Finite Group Databasebeta

← C3×#10593354 #13166799

Subgroups — C3×#10593354 #13166799

Order 1728.

766 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 74 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
2C2×C3×#3775C212yesmaximal, complemented
3C3×#63865C211yesmaximal, characteristic, complemented
4C3×#63778C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593354C311yesmaximal, characteristic, direct product factor
7C22×C3×D32—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12—21no—
10C2×C3×#627C2211yescharacteristic, complemented
11C2×C3×#625C2212yes—
12C3×#3802—24no—
13C2×C3×#622C2212yescomplemented
14C3×#3775—24no—
15C32×Dic3⋊C4—21no—
16C3×#3724C2211yescharacteristic, complemented
17C3×#3722—21no—
18C2×C3×D3×D4—61no—
19C2×#3802C2×C311yescharacteristic
20C2×#3775C2×C312yescomplemented
21C3×#896—61no—
22#63865C2×C311yescharacteristic, complemented
23#63778C2×C311yescharacteristic, complemented
24#63739C2×C312yescomplemented
25C22×C3×C3⋊D3D411yescharacteristic
26C22×C32×D3—22no—
27C2×C3×D32—24no—
28C2×C3×D32—42no—
29C2×C3×D32—42no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4×C3⋊D3—22no—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C32×D12—24no—
36C3×#627—22no—
37C3×#625—22no—
38C3×#625D414yes—
39C3×#622—22no—
40C3×#450—91nomaximal
41C23×C3×D3—62no—
42C22×D32—21no—
43C2×C32×D4—61no—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×#67—62no—
46C3×D3×D4—68no—
47C2×C3×D12—21no—
48C2×C3×D12—21no—
49C2×C3×D12—41no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—61no—
52C2×C3×Dic6—61no—
53C2×#627C22×C311yescharacteristic, complemented
54C2×#625C22×C312yes—
55#3802—24no—
56C2×#622C22×C312yescomplemented
57#3775—24no—
58C32×C4⋊C4—61no—
59C3×(C2×D3)⋊C4—61no—
60C3×(C2×D3)⋊C4—61no—
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×Dic3⋊C4—41no—
64C3×Dic3⋊C4—61no—
65#3724C22×C311yescharacteristic, complemented
66#3722—21no—
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3SD1612yescomplemented
71C2×C32×D3—24no—
72C2×C32×D3—42no—
73C2×C32×D3—42no—
74C3×D32—48no—
75C33×C4—21no—
76C3×C3⋊Dic3C2×D412yes—
77C32×Dic3—41no—
78C32×Dic3—41no—
79C22×C3×D4—91no—
80C2×D3×D4—61no—
81C2×C3×SD16—92no—
82C3×C22⋊Q8—91no—
83#896—61no—
84C3×#102—92no—
85C3×C22⋊C8—91no—
86C23×C32—62no—
87C22×C3⋊D3C3×D411yescharacteristic
88C22×C3×D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—42no—
91C22×C3×D3—61no—
92C22×C3×D3—61no—
93C22×C3×D3—68no—
94C22×C3×D3—124no—
95C22×C3×D3—124no—
96C2×D32—24no—
97C2×D32—42no—
98C2×D32—42no—
99C32×D4—64no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C32×C4C3×D411yescharacteristic
107C2×C3⋊Dic3C23×C311yescharacteristic
108C4×C3⋊D3—22no—
109C3×#67—64no—
110C3×#67—64no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×D12—24no—
120C3×D12—24no—
121C3×D12—44no—
122C3×C4×D3—64no—
123C3×C4×D3—122no—
124C3×Dic6—122no—
125#627—22no—
126#625—22no—
127#625C3×D414yes—
128#622—22no—
129#450—91noSylow
130C2×C33#12011yescharacteristic
131C2×C33C2×SD1611yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C3×C3⋊D3—22no—
134C32×D3—44no—
135C24×C3—181no—
136C23×D3—62no—
137C2×C3×Q8—91no—
138C2×C3×D4—61no—
139C2×C3×D4—61no—
140C2×C3×D4—92no—
141C2×C3×D4—121no—
142C2×C3×D4—184no—
143C22×C3×C4—91no—
144C2×#67—62no—
145D3×D4—68no—
146C2×D12—21no—
147C2×C4×D3—61no—
148C2×C4×D3—61no—
149C2×Dic6—61no—
150C3×SD16—184no—
151C2×C3×C8—92no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—121no—
156C3×C22⋊C4—181no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4—61no—
159Dic3⋊C4—21no—
160Dic3⋊C4—61no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—64no—
166C22×C32—122no—
167C22×C32—122no—
168C22×C32C2×C3×D411yescharacteristic
169C2×C3⋊D3—21no—
170C2×C3⋊D3—21no—
171C2×C3⋊D3C3×SD1612yescomplemented
172C2×C3×D3—24no—
173C2×C3×D3—24no—
174C2×C3×D3—44no—
175C2×C3×D3—42no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—82no—
184C2×C3×D3—82no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C2×C3×D3—1216no—
188D32—48no—
189C32×C4—22no—
190C32×C4—21no—
191C32×C4—41no—
192C32×C4—42no—
193C32×C4—81no—
194C32×C4—121no—
195C32×C4—121no—
196C3⋊Dic3C2×C3×D412yes—
197C3×Dic3—41no—
198C3×Dic3—41no—
199C3×Dic3—42no—
200C3×Dic3—41no—
201C3×Dic3—41no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—81no—
205C22×D4—91no—
206C2×SD16—92no—
207C22⋊Q8—91no—
208#102—92no—
209C22⋊C8—91no—
210C33#45011yescharacteristic, complemented, Sylow
211C23×C3—62no—
212C23×C3—62no—
213C23×C3—91no—
214C23×C3—122no—
215C23×C3—184no—
216C23×C3—362no—
217C23×C3—362no—
218C22×D3—22no—
219C22×D3—61no—
220C22×D3—61no—
221C22×D3—68no—
222C22×D3—124no—
223C22×D3—124no—
224C3×Q8—182no—
225C3×D4—64no—
226C3×D4—64no—
227C3×D4—94no—
228C3×D4—124no—
229C3×D4—182no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—91no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4—182no—
242C2×C3×C4GO+(4,𝔽2)11yescharacteristic
243#67—64no—
244#67—64no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249D12—24no—
250C4×D3—64no—
251C4×D3—122no—
252Dic6—122no—
253C3×C8—182no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—124no—
267C2×C32C3×#12011yescharacteristic
268C2×C32C2×C3×SD1611yescharacteristic
269C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
270C3⋊D3—22no—
271C3×D3—44no—
272C3×D3—44no—
273C3×D3—64no—
274C3×D3—84no—
275C3×D3—122no—
276C24—181no—
277C2×Q8—91no—
278C2×D4—61no—
279C2×D4—92no—
280C2×D4—184no—
281C22×C4—91no—
282SD16—184no—
283C2×C8—92no—
284C4⋊C4—61no—
285C4⋊C4—91no—
286C22⋊C4—181no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—64no—
292C22×C3—64no—
293C22×C3—92no—
294C22×C3—124no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—122no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—242no—
302C22×C3—242no—
303C22×C3—368no—
304C22×C3C2×GO+(4,𝔽2)11yescharacteristic
305C2×D3—24no—
306C2×D3—42no—
307C2×D3—42no—
308C2×D3—62no—
309C2×D3—62no—
310C2×D3—62no—
311C2×D3—62no—
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—1216no—
315C3×C4—22no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—42no—
319C3×C4—81no—
320C3×C4—92no—
321C3×C4—121no—
322C3×C4—121no—
323C3×C4—122no—
324C3×C4—121no—
325C3×C4—121no—
326C3×C4—241no—
327Dic3—41no—
328Dic3—41no—
329Dic3—62no—
330Dic3—62no—
331C32—21no—
332C32—21no—
333C32—41no—
334C32—41no—
335C32C3×#45011yescharacteristic, complemented
336C23—62no—
337C23—91no—
338C23—184no—
339C23—362no—
340C23—362no—
341Q8—182no—
342D4—64no—
343D4—94no—
344D4—182no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—91no—
348C2×C4—182no—
349C2×C4C3×GO+(4,𝔽2)11yescharacteristic
350C8—182no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—124no—
364C2×C3—124no—
365C2×C3—182no—
366C2×C3—244no—
367C2×C3#6422411yescharacteristic
368C2×C3#6422711yescharacteristic
369C2×C3#6424111yescharacteristic
370D3—44no—
371D3—64no—
372D3—122no—
373C22—64no—
374C22—92no—
375C22—122no—
376C22—122no—
377C22—181no—
378C22—181no—
379C22—368no—
380C22C2×C3×GO+(4,𝔽2)11yescharacteristic
381C4—21no—
382C4—92no—
383C4—121no—
384C4—121no—
385C3—21no—
386C3—21no—
387C3—41no—
388C3—41no—
389C3#1059335411yescharacteristic, direct product factor
390C2—124no—
391C2—182no—
392C2C3×#6422411yescharacteristic
393C2C3×#6422711yescharacteristic
394C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#450
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3775C2semidirect
3. C3×#63865C2semidirect
4. C3×#63778C2semidirect
5. C3×#63739C2semidirect
6. #10593354C3direct (N × H)
10. C2×C3×#627C22semidirect
13. C2×C3×#622C22semidirect
16. C3×#3724C22semidirect
20. C2×#3775C2×C3semidirect
22. #63865C2×C3semidirect
23. #63778C2×C3semidirect
24. #63739C2×C3semidirect
53. C2×#627C22×C3semidirect
56. C2×#622C22×C3semidirect
65. #3724C22×C3semidirect
70. C2×C3×C3⋊D3SD16semidirect
171. C2×C3⋊D3C3×SD16semidirect
210. C33#450semidirect
335. C32C3×#450semidirect
389. C3#10593354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×#37752
C3×#638652
C3×#637782
C3×#637392
#105933543
C3×#4509