×Q8—32no—
271C3×Q8—32no—
272C3×Q8—62no—
273C3×D4—24no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—32no—
278C3×D4—32no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—44no—
282C3×D4—62no—
283C3×D4—62no—
284C3×D4—62no—
285C3×D4—122no—
286C3×D4—184no—
287C3×D4—182no—
288C3×D4—364no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—122no—
295C2×C3×C4—182no—
296C2×C3×C4—181no—
297C2×C3×C4C2×D3211yescharacteristic
298C2×C3×C4C22×C3×D312yes—
299#67—22no—
300#67—64no—
301#67—128no—
302#67—128no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305D12—64no—
306D12—62no—
307D12—62no—
308D12—122no—
309D12C3×#6712yescomplemented
310C4×D3—22no—
311C4×D3—64no—
312Dic6C3×#6712yescomplemented
313C3×C8—64no—
314C3×C8—62no—
315C3×C8—62no—
316C3⋊C8—22no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—21no—
322C2×C32—41no—
323C2×C32—42no—
324C2×C32—42no—
325C2×C32—82no—
326C2×C32—84no—
327C2×C32—84no—
328C2×C32—82no—
329C2×C32—81no—
330C2×C32—122no—
331C2×C32#93712yes—
332C2×C32C3×#12011yescharacteristic
333C3⋊D3—42no—
334C3×D3—42no—
335C3×D3—42no—
336C3×D3—82no—
337C3×D3—124no—
338C3×D3—242no—
339#41—32no—
340#41—182no—
341C2×D4—91no—
342C2×D4—182no—
343C2×D4—182no—
344C2×D4—182no—
345C2×D4C3×D3211yescharacteristic, complemented
346SD16—62no—
347D8—62no—
348M16—32no—
349C42—91no—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—44no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—62no—
362C22×C3—62no—
363C22×C3—84no—
364C22×C3—84no—
365C22×C3—122no—
366C22×C3—182no—
367C22×C3—362no—
368C22×C3—722no—
369C22×C3#381111yescharacteristic
370C22×C3C3×D3×D412yes—
371C2×D3—22no—
372C2×D3—64no—
373C2×D3—122no—
374C2×D3—124no—
375C2×D3—124no—
376C2×D3—244no—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—62no—
382C3×C4—62no—
383C3×C4—122no—
384C3×C4—182no—
385C3×C4D3×#6712yes—
386C3×C4C3×D3×D412yes—
387C3×C4C2×C3×#6712yes—
388Dic3—22no—
389Dic3—64no—
390Dic3—64no—
391C32—21no—
392C32—22no—
393C32—41no—
394C32#519812yescomplemented
395C32C3×#45311yescharacteristic, complemented
396C23—21no—
397C23—92no—
398C23—182no—
399Q8—32no—
400D4—22no—
401D4—32no—
402D4—62no—
403D4—184no—
404D4—182no—
405D4—364no—
406C2×C4—62no—
407C2×C4—182no—
408C2×C4—181no—
409C2×C4C2×C3×D3211yescharacteristic
410C8—62no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—22no—
415C2×C3—21no—
416C2×C3—42no—
417C2×C3—42no—
418C2×C3—41no—
419C2×C3—82no—
420C2×C3—84no—
421C2×C3—84no—
422C2×C3—81no—
423C2×C3—82no—
424C2×C3—122no—
425C2×C3—122no—
426C2×C3—242no—
427C2×C3—362no—
428C2×C3#6398311yescharacteristic
429C2×C3C3×#93712yes—
430D3—42no—
431D3—124no—
432D3—242no—
433C22—42no—
434C22—41no—
435C22—62no—
436C22—182no—
437C22—362no—
438C22—722no—
439C22C3×#381111yescharacteristic
440C4—62no—
441C4—182no—
442C4C3×D3×#6712yes—
443C3—21no—
444C3—22no—
445C3—41no—
446C3#1059152211yescharacteristic, direct product factor
447C3C3×#519812yescomplemented
448C2—21no—
449C2—81no—
450C2—122no—
451C2—362no—
452C2C3×#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×#453
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×#64308C2semidirect
2. C3×#64149C2semidirect
3. C32×#932C2semidirect
4. C3×#63568C2semidirect
5. C3×#63559C2semidirect
7. #10591522C3direct (N × H)
8. C2×C33×D4C22semidirect
15. C3×C4×C3⋊Dic3C22semidirect
18. C32×#208C22semidirect
20. #64308C2×C3semidirect
22. #64149C2×C3semidirect
26. C3×#932D3semidirect
27. C3×#932C2×C3semidirect
31. #63568C2×C3semidirect
32. #63559C2×C3semidirect
42. C32×D12D4semidirect
44. C32×Dic6D4semidirect
54. C2×C32×D4C2×D3semidirect
55. C2×C32×D4C22×C3semidirect
76. C4×C3⋊Dic3C22×C3semidirect
86. C3×#208C2×D3semidirect
87. C3×#208C22×C3semidirect
106. #932C3×D3semidirect
154. C3×D12#67semidirect
155. C3×D12C3×D4semidirect
161. C3×Dic6#67semidirect
162. C3×Dic6C3×D4semidirect
186. C2×C3×D4D32semidirect
187. C2×C3×D4C2×C3×D3semidirect
212. #208C2×C3×D3semidirect
259. C33#453semidirect
309. D12C3×#67semidirect
312. Dic6C3×#67semidirect
345. C2×D4C3×D32semidirect
394. C32#5198semidirect
395. C32C3×#453semidirect
446. C3#10591522direct (N × H)
447. C3C3×#5198semidirect

Maximal subgroups

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

subgroupindex
C3×#643082
C3×#641492
C32×#9322
C3×#635682
C3×#635592
C3×#51983
#105915223