Finite Group Databasebeta

← C3×#15073 #10633680

Subgroups — C3×#15073 #10633680

Order 768.

2118 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#385C211yesmaximal, characteristic
3C2×C3×#381C212yesmaximal
4C2×C3×#381C212yesmaximal
5C3×#1442C214yesmaximal, complemented
6C3×#1441C214yesmaximal, complemented
7C3×C22⋊#95C211yesmaximal, characteristic
8#15073C311yesmaximal, characteristic, direct product factor, Sylow
9C24×C3×C4C2212yes—
10C24×C3×C4C2211yescharacteristic
11C22×C3×C4⋊C4—28no—
12C22×C3×C4⋊C4—24no—
13C22×C3×C4⋊C4C2214yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C3×#385—24no—
19C3×#381—28no—
20C3×#381—28no—
21C2×C3×#95C2214yes—
22C2×C3×#95C2218yes—
23C2×C3×#95C2214yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2214yes—
26C23×C4⋊C4C2×C311yescharacteristic
27C2×#385C2×C311yescharacteristic
28C2×#381C2×C312yes—
29C2×#381C2×C312yes—
30#1442C2×C314yescomplemented
31#1441C2×C314yescomplemented
32C22⋊#95C2×C311yescharacteristic
33C25×C3C2311yescharacteristic
34C23×C3×C4—24no—
35C23×C3×C4—28no—
36C23×C3×C4—24no—
37C23×C3×C4—24no—
38C23×C3×C4—24no—
39C23×C3×C4—24no—
40C23×C3×C4—24no—
41C23×C3×C4—24no—
42C23×C3×C4D418yes—
43C23×C3×C4D414yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2314yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2314yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C2×C3×C4⋊C4—232no—
51C2×C3×C4⋊C4—216no—
52C2×C3×C4⋊C4—28no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—28no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4D414yes—
63C3×#95—216no—
64C3×#95—28no—
65C3×#95—28no—
66C3×#95—24no—
67C3×#95—28no—
68C24×C4C22×C311yescharacteristic
69C24×C4C22×C312yes—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4C22×C314yes—
73C22×C4⋊C4C22×C312yes—
74C22×C22⋊C4C22×C312yes—
75C22×C22⋊C4C22×C311yescharacteristic
76C22×C22⋊C4C22×C311yescharacteristic
77#385—24no—
78#381—28no—
79#381—28no—
80C2×#95C22×C314yes—
81C2×#95C22×C314yes—
82C2×#95C22×C318yes—
83C2×#95C22×C314yes—
84C2×#95C22×C312yes—
85C24×C3—24no—
86C24×C3—24no—
87C24×C3C2×D414yes—
88C24×C3C2×D412yes—
89C24×C3C2×D412yes—
90C24×C3#4112yes—
91C24×C3#4114yes—
92C24×C3C2411yescharacteristic
93C22×C3×C4—232no—
94C22×C3×C4—28no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—216no—
98C22×C3×C4—216no—
99C22×C3×C4—216no—
100C22×C3×C4—216no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—24no—
108C22×C3×C4—24no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—21no—
117C22×C3×C4—28no—
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4—24no—
121C22×C3×C4—28no—
122C22×C3×C4—48no—
123C22×C3×C4—48no—
124C22×C3×C4—48no—
125C22×C3×C4—48no—
126C22×C3×C4C2×D414yes—
127C22×C3×C4C2×D418yes—
128C22×C3×C4C2×D414yes—
129C22×C3×C4#4118yes—
130C3×C4⋊C4—232no—
131C3×C4⋊C4—48no—
132C3×C22⋊C4—28no—
133C3×C22⋊C4—44no—
134C3×C22⋊C4—48no—
135C25C23×C311yescharacteristic
136C23×C4—24no—
137C23×C4—28no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4C3×D418yes—
145C23×C4C3×D414yes—
146C23×C4C23×C311yescharacteristic
147C23×C4C23×C311yescharacteristic
148C23×C4C23×C314yes—
149C23×C4C23×C312yes—
150C23×C4C23×C314yes—
151C23×C4C23×C312yes—
152C2×C4⋊C4—232no—
153C2×C4⋊C4—216no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4C3×D414yes—
165#95—216no—
166#95—28no—
167#95—28no—
168#95—24no—
169#95—28no—
170C23×C3—24no—
171C23×C3—28no—
172C23×C3—28no—
173C23×C3—24no—
174C23×C3—28no—
175C23×C3—216no—
176C23×C3—28no—
177C23×C3#12014yes—
178C23×C3#12114yes—
179C23×C3#12118yes—
180C23×C3#12318yes—
181C23×C3#12414yes—
182C23×C3C22×D411yescharacteristic
183C23×C3C22×D412yes—
184C23×C3C22×D411yescharacteristic
185C23×C3C2×#4111yescharacteristic
186C23×C3C2×#4112yes—
187C23×C3C2×#4114yes—
188C23×C32−1+414yes—
189C2×C3×C4—232no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—216no—
193C2×C3×C4—28no—
194C2×C3×C4—24no—
195C2×C3×C4—28no—
196C2×C3×C4—28no—
197C2×C3×C4—28no—
198C2×C3×C4—44no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—44no—
205C2×C3×C4—432no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—48no—
216C2×C3×C4—48no—
217C2×C3×C4#12318yes—
218C24—24no—
219C24—24no—
220C24C2×C3×D414yes—
221C24C2×C3×D412yes—
222C24C2×C3×D412yes—
223C24C3×#4112yes—
224C24C3×#4114yes—
225C24C24×C311yescharacteristic
226C22×C4—232no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—216no—
232C22×C4—216no—
233C22×C4—216no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4C2×C3×D414yes—
260C22×C4C2×C3×D418yes—
261C22×C4C2×C3×D414yes—
262C22×C4C3×#4118yes—
263C4⋊C4—232no—
264C4⋊C4—48no—
265C22⋊C4—28no—
266C22⋊C4—44no—
267C22⋊C4—48no—
268C22×C3—24no—
269C22×C3—24no—
270C22×C3—216no—
271C22×C3—216no—
272C22×C3—28no—
273C22×C3—28no—
274C22×C3#39418yes—
275C22×C3C2×#12011yescharacteristic
276C22×C3C2×#12112yes—
277C22×C3C2×#12111yescharacteristic
278C22×C3C2×#12312yes—
279C22×C3C2×#12312yes—
280C22×C3#52514yes—
281C22×C3C2×#12411yescharacteristic
282C22×C3#53614yes—
283C22×C3#53612yes—
284C22×C3#54714yes—
285C22×C3#55218yes—
286C22×C3#55214yes—
287C3×C4—28no—
288C3×C4—48no—
289C3×C4—44no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4—84no—
293C23—24no—
294C23—28no—
295C23—28no—
296C23—24no—
297C23—28no—
298C23—216no—
299C23—28no—
300C23C3×#12014yes—
301C23C3×#12114yes—
302C23C3×#12118yes—
303C23C3×#12318yes—
304C23C3×#12414yes—
305C23C22×C3×D411yescharacteristic
306C23C22×C3×D412yes—
307C23C22×C3×D411yescharacteristic
308C23C2×C3×#4111yescharacteristic
309C23C2×C3×#4112yes—
310C23C2×C3×#4114yes—
311C23C3×2−1+414yes—
312C2×C4—232no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—216no—
316C2×C4—28no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—432no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4C3×#12318yes—
341C2×C3—28no—
342C2×C3C2×#39411yescharacteristic
343C2×C3#241314yes—
344C2×C3#248414yes—
345C2×C3#248612yes—
346C2×C3#250312yes—
347C2×C3#261411yescharacteristic
348C2×C3#284911yescharacteristic
349C22—216no—
350C22—216no—
351C22—28no—
352C22—28no—
353C22—24no—
354C22—24no—
355C22C3×#39418yes—
356C22C2×C3×#12011yescharacteristic
357C22C2×C3×#12112yes—
358C22C2×C3×#12111yescharacteristic
359C22C2×C3×#12312yes—
360C22C2×C3×#12312yes—
361C22C3×#52514yes—
362C22C2×C3×#12411yescharacteristic
363C22C3×#53614yes—
364C22C3×#53612yes—
365C22C3×#54714yes—
366C22C3×#55218yes—
367C22C3×#55214yes—
368C4—28no—
369C4—48no—
370C4—44no—
371C4—84no—
372C4—81no—
373C4—81no—
374C3#1507311yescharacteristic, direct product factor, Sylow
375C2—28no—
376C2C2×C3×#39411yescharacteristic
377C2C3×#241314yes—
378C2C3×#248414yes—
379C2C3×#248612yes—
380C2C3×#250312yes—
381C2C3×#261411yescharacteristic
382C2C3×#284911yescharacteristic
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
5. C3×#1442C2semidirect
6. C3×#1441C2semidirect
8. #15073C3direct (N × H)
30. #1442C2×C3semidirect
31. #1441C2×C3semidirect
374. C3#15073direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×C3×#3852
C2×C3×#3812
C3×#14422
C3×#14412
C3×C22⋊#952
#150733