Finite Group Databasebeta

← C3×#25824 #10644431

Subgroups — C3×#25824 #10644431

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic
2C3×#3165C212yesmaximal, complemented
3C3×#2542C211yesmaximal, characteristic
4C3×#2060C214yesmaximal, complemented
5C3×#1986C212yesmaximal, complemented
6C3×#1856C214yesmaximal
7C3×C4×#100C211yesmaximal, characteristic
8#25824C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#554—22no—
10C3×#554—24no—
11C3×D4×Q8—22no—
12C3×#544C2212yes—
13C3×#543—22no—
14C3×#533C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C22⋊Q8—22no—
17C2×C3×C22⋊Q8—22no—
18C3×#518C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C3×#501—24no—
21C3×#500—22no—
22C3×#498—22no—
23C3×#428C2218yes—
24C2×C3×#100C2212yes—
25C3×C4×M16C2212yes—
26C3×#399—22no—
27C3×#390C2212yes—
28C3×#384C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×#343C2212yes—
31C3×#339C2218yes—
32#3501C2×C311yescharacteristic
33#3165C2×C312yescomplemented
34#2542C2×C311yescharacteristic
35#2060C2×C314yescomplemented
36#1986C2×C312yescomplemented
37#1856C2×C314yes—
38C4×#100C2×C311yescharacteristic
39C22×C3×Q8—22no—
40C22×C3×Q8—21no—
41C22×C3×D4C2311yescharacteristic
42C23×C3×C4D412yes—
43C2×C3×M16C2314yes—
44C3×C4⋊Q8—24no—
45C3×C4⋊Q8—22no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8D412yes—
48C3×C4⋊Q8D412yes—
49C3×#125—24no—
50C3×#125D414yes—
51C3×C22⋊Q8—24no—
52C3×C22⋊Q8—24no—
53C3×C22⋊Q8—28no—
54C3×C22⋊Q8—28no—
55C3×C22⋊Q8—22no—
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—48no—
59C3×C4×Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×D4—24no—
62C3×C4×D4—24no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×#117C2314yes—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78C3×#107—28no—
79C3×#106—28no—
80C3×#100Q818yescomplemented
81C3×#97—22no—
82C3×C4×C8—22no—
83C3×#95—21no—
84#554—22no—
85#554—24no—
86D4×Q8—22no—
87#544C22×C312yes—
88#543—22no—
89#533C22×C312yes—
90C2×C4⋊Q8C22×C311yescharacteristic
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93#518C22×C312yes—
94C2×C4×D4C22×C311yescharacteristic
95#501—24no—
96#500—22no—
97#498—22no—
98#428C22×C318yes—
99C2×#100C22×C312yes—
100C4×M16C22×C312yes—
101#399—22no—
102#390C22×C312yes—
103#384C22×C312yes—
104C4×C4⋊C4C22×C311yescharacteristic
105#343C22×C312yes—
106#339C22×C318yes—
107C24×C3C2×D412yes—
108C2×C3×Q8—24no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—44no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—84no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4C2×Q814yes—
118C22×C3×C4—22no—
119C22×C3×C4—24no—
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—44no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2411yescharacteristic
129C3×M16—24no—
130C3×M16C2×Q818yescomplemented
131C2×C3×C8—24no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—28no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—48no—
142C3×C4⋊C4—42no—
143C3×C4⋊C4C2×D414yes—
144C3×C4⋊C4C2×D414yes—
145C3×C22⋊C4—28no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—24no—
148C3×C22⋊C4—48no—
149C3×C42—24no—
150C3×C42—24no—
151C3×C42C2×D412yes—
152C3×C42C2×D412yes—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×D4C23×C311yescharacteristic
156C23×C4C3×D412yes—
157C2×M16C23×C314yes—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8C3×D412yes—
162C4⋊Q8C3×D412yes—
163#125—24no—
164#125C3×D414yes—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—28no—
168C22⋊Q8—28no—
169C22⋊Q8—22no—
170C22⋊Q8—24no—
171C22⋊Q8—22no—
172C22⋊Q8—48no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179#117C23×C314yes—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C23×C312yes—
186C2×C4⋊C4C23×C311yescharacteristic
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4C3×D412yes—
190C2×C42C23×C312yes—
191C2×C42C23×C311yescharacteristic
192#107—28no—
193#106—28no—
194#100C3×Q818yescomplemented
195#97—22no—
196C4×C8—22no—
197#95—21no—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3C4⋊Q814yes—
203C23×C3C22×D411yescharacteristic
204C3×Q8—44no—
205C3×Q8—82no—
206C3×Q8—82no—
207C3×Q8—84no—
208C3×Q8—82no—
209C3×D4—28no—
210C2×C3×C4—22no—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—44no—
219C2×C3×C4—42no—
220C2×C3×C4—44no—
221C2×C3×C4—42no—
222C2×C3×C4—82no—
223C2×C3×C4—84no—
224C2×C3×C4—84no—
225C2×C3×C4#12012yes—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4C22×D412yes—
229C2×C3×C4C22×Q812yes—
230C2×C3×C42−1+412yes—
231C3×C8—28no—
232C24C2×C3×D412yes—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—84no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4C2×C3×Q814yes—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—44no—
250C22×C4C2×C3×D411yescharacteristic
251C22×C4C2×C3×D412yes—
252C22×C4C2×C3×D411yescharacteristic
253C22×C4C24×C311yescharacteristic
254M16—24no—
255M16C2×C3×Q818yescomplemented
256C2×C8—24no—
257C4⋊C4—24no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—48no—
267C4⋊C4—42no—
268C4⋊C4C2×C3×D414yes—
269C4⋊C4C2×C3×D414yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—48no—
274C42—24no—
275C42—24no—
276C42C2×C3×D412yes—
277C42C2×C3×D412yes—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—24no—
281C22×C3—44no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—82no—
285C22×C3C2×#12011yescharacteristic
286C22×C3C2×C4⋊Q811yescharacteristic
287C22×C3#53611yescharacteristic
288C3×C4—21no—
289C3×C4—22no—
290C3×C4—44no—
291C3×C4—42no—
292C3×C4—44no—
293C3×C4—44no—
294C3×C4—82no—
295C3×C4#45312yes—
296C3×C4D4×Q814yes—
297C23—22no—
298C23—22no—
299C23—42no—
300C23—42no—
301C23C3×C4⋊Q814yes—
302C23C22×C3×D411yescharacteristic
303Q8—44no—
304Q8—82no—
305Q8—82no—
306Q8—84no—
307Q8—82no—
308D4—28no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4C3×#12012yes—
325C2×C4C3×#12011yescharacteristic
326C2×C4C3×#12011yescharacteristic
327C2×C4C22×C3×D412yes—
328C2×C4C22×C3×Q812yes—
329C2×C4C3×2−1+412yes—
330C8—28no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—44no—
334C2×C3#243411yescharacteristic
335C2×C3C2×#45311yescharacteristic
336C2×C3#302211yescharacteristic
337C22—21no—
338C22—21no—
339C22—24no—
340C22—44no—
341C22—42no—
342C22—42no—
343C22—82no—
344C22C2×C3×#12011yescharacteristic
345C22C2×C3×C4⋊Q811yescharacteristic
346C22C3×#53611yescharacteristic
347C4—21no—
348C4—22no—
349C4—44no—
350C4—42no—
351C4—44no—
352C4—44no—
353C4—82no—
354C4C3×#45312yes—
355C4C3×D4×Q814yes—
356C3#2582411yescharacteristic, direct product factor, Sylow
357C2—21no—
358C2—21no—
359C2—44no—
360C2C3×#243411yescharacteristic
361C2C2×C3×#45311yescharacteristic
362C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3165C2semidirect
4. C3×#2060C2semidirect
5. C3×#1986C2semidirect
8. #25824C3direct (N × H)
33. #3165C2×C3semidirect
35. #2060C2×C3semidirect
36. #1986C2×C3semidirect
80. C3×#100Q8semidirect
130. C3×M16C2×Q8semidirect
194. #100C3×Q8semidirect
255. M16C2×C3×Q8semidirect
356. C3#25824direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35012
C3×#31652
C3×#25422
C3×#20602
C3×#19862
C3×#18562
C3×C4×#1002
#258243