Finite Group Databasebeta

← C3×#10591001 #13158018

Subgroups — C3×#10591001 #13158018

Order 1728.

924 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×C3×#3802C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#63818C211yesmaximal, characteristic, complemented
4C32×#823C212yesmaximal, complemented
5C3×#63558C212yesmaximal, complemented
6C3×D3×#100—32nomaximal
7#10591001C311yesmaximal, characteristic, direct product factor
8C22×C3×D32C412yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C3×#625C412yes—
12C3×#3802—21no—
13C3×#3802—21no—
14C3×#3802—21no—
15C3×#3802—21no—
16C33×M16C2211yescharacteristic, complemented
17C3×#3748C2211yescharacteristic, complemented
18C3×#3743—21no—
19C3×C8×C3⋊D3—21no—
20C32×#208C2212yescomplemented
21C3×#3712—21no—
22C3×#3710—21no—
23C2×C3×D3×D4—32no—
24C2×#3802C2×C311yescharacteristic
25M16×C3⋊D3C2×C311yescharacteristic, complemented
26C3×D3×M16—32no—
27C3×D3×M16—32no—
28C3×D3×M16—61no—
29#63818C2×C311yescharacteristic, complemented
30C32×#100—32no—
31C3×#833—32no—
32C3×#823—22no—
33C3×#823—32no—
34C3×#823D312yescomplemented
35C3×#823C2×C312yescomplemented
36#63558C2×C312yescomplemented
37C22×C3×C3⋊D3C2×C411yescharacteristic
38C22×C32×D3C2×C414yes—
39C2×C3×D32—22no—
40C2×C3×D32—22no—
41C2×C3×D32—44no—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C3×C4×C3⋊D3D411yescharacteristic
47C3×C4×C3⋊D3D411yescharacteristic
48C32×D12—22no—
49C32×D12—22no—
50C3×#625—22no—
51C3×#625—22no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8—22no—
55C2×C3×#100—91no—
56D3×#100—32no—
57C23×C3×D3—34no—
58C22×D32C3×C412yes—
59C2×C32×D4—32no—
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C2×C3×#67—34no—
62C3×D3×D4—62no—
63C3×D3×D4—62no—
64C3×D3×D4—62no—
65C3×D3×D4—62no—
66C2×C3×D12—22no—
67C2×C3×D12C2×D312yes—
68C2×C3×D12C22×C312yes—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—61no—
71C2×#625C3×C412yes—
72#3802—21no—
73#3802—21no—
74#3802—21no—
75#3802—21no—
76C32×M16—21no—
77C32×M16—22no—
78C32×M16—32no—
79C32×M16—41no—
80C32×M16C2×D312yescomplemented
81C32×M16C22×C311yescharacteristic, complemented
82#3748C22×C311yescharacteristic, complemented
83#3743—21no—
84C8×C3⋊D3—21no—
85C3×#208—22no—
86C3×#208—32no—
87C3×#208—61no—
88C3×#208C2×D312yescomplemented
89C3×#208C22×C312yescomplemented
90C3×ΓL(1,𝔽25)—62no—
91C3×ΓL(1,𝔽25)—62no—
92C3×ΓL(1,𝔽25)—62no—
93C3×D3×C8—62no—
94C3×D3×C8—62no—
95C3×D3×C8—62no—
96#3712—21no—
97#3710—21no—
98C22×C33C22×C411yescharacteristic
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3C22⋊C411yescharacteristic
101C2×C3×C3⋊D3C22⋊C411yescharacteristic
102C2×C32×D3—24no—
103C2×C32×D3—44no—
104C3×D32—44no—
105C3×D32—44no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3C22⋊C411yescharacteristic
109C3×C3⋊Dic3C22⋊C411yescharacteristic
110C22×C3×D4—91no—
111C2×D3×D4—32no—
112C2×C3×M16—91no—
113C2×C3×M16—91no—
114D3×M16—32no—
115D3×M16—32no—
116D3×M16—61no—
117C3×#100—32no—
118C3×#100—32no—
119C3×#100—62no—
120C3×#100—92no—
121#833—32no—
122#823—32no—
123#823C3×D312yescomplemented
124C23×C32—34no—
125C22×C3⋊D3C2×C3×C411yescharacteristic
126C22×C3×D3—24no—
127C22×C3×D3—32no—
128C22×C3×D3—61no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—128no—
132C22×C3×D3C4×D314yes—
133C22×C3×D3C2×C3×C414yes—
134C2×D32—22no—
135C2×D32—22no—
136C2×D32—44no—
137C32×D4—62no—
138C32×D4—62no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—41no—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C23×C311yescharacteristic
144C2×C3⋊Dic3C2×C3×C411yescharacteristic
145C4×C3⋊D3C3×D411yescharacteristic
146C4×C3⋊D3C3×D411yescharacteristic
147C4×C3⋊D3C3×D411yescharacteristic
148C4×C3⋊D3C3×D411yescharacteristic
149C3×#67—64no—
150C3×#67—64no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C3×D12—22no—
154C3×D12—22no—
155C3×D12—22no—
156C3×D12—22no—
157C3×D12—42no—
158C3×D12—42no—
159C3×C4×D3—32no—
160C3×C4×D3—32no—
161C3×C4×D3—32no—
162C3×C4×D3—32no—
163C3×C4×D3—61no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167#625—22no—
168#625—22no—
169C32×C8—21no—
170C32×C8—22no—
171C32×C8—22no—
172C32×C8—42no—
173C32×C8—42no—
174C32×C8—62no—
175C3⋊(C3⋊C8)—21no—
176C3×C3⋊C8—22no—
177C3×C3⋊C8—22no—
178C3×C3⋊C8—42no—
179C3×C3⋊C8—62no—
180C3×C3⋊C8—62no—
181C2×#100—91noSylow
182C2×C33—21no—
183C2×C33C2×C22⋊C411yescharacteristic
184C3×C3⋊D3—21no—
185C3×C3⋊D3#10012yescomplemented
186C32×D3—44no—
187C24×C3—92no—
188C23×D3—34no—
189C2×C3×D4—32no—
190C2×C3×D4—32no—
191C2×C3×D4—62no—
192C2×C3×D4—92no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×C3×D4—181no—
196C2×C3×D4—181no—
197C22×C3×C4—91no—
198C2×#67—34no—
199D3×D4—62no—
200D3×D4—62no—
201D3×D4—62no—
202D3×D4—62no—
203C2×D12C2×C3×D312yes—
204C2×C4×D3—32no—
205C2×C4×D3—61no—
206C3×M16—21no—
207C3×M16—22no—
208C3×M16—32no—
209C3×M16—32no—
210C3×M16—41no—
211C3×M16—62no—
212C3×M16—91no—
213C3×M16—181no—
214C3×M16—181no—
215C3×M16D3211yescharacteristic, complemented
216C3×M16C2×C3×D312yescomplemented
217C2×C3×C8—181no—
218C2×C3×C8—181no—
219#208—32no—
220#208—61no—
221#208C2×C3×D312yescomplemented
222ΓL(1,𝔽25)—62no—
223ΓL(1,𝔽25)—62no—
224ΓL(1,𝔽25)—62no—
225D3×C8—62no—
226D3×C8—62no—
227D3×C8—62no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—41no—
231C22×C32—64no—
232C22×C32—124no—
233C22×C32C2×C4×D312yes—
234C22×C32C22×C3×C411yescharacteristic
235C2×C3⋊D3—22no—
236C2×C3⋊D3C3×C22⋊C411yescharacteristic
237C2×C3⋊D3C3×C22⋊C411yescharacteristic
238C2×C3×D3—24no—
239C2×C3×D3—24no—
240C2×C3×D3—32no—
241C2×C3×D3—32no—
242C2×C3×D3—44no—
243C2×C3×D3—44no—
244C2×C3×D3—44no—
245C2×C3×D3—61no—
246C2×C3×D3—61no—
247C2×C3×D3—64no—
248C2×C3×D3—84no—
249C2×C3×D3—122no—
250C2×C3×D3—128no—
251C2×C3×D3—128no—
252D32—44no—
253D32—44no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—41no—
259C32×C4—41no—
260C32×C4D3×D412yes—
261C32×C4D3×D412yes—
262C32×C4C2×C3×D411yescharacteristic
263C32×C4C2×C3×D411yescharacteristic
264C3⋊Dic3C3×C22⋊C411yescharacteristic
265C3⋊Dic3C3×C22⋊C411yescharacteristic
266C3×Dic3—32no—
267C3×Dic3—32no—
268C3×Dic3—61no—
269C3×Dic3—61no—
270C22×D4—91no—
271C2×M16—91no—
272C2×M16—91no—
273#100—32no—
274#100—92no—
275C33C2×#10011yescharacteristic, complemented, Sylow
276C23×C3—34no—
277C23×C3—34no—
278C23×C3—64no—
279C23×C3—91no—
280C23×C3—182no—
281C23×C3—182no—
282C23×C3—364no—
283C22×D3—32no—
284C22×D3—61no—
285C22×D3—64no—
286C22×D3—64no—
287C22×D3—128no—
288C22×D3C3×C4×D314yes—
289C3×D4—62no—
290C3×D4—62no—
291C3×D4—62no—
292C3×D4—62no—
293C3×D4—122no—
294C3×D4—122no—
295C3×D4—182no—
296C3×D4—182no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—91no—
301C2×C3×C4—91no—
302C2×C3×C4—91no—
303C2×C3×C4—91no—
304C2×C3×C4—91no—
305C2×C3×C4C2×D3211yescharacteristic
306C2×C3×C4C22×C3×D312yes—
307#67—64no—
308#67—64no—
309C2×Dic3—32no—
310C2×Dic3—61no—
311D12—22no—
312D12—22no—
313C4×D3—32no—
314C4×D3—32no—
315C4×D3—32no—
316C4×D3—32no—
317C4×D3—61no—
318C4×D3—61no—
319C4×D3—61no—
320C4×D3—61no—
321C3×C8—22no—
322C3×C8—21no—
323C3×C8—22no—
324C3×C8—42no—
325C3×C8—42no—
326C3×C8—62no—
327C3×C8—62no—
328C3×C8—122no—
329C3×C8—181no—
330C3⋊C8—22no—
331C3⋊C8—62no—
332C3⋊C8—62no—
333C2×C32—22no—
334C2×C32—21no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—42no—
339C2×C32—41no—
340C2×C32—81no—
341C2×C32—124no—
342C2×C32D3×C22⋊C412yes—
343C2×C32C2×C3×C22⋊C411yescharacteristic
344C3⋊D3—21no—
345C3⋊D3C3×#10012yescomplemented
346C3×D3—34no—
347C3×D3—44no—
348C3×D3—44no—
349C3×D3—62no—
350C3×D3—62no—
351C3×D3—84no—
352C3×D3—121no—
353C24—92no—
354C2×D4—32no—
355C2×D4—92no—
356C2×D4—181no—
357C2×D4—181no—
358C2×D4—181no—
359C2×D4—181no—
360C22×C4—91no—
361M16—32no—
362M16—91no—
363M16—181no—
364M16—181no—
365M16C3×D3211yescharacteristic, complemented
366C2×C8—181no—
367C2×C8—181no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—41no—
371C22×C3—64no—
372C22×C3—64no—
373C22×C3—91no—
374C22×C3—91no—
375C22×C3—124no—
376C22×C3—124no—
377C22×C3—124no—
378C22×C3—182no—
379C22×C3—244no—
380C22×C3—364no—
381C22×C3—364no—
382C22×C3C4×D3211yescharacteristic
383C22×C3C2×C3×C4×D312yes—
384C2×D3—24no—
385C2×D3—32no—
386C2×D3—32no—
387C2×D3—44no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—64no—
391C2×D3—122no—
392C2×D3—128no—
393C2×D3—128no—
394C3×C4—22no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—22no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—91no—
401C3×C4—91no—
402C3×C4#380211yescharacteristic
403C3×C4#381111yescharacteristic
404C3×C4C3×D3×D412yes—
405C3×C4C3×D3×D412yes—
406Dic3—32no—
407Dic3—32no—
408Dic3—61no—
409Dic3—61no—
410C32—21no—
411C32—22no—
412C32—41no—
413C32D3×#10012yescomplemented
414C32C2×C3×#10011yescharacteristic, complemented
415C23—34no—
416C23—91no—
417C23—182no—
418C23—182no—
419C23—364no—
420D4—62no—
421D4—62no—
422D4—182no—
423D4—182no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4C2×C3×D3211yescharacteristic
430C8—21no—
431C8—62no—
432C8—181no—
433C2×C3—21no—
434C2×C3—22no—
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—41no—
438C2×C3—42no—
439C2×C3—41no—
440C2×C3—81no—
441C2×C3—92no—
442C2×C3—124no—
443C2×C3—124no—
444C2×C3—181no—
445C2×C3—244no—
446C2×C3#6392111yescharacteristic
447C2×C3C3×D3×C22⋊C412yes—
448D3—34no—
449D3—44no—
450D3—62no—
451D3—62no—
452D3—121no—
453C22—64no—
454C22—91no—
455C22—91no—
456C22—124no—
457C22—182no—
458C22—364no—
459C22—364no—
460C22C3×C4×D3211yescharacteristic
461C4—91no—
462C4—91no—
463C4C3×#380211yescharacteristic
464C4C3×#381111yescharacteristic
465C3—21no—
466C3—22no—
467C3—41no—
468C3#1059100111yescharacteristic, direct product factor
469C3C3×D3×#10012yescomplemented
470C2—21no—
471C2—92no—
472C2—124no—
473C2—181no—
474C2C3×#6392111yescharacteristic
subgroupG/H
centerC2×C3#63921
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3802C2
cosocleC2×C32×C4C23×C3
hypercenterC3×M16D32
hypocenterC32C2×C3×#100
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. C3×#63818C2semidirect
4. C32×#823C2semidirect
5. C3×#63558C2semidirect
7. #10591001C3direct (N × H)
16. C33×M16C22semidirect
17. C3×#3748C22semidirect
20. C32×#208C22semidirect
25. M16×C3⋊D3C2×C3semidirect
29. #63818C2×C3semidirect
34. C3×#823D3semidirect
35. C3×#823C2×C3semidirect
36. #63558C2×C3semidirect
80. C32×M16C2×D3semidirect
81. C32×M16C22×C3semidirect
82. #3748C22×C3semidirect
88. C3×#208C2×D3semidirect
89. C3×#208C22×C3semidirect
123. #823C3×D3semidirect
185. C3×C3⋊D3#100semidirect
215. C3×M16D32semidirect
216. C3×M16C2×C3×D3semidirect
221. #208C2×C3×D3semidirect
275. C33C2×#100semidirect
345. C3⋊D3C3×#100semidirect
365. M16C3×D32semidirect
413. C32D3×#100semidirect
414. C32C2×C3×#100semidirect
468. C3#10591001direct (N × H)
469. C3C3×D3×#100semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C3×M16×C3⋊D32
C3×#638182
C32×#8232
C3×#635582
C3×D3×#1003
#105910013