Finite Group Databasebeta

← C3×#10592648 #13162989

Subgroups — C3×#10592648 #13162989

Order 1728.

1644 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C22⋊C4C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal
3C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C2×C3×#3750C212yesmaximal, complemented
5C3×#63658C211yesmaximal, characteristic, complemented
6#10592648C311yesmaximal, characteristic, direct product factor
7C3×#4949—34nomaximal
8C24×C33C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2212yes—
13C33×C22⋊C4—22no—
14C3×#3757—24no—
15C3×C4⋊(C3⋊Dic3)—22no—
16C3×#3750—24no—
17C2×C32×C22⋊C4—24no—
18C2×C32×C22⋊C4D314yes—
19C2×C32×C22⋊C4C2×C311yescharacteristic
20C2×#3757C2×C312yes—
21C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
22C2×#3750C2×C312yescomplemented
23C2×C3×#217—38no—
24C2×C3×C4⋊Dic3—34no—
25C2×C3×Dic3⋊C4—38no—
26#63658C2×C311yescharacteristic, complemented
27C3×C3⋊#95—34no—
28C23×C33—21no—
29C23×C33—22no—
30C23×C33—21no—
31C23×C33—22no—
32C23×C33Q812yes—
33C23×C33C2311yescharacteristic
34C2×C33×C4—22no—
35C2×C33×C4—21no—
36C2×C33×C4—22no—
37C2×C33×C4D412yes—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3D414yes—
44C3×#399—91no—
45#4949—34no—
46C24×C32—24no—
47C24×C32C2×D314yes—
48C24×C32C22×C311yescharacteristic
49C22×C32×C4—24no—
50C22×C32×C4—24no—
51C22×C32×C4C2×D314yes—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C22×C311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C312yes—
56C22×C3⋊Dic3C22×C312yes—
57C22×C3×Dic3—38no—
58C22×C3×Dic3—38no—
59C32×C22⋊C4—216no—
60C32×C22⋊C4—28no—
61C32×C22⋊C4—22no—
62#3757—24no—
63C4⋊(C3⋊Dic3)—22no—
64#3750—24no—
65C3×#217—616no—
66C3×C4⋊Dic3—68no—
67C3×Dic3⋊C4—616no—
68C22×C33—22no—
69C22×C33—24no—
70C22×C33—42no—
71C22×C33—41no—
72C22×C33—41no—
73C22×C33C2×D412yes—
74C22×C33C2×D411yescharacteristic
75C22×C33C2×Q811yescharacteristic
76C22×C33#4111yescharacteristic
77C22×C33#4112yes—
78C33×C4—22no—
79C33×C4—41no—
80C3×C3⋊Dic3—24no—
81C3×C3⋊Dic3—42no—
82C2×C3×C4⋊C4—92no—
83C2×C3×C4⋊C4—91no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—92no—
86C2×C3×C22⋊C4C3×D314yes—
87C2×C3×C22⋊C4C3⋊D311yescharacteristic
88C2×#217—38no—
89C2×C4⋊Dic3—34no—
90C2×Dic3⋊C4—38no—
91C3×#95—91no—
92C3⋊#95—34no—
93C23×C32—28no—
94C23×C32—28no—
95C23×C32—24no—
96C23×C32—24no—
97C23×C32—22no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—22no—
101C23×C32—28no—
102C23×C32—24no—
103C23×C32—28no—
104C23×C32—28no—
105C23×C32—48no—
106C23×C32—48no—
107C23×C32Dic618yes—
108C23×C32C3×Q812yes—
109C23×C32C22×D314yes—
110C23×C32C23×C311yescharacteristic
111C2×C32×C4—22no—
112C2×C32×C4—28no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—28no—
117C2×C32×C4—28no—
118C2×C32×C4—28no—
119C2×C32×C4—48no—
120C2×C32×C4—48no—
121C2×C32×C4#6718yes—
122C2×C32×C4C3×D412yes—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3C3×D414yes—
129C2×C3×Dic3—316no—
130C2×C3×Dic3—68no—
131C2×C3×Dic3—68no—
132C2×C3×Dic3—68no—
133C2×C3×Dic3—68no—
134C2×C3×Dic3—68no—
135#399—91noSylow
136C2×C33—42no—
137C2×C33C22⋊Q812yes—
138C2×C33C22⋊Q811yescharacteristic
139C2×C33#12312yes—
140C2×C33#12311yescharacteristic
141C2×C33#12711yescharacteristic
142C24×C3—24no—
143C24×C3C2×C3×D314yes—
144C24×C3C2×C3⋊D311yescharacteristic
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—92no—
148C22×C3×C4—92no—
149C22×C3×C4C2×C3×D314yes—
150C22×C3×C4C2×C3×D314yes—
151C22×C3×C4C2×C3⋊D311yescharacteristic
152C22×C3×C4C2×C3⋊D311yescharacteristic
153C22×Dic3—38no—
154C22×Dic3—38no—
155C3×C4⋊C4—184no—
156C3×C4⋊C4—182no—
157C3×C22⋊C4—216no—
158C3×C22⋊C4—28no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—184no—
161#217—616no—
162C4⋊Dic3—68no—
163Dic3⋊C4—616no—
164C22×C32—216no—
165C22×C32—28no—
166C22×C32—24no—
167C22×C32—22no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32—28no—
171C22×C32—28no—
172C22×C32—24no—
173C22×C32—216no—
174C22×C32—416no—
175C22×C32—416no—
176C22×C32—48no—
177C22×C32—48no—
178C22×C32—48no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32C2×Dic614yes—
185C22×C32#23514yes—
186C22×C32D3×D418yes—
187C22×C32#23718yes—
188C22×C32C2×#6714yes—
189C22×C32C2×C3×D412yes—
190C22×C32C2×C3×D411yescharacteristic
191C22×C32C2×C3×Q811yescharacteristic
192C22×C32C3×#4111yescharacteristic
193C22×C32C3×#4112yes—
194C32×C4—22no—
195C32×C4—28no—
196C32×C4—48no—
197C32×C4—48no—
198C32×C4—44no—
199C32×C4—41no—
200C3⋊Dic3—24no—
201C3⋊Dic3—42no—
202C3×Dic3—616no—
203C3×Dic3—128no—
204C2×C4⋊C4—92no—
205C2×C4⋊C4—91no—
206C2×C22⋊C4—92no—
207C2×C22⋊C4C3×C3⋊D311yescharacteristic
208#95—91no—
209C33#39911yescharacteristic, complemented, Sylow
210C23×C3—28no—
211C23×C3—28no—
212C23×C3—24no—
213C23×C3—28no—
214C23×C3—28no—
215C23×C3—24no—
216C23×C3—21no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—21no—
220C23×C3—28no—
221C23×C3—24no—
222C23×C3—48no—
223C23×C3—48no—
224C23×C3C3×Dic618yes—
225C23×C3#63412yes—
226C23×C3C22×C3×D314yes—
227C23×C3C22×C3⋊D311yescharacteristic
228C2×C3×C4—28no—
229C2×C3×C4—22no—
230C2×C3×C4—28no—
231C2×C3×C4—28no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—28no—
236C2×C3×C4—48no—
237C2×C3×C4—48no—
238C2×C3×C4—94no—
239C2×C3×C4—182no—
240C2×C3×C4—182no—
241C2×C3×C4—182no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4C3×#6718yes—
245C2×C3×C4#63812yes—
246C2×Dic3—316no—
247C2×Dic3—68no—
248C2×Dic3—68no—
249C2×Dic3—68no—
250C2×Dic3—68no—
251C2×Dic3—68no—
252C2×C32—28no—
253C2×C32—28no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—416no—
258C2×C32—48no—
259C2×C32—42no—
260C2×C32#87818yes—
261C2×C32#88318yes—
262C2×C32#92414yes—
263C2×C32#93514yes—
264C2×C32#94014yes—
265C2×C32C3×C22⋊Q812yes—
266C2×C32C3×C22⋊Q811yescharacteristic
267C2×C32C3×#12312yes—
268C2×C32C3×#12311yescharacteristic
269C2×C32C3×#12711yescharacteristic
270C24C2×C3×C3⋊D311yescharacteristic
271C22×C4—92no—
272C22×C4—92no—
273C22×C4C2×C3×C3⋊D311yescharacteristic
274C22×C4C2×C3×C3⋊D311yescharacteristic
275C4⋊C4—184no—
276C4⋊C4—182no—
277C22⋊C4—22no—
278C22⋊C4—184no—
279C22×C3—216no—
280C22×C3—216no—
281C22×C3—28no—
282C22×C3—24no—
283C22×C3—22no—
284C22×C3—24no—
285C22×C3—24no—
286C22×C3—28no—
287C22×C3—28no—
288C22×C3—24no—
289C22×C3—416no—
290C22×C3—416no—
291C22×C3—48no—
292C22×C3—48no—
293C22×C3—48no—
294C22×C3—44no—
295C22×C3—44no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3C2×C3×Dic614yes—
300C22×C3C3×#23514yes—
301C22×C3C3×D3×D418yes—
302C22×C3C3×#23718yes—
303C22×C3C2×C3×#6714yes—
304C22×C3C2×#63411yescharacteristic
305C22×C3#382811yescharacteristic
306C22×C3D4×C3⋊D312yes—
307C22×C3#383012yes—
308C22×C3C2×#63811yescharacteristic
309C3×C4—28no—
310C3×C4—22no—
311C3×C4—48no—
312C3×C4—48no—
313C3×C4—44no—
314C3×C4—41no—
315C3×C4—184no—
316C3×C4—362no—
317Dic3—616no—
318Dic3—128no—
319C32—24no—
320C32#494914yescomplemented
321C32C3×#39911yescharacteristic, complemented
322C23—21no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23C3×#63412yes—
327C23C22×C3×C3⋊D311yescharacteristic
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—94no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4C3×#63812yes—
338C2×C3—28no—
339C2×C3—28no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—24no—
343C2×C3—416no—
344C2×C3—48no—
345C2×C3—42no—
346C2×C3C3×#87818yes—
347C2×C3C3×#88318yes—
348C2×C3C3×#92414yes—
349C2×C3C3×#93514yes—
350C2×C3C3×#94014yes—
351C2×C3#6408712yes—
352C2×C3#6409212yes—
353C2×C3#6413311yescharacteristic
354C2×C3#6414411yescharacteristic
355C2×C3#6414911yescharacteristic
356C22—24no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22C2×C3×#63411yescharacteristic
362C22C3×#382811yescharacteristic
363C22C3×D4×C3⋊D312yes—
364C22C3×#383012yes—
365C22C2×C3×#63811yescharacteristic
366C4—22no—
367C4—41no—
368C4—184no—
369C4—362no—
370C3—24no—
371C3C3×#494914yescomplemented
372C3#1059264811yescharacteristic, direct product factor
373C2—42no—
374C2C3×#6408712yes—
375C2C3×#6409212yes—
376C2C3×#6413311yescharacteristic
377C2C3×#6414411yescharacteristic
378C2C3×#6414911yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
prime radicalC24×C33C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C3×#399
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
4. C2×C3×#3750C2semidirect
5. C3×#63658C2semidirect
6. #10592648C3direct (N × H)
21. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
22. C2×#3750C2×C3semidirect
26. #63658C2×C3semidirect
209. C33#399semidirect
320. C32#4949semidirect
321. C32C3×#399semidirect
371. C3C3×#4949semidirect
372. C3#10592648direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C22⋊C42
C2×C3×#37572
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37502
C3×#636582
#105926483
C3×#49493