Finite Group Databasebeta

← C3×#10267 #10628874

Subgroups — C3×#10267 #10628874

Order 768.

646 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×#486C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic
4C3×#1954C211yesmaximal, characteristic, complemented
5C2×C3×#333C211yesmaximal, characteristic, complemented
6C2×C3×#331C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10267C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×D8—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#486—22no—
14C3×#486—22no—
15C3×#476—22no—
16C3×#476—22no—
17C3×#459—22no—
18C3×#459—22no—
19C2×C3×#106—21no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102—21no—
24C2×C3×#102—22no—
25C2×C3×#102—21no—
26C2×C3×C22⋊C8—22no—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#333—22no—
29C3×#331—22no—
30C2×#486C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic, complemented
32C2×#459C2×C311yescharacteristic
33#1954C2×C311yescharacteristic, complemented
34C2×#333C2×C311yescharacteristic, complemented
35C2×#331C2×C311yescharacteristic, complemented
36#1277C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×D8—24no—
44C2×C3×D8—42no—
45C2×C3×D8—42no—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C22×C3×C8—21no—
49C22×C3×C8D412yes—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8D412yes—
54C3×#127—21no—
55C3×#127—21no—
56C3×#127—21no—
57C3×#127D412yes—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C42C2311yescharacteristic
67C3×#106—24no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×#102—24no—
75C3×#102—24no—
76C3×#102—28no—
77C3×C22⋊C8—44no—
78C3×C4×C8—21no—
79C3×C4×C8—21no—
80C22×D8—21no—
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×#127C22×C311yescharacteristic
83C2×C4×D4C22×C311yescharacteristic
84#486—22no—
85#486—22no—
86#476—22no—
87#476—22no—
88#459—22no—
89#459—22no—
90C2×#106—21no—
91C2×C4⋊C8C22×C311yescharacteristic
92C2×C4⋊C8C22×C311yescharacteristic
93C2×C4⋊C8C22×C311yescharacteristic
94C2×#102—21no—
95C2×#102—22no—
96C2×#102—21no—
97C2×C22⋊C8—22no—
98C2×C4×C8C22×C311yescharacteristic
99#333—22no—
100#331—22no—
101C24×C3—21no—
102C24×C3—41no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—82no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—82no—
124C2×C3×D4SD1612yes—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C3×D8—48no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—42no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C3×D411yescharacteristic
163C23×C4—21no—
164C2×D8—24no—
165C2×D8—42no—
166C2×D8—42no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C3×D412yes—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C3×D412yes—
175#127—21no—
176#127—21no—
177#127—21no—
178#127C3×D412yes—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C3×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C42C23×C311yescharacteristic
188#106—24no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8C3×D412yes—
194C4⋊C8C3×D412yes—
195#102—24no—
196#102—24no—
197#102—28no—
198C22⋊C8—44no—
199C4×C8—21no—
200C4×C8—21no—
201C23×C3—22no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—82no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—82no—
211C23×C3#12111yescharacteristic
212C3×Q8—82no—
213C3×Q8—82no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—44no—
218C3×D4—42no—
219C3×D4—41no—
220C3×D4—42no—
221C3×D4—82no—
222C3×D4—82no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C3×D4—81no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—82no—
232C2×C3×C4—82no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×C3×C4C2×SD1611yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4#13512yes—
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)12yes—
246C2×C3×C4#13711yescharacteristic
247C3×C8—42no—
248C3×C8—42no—
249C3×C8—42no—
250C3×C8—81no—
251C24—21no—
252C24—41no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—82no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—82no—
274C2×D4C3×SD1612yes—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4C2×C3×D411yescharacteristic
282C22×C4C2×C3×D411yescharacteristic
283C22×C4C3×#4111yescharacteristic
284D8—48no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C42C2×C3×D411yescharacteristic
305C42C2×C3×D411yescharacteristic
306C42C2×C3×D411yescharacteristic
307C42C2×C3×D411yescharacteristic
308C22×C3—44no—
309C22×C3—42no—
310C22×C3—82no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—162no—
317C22×C3—162no—
318C22×C3#39211yescharacteristic
319C22×C3#46312yes—
320C22×C3#46511yescharacteristic
321C22×C3#46811yescharacteristic
322C22×C3#47212yes—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326C3×C4—82no—
327C3×C4#44811yescharacteristic
328C3×C4#44911yescharacteristic
329C3×C4#44912yes—
330C3×C4#45011yescharacteristic
331C3×C4#45211yescharacteristic
332C3×C4#49511yescharacteristic
333C3×C4#49611yescharacteristic
334C23—22no—
335C23—41no—
336C23—42no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—82no—
341C23—81no—
342C23—81no—
343C23—82no—
344C23C3×#12111yescharacteristic
345Q8—82no—
346Q8—82no—
347D4—22no—
348D4—22no—
349D4—41no—
350D4—44no—
351D4—42no—
352D4—41no—
353D4—42no—
354D4—82no—
355D4—82no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4C3×#12011yescharacteristic
369C2×C4C3×#12011yescharacteristic
370C2×C4C3×#12011yescharacteristic
371C2×C4C3×#12111yescharacteristic
372C2×C4C3×#12111yescharacteristic
373C2×C4C3×#12711yescharacteristic
374C2×C4C2×C3×SD1611yescharacteristic
375C2×C4C3×#13511yescharacteristic
376C2×C4C3×#13512yes—
377C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C3×AGL(1,ℤ/8)12yes—
379C2×C4C3×#13711yescharacteristic
380C8—42no—
381C8—42no—
382C8—42no—
383C8—81no—
384C2×C3—82no—
385C2×C3—162no—
386C2×C3#167712yes—
387C2×C3#169712yes—
388C2×C3#200011yescharacteristic
389C2×C3#201211yescharacteristic
390C2×C3#201811yescharacteristic
391C22—42no—
392C22—44no—
393C22—81no—
394C22—81no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—162no—
400C22—162no—
401C22C3×#39211yescharacteristic
402C22C3×#46312yes—
403C22C3×#46511yescharacteristic
404C22C3×#46811yescharacteristic
405C22C3×#47212yes—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—82no—
410C4C3×#44811yescharacteristic
411C4C3×#44911yescharacteristic
412C4C3×#44912yes—
413C4C3×#45011yescharacteristic
414C4C3×#45211yescharacteristic
415C4C3×#49511yescharacteristic
416C4C3×#49611yescharacteristic
417C3#1026711yescharacteristic, direct product factor, Sylow
418C2—82no—
419C2—162no—
420C2C3×#167712yes—
421C2C3×#169712yes—
422C2C3×#200011yescharacteristic
423C2C3×#201211yescharacteristic
424C2C3×#201811yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#459C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#486C2semidirect
2. C2×C3×#476C2semidirect
4. C3×#1954C2semidirect
5. C2×C3×#333C2semidirect
6. C2×C3×#331C2semidirect
7. C3×#1277C2semidirect
8. #10267C3direct (N × H)
30. C2×#486C2×C3semidirect
31. C2×#476C2×C3semidirect
33. #1954C2×C3semidirect
34. C2×#333C2×C3semidirect
35. C2×#331C2×C3semidirect
36. #1277C2×C3semidirect
417. C3#10267direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×#4762
C2×C3×#4592
C3×#19542
C2×C3×#3332
C2×C3×#3312
C3×#12772
#102673