Finite Group Databasebeta

← C3×#10591518 #13158545

Subgroups — C3×#10591518 #13158545

Order 1728.

916 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 136 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×#3797C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic
3C2×C32×#214C212yesmaximal, complemented
4C3×#63566C212yesmaximal, complemented
5C3×#63557C211yesmaximal, characteristic, complemented
6C3×#5170—32nomaximal
7#10591518C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C2×C3×#627—21no—
13C2×C3×#624—21no—
14C3×#3797—24no—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×#214—24no—
18C2×C32×C3⋊C8C2212yes—
19C2×C3×D3×Q8—32no—
20C2×#3797C2×C311yescharacteristic, complemented
21C2×C32×SD16—32no—
22#64147C2×C311yescharacteristic
23C3×#938—32no—
24C3×#938—61no—
25C2×C3×#214—22no—
26C2×C3×#214D312yescomplemented
27C2×C3×#214C2×C312yescomplemented
28C3×#835—32no—
29C3×(C2×D3)⋊C8—32no—
30C3×#816—32no—
31#63566C2×C312yescomplemented
32#63557C2×C311yescharacteristic, complemented
33C23×C33—21no—
34C22×C3×C3⋊D3D411yescharacteristic
35C33×D4—22no—
36C2×C33×C4C2311yescharacteristic
37C3×#638—42no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4×C3⋊D3—22no—
41C2×C32×Dic3—22no—
42C32×Dic6—22no—
43C32×Dic6D414yescomplemented
44C3×#627—24no—
45C3×#624—24no—
46C32×C3⋊C8—22no—
47C3×#448—91no—
48#5170—32no—
49C2×C32×Q8—32no—
50C2×C32×D4—22no—
51C2×C32×D4—21no—
52C2×C32×D4—41no—
53C2×C32×D4C2×D312yes—
54C2×C32×D4C22×C311yescharacteristic
55C2×#638—21no—
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×#67—62no—
58C2×C3×#67—62no—
59C3×D3×Q8—64no—
60C3×D3×Q8—64no—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—62no—
63C2×C3×C4×D3—61no—
64C2×C3×Dic6—22no—
65C2×C3×Dic6—62no—
66C2×C3×Dic6C2×D312yes—
67C2×C3×Dic6C22×C312yes—
68C2×#627—21no—
69C2×#624—21no—
70#3797—24no—
71C32×SD16—64no—
72C2×C32×C8—32no—
73#3757—21no—
74C4⋊(C3⋊Dic3)C22×C311yescharacteristic
75C3×#217—62no—
76C3×#217—62no—
77C3×#214—28no—
78C3×#214—24no—
79C3×#214—24no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—61no—
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8C2×D312yes—
84C2×C3×C3⋊C8C22×C312yes—
85C22×C33—41no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3SD1612yescomplemented
92C33×C4C2×D412yes—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C32×Dic3—24no—
96C22×C3×Q8—91no—
97C2×D3×Q8—32no—
98C2×C3×SD16—32no—
99C2×C3×SD16—32no—
100C2×C3×SD16—62no—
101C3×#121—91no—
102#938—32no—
103#938—61no—
104C2×#214C3×D312yescomplemented
105C3×#103—92no—
106C3×C22⋊C8—91no—
107#835—32no—
108(C2×D3)⋊C8—32no—
109#816—32no—
110C23×C32—21no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—42no—
114C23×C32—42no—
115C22×C3⋊D3C3×D411yescharacteristic
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C32×Q8—34no—
119C32×Q8—62no—
120C32×D4—22no—
121C32×D4—24no—
122C32×D4—22no—
123C32×D4—22no—
124C32×D4—42no—
125C32×D4—42no—
126C32×D4—44no—
127C2×C32×C4—21no—
128C2×C32×C4—22no—
129C2×C32×C4—41no—
130C2×C32×C4—62no—
131C2×C32×C4C22×D312yes—
132C2×C32×C4C23×C311yescharacteristic
133#638—42no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C3×D411yescharacteristic
136C4×C3⋊D3—22no—
137C3×#67—124no—
138C3×#67—124no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—42no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C3×C4×D3—68no—
147C3×C4×D3—62no—
148C3×C4×D3—62no—
149C3×C4×D3—122no—
150C3×Dic6—22no—
151C3×Dic6—22no—
152C3×Dic6—24no—
153C3×Dic6—42no—
154C3×Dic6—68no—
155C3×Dic6#6714yescomplemented
156C3×Dic6C3×D414yescomplemented
157#627—24no—
158#624—24no—
159C32×C8—62no—
160C3×C3⋊C8—24no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—22no—
163#448—91noSylow
164C2×C33—81no—
165C2×C33#12011yescharacteristic
166C2×C33C2×SD1611yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—22no—
169C2×C3×Q8—32no—
170C2×C3×Q8—32no—
171C2×C3×Q8—62no—
172C2×C3×Q8—181no—
173C2×C3×Q8—184no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—41no—
177C2×C3×D4—181no—
178C2×C3×D4D3211yescharacteristic
179C2×C3×D4C2×C3×D312yes—
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—62no—
183C2×#67—62no—
184D3×Q8—64no—
185D3×Q8—64no—
186C2×C4×D3—32no—
187C2×C4×D3—61no—
188C2×C4×D3—62no—
189C2×Dic6—62no—
190C2×Dic6C2×C3×D312yes—
191C3×SD16—64no—
192C3×SD16—64no—
193C3×SD16—68no—
194C2×C3×C8—32no—
195C2×C3×C8—32no—
196C2×C3×C8—62no—
197C3×C4⋊C4—91no—
198C3×C22⋊C4—181no—
199#217—62no—
200#217—62no—
201#214—24no—
202C4⋊Dic3—32no—
203C4⋊Dic3—61no—
204C2×C3⋊C8C2×C3×D312yes—
205C22×C32—21no—
206C22×C32—22no—
207C22×C32—41no—
208C22×C32—44no—
209C22×C32—42no—
210C22×C32—44no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—82no—
220C22×C32—82no—
221C22×C32—82no—
222C22×C32—82no—
223C22×C32D3×D412yes—
224C22×C32C2×C3×D411yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3C3×SD1612yescomplemented
228C2×C3×D3—34no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—42no—
238C32×C4—64no—
239C32×C4D3×D412yes—
240C32×C4C2×#6712yes—
241C32×C4C2×C3×D412yes—
242C3⋊Dic3—21no—
243C3⋊Dic3—41no—
244C3×Dic3—24no—
245C3×Dic3—24no—
246C3×Dic3—44no—
247C3×Dic3—62no—
248C3×Dic3—121no—
249C3×Dic3—122no—
250C3×Dic3—122no—
251C22×Q8—91no—
252C2×SD16—32no—
253#121—91no—
254#103—92no—
255C22⋊C8—91no—
256C33#44811yescharacteristic, complemented, Sylow
257C23×C3—22no—
258C23×C3—21no—
259C23×C3—22no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—91no—
263C22×D3—32no—
264C22×D3—61no—
265C3×Q8—34no—
266C3×Q8—34no—
267C3×Q8—64no—
268C3×Q8—62no—
269C3×Q8—62no—
270C3×Q8—122no—
271C3×Q8—184no—
272C3×D4—24no—
273C3×D4—22no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—44no—
279C3×D4—362no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—41no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—91no—
286C2×C3×C4—122no—
287C2×C3×C4—184no—
288C2×C3×C4—182no—
289C2×C3×C4—181no—
290C2×C3×C4C2×D3211yescharacteristic
291C2×C3×C4C22×C3×D312yes—
292#67—124no—
293#67—124no—
294C2×Dic3—22no—
295C2×Dic3—32no—
296C2×Dic3—61no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C4×D3—68no—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—122no—
303Dic6—22no—
304Dic6—68no—
305Dic6C3×#6714yescomplemented
306C3×C8—64no—
307C3×C8—62no—
308C3×C8—62no—
309C3⋊C8—22no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—82no—
320C2×C32—84no—
321C2×C32—84no—
322C2×C32—82no—
323C2×C32—81no—
324C2×C32D3×SD1612yes—
325C2×C32#91512yes—
326C2×C32#93712yes—
327C2×C32C3×#12011yescharacteristic
328C2×C32C2×C3×SD1611yescharacteristic
329C2×C32C3×#13711yescharacteristic
330C3⋊D3—22no—
331C3×D3—64no—
332C3×D3—122no—
333C2×Q8—32no—
334C2×Q8—181no—
335C2×Q8—184no—
336C2×D4—181no—
337C2×D4C3×D3211yescharacteristic
338C22×C4—91no—
339C22×C4—181no—
340SD16—64no—
341C2×C8—32no—
342C4⋊C4—91no—
343C22⋊C4—181no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—42no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—92no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3#381111yescharacteristic
366C22×C3C3×D3×D412yes—
367C2×D3—34no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—121no—
372C2×D3—121no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—64no—
378C3×C4—64no—
379C3×C4—124no—
380C3×C4—181no—
381C3×C4—361no—
382C3×C4D3×#6712yes—
383C3×C4C3×D3×D412yes—
384C3×C4C2×C3×#6712yes—
385Dic3—24no—
386Dic3—62no—
387Dic3—121no—
388Dic3—122no—
389Dic3—122no—
390C32—22no—
391C32—21no—
392C32—41no—
393C32#517012yescomplemented
394C32C3×#44811yescharacteristic, complemented
395C23—21no—
396C23—91no—
397Q8—34no—
398Q8—62no—
399Q8—184no—
400D4—22no—
401D4—362no—
402C2×C4—62no—
403C2×C4—91no—
404C2×C4—184no—
405C2×C4—182no—
406C2×C4—181no—
407C2×C4C2×C3×D3211yescharacteristic
408C8—62no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—82no—
419C2×C3—84no—
420C2×C3—84no—
421C2×C3—82no—
422C2×C3—81no—
423C2×C3—182no—
424C2×C3#6393011yescharacteristic
425C2×C3#6393111yescharacteristic
426C2×C3#6398311yescharacteristic
427C2×C3C3×D3×SD1612yes—
428C2×C3C3×#91512yes—
429C2×C3C3×#93712yes—
430D3—64no—
431D3—122no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—92no—
436C22—181no—
437C22—181no—
438C22C3×#381111yescharacteristic
439C4—64no—
440C4—181no—
441C4—361no—
442C4C3×D3×#6712yes—
443C3—21no—
444C3—22no—
445C3—41no—
446C3#1059151811yescharacteristic, direct product factor
447C3C3×#517012yescomplemented
448C2—81no—
449C2—182no—
450C2C3×#6393011yescharacteristic
451C2C3×#6393111yescharacteristic
452C2C3×#6398311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#448
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3797C2semidirect
3. C2×C32×#214C2semidirect
4. C3×#63566C2semidirect
5. C3×#63557C2semidirect
7. #10591518C3direct (N × H)
20. C2×#3797C2×C3semidirect
26. C2×C3×#214D3semidirect
27. C2×C3×#214C2×C3semidirect
31. #63566C2×C3semidirect
32. #63557C2×C3semidirect
43. C32×Dic6D4semidirect
91. C2×C3×C3⋊D3SD16semidirect
104. C2×#214C3×D3semidirect
155. C3×Dic6#67semidirect
156. C3×Dic6C3×D4semidirect
227. C2×C3⋊D3C3×SD16semidirect
256. C33#448semidirect
305. Dic6C3×#67semidirect
393. C32#5170semidirect
394. C32C3×#448semidirect
446. C3#10591518direct (N × H)
447. C3C3×#5170semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C3×#641472
C2×C32×#2142
C3×#635662
C3×#635572
C3×#51703
#105915183