Finite Group Databasebeta

← C3×#25650 #10644257

Subgroups — C3×#25650 #10644257

Order 768.

1054 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×#556C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2042C214yesmaximal
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1863C214yesmaximal, complemented
9C3×C4×#102C211yesmaximal, characteristic
10#25650C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#556—24no—
12C3×#556—24no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#124C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C3×#487—24no—
24C3×#474—24no—
25C2×C3×#107C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2214yes—
28C2×C3×#103C2214yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#398—22no—
33C3×#391—21no—
34C3×#389C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#556C2×C311yescharacteristic
39C2×#487C2×C311yescharacteristic, complemented
40C2×#474C2×C311yescharacteristic
41#2546C2×C311yescharacteristic, complemented
42#2042C2×C314yes—
43#1984C2×C311yescharacteristic, complemented
44#1981C2×C311yescharacteristic, complemented
45#1863C2×C314yescomplemented
46C4×#102C2×C311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8D411yescharacteristic
49C22×C3×Q8C2312yes—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—44no—
58C3×C4⋊Q8D414yes—
59C3×#126—216no—
60C3×#124—24no—
61C3×#124—24no—
62C3×C22⋊Q8—28no—
63C3×C4×Q8—24no—
64C3×C4×Q8—24no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×D4—22no—
68C3×C4×D4D414yes—
69C3×#117—28no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
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—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—22no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#107—24no—
88C3×C4⋊C8—22no—
89C3×#103—28no—
90C3×#103—28no—
91C3×#102—24no—
92C3×#102—24no—
93C3×C4×C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96#556—24no—
97#556—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×#126—22no—
102C2×#124C22×C312yes—
103C2×C22⋊Q8—21no—
104C2×C4×Q8C22×C312yes—
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107C2×#117—21no—
108#487—24no—
109#474—24no—
110C2×#107C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#103C22×C314yes—
113C2×#103C22×C314yes—
114C2×#102C22×C312yes—
115C2×#102C22×C312yes—
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—21no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—24no—
129C2×C3×Q8—24no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8C2×D414yes—
134C2×C3×D4—21no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4C2×D412yes—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—24no—
154C2×C3×C8—22no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—24no—
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—48no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4C2×D412yes—
167C3×C4⋊C4#4114yes—
168C3×C22⋊C4—28no—
169C3×C22⋊C4—28no—
170C3×C22⋊C4—24no—
171C3×C42—28no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42—42no—
175C3×C42C2×D412yes—
176C3×C42C2×D412yes—
177C22×Q8—21no—
178C22×Q8C3×D411yescharacteristic
179C22×Q8C23×C312yes—
180C22×D4C23×C311yescharacteristic
181C23×C4—21no—
182C22×C8C23×C312yes—
183C22×C8C23×C312yes—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—44no—
188C4⋊Q8C3×D414yes—
189#126—216no—
190#124—24no—
191#124—24no—
192C22⋊Q8—28no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×D4—22no—
198C4×D4C3×D414yes—
199#117—28no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C3×D411yescharacteristic
206C2×C4⋊C4C3×D412yes—
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42C23×C312yes—
216C2×C42C23×C311yescharacteristic
217#107—24no—
218C4⋊C8—22no—
219#103—28no—
220#103—28no—
221#102—24no—
222#102—24no—
223C4×C8—22no—
224#95—21no—
225#95—24no—
226C23×C3—22no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3C22×D411yescharacteristic
232C3×Q8—28no—
233C3×Q8—42no—
234C3×Q8—42no—
235C3×Q8—44no—
236C3×Q8—84no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—42no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—44no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—48no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4#12012yes—
259C2×C3×C4#12012yes—
260C2×C3×C4#13512yes—
261C2×C3×C4AGL(1,ℤ/8)12yes—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4112yes—
265C2×C3×C42−1+412yes—
266C3×C8—42no—
267C3×C8—42no—
268C24—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8C2×C3×D414yes—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4C2×C3×D412yes—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C2×C3×D412yes—
291C22×C4C2×C3×D412yes—
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C24×C311yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—22no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—48no—
310C4⋊C4—42no—
311C4⋊C4C2×C3×D412yes—
312C4⋊C4C3×#4114yes—
313C22⋊C4—28no—
314C22⋊C4—28no—
315C22⋊C4—24no—
316C42—28no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42C2×C3×D412yes—
321C42C2×C3×D412yes—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3C2×#12011yescharacteristic
328C22×C3#54812yes—
329C22×C3C2×#13511yescharacteristic
330C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
331C22×C3#57812yes—
332C3×C4—22no—
333C3×C4—44no—
334C3×C4—44no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—42no—
338C3×C4—82no—
339C3×C4#44914yes—
340C3×C4#53211yescharacteristic
341C3×C4#53611yescharacteristic
342C3×C4#54712yes—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23C22×C3×D411yescharacteristic
349Q8—28no—
350Q8—44no—
351Q8—42no—
352Q8—42no—
353Q8—84no—
354D4—22no—
355D4—22no—
356D4—42no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4C3×#12012yes—
376C2×C4C3×#12012yes—
377C2×C4C3×#13512yes—
378C2×C4C3×AGL(1,ℤ/8)12yes—
379C2×C4C22×C3×D411yescharacteristic
380C2×C4C22×C3×D411yescharacteristic
381C2×C4C2×C3×#4112yes—
382C2×C4C3×2−1+412yes—
383C8—42no—
384C8—42no—
385C2×C3—82no—
386C2×C3#243211yescharacteristic
387C2×C3C2×#44911yescharacteristic
388C2×C3#301111yescharacteristic
389C2×C3#336412yes—
390C2×C3#336712yes—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—82no—
396C22C2×C3×#12011yescharacteristic
397C22C3×#54812yes—
398C22C2×C3×#13511yescharacteristic
399C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
400C22C3×#57812yes—
401C4—22no—
402C4—44no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—44no—
407C4—82no—
408C4C3×#44914yes—
409C4C3×#53211yescharacteristic
410C4C3×#53611yescharacteristic
411C4C3×#54712yes—
412C3#2565011yescharacteristic, direct product factor, Sylow
413C2—82no—
414C2C3×#243211yescharacteristic
415C2C2×C3×#44911yescharacteristic
416C2C3×#301111yescharacteristic
417C2C3×#336412yes—
418C2C3×#336712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
2. C2×C3×#487C2semidirect
4. C3×#2546C2semidirect
6. C3×#1984C2semidirect
7. C3×#1981C2semidirect
8. C3×#1863C2semidirect
10. #25650C3direct (N × H)
39. C2×#487C2×C3semidirect
41. #2546C2×C3semidirect
43. #1984C2×C3semidirect
44. #1981C2×C3semidirect
45. #1863C2×C3semidirect
412. C3#25650direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4872
C2×C3×#4742
C3×#25462
C3×#20422
C3×#19842
C3×#19812
C3×#18632
C3×C4×#1022
#256503