Finite Group Databasebeta

← C3×#10591119 #13158151

Subgroups — C3×#10591119 #13158151

Order 1728.

730 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C2×C32×#208C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×#63572C214yesmaximal, complemented
5C3×#5119—31nomaximal
6C3×#5102—31nomaximal
7#10591119C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C3×#3757C412yes—
10C3×C4⋊(C3⋊Dic3)C412yes—
11C3×C4×C3⋊Dic3C2212yes—
12C32×#208—22no—
13C2×C32×C3⋊C8—22no—
14C2×C32×C3⋊C8C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C2×C32×M16—31no—
17C22×C32×C8—31no—
18#64135C2×C311yescharacteristic
19C3×#926—31no—
20C3×#926—31no—
21C3×#926—61no—
22C2×C3×#208—21no—
23C2×C3×#208D311yescharacteristic
24C2×C3×#208C2×C311yescharacteristic
25C22×C3×C3⋊C8—21no—
26C22×C3×C3⋊C8D311yescharacteristic
27C22×C3×C3⋊C8C2×C311yescharacteristic
28C3×Dic3⋊C8—34no—
29C3×Dic3⋊C8—34no—
30#63572C2×C314yescomplemented
31C23×C33C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4D412yes—
34C2×C33×C4Q812yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3C2×C414yes—
37C32×C3⋊C8—22no—
38C32×C3⋊C8—22no—
39C3×#424—91no—
40#5119—31no—
41#5102—31no—
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4—41no—
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C22×C311yescharacteristic
49C32×M16—62no—
50C2×C32×C8—32no—
51C2×C32×C8—32no—
52C2×C32×C8—62no—
53#3757C3×C412yes—
54C4⋊(C3⋊Dic3)C3×C412yes—
55C4×C3⋊Dic3C22×C312yes—
56C3×#217—32no—
57C3×#217—32no—
58C3×#217—62no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—62no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—62no—
65C3×#208—24no—
66C3×#208—22no—
67C3×#208—22no—
68C2×C3×C3⋊C8—24no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8—22no—
73C2×C3×C3⋊C8C2×D312yes—
74C2×C3×C3⋊C8C2×D312yes—
75C2×C3×C3⋊C8C22×C312yes—
76C2×C3×C3⋊C8C22×C312yes—
77C22×C33—22no—
78C22×C33C4⋊C412yes—
79C22×C33C22×C411yescharacteristic
80C33×C4C4⋊C412yes—
81C33×C4C2×D411yescharacteristic
82C33×C4C2×Q811yescharacteristic
83C3×C3⋊Dic3—24no—
84C2×C3×M16—31no—
85C2×C3×M16—31no—
86C2×C3×M16—61no—
87C22×C3×C8—31no—
88C22×C3×C8—31no—
89C22×C3×C8—61no—
90C3×#117—91no—
91#926—31no—
92#926—31no—
93#926—61no—
94C2×#208C3×D311yescharacteristic
95C22×C3⋊C8C3×D311yescharacteristic
96C3×C4⋊C8—94no—
97Dic3⋊C8—34no—
98Dic3⋊C8—34no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32C4×D311yescharacteristic
104C23×C32C4×D311yescharacteristic
105C23×C32C2×C3×C411yescharacteristic
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—41no—
122C2×C32×C4Dic612yes—
123C2×C32×C4Dic612yes—
124C2×C32×C4C4×D311yescharacteristic
125C2×C32×C4C4×D311yescharacteristic
126C2×C32×C4#6712yes—
127C2×C32×C4#6712yes—
128C2×C32×C4C2×C3×C411yescharacteristic
129C2×C32×C4C3×D412yes—
130C2×C32×C4C3×Q812yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3C2×C3×C414yes—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—64no—
138C32×C8—62no—
139C32×C8—62no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8—24no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—22no—
146#424—91noSylow
147C2×C33—22no—
148C2×C33C2×C4⋊C411yescharacteristic
149C2×C33#13112yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—41no—
154C22×C3×C4D3211yescharacteristic
155C22×C3×C4C2×C3×D311yescharacteristic
156C22×C3×C4C2×C3×D311yescharacteristic
157C3×M16—64no—
158C3×M16—62no—
159C3×M16—62no—
160C2×C3×C8—32no—
161C2×C3×C8—32no—
162C2×C3×C8—32no—
163C2×C3×C8—32no—
164C2×C3×C8—62no—
165C2×C3×C8—62no—
166C2×C3×C8—64no—
167C2×C3×C8—62no—
168C2×C3×C8—62no—
169C3×C4⋊C4—92no—
170C3×C22⋊C4—92no—
171C3×C42—92no—
172#217—32no—
173#217—32no—
174#217—62no—
175C4⋊Dic3—32no—
176C4⋊Dic3—32no—
177C4⋊Dic3—62no—
178C4×Dic3—32no—
179C4×Dic3—32no—
180C4×Dic3—62no—
181#208—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8C2×C3×D312yes—
184C2×C3⋊C8C2×C3×D312yes—
185C22×C32—22no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—42no—
196C22×C32—41no—
197C22×C32—42no—
198C22×C32—42no—
199C22×C32—44no—
200C22×C32Dic3⋊C412yes—
201C22×C32Dic3⋊C412yes—
202C22×C32C3×C4⋊C412yes—
203C22×C32C2×C4×D311yescharacteristic
204C22×C32C2×C4×D311yescharacteristic
205C22×C32C22×C3×C411yescharacteristic
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—42no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4Dic3⋊C412yes—
219C32×C4Dic3⋊C412yes—
220C32×C4C3×C4⋊C412yes—
221C32×C4C2×Dic611yescharacteristic
222C32×C4C2×Dic611yescharacteristic
223C32×C4C2×#6711yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C32×C4C2×C3×D411yescharacteristic
226C32×C4C2×C3×Q811yescharacteristic
227C3⋊Dic3—24no—
228C3×Dic3—64no—
229C3×Dic3—64no—
230C3×Dic3—124no—
231C2×M16—31no—
232C22×C8—31no—
233#117—91no—
234C4⋊C8—94no—
235C33#42411yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3#62411yescharacteristic
241C23×C3C3×C4×D311yescharacteristic
242C23×C3C3×C4×D311yescharacteristic
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—94no—
260C2×C3×C4#62411yescharacteristic
261C2×C3×C4#62512yes—
262C2×C3×C4#62712yes—
263C2×C3×C4C3×Dic612yes—
264C2×C3×C4C3×Dic612yes—
265C2×C3×C4C3×C4×D311yescharacteristic
266C2×C3×C4C3×C4×D311yescharacteristic
267C2×C3×C4C3×#6712yes—
268C2×C3×C4C3×#6712yes—
269C2×C3×C4C2×D3211yescharacteristic
270C2×C3×C4C22×C3×D311yescharacteristic
271C2×C3×C4C22×C3×D311yescharacteristic
272C2×Dic3—34no—
273C2×Dic3—34no—
274C2×Dic3—64no—
275C3×C8—64no—
276C3×C8—64no—
277C3×C8—62no—
278C3×C8—62no—
279C3×C8—62no—
280C3×C8—62no—
281C3⋊C8—22no—
282C3⋊C8—22no—
283C2×C32—22no—
284C2×C32—24no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—42no—
295C2×C32—42no—
296C2×C32—42no—
297C2×C32—44no—
298C2×C32#90112yes—
299C2×C32#90712yes—
300C2×C32C2×Dic3⋊C411yescharacteristic
301C2×C32C2×Dic3⋊C411yescharacteristic
302C2×C32C2×C3×C4⋊C411yescharacteristic
303C2×C32C3×#13112yes—
304C22×C4C3×D3211yescharacteristic
305M16—62no—
306C2×C8—32no—
307C2×C8—32no—
308C2×C8—62no—
309C4⋊C4—92no—
310C22⋊C4—92no—
311C42—92no—
312C22×C3—24no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—44no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3#372412yes—
328C22×C3C3×Dic3⋊C412yes—
329C22×C3C3×Dic3⋊C412yes—
330C22×C3C2×#62411yescharacteristic
331C22×C3C2×C3×C4×D311yescharacteristic
332C22×C3C2×C3×C4×D311yescharacteristic
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—42no—
345C3×C4—184no—
346C3×C4#372412yes—
347C3×C4C3×Dic3⋊C412yes—
348C3×C4C3×Dic3⋊C412yes—
349C3×C4C2×#62511yescharacteristic
350C3×C4C2×#62711yescharacteristic
351C3×C4C2×C3×Dic611yescharacteristic
352C3×C4C2×C3×Dic611yescharacteristic
353C3×C4C2×C3×#6711yescharacteristic
354C3×C4C2×C3×#6711yescharacteristic
355Dic3—64no—
356Dic3—64no—
357Dic3—124no—
358C32—21no—
359C32—21no—
360C32—21no—
361C32—41no—
362C32#510211yescharacteristic, complemented
363C32#511911yescharacteristic, complemented
364C32C3×#42411yescharacteristic, complemented
365C23C3×#62411yescharacteristic
366C2×C4—94no—
367C2×C4C3×#62411yescharacteristic
368C2×C4C3×#62512yes—
369C2×C4C3×#62712yes—
370C2×C4C2×C3×D3211yescharacteristic
371C8—62no—
372C8—62no—
373C2×C3—24no—
374C2×C3—22no—
375C2×C3—22no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—22no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—44no—
386C2×C3—41no—
387C2×C3—42no—
388C2×C3#6381712yes—
389C2×C3C2×#372411yescharacteristic
390C2×C3C3×#90112yes—
391C2×C3C3×#90712yes—
392C2×C3C2×C3×Dic3⋊C411yescharacteristic
393C2×C3C2×C3×Dic3⋊C411yescharacteristic
394C22—22no—
395C22C3×#372412yes—
396C22C2×C3×#62411yescharacteristic
397C4—184no—
398C4C3×#372412yes—
399C4C2×C3×#62511yescharacteristic
400C4C2×C3×#62711yescharacteristic
401C3—21no—
402C3—21no—
403C3—21no—
404C3—41no—
405C3#1059111911yescharacteristic, direct product factor
406C3C3×#510211yescharacteristic, complemented
407C3C3×#511911yescharacteristic, complemented
408C2—22no—
409C2C3×#6381712yes—
410C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#424
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63572C2semidirect
7. #10591119C3direct (N × H)
30. #63572C2×C3semidirect
235. C33#424semidirect
362. C32#5102semidirect
363. C32#5119semidirect
364. C32C3×#424semidirect
405. C3#10591119direct (N × H)
406. C3C3×#5102semidirect
407. C3C3×#5119semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C2×C32×#2082
C22×C32×C3⋊C82
C3×#635722
C3×#51193
C3×#51023
#105911193