Finite Group Databasebeta

← C3×#10593758 #13173229

Subgroups — C3×#10593758 #13173229

Order 1728.

544 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊#157C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic
3C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5C3×#63781C211yesmaximal, characteristic
6C3×#63781C211yesmaximal, characteristic
7C3×#63781C211yesmaximal, characteristic
8#10593758C311yesmaximal, characteristic, direct product factor
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C3×C4⋊#157—22no—
17C3×C4×#157—22no—
18C33×C4⋊C4C411yescharacteristic, complemented
19C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
20C3×#3750C411yescharacteristic, complemented
21C3×#3750C411yescharacteristic, complemented
22C2×C4⋊#157C2×C311yescharacteristic
23C2×C4×#157C2×C311yescharacteristic
24C4⋊C4×C3⋊D3C2×C311yescharacteristic
25C3×D3×C4⋊C4—62no—
26#63781C2×C311yescharacteristic
27#63781C2×C311yescharacteristic
28#63781C2×C311yescharacteristic
29#63781C2×C311yescharacteristic
30C22×C3×C3⋊D3C2311yescharacteristic
31C2×C3×#157—21no—
32C2×C3×#157—21no—
33C2×C3×#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157—21no—
37C2×C3×#157—21no—
38C2×C3×#157—21no—
39C2×C3×#157—21no—
40C2×C3×#157—21no—
41C2×C3×#157D412yes—
42C2×C3×#157Q812yes—
43C2×C33×C4C2×C411yescharacteristic
44C2×C33×C4C2×C411yescharacteristic
45C2×C33×C4C2×C411yescharacteristic
46C2×C3×C3⋊Dic3C2×C411yescharacteristic
47C2×C3×C3⋊Dic3C2×C411yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3×#387—91nomaximal
56C22×#157C22×C311yescharacteristic
57C22×#157C22×C311yescharacteristic
58C22×#157C22×C311yescharacteristic
59C22×#157C22×C311yescharacteristic
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C2×C4×C3⋊D3C22×C311yescharacteristic
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—62no—
65C2×C3×C4×D3—62no—
66C4⋊#157—22no—
67C4×#157—22no—
68C32×C4⋊C4—22no—
69C32×C4⋊C4—42no—
70C32×C4⋊C4C3×C411yescharacteristic, complemented
71C4⋊(C3⋊Dic3)C3×C411yescharacteristic, complemented
72#3750C3×C411yescharacteristic, complemented
73#3750C3×C411yescharacteristic, complemented
74C3×C4⋊Dic3—62no—
75C3×Dic3⋊C4—62no—
76C3×Dic3⋊C4—62no—
77C22×C33C22×C411yescharacteristic
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C2×C3×C3⋊D3C2×Q811yescharacteristic
80C2×C3×C3⋊D3#4111yescharacteristic
81C2×C3×C3⋊D3#4111yescharacteristic
82C2×C3×C3⋊D3#4111yescharacteristic
83C2×C3×C3⋊D3#4111yescharacteristic
84C3×#157—22no—
85C3×#157—22no—
86C3×#157—41no—
87C3×#157—41no—
88C33×C4—21no—
89C33×C4—21no—
90C33×C4—21no—
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—21no—
94C2×C3×C4⋊C4—91no—
95C2×C3×C4⋊C4—91no—
96C2×C3×C42—91no—
97D3×C4⋊C4—62no—
98C3×#95—91no—
99C3×#95—91no—
100C3×#95—91no—
101C3×#95—91no—
102C22×C3⋊D3C23×C311yescharacteristic
103C22×C3×D3—62no—
104C2×#157—21no—
105C2×#157—21no—
106C2×#157—21no—
107C2×#157—21no—
108C2×#157—21no—
109C2×#157—21no—
110C2×#157—21no—
111C2×#157—21no—
112C2×#157—21no—
113C2×#157—21no—
114C2×#157C3×D412yes—
115C2×#157C3×Q812yes—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—42no—
122C2×C32×C4C2×C3×C411yescharacteristic
123C2×C32×C4C2×C3×C411yescharacteristic
124C2×C32×C4C2×C3×C411yescharacteristic
125C2×C3⋊Dic3C2×C3×C411yescharacteristic
126C2×C3⋊Dic3C2×C3×C411yescharacteristic
127C2×C3⋊Dic3C2×C3×C411yescharacteristic
128C4×C3⋊D3—21no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139C3×C4×D3—122no—
140C3×C4×D3—122no—
141C3×C4×D3—122no—
142C3×C4×D3—122no—
143#387—91noSylow
144C2×C33#11711yescharacteristic
145C2×C33C4×D411yescharacteristic
146C2×C33C4×Q811yescharacteristic
147C3×C3⋊D3C22⋊Q811yescharacteristic
148C3×C3⋊D3#12311yescharacteristic
149C3×C3⋊D3#12511yescharacteristic
150C3×C3⋊D3#12611yescharacteristic
151C22×C3×C4—91no—
152C22×C3×C4—91no—
153C22×C3×C4—91no—
154C22×C3×C4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—91no—
157C22×C3×C4—91no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×C4×D3—62no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—182no—
167C3×C4⋊C4#15711yescharacteristic, complemented
168C3×C42—182no—
169C4⋊Dic3—62no—
170Dic3⋊C4—62no—
171Dic3⋊C4—62no—
172C22×C32—22no—
173C22×C32—42no—
174C22×C32C22×C3×C411yescharacteristic
175C2×C3⋊D3C2×C3×D411yescharacteristic
176C2×C3⋊D3C2×C3×Q811yescharacteristic
177C2×C3⋊D3C3×#4111yescharacteristic
178C2×C3⋊D3C3×#4111yescharacteristic
179C2×C3⋊D3C3×#4111yescharacteristic
180C2×C3⋊D3C3×#4111yescharacteristic
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187#157—22no—
188#157—22no—
189#157—41no—
190#157—41no—
191C32×C4—21no—
192C32×C4—24no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—44no—
196C32×C4—42no—
197C32×C4—42no—
198C32×C4—82no—
199C32×C4—82no—
200C3⋊Dic3—21no—
201C3⋊Dic3—21no—
202C3⋊Dic3—21no—
203C3×Dic3—64no—
204C3×Dic3—122no—
205C3×Dic3—122no—
206C2×C4⋊C4—91no—
207C2×C4⋊C4—91no—
208C2×C42—91no—
209#95—91no—
210#95—91no—
211#95—91no—
212#95—91no—
213C33#38711yescharacteristic, complemented, Sylow
214C23×C3—91no—
215C22×D3—62no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—92no—
223C2×C3×C4—92no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×#15711yescharacteristic
244C2×C3×C4C2×#15711yescharacteristic
245C2×C3×C4C2×#15711yescharacteristic
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C4×D3—64no—
250C4×D3—64no—
251C4×D3—122no—
252C4×D3—122no—
253C4×D3—122no—
254C4×D3—122no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—42no—
259C2×C32—42no—
260C2×C32—42no—
261C2×C32C3×#11711yescharacteristic
262C2×C32C3×C4×D411yescharacteristic
263C2×C32C3×C4×Q811yescharacteristic
264C3⋊D3C3×C22⋊Q811yescharacteristic
265C3⋊D3C3×#12311yescharacteristic
266C3⋊D3C3×#12511yescharacteristic
267C3⋊D3C3×#12611yescharacteristic
268C3×D3—62no—
269C3×D3—62no—
270C3×D3—62no—
271C3×D3—62no—
272C22×C4—91no—
273C22×C4—91no—
274C22×C4—91no—
275C22×C4—91no—
276C22×C4—91no—
277C22×C4—91no—
278C22×C4—91no—
279C4⋊C4—91no—
280C4⋊C4—91no—
281C4⋊C4—91no—
282C4⋊C4—182no—
283C4⋊C4C3×#15711yescharacteristic, complemented
284C42—182no—
285C22×C3—22no—
286C22×C3—42no—
287C22×C3—91no—
288C22×C3—91no—
289C22×C3—91no—
290C22×C3—91no—
291C22×C3—91no—
292C22×C3—91no—
293C22×C3C22×#15711yescharacteristic
294C2×D3—62no—
295C2×D3—62no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—62no—
299C2×D3—62no—
300C3×C4—24no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—44no—
307C3×C4—82no—
308C3×C4—82no—
309C3×C4—182no—
310C3×C4—182no—
311C3×C4—181no—
312C3×C4—181no—
313C3×C4—181no—
314C3×C4—361no—
315C3×C4—361no—
316Dic3—64no—
317Dic3—122no—
318Dic3—122no—
319C32—22no—
320C32—42no—
321C32C3×#38711yescharacteristic, complemented
322C23—91no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4C2×C3×#15711yescharacteristic
345C2×C4C2×C3×#15711yescharacteristic
346C2×C4C2×C3×#15711yescharacteristic
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—22no—
350C2×C3—42no—
351C2×C3—42no—
352C2×C3—42no—
353C2×C3—91no—
354C2×C3—91no—
355C2×C3—91no—
356C2×C3—91no—
357C2×C3#6428611yescharacteristic
358C2×C3D4×#15711yescharacteristic
359C2×C3Q8×#15711yescharacteristic
360D3—62no—
361D3—62no—
362D3—62no—
363D3—62no—
364C22—91no—
365C22—91no—
366C22—91no—
367C22—91no—
368C22—91no—
369C22—91no—
370C22C22×C3×#15711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—182no—
375C4—182no—
376C4—181no—
377C4—181no—
378C4—181no—
379C4—361no—
380C4—361no—
381C3—22no—
382C3—42no—
383C3#1059375811yescharacteristic, direct product factor
384C2—91no—
385C2—91no—
386C2—91no—
387C2—91no—
388C2C3×#6428611yescharacteristic
389C2C3×D4×#15711yescharacteristic
390C2C3×Q8×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#387
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #10593758C3direct (N × H)
18. C33×C4⋊C4C4semidirect
19. C3×C4⋊(C3⋊Dic3)C4semidirect
20. C3×#3750C4semidirect
21. C3×#3750C4semidirect
70. C32×C4⋊C4C3×C4semidirect
71. C4⋊(C3⋊Dic3)C3×C4semidirect
72. #3750C3×C4semidirect
73. #3750C3×C4semidirect
167. C3×C4⋊C4#157semidirect
213. C33#387semidirect
283. C4⋊C4C3×#157semidirect
321. C32C3×#387semidirect
383. C3#10593758direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C3×C4⋊C4×C3⋊D32
C3×#637812
#105937583
C3×#3879