Finite Group Databasebeta

← C3×#15092 #10633699

Subgroups — C3×#15092 #10633699

Order 768.

2150 conjugacy classes of subgroups.

The subgroup lattice has 366 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×#385C212yesmaximal
4C3×#1444C214yesmaximal, complemented
5C3×#1441C214yesmaximal, complemented
6C3×C22⋊#95C211yesmaximal, characteristic
7C3×C22⋊#95C212yesmaximal
8#15092C311yesmaximal, characteristic, direct product factor, Sylow
9C24×C3×C4C2211yescharacteristic
10C24×C3×C4C2212yes—
11C22×C3×C4⋊C4—28no—
12C22×C3×C4⋊C4—24no—
13C22×C3×C4⋊C4C2214yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2214yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C3×#385—24no—
20C3×#385—28no—
21C2×C3×#95C2214yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2214yes—
24C2×C3×#95C2218yes—
25C23×C4⋊C4C2×C311yescharacteristic
26C2×#385C2×C311yescharacteristic
27C2×#385C2×C312yes—
28#1444C2×C314yescomplemented
29#1441C2×C314yescomplemented
30C22⋊#95C2×C311yescharacteristic
31C22⋊#95C2×C312yes—
32C25×C3C2311yescharacteristic
33C23×C3×C4—24no—
34C23×C3×C4—28no—
35C23×C3×C4—24no—
36C23×C3×C4—24no—
37C23×C3×C4—48no—
38C23×C3×C4D414yes—
39C23×C3×C4D418yes—
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2314yes—
43C23×C3×C4C2314yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C2×C3×C4⋊C4—232no—
47C2×C3×C4⋊C4—216no—
48C2×C3×C4⋊C4—28no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4—28no—
52C2×C3×C4⋊C4—48no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4Q818yes—
60C2×C3×C22⋊C4Q814yes—
61C3×#95—28no—
62C3×#95—24no—
63C3×#95—28no—
64C3×#95—48no—
65C24×C4C22×C311yescharacteristic
66C24×C4C22×C312yes—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4C22×C314yes—
70C22×C4⋊C4C22×C312yes—
71C22×C4⋊C4C22×C314yes—
72C22×C22⋊C4C22×C311yescharacteristic
73C22×C22⋊C4C22×C312yes—
74C22×C22⋊C4C22×C311yescharacteristic
75#385—24no—
76#385—28no—
77C2×#95C22×C314yes—
78C2×#95C22×C314yes—
79C2×#95C22×C312yes—
80C2×#95C22×C318yes—
81C24×C3—24no—
82C24×C3—24no—
83C24×C3C2×D412yes—
84C24×C3C2×D414yes—
85C24×C3C2×Q814yes—
86C24×C3C2×Q812yes—
87C24×C3#4112yes—
88C24×C3C2411yescharacteristic
89C22×C3×C4—232no—
90C22×C3×C4—28no—
91C22×C3×C4—24no—
92C22×C3×C4—24no—
93C22×C3×C4—216no—
94C22×C3×C4—216no—
95C22×C3×C4—22no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—28no—
100C22×C3×C4—24no—
101C22×C3×C4—22no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—28no—
105C22×C3×C4—28no—
106C22×C3×C4—28no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—416no—
114C22×C3×C4—416no—
115C22×C3×C4—416no—
116C22×C3×C4C2×D418yes—
117C22×C3×C4C2×D414yes—
118C22×C3×C4C2×Q818yes—
119C22×C3×C4C2×Q814yes—
120C22×C3×C4#4118yes—
121C3×C4⋊C4—232no—
122C3×C4⋊C4—416no—
123C3×C4⋊C4—48no—
124C3×C22⋊C4—216no—
125C3×C22⋊C4—28no—
126C3×C22⋊C4—44no—
127C25C23×C311yescharacteristic
128C23×C4—24no—
129C23×C4—28no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—48no—
133C23×C4C3×D414yes—
134C23×C4C3×D418yes—
135C23×C4C23×C311yescharacteristic
136C23×C4C23×C311yescharacteristic
137C23×C4C23×C314yes—
138C23×C4C23×C314yes—
139C23×C4C23×C312yes—
140C23×C4C23×C312yes—
141C2×C4⋊C4—232no—
142C2×C4⋊C4—216no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—48no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C3×Q814yes—
155C2×C22⋊C4C3×Q818yes—
156#95—28no—
157#95—24no—
158#95—28no—
159#95—48no—
160C23×C3—24no—
161C23×C3—28no—
162C23×C3—28no—
163C23×C3—24no—
164C23×C3—28no—
165C23×C3—216no—
166C23×C3—28no—
167C23×C3#12014yes—
168C23×C3C22⋊Q818yes—
169C23×C3C22⋊Q814yes—
170C23×C3C4⋊Q818yes—
171C23×C3C4⋊Q814yes—
172C23×C3C22×D411yescharacteristic
173C23×C3C22×D412yes—
174C23×C3C22×Q812yes—
175C23×C3C22×Q811yescharacteristic
176C23×C3C2×#4111yescharacteristic
177C23×C3C2×#4114yes—
178C23×C3H(5,𝔽2)14yes—
179C2×C3×C4—232no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—216no—
183C2×C3×C4—28no—
184C2×C3×C4—28no—
185C2×C3×C4—24no—
186C2×C3×C4—28no—
187C2×C3×C4—28no—
188C2×C3×C4—216no—
189C2×C3×C4—28no—
190C2×C3×C4—432no—
191C2×C3×C4—42no—
192C2×C3×C4—44no—
193C2×C3×C4—44no—
194C2×C3×C4—48no—
195C2×C3×C4—42no—
196C2×C3×C4—48no—
197C2×C3×C4—48no—
198C2×C3×C4—44no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—44no—
203C2×C3×C4—44no—
204C2×C3×C4—42no—
205C2×C3×C4#12318yes—
206C24—24no—
207C24—24no—
208C24C2×C3×D412yes—
209C24C2×C3×D414yes—
210C24C2×C3×Q812yes—
211C24C2×C3×Q814yes—
212C24C3×#4112yes—
213C24C24×C311yescharacteristic
214C22×C4—232no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—216no—
219C22×C4—216no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—416no—
239C22×C4—416no—
240C22×C4—416no—
241C22×C4C2×C3×D418yes—
242C22×C4C2×C3×D414yes—
243C22×C4C2×C3×Q814yes—
244C22×C4C2×C3×Q818yes—
245C22×C4C3×#4118yes—
246C4⋊C4—232no—
247C4⋊C4—416no—
248C4⋊C4—48no—
249C22⋊C4—28no—
250C22⋊C4—216no—
251C22⋊C4—44no—
252C22×C3—216no—
253C22×C3—216no—
254C22×C3—24no—
255C22×C3—24no—
256C22×C3—28no—
257C22×C3—28no—
258C22×C3#39518yes—
259C22×C3C2×#12011yescharacteristic
260C22×C3C2×C22⋊Q812yes—
261C22×C3C2×C22⋊Q811yescharacteristic
262C22×C3C2×#12312yes—
263C22×C3#52514yes—
264C22×C3C2×C4⋊Q812yes—
265C22×C3C2×C4⋊Q811yescharacteristic
266C22×C3#53414yes—
267C22×C3#53512yes—
268C22×C3#55314yes—
269C22×C3#55418yes—
270C22×C3#55414yes—
271C3×C4—28no—
272C3×C4—48no—
273C3×C4—48no—
274C3×C4—44no—
275C3×C4—81no—
276C3×C4—81no—
277C23—24no—
278C23—28no—
279C23—28no—
280C23—24no—
281C23—28no—
282C23—28no—
283C23—216no—
284C23C3×#12014yes—
285C23C3×C22⋊Q814yes—
286C23C3×C22⋊Q818yes—
287C23C3×C4⋊Q814yes—
288C23C3×C4⋊Q818yes—
289C23C22×C3×D411yescharacteristic
290C23C22×C3×D412yes—
291C23C22×C3×Q811yescharacteristic
292C23C22×C3×Q812yes—
293C23C2×C3×#4111yescharacteristic
294C23C2×C3×#4114yes—
295C23C3×H(5,𝔽2)14yes—
296C2×C4—232no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—216no—
300C2×C4—28no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—216no—
307C2×C4—432no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—48no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4C3×#12318yes—
323C2×C3—28no—
324C2×C3C2×#39511yescharacteristic
325C2×C3#248712yes—
326C2×C3#250212yes—
327C2×C3#257914yes—
328C2×C3#258314yes—
329C2×C3#261311yescharacteristic
330C2×C3#284911yescharacteristic
331C22—28no—
332C22—28no—
333C22—216no—
334C22—216no—
335C22—24no—
336C22—24no—
337C22C3×#39518yes—
338C22C2×C3×#12011yescharacteristic
339C22C2×C3×C22⋊Q811yescharacteristic
340C22C2×C3×C22⋊Q812yes—
341C22C2×C3×#12312yes—
342C22C3×#52514yes—
343C22C2×C3×C4⋊Q811yescharacteristic
344C22C2×C3×C4⋊Q812yes—
345C22C3×#53414yes—
346C22C3×#53512yes—
347C22C3×#55314yes—
348C22C3×#55414yes—
349C22C3×#55418yes—
350C4—28no—
351C4—44no—
352C4—48no—
353C4—48no—
354C4—81no—
355C4—81no—
356C3#1509211yescharacteristic, direct product factor, Sylow
357C2—28no—
358C2C2×C3×#39511yescharacteristic
359C2C3×#248712yes—
360C2C3×#250212yes—
361C2C3×#257914yes—
362C2C3×#258314yes—
363C2C3×#261311yescharacteristic
364C2C3×#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
4. C3×#1444C2semidirect
5. C3×#1441C2semidirect
8. #15092C3direct (N × H)
28. #1444C2×C3semidirect
29. #1441C2×C3semidirect
356. C3#15092direct (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
C3×#14442
C3×#14412
C3×C22⋊#952
#150923