Finite Group Databasebeta

← C3×#14955 #10633562

Subgroups — C3×#14955 #10633562

Order 768.

1846 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#385C212yesmaximal
2C2×C3×#385C211yesmaximal, characteristic
3C22×C3×#95C211yesmaximal, characteristic
4C3×#1442C214yesmaximal, complemented
5C3×#1441C214yesmaximal, complemented
6C3×C22⋊#95C212yesmaximal
7C3×C22⋊#95C211yesmaximal, characteristic
8#14955C311yesmaximal, characteristic, direct product factor, Sylow
9C24×C3×C4C2212yes—
10C24×C3×C4C2211yescharacteristic
11C22×C3×C4⋊C4C2214yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C3×#385—24no—
17C3×#385—28no—
18C2×C3×#95—24no—
19C2×C3×#95C2214yes—
20C2×C3×#95C2214yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2214yes—
23C2×C3×#95C2214yes—
24C2×C3×#95C2214yes—
25C2×#385C2×C311yescharacteristic
26C2×#385C2×C312yes—
27C22×#95C2×C311yescharacteristic
28#1442C2×C314yescomplemented
29#1441C2×C314yescomplemented
30C22⋊#95C2×C312yes—
31C22⋊#95C2×C311yescharacteristic
32C25×C3C2311yescharacteristic
33C23×C3×C4—24no—
34C23×C3×C4—24no—
35C23×C3×C4—28no—
36C23×C3×C4—24no—
37C23×C3×C4—24no—
38C23×C3×C4—44no—
39C23×C3×C4—44no—
40C23×C3×C4D414yes—
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2314yes—
46C23×C3×C4C2314yes—
47C2×C3×C4⋊C4—28no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C4⋊C4—44no—
51C2×C3×C4⋊C4—44no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4D414yes—
59C2×C3×C22⋊C4Q818yes—
60C3×#95—28no—
61C3×#95—28no—
62C3×#95—24no—
63C3×#95—28no—
64C3×#95—44no—
65C3×#95—44no—
66C24×C4C22×C312yes—
67C24×C4C22×C311yescharacteristic
68C22×C4⋊C4C22×C314yes—
69C22×C4⋊C4C22×C312yes—
70C22×C22⋊C4C22×C311yescharacteristic
71C22×C22⋊C4C22×C311yescharacteristic
72C22×C22⋊C4C22×C312yes—
73#385—24no—
74#385—28no—
75C2×#95—24no—
76C2×#95C22×C314yes—
77C2×#95C22×C314yes—
78C2×#95C22×C314yes—
79C2×#95C22×C312yes—
80C2×#95C22×C314yes—
81C2×#95C22×C314yes—
82C24×C3—24no—
83C24×C3—24no—
84C24×C3C2×D412yes—
85C24×C3C2×D412yes—
86C24×C3C2×Q814yes—
87C24×C3#4112yes—
88C24×C3#4114yes—
89C24×C3C2411yescharacteristic
90C22×C3×C4—216no—
91C22×C3×C4—28no—
92C22×C3×C4—24no—
93C22×C3×C4—22no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—22no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—24no—
108C22×C3×C4—24no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—24no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—24no—
117C22×C3×C4—48no—
118C22×C3×C4—416no—
119C22×C3×C4—416no—
120C22×C3×C4—48no—
121C22×C3×C4—416no—
122C22×C3×C4—48no—
123C22×C3×C4C2×D414yes—
124C22×C3×C4C2×D414yes—
125C22×C3×C4C2×Q818yes—
126C22×C3×C4#4118yes—
127C3×C4⋊C4—416no—
128C3×C4⋊C4—48no—
129C3×C22⋊C4—28no—
130C3×C22⋊C4—216no—
131C3×C22⋊C4—44no—
132C25C23×C311yescharacteristic
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—28no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4C3×D414yes—
141C23×C4C23×C311yescharacteristic
142C23×C4C23×C311yescharacteristic
143C23×C4C23×C312yes—
144C23×C4C23×C312yes—
145C23×C4C23×C314yes—
146C23×C4C23×C314yes—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4C3×D414yes—
159C2×C22⋊C4C3×Q818yes—
160#95—28no—
161#95—28no—
162#95—24no—
163#95—28no—
164#95—44no—
165#95—44no—
166C23×C3—24no—
167C23×C3—28no—
168C23×C3—24no—
169C23×C3—28no—
170C23×C3—28no—
171C23×C3—28no—
172C23×C3—28no—
173C23×C3—28no—
174C23×C3#12114yes—
175C23×C3C22⋊Q818yes—
176C23×C3#12314yes—
177C23×C3#12518yes—
178C23×C3C4⋊Q814yes—
179C23×C3C22×D411yescharacteristic
180C23×C3C22×D411yescharacteristic
181C23×C3C22×Q812yes—
182C23×C3C2×#4111yescharacteristic
183C23×C3C2×#4112yes—
184C23×C3C2×#4114yes—
185C23×C3H(5,𝔽2)14yes—
186C2×C3×C4—24no—
187C2×C3×C4—28no—
188C2×C3×C4—24no—
189C2×C3×C4—28no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—28no—
195C2×C3×C4—28no—
196C2×C3×C4—28no—
197C2×C3×C4—416no—
198C2×C3×C4—432no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—42no—
203C2×C3×C4—44no—
204C2×C3×C4—44no—
205C2×C3×C4—48no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—44no—
217C24—24no—
218C24—24no—
219C24C2×C3×D412yes—
220C24C2×C3×D412yes—
221C24C2×C3×Q814yes—
222C24C3×#4112yes—
223C24C3×#4114yes—
224C24C24×C311yescharacteristic
225C22×C4—216no—
226C22×C4—28no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—48no—
253C22×C4—416no—
254C22×C4—416no—
255C22×C4—48no—
256C22×C4—416no—
257C22×C4—48no—
258C22×C4C2×C3×D414yes—
259C22×C4C2×C3×D414yes—
260C22×C4C2×C3×Q818yes—
261C22×C4C3×#4118yes—
262C4⋊C4—416no—
263C4⋊C4—48no—
264C22⋊C4—28no—
265C22⋊C4—216no—
266C22⋊C4—44no—
267C22×C3—28no—
268C22×C3—24no—
269C22×C3—24no—
270C22×C3—216no—
271C22×C3—216no—
272C22×C3—28no—
273C22×C3#39818yes—
274C22×C3C2×#12111yescharacteristic
275C22×C3C2×C22⋊Q812yes—
276C22×C3C2×#12311yescharacteristic
277C22×C3#52512yes—
278C22×C3C2×#12512yes—
279C22×C3#52914yes—
280C22×C3C2×C4⋊Q811yescharacteristic
281C22×C3#53412yes—
282C22×C3#53814yes—
283C22×C3#54614yes—
284C22×C3#55114yes—
285C22×C3#55414yes—
286C22×C3#55414yes—
287C3×C4—44no—
288C3×C4—48no—
289C3×C4—44no—
290C3×C4—48no—
291C3×C4—81no—
292C3×C4—81no—
293C23—24no—
294C23—28no—
295C23—24no—
296C23—28no—
297C23—28no—
298C23—28no—
299C23—28no—
300C23—28no—
301C23C3×#12114yes—
302C23C3×C22⋊Q818yes—
303C23C3×#12314yes—
304C23C3×#12518yes—
305C23C3×C4⋊Q814yes—
306C23C22×C3×D411yescharacteristic
307C23C22×C3×D411yescharacteristic
308C23C22×C3×Q812yes—
309C23C2×C3×#4111yescharacteristic
310C23C2×C3×#4112yes—
311C23C2×C3×#4114yes—
312C23C3×H(5,𝔽2)14yes—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—416no—
325C2×C4—432no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C3—28no—
345C2×C3C2×#39811yescharacteristic
346C2×C3#245014yes—
347C2×C3#250514yes—
348C2×C3#253412yes—
349C2×C3#258312yes—
350C2×C3#262211yescharacteristic
351C2×C3#285311yescharacteristic
352C22—24no—
353C22—216no—
354C22—216no—
355C22—28no—
356C22—28no—
357C22—24no—
358C22C3×#39818yes—
359C22C2×C3×#12111yescharacteristic
360C22C2×C3×C22⋊Q812yes—
361C22C2×C3×#12311yescharacteristic
362C22C3×#52512yes—
363C22C2×C3×#12512yes—
364C22C3×#52914yes—
365C22C2×C3×C4⋊Q811yescharacteristic
366C22C3×#53412yes—
367C22C3×#53814yes—
368C22C3×#54614yes—
369C22C3×#55114yes—
370C22C3×#55414yes—
371C22C3×#55414yes—
372C4—44no—
373C4—48no—
374C4—48no—
375C4—44no—
376C4—81no—
377C4—81no—
378C3#1495511yescharacteristic, direct product factor, Sylow
379C2—28no—
380C2C2×C3×#39811yescharacteristic
381C2C3×#245014yes—
382C2C3×#250514yes—
383C2C3×#253412yes—
384C2C3×#258312yes—
385C2C3×#262211yescharacteristic
386C2C3×#285311yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1442C2semidirect
5. C3×#1441C2semidirect
8. #14955C3direct (N × H)
28. #1442C2×C3semidirect
29. #1441C2×C3semidirect
378. C3#14955direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3852
C22×C3×#952
C3×#14422
C3×#14412
C3×C22⋊#952
#149553