Finite Group Databasebeta

← C3×#25648 #10644255

Subgroups — C3×#25648 #10644255

Order 768.

1098 conjugacy classes of subgroups.

The subgroup lattice has 420 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×#477C211yesmaximal, characteristic, complemented
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2042C214yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1982C211yesmaximal, characteristic
8C3×#1862C214yesmaximal, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25648C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—24no—
12C3×#559—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#124—21no—
16C2×C3×#124C2212yes—
17C2×C3×#123—22no—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2212yes—
22C3×#486—24no—
23C3×#477—24no—
24C2×C3×#107C2212yes—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2214yes—
29C2×C3×#102C2214yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#395—22no—
32C3×#391—21no—
33C3×#389C2214yes—
34C3×#384—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#559C2×C311yescharacteristic
39C2×#486C2×C311yescharacteristic
40C2×#477C2×C311yescharacteristic, complemented
41#2567C2×C311yescharacteristic, complemented
42#2042C2×C314yescomplemented
43#1984C2×C311yescharacteristic, complemented
44#1982C2×C311yescharacteristic
45#1862C2×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×#127—22no—
59C3×#127D414yes—
60C3×#124—28no—
61C3×#124—24no—
62C3×#124—24no—
63C3×#123—216no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8D414yes—
68C3×C4×D4—28no—
69C3×C4×D4—24no—
70C3×C4×D4—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D412yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—21no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×C4⋊C8—22no—
88C3×#103—24no—
89C3×#103—24no—
90C3×#102—28no—
91C3×#102—28no—
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—24no—
95#559—24no—
96#559—24no—
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×#127C22×C311yescharacteristic
99C2×#124—21no—
100C2×#124C22×C312yes—
101C2×#123—22no—
102C2×C4×Q8C22×C311yescharacteristic
103C2×C4×Q8C22×C311yescharacteristic
104C2×C4×D4—21no—
105C2×C4×D4C22×C312yes—
106#486—24no—
107#477—24no—
108C2×#107C22×C312yes—
109C2×C4⋊C8C22×C311yescharacteristic
110C2×#103C22×C312yes—
111C2×#103C22×C312yes—
112C2×#102C22×C314yes—
113C2×#102C22×C314yes—
114C2×C4×C8C22×C311yescharacteristic
115#395—22no—
116#391—21no—
117#389C22×C314yes—
118#384—21no—
119C4×C4⋊C4C22×C311yescharacteristic
120#340C22×C312yes—
121#336C22×C314yes—
122C24×C3—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—21no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—42no—
130C2×C3×Q8—82no—
131C2×C3×Q8C2×D412yes—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4—22no—
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—82no—
140C2×C3×D4C2×D414yes—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—28no—
145C22×C3×C4—22no—
146C22×C3×C4—84no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—24no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—84no—
166C3×C4⋊C4C2×D412yes—
167C3×C4⋊C4#4114yes—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—28no—
170C3×C22⋊C4—28no—
171C3×C42—24no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42—42no—
175C3×C42C2×D412yes—
176C3×C42C2×D412yes—
177C22×Q8—21no—
178C22×Q8C3×D411yescharacteristic
179C22×Q8C23×C311yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C23×C312yes—
183C23×C4—22no—
184C22×C8C23×C312yes—
185C22×C8C23×C312yes—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188#127—22no—
189#127C3×D414yes—
190#124—28no—
191#124—24no—
192#124—24no—
193#123—216no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8C3×D414yes—
198C4×D4—28no—
199C4×D4—24no—
200C4×D4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4C3×D411yescharacteristic
205C2×C4⋊C4C3×D412yes—
206C2×C4⋊C4C23×C312yes—
207C2×C4⋊C4C23×C311yescharacteristic
208C2×C4⋊C4C23×C312yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C23×C312yes—
215C2×C42C23×C311yescharacteristic
216#107—24no—
217C4⋊C8—22no—
218#103—24no—
219#103—24no—
220#102—28no—
221#102—28no—
222C4×C8—22no—
223#95—21no—
224#95—24no—
225C23×C3—24no—
226C23×C3—44no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—44no—
230C23×C3C22×D411yescharacteristic
231C3×Q8—22no—
232C3×Q8—22no—
233C3×Q8—42no—
234C3×Q8—44no—
235C3×Q8—84no—
236C3×D4—28no—
237C3×D4—42no—
238C3×D4—42no—
239C3×D4—48no—
240C3×D4—84no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—88no—
257C2×C3×C4#12012yes—
258C2×C3×C4#12012yes—
259C2×C3×C4#13512yes—
260C2×C3×C4#13712yes—
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C42−1+412yes—
265C3×C8—42no—
266C3×C8—42no—
267C24—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8C2×C3×D412yes—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—82no—
285C2×D4C2×C3×D414yes—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—84no—
292C22×C4C2×C3×D412yes—
293C22×C4C2×C3×D412yes—
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C24×C311yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C4⋊C4—84no—
311C4⋊C4C2×C3×D412yes—
312C4⋊C4C3×#4114yes—
313C22⋊C4—28no—
314C22⋊C4—28no—
315C22⋊C4—28no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42C2×C3×D412yes—
321C42C2×C3×D412yes—
322C22×C3—44no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—84no—
326C22×C3—84no—
327C22×C3C2×#12011yescharacteristic
328C22×C3#54812yes—
329C22×C3C2×#13511yescharacteristic
330C22×C3C2×#13711yescharacteristic
331C22×C3#57612yes—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—42no—
338C3×C4—82no—
339C3×C4#45214yes—
340C3×C4#53211yescharacteristic
341C3×C4#53611yescharacteristic
342C3×C4#54712yes—
343C23—24no—
344C23—42no—
345C23—42no—
346C23—44no—
347C23—44no—
348C23C22×C3×D411yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—44no—
352Q8—42no—
353Q8—84no—
354D4—28no—
355D4—48no—
356D4—42no—
357D4—42no—
358D4—84no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—88no—
375C2×C4C3×#12012yes—
376C2×C4C3×#12012yes—
377C2×C4C3×#13512yes—
378C2×C4C3×#13712yes—
379C2×C4C22×C3×D411yescharacteristic
380C2×C4C22×C3×D411yescharacteristic
381C2×C4C2×C3×#4112yes—
382C2×C4C3×2−1+412yes—
383C8—42no—
384C8—42no—
385C2×C3—84no—
386C2×C3#243211yescharacteristic
387C2×C3C2×#45211yescharacteristic
388C2×C3#300811yescharacteristic
389C2×C3#336112yes—
390C2×C3#337512yes—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—84no—
395C22—84no—
396C22C2×C3×#12011yescharacteristic
397C22C3×#54812yes—
398C22C2×C3×#13511yescharacteristic
399C22C2×C3×#13711yescharacteristic
400C22C3×#57612yes—
401C4—22no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—44no—
407C4—82no—
408C4C3×#45214yes—
409C4C3×#53211yescharacteristic
410C4C3×#53611yescharacteristic
411C4C3×#54712yes—
412C3#2564811yescharacteristic, direct product factor, Sylow
413C2—84no—
414C2C3×#243211yescharacteristic
415C2C2×C3×#45211yescharacteristic
416C2C3×#300811yescharacteristic
417C2C3×#336112yes—
418C2C3×#337512yes—
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×#477C2semidirect
4. C3×#2567C2semidirect
5. C3×#2042C2semidirect
6. C3×#1984C2semidirect
8. C3×#1862C2semidirect
9. C3×C4×#103C2semidirect
10. #25648C3direct (N × H)
40. C2×#477C2×C3semidirect
41. #2567C2×C3semidirect
42. #2042C2×C3semidirect
43. #1984C2×C3semidirect
45. #1862C2×C3semidirect
46. C4×#103C2×C3semidirect
412. C3#25648direct (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×#4772
C3×#25672
C3×#20422
C3×#19842
C3×#19822
C3×#18622
C3×C4×#1032
#256483