Finite Group Databasebeta

← C3×#10591513 #13158539

Subgroups — C3×#10591513 #13158539

Order 1728.

950 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C32×#1004C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C3×#63831C212yesmaximal, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5C3×#63568C212yesmaximal, complemented
6C3×#5242—31nomaximal
7C3×#4746—31nomaximal
8#10591513C311yesmaximal, characteristic, direct product factor
9C2×C33×D4C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic, complemented
11C2×C32×#67—22no—
12C32×#237—22no—
13C32×D3×D4—22no—
14C32×#235C2212yes—
15C3×#3757—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
17C32×#208C2212yescomplemented
18C3×#3719—22no—
19C3×#3716—22no—
20C32×H(5,𝔽2)—31no—
21C3×#1004—21no—
22C3×#1004D311yescharacteristic
23C3×#1004C2×C311yescharacteristic
24#64145C2×C311yescharacteristic, complemented
25C3×#951—32no—
26C3×#936—31no—
27C3×#936—31no—
28C3×#936—61no—
29C3×#909—32no—
30#63831C2×C312yescomplemented
31C32×#100—31no—
32C3×#837—32no—
33C3×#833—21no—
34C3×#833D311yescharacteristic, complemented
35C3×#833C2×C311yescharacteristic, complemented
36C3×#825—32no—
37#63568C2×C312yescomplemented
38C23×C33D412yes—
39C22×C32×D3—22no—
40C33×D4—22no—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—22no—
45C32×#67—24no—
46C32×#67—22no—
47C2×C32×Dic3—22no—
48C32×D12D412yes—
49C32×C4×D3—22no—
50C32×Dic6D412yes—
51C32×C3⋊C8—22no—
52C3×#455—91no—
53#5242—31no—
54#4746—31no—
55C32×#41—32no—
56C32×#41—62no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4—41no—
61C2×C32×D4—62no—
62C2×C32×D4—62no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C22×C311yescharacteristic
66C2×#634C22×C311yescharacteristic, complemented
67C2×C3×#67—24no—
68C2×C3×#67—22no—
69C2×C3×#67—22no—
70C3×#237—24no—
71C3×#237—22no—
72C3×#237—22no—
73C3×D3×D4—24no—
74C3×D3×D4—22no—
75C3×D3×D4—22no—
76C3×#235—22no—
77C3×#235C2×D312yes—
78C3×#235C22×C312yes—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C32×M16—32no—
83#3757—22no—
84C4×C3⋊Dic3C22×C311yescharacteristic, complemented
85C3×#217—64no—
86C3×#217—62no—
87C3×#217—62no—
88C3×#216—62no—
89C3×#214—62no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—61no—
93C3×#208—22no—
94C3×#208C2×D312yescomplemented
95C3×#208C22×C312yescomplemented
96C3×Dic12—62no—
97C3×#204—62no—
98#3719—22no—
99#3716—22no—
100C22×C33—22no—
101C22×C33—42no—
102C22×C33C2×D411yescharacteristic
103C2×C32×D3—22no—
104C2×C32×D3—82no—
105C33×C4C2×D412yes—
106C3×C3⋊Dic3—41no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—22no—
109C3×H(5,𝔽2)—31no—
110C3×H(5,𝔽2)—31no—
111C3×H(5,𝔽2)—61no—
112#1004C3×D311yescharacteristic
113C3×#137—92no—
114C3×#124—91no—
115#951—32no—
116#936—31no—
117#936—31no—
118#936—61no—
119#909—32no—
120C3×#104—92no—
121C3×#100—31no—
122C3×#100—31no—
123C3×#100—61no—
124#837—32no—
125#833C3×D311yescharacteristic, complemented
126#825—32no—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32—22no—
130C23×C32—42no—
131C23×C32—62no—
132C23×C32D1212yes—
133C23×C32#6712yes—
134C23×C32C3×D412yes—
135C22×C3×D3—24no—
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C32×Q8—32no—
139C32×D4—22no—
140C32×D4—24no—
141C32×D4—22no—
142C32×D4—22no—
143C32×D4—32no—
144C32×D4—42no—
145C32×D4—42no—
146C32×D4—44no—
147C32×D4—64no—
148C32×D4—62no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—41no—
153C2×C32×C4—62no—
154C2×C32×C4—62no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160#634—22no—
161C3×#67—24no—
162C3×#67—28no—
163C3×#67—24no—
164C3×#67—24no—
165C3×#67—22no—
166C3×#67—22no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C3×D12—22no—
177C3×D12#6712yes—
178C3×D12C3×D412yes—
179C3×C4×D3—24no—
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×Dic6—22no—
183C3×Dic6—62no—
184C3×Dic6—62no—
185C3×Dic6—122no—
186C3×Dic6#6712yes—
187C3×Dic6C3×D412yes—
188C32×C8—62no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—42no—
192#455—91noSylow
193C2×C33—21no—
194C2×C33—42no—
195C2×C33#12011yescharacteristic
196C32×D3—42no—
197C3×#41—32no—
198C3×#41—32no—
199C3×#41—62no—
200C3×#41—62no—
201C3×#41—62no—
202C3×#41—64no—
203C2×C3×Q8—91no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—41no—
208C2×C3×D4—62no—
209C2×C3×D4—62no—
210C2×C3×D4—64no—
211C2×C3×D4—64no—
212C2×C3×D4—62no—
213C2×C3×D4—62no—
214C2×C3×D4D3211yescharacteristic
215C2×C3×D4C2×C3×D311yescharacteristic
216C2×C3×D4C2×C3×D311yescharacteristic
217C2×#67—22no—
218#237—22no—
219D3×D4—22no—
220#235C2×C3×D312yes—
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C3×Q16—182no—
225C3×SD16—182no—
226C3×M16—32no—
227C3×M16—32no—
228C3×M16—62no—
229C3×C22⋊C4—182no—
230C3×C42—91no—
231#217—64no—
232#217—62no—
233#217—62no—
234#216—62no—
235#214—62no—
236C4×Dic3—31no—
237C4×Dic3—31no—
238C4×Dic3—61no—
239#208C2×C3×D312yescomplemented
240Dic12—62no—
241#204—62no—
242C22×C32—22no—
243C22×C32—24no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—42no—
251C22×C32—42no—
252C22×C32—44no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—44no—
257C22×C32—62no—
258C22×C32—82no—
259C22×C32—82no—
260C22×C32—84no—
261C22×C32—242no—
262C22×C32C2×D1211yescharacteristic
263C22×C32C2×#6711yescharacteristic
264C22×C32C2×C3×D411yescharacteristic
265C2×C3×D3—24no—
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—84no—
269C2×C3×D3—82no—
270C2×C3×D3—82no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—42no—
275C32×C4—62no—
276C32×C4D3×D412yes—
277C32×C4C2×#6712yes—
278C32×C4C2×C3×D412yes—
279C3⋊Dic3—41no—
280C3⋊Dic3—41no—
281C3×Dic3—24no—
282C3×Dic3—22no—
283C3×Dic3—22no—
284C3×Dic3—122no—
285C3×Dic3—122no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C3×Dic3—121no—
290H(5,𝔽2)—31no—
291#137—92no—
292#124—91no—
293#104—92no—
294#100—31no—
295C33#45511yescharacteristic, complemented, Sylow
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—64no—
301C23×C3—62no—
302C23×C3—62no—
303C23×C3#62612yes—
304C23×C3C3×D1212yes—
305C23×C3C3×#6712yes—
306C22×D3—22no—
307C3×Q8—32no—
308C3×Q8—32no—
309C3×Q8—62no—
310C3×Q8—182no—
311C3×D4—24no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—32no—
316C3×D4—32no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—44no—
320C3×D4—62no—
321C3×D4—68no—
322C3×D4—64no—
323C3×D4—64no—
324C3×D4—64no—
325C3×D4—62no—
326C3×D4—62no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—64no—
332C2×C3×C4—64no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C22×C3×D311yescharacteristic
341C2×C3×C4C22×C3×D311yescharacteristic
342#67—24no—
343#67—22no—
344C2×Dic3—22no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351D12C3×#6712yes—
352C4×D3—22no—
353Dic6—62no—
354Dic6—62no—
355Dic6—122no—
356Dic6C3×#6712yes—
357C3×C8—62no—
358C3×C8—62no—
359C3×C8—122no—
360C3⋊C8—22no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32—44no—
374C2×C32—42no—
375C2×C32—42no—
376C2×C32—82no—
377C2×C32—82no—
378C2×C32—84no—
379C2×C32—122no—
380C2×C32#88211yescharacteristic
381C2×C32#95311yescharacteristic
382C2×C32C3×#12011yescharacteristic
383C3×D3—44no—
384C3×D3—42no—
385C3×D3—42no—
386#41—32no—
387#41—62no—
388C2×Q8—91no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4C3×D3211yescharacteristic
392Q16—182no—
393SD16—182no—
394M16—32no—
395C22⋊C4—182no—
396C42—91no—
397C22×C3—24no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—41no—
412C22×C3—64no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—84no—
418C22×C3—244no—
419C22×C3—242no—
420C22×C3—242no—
421C22×C3C2×#62611yescharacteristic
422C22×C3C2×C3×D1211yescharacteristic
423C22×C3C2×C3×#6711yescharacteristic
424C2×D3—22no—
425C2×D3—82no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—42no—
430C3×C4—64no—
431C3×C4—62no—
432C3×C4—62no—
433C3×C4—361no—
434C3×C4—361no—
435C3×C4D3×#6712yes—
436C3×C4C3×D3×D412yes—
437C3×C4C2×C3×#6712yes—
438Dic3—22no—
439Dic3—122no—
440Dic3—122no—
441Dic3—121no—
442Dic3—121no—
443Dic3—121no—
444Dic3—121no—
445C32—21no—
446C32—21no—
447C32—21no—
448C32—41no—
449C32#474611yescharacteristic, complemented
450C32#524211yescharacteristic, complemented
451C32C3×#45511yescharacteristic, complemented
452C23—62no—
453C23C3×#62612yes—
454Q8—32no—
455Q8—182no—
456D4—22no—
457D4—32no—
458D4—64no—
459D4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4C2×C3×D3211yescharacteristic
465C8—62no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—42no—
476C2×C3—44no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—41no—
481C2×C3—82no—
482C2×C3—82no—
483C2×C3—84no—
484C2×C3—124no—
485C2×C3—122no—
486C2×C3—122no—
487C2×C3#6397711yescharacteristic
488C2×C3C3×#88211yescharacteristic
489C2×C3C3×#95311yescharacteristic
490D3—42no—
491C22—22no—
492C22—42no—
493C22—62no—
494C22—242no—
495C22C2×C3×#62611yescharacteristic
496C4—62no—
497C4—361no—
498C4—361no—
499C4C3×D3×#6712yes—
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3#1059151311yescharacteristic, direct product factor
505C3C3×#474611yescharacteristic, complemented
506C3C3×#524211yescharacteristic, complemented
507C2—21no—
508C2—42no—
509C2—122no—
510C2C3×#6397711yescharacteristic
subgroupG/H
centerC2×C3#63977
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#455
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
2. C3×#64145C2semidirect
3. C3×#63831C2semidirect
4. C32×#833C2semidirect
5. C3×#63568C2semidirect
8. #10591513C3direct (N × H)
10. C2×C3×#634C22semidirect
16. C3×C4×C3⋊Dic3C22semidirect
17. C32×#208C22semidirect
24. #64145C2×C3semidirect
30. #63831C2×C3semidirect
34. C3×#833D3semidirect
35. C3×#833C2×C3semidirect
37. #63568C2×C3semidirect
66. C2×#634C22×C3semidirect
84. C4×C3⋊Dic3C22×C3semidirect
94. C3×#208C2×D3semidirect
95. C3×#208C22×C3semidirect
125. #833C3×D3semidirect
239. #208C2×C3×D3semidirect
295. C33#455semidirect
449. C32#4746semidirect
450. C32#5242semidirect
451. C32C3×#455semidirect
504. C3#10591513direct (N × H)
505. C3C3×#4746semidirect
506. C3C3×#5242semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C3×#641452
C3×#638312
C32×#8332
C3×#635682
C3×#52423
C3×#47463
#105915133