Finite Group Databasebeta

← C3×#10592752 #13163091

Subgroups — C3×#10592752 #13163091

Order 1728.

1272 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64361C211yesmaximal, characteristic
2C33×#123C211yesmaximal, characteristic, complemented
3C3×#64138C212yesmaximal, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5C3×#63633C212yesmaximal, complemented
6#10592752C311yesmaximal, characteristic, direct product factor
7C3×#5053—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic, complemented
10C2×C3×#638—21no—
11C2×C3×#638C2212yes—
12C3×#3830—22no—
13C3×D4×C3⋊D3—22no—
14C3×#3828—21no—
15C33×C4⋊C4—21no—
16C33×C22⋊C4—21no—
17C33×C22⋊C4C2211yescharacteristic, complemented
18C3×#3757C2212yescomplemented
19C3×#3752—22no—
20C3×#3750—22no—
21#64361C2×C311yescharacteristic
22C3×#1004—34no—
23C32×#123—24no—
24C32×#123D314yescomplemented
25C32×#123C2×C311yescharacteristic, complemented
26#64138C2×C312yescomplemented
27C3×#929—38no—
28#63661C2×C311yescharacteristic, complemented
29#63633C2×C312yescomplemented
30C3×#834—34no—
31C3×#806—38no—
32C23×C33D411yescharacteristic
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3D412yes—
36C33×D4—22no—
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4—22no—
40C2×C33×C4D411yescharacteristic
41C3×#638—22no—
42C3×#638—22no—
43C3×#638—22no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3D412yes—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4×C3⋊D3—22no—
49C3×#634—21no—
50C3×#458—91no—
51#5053—34no—
52C2×C32×D4—24no—
53C2×C32×D4C2×D314yes—
54C2×C32×D4C22×C311yescharacteristic
55C22×C32×C4—24no—
56C22×C32×C4C2×D314yescomplemented
57C22×C32×C4C22×C311yescharacteristic, complemented
58C2×#638—21no—
59C2×#638C22×C312yes—
60#3830—22no—
61D4×C3⋊D3—22no—
62#3828—21no—
63C2×C3×#67—38no—
64C2×C3×#67—64no—
65C3×#237—68no—
66C3×D3×D4—68no—
67C3×#235—64no—
68C32×C4⋊C4—28no—
69C32×C4⋊C4—24no—
70C32×C4⋊C4—21no—
71C32×C22⋊C4—28no—
72C32×C22⋊C4—24no—
73C32×C22⋊C4—24no—
74C32×C22⋊C4—21no—
75C32×C22⋊C4C2×D314yescomplemented
76C32×C22⋊C4C22×C311yescharacteristic, complemented
77#3757C22×C312yescomplemented
78#3752—22no—
79#3750—22no—
80C3×#217—38no—
81C3×(C2×D3)⋊C4—68no—
82C3×Dic3⋊C4—68no—
83C22×C33—21no—
84C22×C33—41no—
85C22×C33—41no—
86C22×C33C2×D412yes—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—42no—
90C2×C3×C3⋊D3—81no—
91C33×C4—21no—
92C33×C4—41no—
93C33×C4—41no—
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3—42no—
96C3×H(5,𝔽2)—91no—
97#1004—34no—
98C3×#123—24no—
99C3×#123—92no—
100C3×#123C3×D314yescomplemented
101C3×#123C3⋊D311yescharacteristic, complemented
102#929—38no—
103C3×#99—91no—
104C3×#99—92no—
105#834—34no—
106#806—38no—
107C23×C32—24no—
108C23×C32—24no—
109C23×C32#6714yes—
110C23×C32C3×D411yescharacteristic
111C23×C32C22×D314yes—
112C23×C32C23×C311yescharacteristic
113C22×C3⋊D3—21no—
114C22×C3⋊D3C3×D412yes—
115C22×C3×D3—38no—
116C22×C3×D3—64no—
117C32×D4—22no—
118C32×D4—28no—
119C32×D4—48no—
120C2×C32×C4—28no—
121C2×C32×C4—28no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—28no—
129C2×C32×C4—48no—
130C2×C32×C4#6714yes—
131C2×C32×C4C3×D411yescharacteristic
132#638—22no—
133#638—22no—
134#638—22no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3C3×D412yes—
138C4⋊(C3⋊D3)—21no—
139C4×C3⋊D3—22no—
140#634—21no—
141C3×#67—68no—
142C3×#67—68no—
143C3×#67—68no—
144C2×C3×Dic3—38no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—68no—
147C3×D12—64no—
148C3×C4×D3—68no—
149C3×Dic6—64no—
150#458—91noSylow
151C2×C33—21no—
152C2×C33—22no—
153C2×C33—41no—
154C2×C33#12011yescharacteristic
155C3×C3⋊D3—42no—
156C3×#41—182no—
157C3×#41—181no—
158C2×C3×D4—24no—
159C2×C3×D4—92no—
160C2×C3×D4—182no—
161C2×C3×D4—181no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C22×C3×C4—24no—
165C22×C3×C4C2×C3×D314yescomplemented
166C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
167C2×#67—38no—
168C2×#67—64no—
169#237—68no—
170D3×D4—68no—
171#235—64no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—182no—
176C3×C22⋊C4—28no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—92no—
181C3×C22⋊C4—182no—
182C3×C22⋊C4C2×C3×D314yescomplemented
183C3×C22⋊C4C2×C3⋊D311yescharacteristic, complemented
184#217—38no—
185(C2×D3)⋊C4—68no—
186Dic3⋊C4—68no—
187C22×C32—21no—
188C22×C32—28no—
189C22×C32—24no—
190C22×C32—24no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32—41no—
194C22×C32—44no—
195C22×C32—41no—
196C22×C32—48no—
197C22×C32—48no—
198C22×C32D3×D418yes—
199C22×C32C2×#6714yes—
200C22×C32C2×C3×D412yes—
201C22×C32C2×C3×D411yescharacteristic
202C2×C3⋊D3—22no—
203C2×C3⋊D3—42no—
204C2×C3⋊D3—81no—
205C2×C3×D3—68no—
206C2×C3×D3—128no—
207C2×C3×D3—244no—
208C32×C4—21no—
209C32×C4—24no—
210C32×C4—44no—
211C32×C4—48no—
212C32×C4—44no—
213C32×C4—41no—
214C32×C4—48no—
215C32×C4—44no—
216C32×C4—41no—
217C3⋊Dic3—22no—
218C3⋊Dic3—42no—
219C3×Dic3—68no—
220C3×Dic3—128no—
221H(5,𝔽2)—91no—
222#123—92no—
223#123C3×C3⋊D311yescharacteristic, complemented
224#99—91no—
225#99—92no—
226C33#45811yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—92no—
230C23×C3—181no—
231C23×C3C3×#6714yes—
232C23×C3#63811yescharacteristic
233C23×C3C22×C3×D314yes—
234C23×C3C22×C3⋊D311yescharacteristic
235C22×D3—38no—
236C22×D3—64no—
237C3×Q8—181no—
238C3×D4—28no—
239C3×D4—22no—
240C3×D4—48no—
241C3×D4—181no—
242C3×D4—182no—
243C3×D4—182no—
244C3×D4—182no—
245C2×C3×C4—28no—
246C2×C3×C4—28no—
247C2×C3×C4—28no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—48no—
255C2×C3×C4—92no—
256C2×C3×C4—182no—
257C2×C3×C4—181no—
258C2×C3×C4—182no—
259C2×C3×C4C3×#6714yes—
260C2×C3×C4#63811yescharacteristic
261#67—68no—
262#67—68no—
263#67—68no—
264C2×Dic3—38no—
265C2×Dic3—64no—
266C2×Dic3—68no—
267D12—64no—
268C4×D3—68no—
269Dic6—64no—
270C2×C32—28no—
271C2×C32—24no—
272C2×C32—22no—
273C2×C32—21no—
274C2×C32—24no—
275C2×C32—28no—
276C2×C32—48no—
277C2×C32—41no—
278C2×C32—44no—
279C2×C32—84no—
280C2×C32#93714yes—
281C2×C32C3×#12011yescharacteristic
282C3⋊D3—42no—
283C3×D3—128no—
284#41—182no—
285#41—181no—
286C2×D4—92no—
287C2×D4—182no—
288C2×D4—181no—
289C2×D4C2×C3×C3⋊D311yescharacteristic
290C22×C4C2×C3×C3⋊D311yescharacteristic, complemented
291C4⋊C4—21no—
292C4⋊C4—182no—
293C22⋊C4—21no—
294C22⋊C4—92no—
295C22⋊C4—182no—
296C22⋊C4C2×C3×C3⋊D311yescharacteristic, complemented
297C22×C3—24no—
298C22×C3—21no—
299C22×C3—28no—
300C22×C3—24no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—41no—
304C22×C3—44no—
305C22×C3—41no—
306C22×C3—48no—
307C22×C3—48no—
308C22×C3—182no—
309C22×C3—362no—
310C22×C3—721no—
311C22×C3C3×D3×D418yes—
312C22×C3C2×C3×#6714yes—
313C22×C3D4×C3⋊D312yes—
314C22×C3C2×#63811yescharacteristic
315C2×D3—68no—
316C2×D3—128no—
317C2×D3—244no—
318C3×C4—24no—
319C3×C4—21no—
320C3×C4—44no—
321C3×C4—48no—
322C3×C4—48no—
323C3×C4—44no—
324C3×C4—41no—
325C3×C4—44no—
326C3×C4—41no—
327C3×C4—182no—
328C3×C4—362no—
329Dic3—68no—
330Dic3—128no—
331C32—24no—
332C32#505314yescomplemented
333C32C3×#45811yescharacteristic, complemented
334C23—92no—
335C23—181no—
336C23C3×#63811yescharacteristic
337C23C22×C3×C3⋊D311yescharacteristic
338Q8—181no—
339D4—22no—
340D4—181no—
341D4—182no—
342D4—182no—
343D4—182no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—92no—
348C2×C4—182no—
349C2×C4—181no—
350C2×C4—182no—
351C2×C4C3×#63811yescharacteristic
352C2×C3—28no—
353C2×C3—28no—
354C2×C3—24no—
355C2×C3—21no—
356C2×C3—22no—
357C2×C3—24no—
358C2×C3—48no—
359C2×C3—44no—
360C2×C3—41no—
361C2×C3—84no—
362C2×C3—362no—
363C2×C3C3×#93714yes—
364C2×C3#6414611yescharacteristic
365D3—128no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—182no—
370C22—362no—
371C22—721no—
372C22C3×D4×C3⋊D312yes—
373C22C2×C3×#63811yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—182no—
378C4—362no—
379C3—24no—
380C3C3×#505314yescomplemented
381C3#1059275211yescharacteristic, direct product factor
382C2—21no—
383C2—22no—
384C2—41no—
385C2—362no—
386C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC33×#123C2
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC3×#123C3⋊D3
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
2. C33×#123C2semidirect
3. C3×#64138C2semidirect
4. C3×#63661C2semidirect
5. C3×#63633C2semidirect
6. #10592752C3direct (N × H)
9. C22×C33×C4C22semidirect
17. C33×C22⋊C4C22semidirect
18. C3×#3757C22semidirect
24. C32×#123D3semidirect
25. C32×#123C2×C3semidirect
26. #64138C2×C3semidirect
28. #63661C2×C3semidirect
29. #63633C2×C3semidirect
56. C22×C32×C4C2×D3semidirect
57. C22×C32×C4C22×C3semidirect
75. C32×C22⋊C4C2×D3semidirect
76. C32×C22⋊C4C22×C3semidirect
77. #3757C22×C3semidirect
100. C3×#123C3×D3semidirect
101. C3×#123C3⋊D3semidirect
165. C22×C3×C4C2×C3×D3semidirect
166. C22×C3×C4C2×C3⋊D3semidirect
182. C3×C22⋊C4C2×C3×D3semidirect
183. C3×C22⋊C4C2×C3⋊D3semidirect
223. #123C3×C3⋊D3semidirect
226. C33#458semidirect
290. C22×C4C2×C3×C3⋊D3semidirect
296. C22⋊C4C2×C3×C3⋊D3semidirect
332. C32#5053semidirect
333. C32C3×#458semidirect
380. C3C3×#5053semidirect
381. C3#10592752direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C33×#1232
C3×#641382
C3×#636612
C3×#636332
#105927523
C3×#50533