Finite Group Databasebeta

← C3×#10593349 #13166693

Subgroups — C3×#10593349 #13166693

Order 1728.

640 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3798C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C3×#63874C211yesmaximal, characteristic, complemented
4C3×#63740C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593349C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×C4×D3—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3×#626—21no—
11C2×C3×#625C2211yescharacteristic
12C3×#3798—22no—
13C3×#3798C414yescomplemented
14C2×C3×#622C2212yes—
15C3×#3787—22no—
16C32×C4×Dic3—21no—
17C3×#3724C2212yes—
18C3×#3722—21no—
19C2×C3×#235—61no—
20C2×#3798C2×C311yescharacteristic
21C2×#3787C2×C311yescharacteristic, complemented
22C3×#872—61no—
23#63874C2×C311yescharacteristic, complemented
24#63740C2×C312yescomplemented
25#63739C2×C312yescomplemented
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—21no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4×C3⋊D3C2×C412yes—
31C3×C4×C3⋊D3D412yes—
32C2×C32×Dic3—21no—
33C2×C32×Dic3—22no—
34C32×C4×D3—24no—
35C3×#627—21no—
36C3×#627C2×C412yes—
37C3×#626—24no—
38C3×#625—21no—
39C3×#625C2×C412yes—
40C3×#622—22no—
41C3×#416—91nomaximal
42C22×C32×C4—61no—
43C2×C4×C3⋊D3C22×C311yescharacteristic
44C2×C3×#67—61no—
45C2×C3×#67—61no—
46C3×#235—64no—
47C3×#235—64no—
48C2×C3×D12—61no—
49C2×C3×C4×D3—21no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—41no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—61no—
55C2×#627C22×C311yescharacteristic
56C2×#626—21no—
57C2×#625C22×C311yescharacteristic
58#3798—22no—
59#3798C3×C414yescomplemented
60C2×#622C22×C312yes—
61#3787—22no—
62C32×C42—61no—
63C3×(C2×D3)⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3—21no—
67C3×C4×Dic3—41no—
68#3724C22×C312yes—
69#3722—21no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3C22⋊C412yes—
73C2×C32×D3—22no—
74C2×C32×D3—42no—
75C33×C4C22⋊C412yes—
76C3×C3⋊Dic3C22×C411yescharacteristic
77C3×C3⋊Dic3C2×D411yescharacteristic
78C32×Dic3—22no—
79C32×Dic3—42no—
80C2×C3×#41—91no—
81C2×#235—61no—
82C22×C3×C8—91no—
83C3×#117—91no—
84#872—61no—
85C3×#103—92no—
86C3×#102—92no—
87C23×C32—61no—
88C22×C3⋊D3C3×D411yescharacteristic
89C22×C3×D3—21no—
90C22×C3×D3—21no—
91C22×C3×D3—41no—
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—64no—
99C2×C32×C4—61no—
100C2×C32×C4—62no—
101C2×C32×C4C3×D411yescharacteristic
102C2×C3⋊Dic3C23×C311yescharacteristic
103C4×C3⋊D3C2×C3×C412yes—
104C4×C3⋊D3C3×D412yes—
105C3×#67—62no—
106C3×#67—64no—
107C3×#67—62no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—42no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C3×D12—64no—
117C3×C4×D3—24no—
118C3×C4×D3—24no—
119C3×C4×D3—44no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126#627—21no—
127#627C2×C3×C412yes—
128#626—24no—
129#625—21no—
130#625C2×C3×C412yes—
131#622—22no—
132#416—91noSylow
133C2×C33C2×C22⋊C411yescharacteristic
134C2×C33#13512yes—
135C3×C3⋊D3—22no—
136C32×D3—42no—
137C3×#41—94no—
138C3×#41—182no—
139C2×C3×Q8—91no—
140C2×C3×D4—91no—
141C2×C3×D4—181no—
142C22×C3×C4—61no—
143C22×C3×C4—61no—
144C22×C3×C4—91no—
145C22×C3×C4—121no—
146C2×#67—61no—
147C2×#67—61no—
148#235—64no—
149#235—64no—
150C2×D12—61no—
151C2×C4×D3—21no—
152C2×C4×D3—61no—
153C2×C4×D3—61no—
154C2×Dic6—61no—
155C2×C3×C8—92no—
156C2×C3×C8—182no—
157C3×C4⋊C4—92no—
158C3×C22⋊C4—181no—
159C3×C42—61no—
160C3×C42—61no—
161C3×C42—121no—
162(C2×D3)⋊C4—62no—
163Dic3⋊C4—62no—
164C4×Dic3—21no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—62no—
170C22×C32—122no—
171C22×C32C2×C3×D411yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3C3×C22⋊C412yes—
174C2×C3×D3—22no—
175C2×C3×D3—22no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—44no—
179C2×C3×D3—42no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—122no—
183C2×C3×D3—122no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—42no—
188C32×C4—62no—
189C32×C4—122no—
190C32×C4C3×C22⋊C412yes—
191C3⋊Dic3C22×C3×C411yescharacteristic
192C3⋊Dic3C2×C3×D411yescharacteristic
193C3×Dic3—22no—
194C3×Dic3—22no—
195C3×Dic3—42no—
196C3×Dic3—42no—
197C3×Dic3—44no—
198C3×Dic3—42no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C2×#41—91no—
204C22×C8—91no—
205#117—91no—
206#103—92no—
207#102—92no—
208C33#41611yescharacteristic, complemented, Sylow
209C23×C3—61no—
210C23×C3—61no—
211C23×C3—91no—
212C23×C3—121no—
213C22×D3—21no—
214C22×D3—61no—
215C22×D3—61no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×D4—92no—
219C3×D4—184no—
220C3×D4—181no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—64no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—64no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—91no—
232C2×C3×C4—92no—
233C2×C3×C4—92no—
234C2×C3×C4—124no—
235C2×C3×C4—122no—
236C2×C3×C4—121no—
237C2×C3×C4GO+(4,𝔽2)11yescharacteristic
238#67—62no—
239#67—64no—
240#67—62no—
241C2×Dic3—21no—
242C2×Dic3—22no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245D12—64no—
246C4×D3—24no—
247C4×D3—62no—
248C4×D3—62no—
249C4×D3—62no—
250C4×D3—62no—
251Dic6—62no—
252Dic6—62no—
253C3×C8—182no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—42no—
261C2×C32—42no—
262C2×C32—122no—
263C2×C32C2×C3×C22⋊C411yescharacteristic
264C2×C32C3×#13512yes—
265C3⋊D3—22no—
266C3×D3—42no—
267C3×D3—44no—
268C3×D3—42no—
269C3×D3—122no—
270C3×D3—122no—
271#41—94no—
272#41—182no—
273C2×Q8—91no—
274C2×D4—91no—
275C2×D4—181no—
276C22×C4—61no—
277C22×C4—91no—
278C2×C8—92no—
279C2×C8—182no—
280C4⋊C4—92no—
281C22⋊C4—181no—
282C42—61no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—92no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—124no—
293C22×C3—122no—
294C22×C3—182no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—22no—
297C2×D3—42no—
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—122no—
301C2×D3—122no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—62no—
307C3×C4—62no—
308C3×C4—91no—
309C3×C4—91no—
310C3×C4—122no—
311C3×C4—122no—
312C3×C4—124no—
313C3×C4—122no—
314C3×C4D32⋊C412yes—
315Dic3—22no—
316Dic3—42no—
317Dic3—61no—
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321C32—21no—
322C32—21no—
323C32—41no—
324C32—41no—
325C32C3×#41611yescharacteristic, complemented
326C23—61no—
327C23—91no—
328Q8—92no—
329Q8—181no—
330D4—92no—
331D4—184no—
332D4—181no—
333C2×C4—61no—
334C2×C4—64no—
335C2×C4—62no—
336C2×C4—91no—
337C2×C4—92no—
338C2×C4—92no—
339C2×C4C3×GO+(4,𝔽2)11yescharacteristic
340C8—182no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3—42no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—122no—
350C2×C3—124no—
351C2×C3—122no—
352C2×C3—182no—
353C2×C3#6422312yes—
354C2×C3C2×D32⋊C411yescharacteristic
355D3—42no—
356D3—122no—
357D3—122no—
358C22—62no—
359C22—92no—
360C22—122no—
361C22—182no—
362C22C2×C3×GO+(4,𝔽2)11yescharacteristic
363C4—62no—
364C4—91no—
365C4—91no—
366C4—122no—
367C4C3×D32⋊C412yes—
368C3—21no—
369C3—21no—
370C3—41no—
371C3—41no—
372C3#1059334911yescharacteristic, direct product factor
373C2—122no—
374C2—182no—
375C2C3×#6422312yes—
376C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3798C2
cosocleC2×C3⋊Dic3C23×C3
hypocenterC32C3×#416
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3787C2semidirect
3. C3×#63874C2semidirect
4. C3×#63740C2semidirect
5. C3×#63739C2semidirect
6. #10593349C3direct (N × H)
13. C3×#3798C4semidirect
21. C2×#3787C2×C3semidirect
23. #63874C2×C3semidirect
24. #63740C2×C3semidirect
25. #63739C2×C3semidirect
59. #3798C3×C4semidirect
208. C33#416semidirect
325. C32C3×#416semidirect
372. C3#10593349direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×#37872
C3×#638742
C3×#637402
C3×#637392
#105933493
C3×#4169