Finite Group Databasebeta

← C3×#10593529 #13166898

Subgroups — C3×#10593529 #13166898

Order 1728.

634 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 62 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×#64330C211yesmaximal, characteristic
2C3×#64239C212yesmaximal, complemented
3C3×#63953C211yesmaximal, characteristic, complemented
4C3×#63788C211yesmaximal, characteristic, complemented
5C3×#63741C212yesmaximal, complemented
6#10593529C311yesmaximal, characteristic, direct product factor
7C2×C3×#638C2211yescharacteristic
8C2×C32×#67—21no—
9C3×#3811—21no—
10C3×D3×#67—22no—
11C2×C3×#627C2211yescharacteristic, complemented
12C3×#3805C2212yes—
13C3×#3804—22no—
14C3×#3792C2212yescomplemented
15C3×#3776—22no—
16C3×#3775—22no—
17C32×#217—21no—
18C3×#3722—21no—
19C3×Dic32C2211yescharacteristic, complemented
20C3×#1004—61no—
21#64330C2×C311yescharacteristic
22#64239C2×C312yescomplemented
23C3×#885—61no—
24#63953C2×C311yescharacteristic, complemented
25#63788C2×C311yescharacteristic, complemented
26#63741C2×C312yescomplemented
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3D411yescharacteristic
29C22×C32×D3—21no—
30C2×C3×D32—21no—
31C3×#638—22no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C32×#67—22no—
34C32×#67—22no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C3×#627—22no—
39C3×#627D412yes—
40C3×#626—22no—
41C3×#625D412yes—
42C3×#624—21no—
43C3×D3×Dic3—22no—
44C3×#622—22no—
45C3×#455—91nomaximal
46C2×C32×D4—61no—
47C2×#638C22×C311yescharacteristic
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—41no—
51C2×C3×#67—62no—
52C2×C3×#67—61no—
53C2×C3×#67—61no—
54C3×#237—62no—
55C3×#237—62no—
56C3×D3×D4—62no—
57C3×D3×D4—62no—
58C3×#235—62no—
59C2×C3×Dic6—61no—
60#3811—21no—
61D3×#67—22no—
62C2×#627C22×C311yescharacteristic, complemented
63#3805C22×C312yes—
64#3804—22no—
65#3792C22×C312yescomplemented
66#3776—22no—
67#3775—22no—
68C32×C22⋊C4—61no—
69C3×#217—21no—
70C3×#217—21no—
71C3×#217—41no—
72C3×(C2×D3)⋊C4—61no—
73C3×(C2×D3)⋊C4—61no—
74C3×C4×Dic3—61no—
75#3722—21no—
76Dic32C22×C311yescharacteristic, complemented
77C22×C33—21no—
78C22×C33—41no—
79C22×C33C2×D411yescharacteristic
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—41no—
82C2×C32×D3—22no—
83C2×C32×D3—81no—
84C3×D32—81no—
85C3×C3⋊Dic3C2×D412yes—
86C32×Dic3—22no—
87C32×Dic3—41no—
88C32×Dic3—41no—
89C3×H(5,𝔽2)—91no—
90#1004—61no—
91C3×#137—92no—
92C3×#124—91no—
93#885—61no—
94C3×#104—92no—
95C3×#100—91no—
96C23×C32—21no—
97C23×C32—21no—
98C23×C32—41no—
99C23×C32—41no—
100C23×C32—61no—
101C23×C32C3×D411yescharacteristic
102C22×C3⋊D3C3×D411yescharacteristic
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C22×C3×D3—41no—
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C22×C3×D3—62no—
109C2×D32—21no—
110C32×D4—62no—
111C32×D4—62no—
112C2×C32×C4—61no—
113C2×C32×C4—61no—
114C2×C32×C4—61no—
115#638—22no—
116C2×C3⋊Dic3C23×C311yescharacteristic
117C3×#67—22no—
118C3×#67—22no—
119C3×#67—22no—
120C3×#67—22no—
121C3×#67—42no—
122C3×#67—42no—
123C3×#67—62no—
124C3×#67—62no—
125C3×#67—62no—
126C3×#67—62no—
127C3×#67—122no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—62no—
140C3×D12—62no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×Dic6—62no—
144C3×Dic6—122no—
145#627—22no—
146#627C3×D412yes—
147#626—22no—
148#625C3×D412yes—
149#624—21no—
150D3×Dic3—22no—
151#622—22no—
152#455—91noSylow
153C2×C33—21no—
154C2×C33—41no—
155C2×C33#12011yescharacteristic
156C3×C3⋊D3—41no—
157C32×D3—42no—
158C3×#41—92no—
159C3×#41—182no—
160C2×C3×Q8—91no—
161C2×C3×D4—61no—
162C2×C3×D4—61no—
163C2×C3×D4—91no—
164C2×C3×D4—121no—
165C2×C3×D4—181no—
166C2×C3×D4—182no—
167C2×#67—21no—
168C2×#67—61no—
169C2×#67—61no—
170C2×#67—62no—
171#237—62no—
172#237—62no—
173D3×D4—62no—
174D3×D4—62no—
175#235—62no—
176C2×Dic6—61no—
177C3×Q16—182no—
178C3×SD16—182no—
179C3×M16—92no—
180C3×C22⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—121no—
183C3×C22⋊C4—181no—
184C3×C42—91no—
185#217—21no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188C4×Dic3—61no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—42no—
203C22×C32—62no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32—82no—
208C22×C32—241no—
209C22×C32C2×C3×D411yescharacteristic
210C2×C3⋊D3—21no—
211C2×C3⋊D3—41no—
212C2×C3×D3—22no—
213C2×C3×D3—22no—
214C2×C3×D3—42no—
215C2×C3×D3—62no—
216C2×C3×D3—81no—
217C2×C3×D3—81no—
218C2×C3×D3—82no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—241no—
223C2×C3×D3—242no—
224D32—81no—
225C32×C4—62no—
226C32×C4—121no—
227C32×C4—121no—
228C3⋊Dic3C2×C3×D412yes—
229C3×Dic3—22no—
230C3×Dic3—22no—
231C3×Dic3—42no—
232C3×Dic3—41no—
233C3×Dic3—41no—
234C3×Dic3—41no—
235C3×Dic3—41no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—81no—
239C3×Dic3—81no—
240H(5,𝔽2)—91no—
241#137—92no—
242#124—91no—
243#104—92no—
244#100—91no—
245C33#45511yescharacteristic, complemented, Sylow
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—61no—
251C23×C3—61no—
252C23×C3—91no—
253C23×C3—121no—
254C23×C3—181no—
255C23×C3GO+(4,𝔽2)11yescharacteristic
256C22×D3—21no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—62no—
260C3×Q8—92no—
261C3×Q8—182no—
262C3×D4—62no—
263C3×D4—62no—
264C3×D4—62no—
265C3×D4—62no—
266C3×D4—92no—
267C3×D4—122no—
268C3×D4—122no—
269C3×D4—182no—
270C3×D4—182no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—91no—
278C2×C3×C4—121no—
279C2×C3×C4—121no—
280C2×C3×C4—121no—
281C2×C3×C4—181no—
282C2×C3×C4—182no—
283#67—22no—
284#67—22no—
285#67—62no—
286#67—62no—
287#67—62no—
288#67—62no—
289#67—122no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—62no—
296D12—62no—
297C4×D3—62no—
298C4×D3—62no—
299Dic6—62no—
300Dic6—122no—
301C3×C8—182no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—42no—
315C2×C32—41no—
316C2×C32—81no—
317C2×C32—81no—
318C2×C32—81no—
319C2×C32—82no—
320C2×C32—122no—
321C2×C32C3×#12011yescharacteristic
322C3⋊D3—41no—
323C3×D3—42no—
324C3×D3—42no—
325C3×D3—82no—
326C3×D3—122no—
327C3×D3—241no—
328#41—92no—
329#41—182no—
330C2×Q8—91no—
331C2×D4—61no—
332C2×D4—91no—
333C2×D4—181no—
334C2×D4—182no—
335Q16—182no—
336SD16—182no—
337M16—92no—
338C22⋊C4—61no—
339C22⋊C4—181no—
340C42—91no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—62no—
356C22×C3—62no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—82no—
361C22×C3—122no—
362C22×C3—181no—
363C22×C3—241no—
364C22×C3—241no—
365C22×C3—242no—
366C22×C3—361no—
367C22×C3—721no—
368C22×C3C2×GO+(4,𝔽2)11yescharacteristic
369C2×D3—22no—
370C2×D3—62no—
371C2×D3—81no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—241no—
376C2×D3—242no—
377C3×C4—62no—
378C3×C4—62no—
379C3×C4—92no—
380C3×C4—122no—
381C3×C4—121no—
382C3×C4—121no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—241no—
386C3×C4—241no—
387Dic3—22no—
388Dic3—41no—
389Dic3—41no—
390Dic3—62no—
391Dic3—62no—
392C32—21no—
393C32—21no—
394C32—41no—
395C32—41no—
396C32C3×#45511yescharacteristic, complemented
397C23—61no—
398C23—91no—
399C23—181no—
400C23C3×GO+(4,𝔽2)11yescharacteristic
401Q8—92no—
402Q8—182no—
403D4—62no—
404D4—62no—
405D4—92no—
406D4—182no—
407D4—182no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—91no—
412C2×C4—181no—
413C2×C4—182no—
414C8—182no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—42no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—81no—
430C2×C3—81no—
431C2×C3—81no—
432C2×C3—82no—
433C2×C3—122no—
434C2×C3—122no—
435C2×C3—242no—
436C2×C3—361no—
437C2×C3#6424111yescharacteristic
438D3—42no—
439D3—122no—
440D3—241no—
441C22—21no—
442C22—41no—
443C22—62no—
444C22—181no—
445C22—241no—
446C22—361no—
447C22—721no—
448C22C2×C3×GO+(4,𝔽2)11yescharacteristic
449C4—62no—
450C4—92no—
451C4—121no—
452C4—121no—
453C3—21no—
454C3—21no—
455C3—41no—
456C3—41no—
457C3#1059352911yescharacteristic, direct product factor
458C2—21no—
459C2—41no—
460C2—122no—
461C2—361no—
462C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33#120
prime radicalC3×#64330C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#455
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C3×#64239C2semidirect
3. C3×#63953C2semidirect
4. C3×#63788C2semidirect
5. C3×#63741C2semidirect
6. #10593529C3direct (N × H)
11. C2×C3×#627C22semidirect
14. C3×#3792C22semidirect
19. C3×Dic32C22semidirect
22. #64239C2×C3semidirect
24. #63953C2×C3semidirect
25. #63788C2×C3semidirect
26. #63741C2×C3semidirect
62. C2×#627C22×C3semidirect
65. #3792C22×C3semidirect
76. Dic32C22×C3semidirect
245. C33#455semidirect
396. C32C3×#455semidirect
457. C3#10593529direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643302
C3×#642392
C3×#639532
C3×#637882
C3×#637412
#105935293
C3×#4559