Finite Group Databasebeta

← C3×#10590299 #13157315

Subgroups — C3×#10590299 #13157315

Order 1728.

658 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2C32×#803C212yesmaximal, complemented
3C3×#63572C212yesmaximal, complemented
4C3×#63557C212yesmaximal
5C3×#4705—32nomaximal
6#10590299C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C411yescharacteristic
8C2×C3×C4×C3⋊D3C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2214yes—
15C42×C3⋊D3C2×C311yescharacteristic
16C3×C42×D3—32no—
17C3×C42×D3—61no—
18C32×#97—32no—
19C3×(C2×D3)⋊C8—34no—
20C3×Dic3⋊C8—34no—
21C3×#803—22no—
22C3×#803D312yescomplemented
23C3×#803C2×C312yescomplemented
24#63572C2×C312yescomplemented
25#63557C2×C312yes—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4C2×C411yescharacteristic
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3—22no—
36C32×C3⋊C8—24no—
37C3×#432—91no—
38#4705—32no—
39C2×C4×C3⋊D3C3×C411yescharacteristic
40C2×C4×C3⋊D3C3×C411yescharacteristic
41C2×C4×C3⋊D3C22×C311yescharacteristic
42C2×C3×C4×D3—32no—
43C2×C3×C4×D3—32no—
44C2×C3×C4×D3—32no—
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—61no—
48C2×C32×C8—34no—
49C32×C42—21no—
50C32×C42—22no—
51C32×C42—41no—
52C32×C42C2×D312yes—
53C32×C42C22×C311yescharacteristic
54C4×C3⋊Dic3C3×C411yescharacteristic
55C4×C3⋊Dic3C3×C411yescharacteristic
56C4×C3⋊Dic3C22×C311yescharacteristic
57C3×C4×Dic3—32no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—32no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—24no—
64C2×C3×C3⋊C8C2×D314yes—
65C2×C3×C3⋊C8C22×C314yes—
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3M1612yes—
70C33×C4—21no—
71C33×C4—21no—
72C33×C4#4112yes—
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3M1612yes—
76C2×C3×C42—91no—
77C42×D3—32no—
78C42×D3—61no—
79C3×C4⋊C8—92no—
80C3×C22⋊C8—92no—
81C3×#97—32no—
82C3×#97—32no—
83C3×#97—62no—
84(C2×D3)⋊C8—34no—
85Dic3⋊C8—34no—
86#803C3×D312yescomplemented
87C22×C3⋊D3C2×C3×C411yescharacteristic
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4C4×D312yes—
100C2×C32×C4C4×D312yes—
101C2×C32×C4C2×C3×C411yescharacteristic
102C2×C32×C4C2×C3×C411yescharacteristic
103C2×C32×C4C22×D312yes—
104C2×C32×C4C23×C311yescharacteristic
105C2×C3⋊Dic3C2×C3×C411yescharacteristic
106C2×C3⋊Dic3C2×C3×C411yescharacteristic
107C2×C3⋊Dic3C2×C3×C411yescharacteristic
108C4×C3⋊D3—22no—
109C4×C3⋊D3—22no—
110C4×C3⋊D3—22no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C3×C4×D3—64no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×C4×D3—64no—
121C3×C4×D3—64no—
122C3×C4×D3—64no—
123C32×C8—64no—
124C3×C3⋊C8—28no—
125C3×C3⋊C8—24no—
126C3×C3⋊C8—24no—
127#432—91noSylow
128C2×C33#11711yescharacteristic
129C2×C33C2×M1611yescharacteristic
130C2×C33C2×M1611yescharacteristic
131C3×C3⋊D3—22no—
132C22×C3×C4—91no—
133C22×C3×C4—91no—
134C22×C3×C4—91no—
135C2×C4×D3—32no—
136C2×C4×D3—32no—
137C2×C4×D3—32no—
138C2×C4×D3—61no—
139C2×C4×D3—61no—
140C2×C4×D3—61no—
141C2×C3×C8—34no—
142C2×C3×C8—34no—
143C2×C3×C8—64no—
144C3×C42—22no—
145C3×C42—21no—
146C3×C42—41no—
147C3×C42—91no—
148C3×C42—91no—
149C3×C42—91no—
150C3×C42D3211yescharacteristic
151C3×C42C2×C3×D312yes—
152C4×Dic3—32no—
153C4×Dic3—32no—
154C4×Dic3—32no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C4×Dic3—61no—
158C2×C3⋊C8C2×C3×D314yes—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—41no—
162C22×C32C2×C4×D312yes—
163C22×C32C22×C3×C411yescharacteristic
164C2×C3⋊D3—21no—
165C2×C3⋊D3—21no—
166C2×C3⋊D3C3×M1612yes—
167C2×C3×D3—34no—
168C2×C3×D3—62no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—24no—
181C32×C4—42no—
182C32×C4—42no—
183C32×C4—42no—
184C32×C4—42no—
185C32×C4—42no—
186C32×C4#23514yes—
187C32×C4C3×#4112yes—
188C3⋊Dic3—21no—
189C3⋊Dic3—21no—
190C3⋊Dic3C3×M1612yes—
191C3×Dic3—34no—
192C3×Dic3—62no—
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C2×C42—91no—
198C4⋊C8—92no—
199C22⋊C8—92no—
200#97—32no—
201C33#43211yescharacteristic, complemented, Sylow
202C23×C3—91no—
203C22×D3—32no—
204C22×D3—61no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4#62411yescharacteristic
221C2×C3×C4#62411yescharacteristic
222C2×C3×C4C3×C4×D312yes—
223C2×C3×C4C3×C4×D312yes—
224C2×C3×C4C2×D3211yescharacteristic
225C2×C3×C4C22×C3×D312yes—
226C2×Dic3—32no—
227C2×Dic3—32no—
228C2×Dic3—32no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C4×D3—64no—
233C4×D3—64no—
234C4×D3—64no—
235C4×D3—64no—
236C4×D3—64no—
237C4×D3—64no—
238C3×C8—68no—
239C3×C8—64no—
240C3×C8—64no—
241C3⋊C8—24no—
242C2×C32—22no—
243C2×C32—22no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32#87212yes—
252C2×C32D3×M1612yes—
253C2×C32D3×M1612yes—
254C2×C32C3×#11711yescharacteristic
255C2×C32C2×C3×M1611yescharacteristic
256C2×C32C2×C3×M1611yescharacteristic
257C3⋊D3—22no—
258C3×D3—64no—
259C3×D3—64no—
260C22×C4—91no—
261C22×C4—91no—
262C22×C4—91no—
263C2×C8—34no—
264C42—91no—
265C42—91no—
266C42—91no—
267C42C3×D3211yescharacteristic
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3—92no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3C2×#62411yescharacteristic
275C22×C3C2×C3×C4×D312yes—
276C2×D3—34no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—62no—
282C3×C4—22no—
283C3×C4—22no—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—24no—
289C3×C4—22no—
290C3×C4—42no—
291C3×C4—42no—
292C3×C4—42no—
293C3×C4—42no—
294C3×C4—42no—
295C3×C4—92no—
296C3×C4—181no—
297C3×C4—181no—
298C3×C4#379812yes—
299C3×C4C3×#23514yes—
300Dic3—34no—
301Dic3—62no—
302Dic3—62no—
303Dic3—62no—
304Dic3—62no—
305Dic3—62no—
306C32—22no—
307C32—21no—
308C32—41no—
309C32#470512yescomplemented
310C32C3×#43211yescharacteristic, complemented
311C23—91no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—182no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4C3×#62411yescharacteristic
319C2×C4C3×#62411yescharacteristic
320C2×C4C2×C3×D3211yescharacteristic
321C8—64no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—22no—
325C2×C3—22no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—182no—
332C2×C3#6381811yescharacteristic
333C2×C3#6381811yescharacteristic
334C2×C3#6387411yescharacteristic
335C2×C3C3×#87212yes—
336C2×C3C3×D3×M1612yes—
337C2×C3C3×D3×M1612yes—
338D3—64no—
339D3—64no—
340C22—92no—
341C22—181no—
342C22—181no—
343C22C2×C3×#62411yescharacteristic
344C4—21no—
345C4—21no—
346C4—92no—
347C4—181no—
348C4—181no—
349C4C3×#379812yes—
350C3—21no—
351C3—22no—
352C3—41no—
353C3#1059029911yescharacteristic, direct product factor
354C3C3×#470512yescomplemented
355C2—182no—
356C2C3×#6381811yescharacteristic
357C2C3×#6381811yescharacteristic
358C2C3×#6387411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#432
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#803C2semidirect
3. C3×#63572C2semidirect
6. #10590299C3direct (N × H)
22. C3×#803D3semidirect
23. C3×#803C2×C3semidirect
24. #63572C2×C3semidirect
86. #803C3×D3semidirect
201. C33#432semidirect
309. C32#4705semidirect
310. C32C3×#432semidirect
353. C3#10590299direct (N × H)
354. C3C3×#4705semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
C32×#8032
C3×#635722
C3×#635572
C3×#47053
#105902993