Finite Group Databasebeta

← C3×#10591645 #13158658

Subgroups — C3×#10591645 #13158658

Order 1728.

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

normal subgroups only

This group has 882 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×#1007C211yesmaximal, characteristic, complemented
2C32×#947C211yesmaximal, characteristic, complemented
3C3×#63823C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5C3×#63569C212yesmaximal, complemented
6C3×#5245—31nomaximal
7C3×#5062—31nomaximal
8#10591645C311yesmaximal, characteristic, direct product factor
9C2×C33×Q8C2211yescharacteristic
10C32×#239—22no—
11C32×D3×Q8—22no—
12C32×#235—22no—
13C32×#235C2212yes—
14C2×C32×D12C2211yescharacteristic
15C3×#3748C2212yes—
16C32×(C2×D3)⋊C4—22no—
17C32×C4×Dic3C2211yescharacteristic
18C3×#3715—22no—
19C3×#3713—22no—
20C32×2−1+4—31no—
21C3×#1007—21no—
22C3×#1007D311yescharacteristic, complemented
23C3×#1007C2×C311yescharacteristic, complemented
24C32×#124—31no—
25C3×#949—32no—
26C3×#947—21no—
27C3×#947D311yescharacteristic, complemented
28C3×#947C2×C311yescharacteristic, complemented
29C3×#932—32no—
30#63823C2×C312yescomplemented
31#63663C2×C311yescharacteristic, complemented
32C3×#837—32no—
33C3×#836—31no—
34C3×#836—31no—
35C3×#836—61no—
36C3×#805—32no—
37#63569C2×C312yescomplemented
38C22×C32×D3—21no—
39C33×Q8—22no—
40C2×C33×C4D412yes—
41C2×C33×C4C2311yescharacteristic
42C32×#67—22no—
43C2×C32×Dic3—21no—
44C32×D12—22no—
45C32×D12—22no—
46C32×D12D412yes—
47C32×C4×D3—24no—
48C32×C4×D3—22no—
49C32×Dic6—22no—
50C32×Dic6D412yes—
51C3×C3⋊(C3⋊C8)—22no—
52C3×#454—91no—
53#5245—31no—
54#5062—31no—
55C32×#41—32no—
56C32×#41—62no—
57C32×#41—62no—
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8—21no—
61C2×C32×Q8—41no—
62C2×C32×Q8—62no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C22×C311yescharacteristic
66C2×C32×D4—31no—
67C3×#239—24no—
68C3×#239—22no—
69C3×#239—22no—
70C3×D3×Q8—24no—
71C3×D3×Q8—22no—
72C3×D3×Q8—22no—
73C3×#235—24no—
74C3×#235—22no—
75C3×#235—22no—
76C3×#235—22no—
77C3×#235C2×D312yes—
78C3×#235C22×C312yes—
79C2×C3×D12—21no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C22×C311yescharacteristic
82C32×C22⋊C4—62no—
83C32×C42—31no—
84#3748C22×C312yes—
85C3×#215—62no—
86C3×#214—62no—
87C3×#213—62no—
88C3×#213—62no—
89C3×(C2×D3)⋊C4—24no—
90C3×(C2×D3)⋊C4—22no—
91C3×(C2×D3)⋊C4—22no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C22×C311yescharacteristic
95C3×#208—32no—
96C3×#208—32no—
97C3×#208—62no—
98#3715—22no—
99#3713—22no—
100C22×C33C2×D411yescharacteristic
101C2×C32×D3—22no—
102C2×C32×D3—42no—
103C2×C32×D3—41no—
104C33×C4—22no—
105C33×C4C2×D412yes—
106C32×Dic3—22no—
107C32×Dic3—41no—
108C3×2−1+4—31no—
109C3×2−1+4—31no—
110C3×2−1+4—61no—
111#1007C3×D311yescharacteristic, complemented
112C3×AGL(1,ℤ/8)—92no—
113C3×#124—31no—
114C3×#124—31no—
115C3×#124—61no—
116#949—32no—
117#947C3×D311yescharacteristic, complemented
118#932—32no—
119C3×#104—92no—
120C3×#101—91no—
121#837—32no—
122#836—31no—
123#836—31no—
124#836—61no—
125#805—32no—
126C23×C32—61no—
127C22×C3×D3—22no—
128C22×C3×D3—21no—
129C22×C3×D3—21no—
130C32×Q8—22no—
131C32×Q8—24no—
132C32×Q8—22no—
133C32×Q8—22no—
134C32×Q8—32no—
135C32×Q8—42no—
136C32×Q8—42no—
137C32×Q8—44no—
138C32×Q8—62no—
139C32×D4—32no—
140C32×D4—62no—
141C32×D4—62no—
142C32×D4—62no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—42no—
150C2×C32×C4—41no—
151C2×C32×C4—64no—
152C2×C32×C4—62no—
153C2×C32×C4—61no—
154C2×C32×C4#6712yes—
155C2×C32×C4#6712yes—
156C2×C32×C4C3×D412yes—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160C3×#67—24no—
161C3×#67—22no—
162C3×#67—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C3×D12—24no—
167C3×D12—22no—
168C3×D12—22no—
169C3×D12—22no—
170C3×D12—22no—
171C3×D12—22no—
172C3×D12—42no—
173C3×D12#6712yes—
174C3×D12C3×D412yes—
175C3×C4×D3—28no—
176C3×C4×D3—24no—
177C3×C4×D3—24no—
178C3×C4×D3—24no—
179C3×C4×D3—22no—
180C3×C4×D3—22no—
181C3×Dic6—24no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C3×Dic6—22no—
185C3×Dic6#6712yes—
186C3×Dic6C3×D412yes—
187C3⋊(C3⋊C8)—22no—
188C3×C3⋊C8—62no—
189C3×C3⋊C8—62no—
190C3×C3⋊C8—122no—
191#454—91noSylow
192C2×C33—21no—
193C2×C33#12011yescharacteristic
194C32×D3—42no—
195C32×D3—81no—
196C3×#41—32no—
197C3×#41—32no—
198C3×#41—62no—
199C3×#41—64no—
200C3×#41—62no—
201C3×#41—62no—
202C3×#41—64no—
203C3×#41—62no—
204C3×#41—62no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—41no—
209C2×C3×Q8—62no—
210C2×C3×Q8—62no—
211C2×C3×Q8—64no—
212C2×C3×Q8D3211yescharacteristic
213C2×C3×Q8C2×C3×D311yescharacteristic
214C2×C3×Q8C2×C3×D311yescharacteristic
215C2×C3×D4—31no—
216C2×C3×D4—31no—
217C2×C3×D4—61no—
218#239—22no—
219D3×Q8—22no—
220#235—22no—
221#235C2×C3×D312yes—
222C2×D12C2×C3×D311yescharacteristic
223C3×SD16—182no—
224C3×D8—182no—
225C3×M16—92no—
226C3×C22⋊C4—64no—
227C3×C22⋊C4—62no—
228C3×C22⋊C4—62no—
229C3×C42—31no—
230C3×C42—31no—
231C3×C42—61no—
232#215—62no—
233#214—62no—
234#213—62no—
235#213—62no—
236(C2×D3)⋊C4—22no—
237C4×Dic3C2×C3×D311yescharacteristic
238#208—32no—
239#208—32no—
240#208—62no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32—62no—
246C22×C32—122no—
247C22×C32—121no—
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C22×C32C2×C3×D411yescharacteristic
251C2×C3×D3—24no—
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—42no—
255C2×C3×D3—44no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C32×C4—22no—
261C32×C4—24no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—42no—
268C32×C4—42no—
269C32×C4—42no—
270C32×C4—44no—
271C32×C4—62no—
272C32×C4—121no—
273C32×C4D3×D412yes—
274C32×C4C2×#6712yes—
275C32×C4C2×C3×D412yes—
276C3×Dic3—24no—
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C3×Dic3—41no—
2822−1+4—31no—
283AGL(1,ℤ/8)—92no—
284#124—31no—
285#104—92no—
286#101—91no—
287C33#45411yescharacteristic, complemented, Sylow
288C23×C3—62no—
289C23×C3—61no—
290C23×C3—61no—
291C22×D3—21no—
292C3×Q8—24no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×Q8—32no—
297C3×Q8—32no—
298C3×Q8—42no—
299C3×Q8—42no—
300C3×Q8—44no—
301C3×Q8—62no—
302C3×Q8—64no—
303C3×Q8—62no—
304C3×Q8—62no—
305C3×D4—32no—
306C3×D4—32no—
307C3×D4—62no—
308C3×D4—64no—
309C3×D4—62no—
310C3×D4—62no—
311C3×D4—64no—
312C3×D4—62no—
313C3×D4—62no—
314C3×D4—62no—
315C3×D4—62no—
316C3×D4—122no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—68no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4#62512yes—
335C2×C3×C4C3×#6712yes—
336C2×C3×C4C3×#6712yes—
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C22×C3×D311yescharacteristic
339C2×C3×C4C22×C3×D311yescharacteristic
340#67—22no—
341C2×Dic3—21no—
342D12—22no—
343D12—22no—
344D12C3×#6712yes—
345C4×D3—24no—
346C4×D3—22no—
347Dic6—22no—
348Dic6C3×#6712yes—
349C3×C8—182no—
350C3⋊C8—62no—
351C3⋊C8—62no—
352C3⋊C8—122no—
353C2×C32—21no—
354C2×C32—22no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—42no—
364C2×C32—122no—
365C2×C32—241no—
366C2×C32#93711yescharacteristic
367C2×C32#95311yescharacteristic
368C2×C32C3×#12011yescharacteristic
369C3×D3—44no—
370C3×D3—42no—
371C3×D3—42no—
372C3×D3—82no—
373C3×D3—81no—
374C3×D3—81no—
375#41—32no—
376#41—62no—
377#41—62no—
378C2×Q8—62no—
379C2×Q8C3×D3211yescharacteristic
380C2×D4—31no—
381SD16—182no—
382D8—182no—
383M16—92no—
384C22⋊C4—62no—
385C42—31no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3—64no—
391C22×C3—62no—
392C22×C3—62no—
393C22×C3—122no—
394C22×C3—124no—
395C22×C3—122no—
396C22×C3—122no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3C2×#62511yescharacteristic
400C22×C3C2×C3×#6711yescharacteristic
401C22×C3C2×C3×#6711yescharacteristic
402C2×D3—22no—
403C2×D3—42no—
404C2×D3—41no—
405C3×C4—24no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—44no—
415C3×C4—42no—
416C3×C4—64no—
417C3×C4—62no—
418C3×C4—62no—
419C3×C4—122no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4D3×#6712yes—
423C3×C4C3×D3×D412yes—
424C3×C4C2×C3×#6712yes—
425Dic3—22no—
426Dic3—41no—
427C32—21no—
428C32—21no—
429C32—21no—
430C32—41no—
431C32#506211yescharacteristic, complemented
432C32#524511yescharacteristic, complemented
433C32C3×#45411yescharacteristic, complemented
434C23—61no—
435Q8—22no—
436Q8—32no—
437Q8—62no—
438D4—32no—
439D4—62no—
440D4—62no—
441D4—62no—
442C2×C4—64no—
443C2×C4—62no—
444C2×C4—61no—
445C2×C4C3×#62512yes—
446C2×C4C2×C3×D3211yescharacteristic
447C8—182no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—42no—
458C2×C3—41no—
459C2×C3—124no—
460C2×C3—122no—
461C2×C3—122no—
462C2×C3—242no—
463C2×C3—241no—
464C2×C3—241no—
465C2×C3#6397611yescharacteristic
466C2×C3C3×#93711yescharacteristic
467C2×C3C3×#95311yescharacteristic
468D3—42no—
469D3—81no—
470C22—62no—
471C22—122no—
472C22—121no—
473C22C2×C3×#62511yescharacteristic
474C4—22no—
475C4—62no—
476C4—121no—
477C4C3×D3×#6712yes—
478C3—21no—
479C3—21no—
480C3—21no—
481C3—41no—
482C3#1059164511yescharacteristic, direct product factor
483C3C3×#506211yescharacteristic, complemented
484C3C3×#524511yescharacteristic, complemented
485C2—21no—
486C2—122no—
487C2—241no—
488C2C3×#6397611yescharacteristic
subgroupG/H
centerC2×C3#63976
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC2×C33#120
hypercenterC2×C3×Q8D32
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×#1007C2semidirect
2. C32×#947C2semidirect
3. C3×#63823C2semidirect
4. C3×#63663C2semidirect
5. C3×#63569C2semidirect
8. #10591645C3direct (N × H)
22. C3×#1007D3semidirect
23. C3×#1007C2×C3semidirect
27. C3×#947D3semidirect
28. C3×#947C2×C3semidirect
30. #63823C2×C3semidirect
31. #63663C2×C3semidirect
37. #63569C2×C3semidirect
111. #1007C3×D3semidirect
117. #947C3×D3semidirect
287. C33#454semidirect
431. C32#5062semidirect
432. C32#5245semidirect
433. C32C3×#454semidirect
482. C3#10591645direct (N × H)
483. C3C3×#5062semidirect
484. C3C3×#5245semidirect

Maximal subgroups

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

subgroupindex
C32×#10072
C32×#9472
C3×#638232
C3×#636632
C3×#635692
C3×#52453
C3×#50623
#105916453