Finite Group Databasebeta

← C3×#10590891 #13157856

Subgroups — C3×#10590891 #13157856

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C212yesmaximal
2C3×#63645C211yesmaximal, characteristic
3C32×#818C211yesmaximal, characteristic
4C32×C8×Dic3C211yesmaximal, characteristic
5C3×#63575C212yesmaximal
6C3×#4874—31nomaximal
7C3×#4682—31nomaximal
8#10590891C311yesmaximal, characteristic, direct product factor
9C2×C3×#627—22no—
10C2×C33×C8C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C32×C4⋊Dic3C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15C3×Dic3⋊Dic3—22no—
16C3×#888—32no—
17C3×#869—32no—
18#63918C2×C312yes—
19C32×#107—31no—
20C32×C4×C8—31no—
21#63645C2×C311yescharacteristic
22C3×#818—21no—
23C3×#818—31no—
24C3×#818—31no—
25C3×#818—32no—
26C3×#818—61no—
27C3×#818D311yescharacteristic
28C3×#818C2×C311yescharacteristic
29C3×C8×Dic3—21no—
30C3×C8×Dic3D311yescharacteristic
31C3×C8×Dic3C2×C311yescharacteristic
32C3×#807—32no—
33#63575C2×C312yes—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3D411yescharacteristic
38C2×C32×Dic3D411yescharacteristic
39C3×#627—44no—
40C33×C8Q812yescomplemented
41C32×C3⋊C8Q812yescomplemented
42C3×#500—91no—
43#4874—31no—
44#4682—31no—
45C2×C3×Dic6—62no—
46C2×C3×Dic6—62no—
47C2×#627—22no—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C2×C32×C8—21no—
51C2×C32×C8—31no—
52C2×C32×C8—41no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C2×D311yescharacteristic
55C2×C32×C8C22×C311yescharacteristic
56C32×C4⋊C4—32no—
57C32×C42—31no—
58C4⋊(C3⋊Dic3)C22×C312yes—
59C3×C4⋊Dic3—22no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—62no—
63C3×C4⋊Dic3—62no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3C2×D311yescharacteristic
69C3×C4×Dic3C22×C311yescharacteristic
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8C2×D311yescharacteristic
72C2×C3×C3⋊C8C22×C311yescharacteristic
73Dic3⋊Dic3—22no—
74C22×C33C2×D411yescharacteristic
75C33×C4C2×Q811yescharacteristic
76C33×C4C2×Q811yescharacteristic
77C3×C3⋊Dic3—42no—
78C32×Dic3—42no—
79C32×Dic3D812yes—
80C32×Dic3Q1612yes—
81C3×C4⋊Q8—92no—
82#888—32no—
83#869—32no—
84C3×#107—31no—
85C3×#107—31no—
86C3×#107—61no—
87C3×#107—92no—
88C3×#107—91no—
89C3×C4×C8—31no—
90C3×C4×C8—31no—
91C3×C4×C8—61no—
92#818—31no—
93#818—31no—
94#818—32no—
95#818—61no—
96#818C3×D311yescharacteristic
97C8×Dic3C3×D311yescharacteristic
98#807—32no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—31no—
103C2×C32×C4—31no—
104C2×C32×C4—41no—
105C2×C32×C4—62no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C23×C311yescharacteristic
109C2×C3⋊Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—122no—
118C2×C3×Dic3D1211yescharacteristic
119C2×C3×Dic3D1211yescharacteristic
120C2×C3×Dic3C3×D411yescharacteristic
121C2×C3×Dic3C3×D411yescharacteristic
122C3×Dic6—124no—
123C3×Dic6—124no—
124#627—44no—
125C32×C8—22no—
126C32×C8—22no—
127C32×C8—22no—
128C32×C8—32no—
129C32×C8—42no—
130C32×C8Dic612yescomplemented
131C32×C8Dic612yescomplemented
132C32×C8C3×Q812yescomplemented
133C3×C3⋊C8—22no—
134C3×C3⋊C8Dic612yescomplemented
135C3×C3⋊C8C3×Q812yescomplemented
136#500—91noSylow
137C2×C33C4⋊Q811yescharacteristic
138C2×C33C2×D811yescharacteristic
139C2×C33C2×Q1611yescharacteristic
140C2×C3×Q8—182no—
141C2×Dic6—62no—
142C2×Dic6—62no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—21no—
146C2×C3×C8—31no—
147C2×C3×C8—31no—
148C2×C3×C8—41no—
149C2×C3×C8—61no—
150C2×C3×C8D3211yescharacteristic
151C2×C3×C8C2×C3×D311yescharacteristic
152C2×C3×C8C2×C3×D311yescharacteristic
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—92no—
157C3×C4⋊C4—182no—
158C3×C42—31no—
159C3×C42—31no—
160C3×C42—61no—
161C4⋊Dic3—32no—
162C4⋊Dic3—32no—
163C4⋊Dic3—62no—
164C4⋊Dic3—62no—
165C4⋊Dic3C2×C3×D312yes—
166Dic3⋊C4—62no—
167C4×Dic3C2×C3×D311yescharacteristic
168C2×C3⋊C8C2×C3×D311yescharacteristic
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32C2×D1211yescharacteristic
174C22×C32D3×D411yescharacteristic
175C22×C32C2×C3×D411yescharacteristic
176C32×C4—21no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—32no—
183C32×C4—32no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4—122no—
187C32×C4C2×Dic611yescharacteristic
188C32×C4C2×Dic611yescharacteristic
189C32×C4C2×Dic611yescharacteristic
190C32×C4D3×Q811yescharacteristic
191C32×C4C2×C3×Q811yescharacteristic
192C32×C4C2×C3×Q811yescharacteristic
193C3⋊Dic3—42no—
194C3×Dic3—22no—
195C3×Dic3—22no—
196C3×Dic3—42no—
197C3×Dic3—42no—
198C3×Dic3—82no—
199C3×Dic3—122no—
200C3×Dic3—122no—
201C3×Dic3—242no—
202C3×Dic3D2412yes—
203C3×Dic3Dic1212yes—
204C3×Dic3C3×D812yes—
205C3×Dic3C3×Q1612yes—
206C4⋊Q8—92no—
207#107—31no—
208#107—92no—
209#107—91no—
210C4×C8—31no—
211C33#50011yescharacteristic, complemented, Sylow
212C3×Q8—364no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—41no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—122no—
226C2×C3×C4—182no—
227C2×C3×C4C2×D3211yescharacteristic
228C2×C3×C4C22×C3×D311yescharacteristic
229C2×C3×C4C22×C3×D311yescharacteristic
230C2×Dic3—22no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—122no—
234C2×Dic3C3×D1211yescharacteristic
235C2×Dic3C3×D1211yescharacteristic
236Dic6—124no—
237Dic6—124no—
238C3×C8—22no—
239C3×C8—22no—
240C3×C8—22no—
241C3×C8—32no—
242C3×C8—32no—
243C3×C8—42no—
244C3×C8—62no—
245C3×C8#62712yescomplemented
246C3×C8C3×Dic612yescomplemented
247C3×C8C3×Dic612yescomplemented
248C3⋊C8C3×Dic612yescomplemented
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32#86911yescharacteristic
262C2×C32#88811yescharacteristic
263C2×C32C2×D2411yescharacteristic
264C2×C32C2×Dic1211yescharacteristic
265C2×C32D3×D811yescharacteristic
266C2×C32D3×Q1611yescharacteristic
267C2×C32C3×C4⋊Q811yescharacteristic
268C2×C32C2×C3×D811yescharacteristic
269C2×C32C2×C3×Q1611yescharacteristic
270C2×Q8—182no—
271C2×C8—31no—
272C2×C8C3×D3211yescharacteristic
273C4⋊C4—32no—
274C4⋊C4—92no—
275C4⋊C4—182no—
276C42—31no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3D3×D1211yescharacteristic
282C22×C3C2×C3×D1211yescharacteristic
283C22×C3C3×D3×D411yescharacteristic
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—32no—
291C3×C4—32no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—62no—
297C3×C4—62no—
298C3×C4—122no—
299C3×C4—122no—
300C3×C4—242no—
301C3×C4—362no—
302C3×C4D3×Dic611yescharacteristic
303C3×C4C2×#62711yescharacteristic
304C3×C4C2×C3×Dic611yescharacteristic
305C3×C4C2×C3×Dic611yescharacteristic
306C3×C4C2×C3×Dic611yescharacteristic
307C3×C4C3×D3×Q811yescharacteristic
308Dic3—42no—
309Dic3—122no—
310Dic3—122no—
311Dic3—242no—
312Dic3C3×D2412yes—
313Dic3C3×Dic1212yes—
314C32—21no—
315C32—21no—
316C32—21no—
317C32—41no—
318C32#468211yescharacteristic, complemented
319C32#487411yescharacteristic, complemented
320C32C3×#50011yescharacteristic, complemented
321Q8—364no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—62no—
325C2×C4—182no—
326C2×C4C2×C3×D3211yescharacteristic
327C8—32no—
328C8C3×#62712yescomplemented
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3D3×D2411yescharacteristic
342C2×C3D3×Dic1211yescharacteristic
343C2×C3#6391811yescharacteristic
344C2×C3C3×#86911yescharacteristic
345C2×C3C3×#88811yescharacteristic
346C2×C3C2×C3×D2411yescharacteristic
347C2×C3C2×C3×Dic1211yescharacteristic
348C2×C3C3×D3×D811yescharacteristic
349C2×C3C3×D3×Q1611yescharacteristic
350C22C3×D3×D1211yescharacteristic
351C4—32no—
352C4—32no—
353C4—122no—
354C4—362no—
355C4C3×D3×Dic611yescharacteristic
356C4C2×C3×#62711yescharacteristic
357C3—21no—
358C3—21no—
359C3—21no—
360C3—41no—
361C3#1059089111yescharacteristic, direct product factor
362C3C3×#468211yescharacteristic, complemented
363C3C3×#487411yescharacteristic, complemented
364C2C3×D3×D2411yescharacteristic
365C2C3×D3×Dic1211yescharacteristic
366C2C3×#6391811yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#500
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
8. #10590891C3direct (N × H)
40. C33×C8Q8semidirect
41. C32×C3⋊C8Q8semidirect
130. C32×C8Dic6semidirect
131. C32×C8Dic6semidirect
132. C32×C8C3×Q8semidirect
134. C3×C3⋊C8Dic6semidirect
135. C3×C3⋊C8C3×Q8semidirect
211. C33#500semidirect
245. C3×C8#627semidirect
246. C3×C8C3×Dic6semidirect
247. C3×C8C3×Dic6semidirect
248. C3⋊C8C3×Dic6semidirect
318. C32#4682semidirect
319. C32#4874semidirect
320. C32C3×#500semidirect
328. C8C3×#627semidirect
361. C3#10590891direct (N × H)
362. C3C3×#4682semidirect
363. C3C3×#4874semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#636452
C32×#8182
C32×C8×Dic32
C3×#635752
C3×#48743
C3×#46823
#105908913