Finite Group Databasebeta

← C3×#25652 #10644259

Subgroups — C3×#25652 #10644259

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×C3×#486C211yesmaximal, characteristic
3C2×C3×#475C211yesmaximal, characteristic, complemented
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2042C214yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1982C211yesmaximal, characteristic
8C3×#1864C214yesmaximal, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25652C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—24no—
12C3×#559—24no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C3×#486—24no—
24C3×#475—24no—
25C2×C3×#106C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2214yes—
30C2×C3×#102C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#398—22no—
33C3×#391—21no—
34C3×#389C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#559C2×C311yescharacteristic
39C2×#486C2×C311yescharacteristic
40C2×#475C2×C311yescharacteristic, complemented
41#2546C2×C311yescharacteristic, complemented
42#2042C2×C314yescomplemented
43#1984C2×C311yescharacteristic, complemented
44#1982C2×C311yescharacteristic
45#1864C2×C314yescomplemented
46C4×#103C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8D411yescharacteristic
49C22×C3×Q8C2311yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4—22no—
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—44no—
59C3×#127—22no—
60C3×#127D414yes—
61C3×#124—28no—
62C3×#124—24no—
63C3×#124—24no—
64C3×#123—216no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C3×C4×D4—28no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#106—24no—
88C3×C4⋊C8—22no—
89C3×#103—24no—
90C3×#103—24no—
91C3×#102—28no—
92C3×#102—28no—
93C3×C4×C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96#559—24no—
97#559—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#127C22×C311yescharacteristic
101C2×#124—21no—
102C2×#124C22×C312yes—
103C2×#123—22no—
104C2×C4×Q8C22×C311yescharacteristic
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4—21no—
107C2×C4×D4C22×C312yes—
108#486—24no—
109#475—24no—
110C2×#106C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#103C22×C312yes—
113C2×#103C22×C312yes—
114C2×#102C22×C314yes—
115C2×#102C22×C314yes—
116C2×C4×C8C22×C311yescharacteristic
117#398—22no—
118#391—21no—
119#389C22×C314yes—
120C4×C4⋊C4C22×C311yescharacteristic
121#340C22×C312yes—
122#336C22×C314yes—
123C24×C3—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—21no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8C2×D412yes—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—82no—
142C2×C3×D4C2×D414yes—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—28no—
147C22×C3×C4—22no—
148C22×C3×C4—84no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—24no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—48no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4C2×D412yes—
169C3×C4⋊C4#4114yes—
170C3×C22⋊C4—28no—
171C3×C22⋊C4—28no—
172C3×C22⋊C4—28no—
173C3×C42—24no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42—42no—
177C3×C42C2×D412yes—
178C3×C42C2×D412yes—
179C22×Q8—21no—
180C22×Q8C3×D411yescharacteristic
181C22×Q8C23×C311yescharacteristic
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C23×C312yes—
185C23×C4—22no—
186C22×C8C23×C312yes—
187C22×C8C23×C312yes—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—44no—
191#127—22no—
192#127C3×D414yes—
193#124—28no—
194#124—24no—
195#124—24no—
196#123—216no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8C3×D414yes—
201C4×D4—28no—
202C4×D4—24no—
203C4×D4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C3×D411yescharacteristic
208C2×C4⋊C4C3×D412yes—
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C23×C312yes—
218C2×C42C23×C311yescharacteristic
219#106—24no—
220C4⋊C8—22no—
221#103—24no—
222#103—24no—
223#102—28no—
224#102—28no—
225C4×C8—22no—
226#95—21no—
227#95—24no—
228C23×C3—24no—
229C23×C3—44no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—44no—
233C23×C3C22×D411yescharacteristic
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×Q8—84no—
239C3×D4—28no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—48no—
243C3×D4—84no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—88no—
261C2×C3×C4#12012yes—
262C2×C3×C4#12012yes—
263C2×C3×C4#13512yes—
264C2×C3×C4AGL(1,ℤ/8)12yes—
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C22×D411yescharacteristic
267C2×C3×C4C2×#4112yes—
268C2×C3×C42−1+412yes—
269C3×C8—42no—
270C3×C8—42no—
271C24—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C2×C3×D412yes—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—82no—
290C2×D4C2×C3×D414yes—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—28no—
295C22×C4—22no—
296C22×C4—84no—
297C22×C4C2×C3×D412yes—
298C22×C4C2×C3×D412yes—
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C24×C311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4C2×C3×D412yes—
317C4⋊C4C3×#4114yes—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C42—24no—
322C42—22no—
323C42—22no—
324C42—42no—
325C42C2×C3×D412yes—
326C42C2×C3×D412yes—
327C22×C3—44no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—84no—
331C22×C3—84no—
332C22×C3C2×#12011yescharacteristic
333C22×C3#54812yes—
334C22×C3C2×#13511yescharacteristic
335C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
336C22×C3#57712yes—
337C3×C4—22no—
338C3×C4—44no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—82no—
344C3×C4#44914yes—
345C3×C4#53211yescharacteristic
346C3×C4#53611yescharacteristic
347C3×C4#54712yes—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—44no—
352C23—44no—
353C23C22×C3×D411yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—44no—
357Q8—42no—
358Q8—84no—
359D4—28no—
360D4—48no—
361D4—42no—
362D4—42no—
363D4—84no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—88no—
381C2×C4C3×#12012yes—
382C2×C4C3×#12012yes—
383C2×C4C3×#13512yes—
384C2×C4C3×AGL(1,ℤ/8)12yes—
385C2×C4C22×C3×D411yescharacteristic
386C2×C4C22×C3×D411yescharacteristic
387C2×C4C2×C3×#4112yes—
388C2×C4C3×2−1+412yes—
389C8—42no—
390C8—42no—
391C2×C3—84no—
392C2×C3#243211yescharacteristic
393C2×C3C2×#44911yescharacteristic
394C2×C3#300911yescharacteristic
395C2×C3#336512yes—
396C2×C3#337312yes—
397C22—44no—
398C22—44no—
399C22—44no—
400C22—84no—
401C22—84no—
402C22C2×C3×#12011yescharacteristic
403C22C3×#54812yes—
404C22C2×C3×#13511yescharacteristic
405C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
406C22C3×#57712yes—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—44no—
413C4—82no—
414C4C3×#44914yes—
415C4C3×#53211yescharacteristic
416C4C3×#53611yescharacteristic
417C4C3×#54712yes—
418C3#2565211yescharacteristic, direct product factor, Sylow
419C2—84no—
420C2C3×#243211yescharacteristic
421C2C2×C3×#44911yescharacteristic
422C2C3×#300911yescharacteristic
423C2C3×#336512yes—
424C2C3×#337312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#475C2semidirect
4. C3×#2546C2semidirect
5. C3×#2042C2semidirect
6. C3×#1984C2semidirect
8. C3×#1864C2semidirect
9. C3×C4×#103C2semidirect
10. #25652C3direct (N × H)
40. C2×#475C2×C3semidirect
41. #2546C2×C3semidirect
42. #2042C2×C3semidirect
43. #1984C2×C3semidirect
45. #1864C2×C3semidirect
46. C4×#103C2×C3semidirect
418. C3#25652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4862
C2×C3×#4752
C3×#25462
C3×#20422
C3×#19842
C3×#19822
C3×#18642
C3×C4×#1032
#256523