Finite Group Databasebeta

← C3×#10256 #10628863

Subgroups — C3×#10256 #10628863

Order 768.

580 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3246C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic, complemented
3C3×#3036C211yesmaximal, characteristic
4C3×#1954C211yesmaximal, characteristic, complemented
5C3×#1493C211yesmaximal, characteristic, complemented
6C3×#1492C211yesmaximal, characteristic, complemented
7C3×#1292C211yesmaximal, characteristic, complemented
8#10256C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#533C2211yescharacteristic
11C3×#532C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#478—22no—
14C3×#476—21no—
15C3×#476—21no—
16C3×#475—21no—
17C3×#474—21no—
18C3×#465—22no—
19C3×#463—22no—
20C3×#461—21no—
21C3×#461—21no—
22C3×#460—21no—
23C3×#459—21no—
24C3×#428—21no—
25C3×#423C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#418—21no—
29C3×#417—21no—
30C2×C3×C22⋊C8—22no—
31C2×C3×C4×C8C2211yescharacteristic
32C3×Q8⋊C8—22no—
33C3×D4⋊C8—22no—
34#3246C2×C311yescharacteristic, complemented
35#3077C2×C311yescharacteristic, complemented
36#3036C2×C311yescharacteristic
37#1954C2×C311yescharacteristic, complemented
38#1493C2×C311yescharacteristic, complemented
39#1492C2×C311yescharacteristic, complemented
40#1292C2×C311yescharacteristic, complemented
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C3×AGL(1,ℤ/8)—42no—
45C3×AGL(1,ℤ/8)—42no—
46C2×C3×SD16—22no—
47C2×C3×SD16—22no—
48C2×C3×D8—22no—
49C2×C3×M16—21no—
50C2×C3×M16—21no—
51C22×C3×C8—21no—
52C22×C3×C8D412yes—
53C3×C4⋊Q8D411yescharacteristic
54C3×C4⋊Q8D411yescharacteristic
55C3×C4⋊Q8D411yescharacteristic
56C3×#127D411yescharacteristic
57C3×#125—21no—
58C3×#124—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×C4×Q8—21no—
64C3×C4×Q8—21no—
65C3×C4×D4—21no—
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4—21no—
70C3×C4×D4—21no—
71C3×#117—21no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C3×#107—22no—
76C3×#106—22no—
77C3×#106—22no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C4⋊C8—21no—
81C3×C4⋊C8—21no—
82C3×C4⋊C8D412yes—
83C3×C4⋊C8D412yes—
84C3×#103—22no—
85C3×#103—22no—
86C3×#102—22no—
87C3×#102—22no—
88C3×#102—22no—
89C3×#102—22no—
90C3×C22⋊C8—44no—
91C3×C4×C8—21no—
92C3×C4×C8—21no—
93C2×AGL(1,ℤ/8)—21no—
94#533C22×C311yescharacteristic
95#532C22×C311yescharacteristic
96C2×C4×D4C22×C311yescharacteristic
97#478—22no—
98#476—21no—
99#476—21no—
100#475—21no—
101#474—21no—
102#465—22no—
103#463—22no—
104#461—21no—
105#461—21no—
106#460—21no—
107#459—21no—
108#428—21no—
109#423C22×C311yescharacteristic
110#423C22×C311yescharacteristic
111C2×C4⋊C8C22×C311yescharacteristic
112#418—21no—
113#417—21no—
114C2×C22⋊C8—22no—
115C2×C4×C8C22×C311yescharacteristic
116Q8⋊C8—22no—
117D4⋊C8—22no—
118C24×C3—21no—
119C3×#41—41no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—41no—
135C2×C3×D4SD1612yescomplemented
136C22×C3×C4—21no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C3×SD16—44no—
146C3×SD16—82no—
147C3×D8—44no—
148C3×M16—41no—
149C3×M16—41no—
150C3×M16—41no—
151C3×M16—41no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—41no—
157C2×C3×C8—41no—
158C2×C3×C8—42no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—41no—
174C3×C42—41no—
175C3×C42C2×D411yescharacteristic
176C3×C42C2×D411yescharacteristic
177C3×C42C2×D411yescharacteristic
178C3×C42C2×D411yescharacteristic
179C2×#41—21no—
180C22×D4C3×D411yescharacteristic
181C23×C4—21no—
182AGL(1,ℤ/8)—42no—
183AGL(1,ℤ/8)—42no—
184C2×SD16—22no—
185C2×SD16—22no—
186C2×D8—22no—
187C2×M16—21no—
188C2×M16—21no—
189C22×C8—21no—
190C22×C8C3×D412yes—
191C4⋊Q8C3×D411yescharacteristic
192C4⋊Q8C3×D411yescharacteristic
193C4⋊Q8C3×D411yescharacteristic
194#127C3×D411yescharacteristic
195#125—21no—
196#124—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199#121—21no—
200#121—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209#117—21no—
210C2×C4⋊C4C3×D411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C23×C311yescharacteristic
213#107—22no—
214#106—22no—
215#106—22no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8C3×D412yes—
221C4⋊C8C3×D412yes—
222#103—22no—
223#103—22no—
224#102—22no—
225#102—22no—
226#102—22no—
227#102—22no—
228C22⋊C8—44no—
229C4×C8—21no—
230C4×C8—21no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3#12111yescharacteristic
238C3×Q8—42no—
239C3×Q8—41no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8—41no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—42no—
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—81no—
253C3×D4—81no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—82no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12711yescharacteristic
273C2×C3×C4C2×SD1611yescharacteristic
274C2×C3×C4#13511yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4#13711yescharacteristic
277C3×C8—42no—
278C3×C8—42no—
279C3×C8—42no—
280C3×C8—42no—
281C24—21no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4C3×SD1612yescomplemented
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C2×C3×D411yescharacteristic
307C22×C4C3×#4111yescharacteristic
308SD16—44no—
309SD16—82no—
310D8—44no—
311M16—41no—
312M16—41no—
313M16—41no—
314M16—41no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42—41no—
338C42C2×C3×D411yescharacteristic
339C42C2×C3×D411yescharacteristic
340C42C2×C3×D411yescharacteristic
341C42C2×C3×D411yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3#39211yescharacteristic
353C22×C3#46511yescharacteristic
354C22×C3#46811yescharacteristic
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4#44811yescharacteristic
363C3×C4#44911yescharacteristic
364C3×C4#45011yescharacteristic
365C3×C4#45211yescharacteristic
366C3×C4#45311yescharacteristic
367C3×C4#45411yescharacteristic
368C23—22no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23C3×#12111yescharacteristic
375Q8—42no—
376Q8—41no—
377Q8—41no—
378Q8—41no—
379Q8—41no—
380D4—22no—
381D4—22no—
382D4—42no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C3×#12011yescharacteristic
405C2×C4C3×#12011yescharacteristic
406C2×C4C3×#12011yescharacteristic
407C2×C4C3×#12111yescharacteristic
408C2×C4C3×#12111yescharacteristic
409C2×C4C3×#12711yescharacteristic
410C2×C4C2×C3×SD1611yescharacteristic
411C2×C4C3×#13511yescharacteristic
412C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
413C2×C4C3×#13711yescharacteristic
414C8—42no—
415C8—42no—
416C8—42no—
417C8—42no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—82no—
421C2×C3—161no—
422C2×C3#200011yescharacteristic
423C2×C3#200611yescharacteristic
424C2×C3#201711yescharacteristic
425C22—21no—
426C22—21no—
427C22—44no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22C3×#39211yescharacteristic
436C22C3×#46511yescharacteristic
437C22C3×#46811yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4C3×#44811yescharacteristic
446C4C3×#44911yescharacteristic
447C4C3×#45011yescharacteristic
448C4C3×#45211yescharacteristic
449C4C3×#45311yescharacteristic
450C4C3×#45411yescharacteristic
451C3#1025611yescharacteristic, direct product factor, Sylow
452C2—21no—
453C2—21no—
454C2—82no—
455C2—161no—
456C2C3×#200011yescharacteristic
457C2C3×#200611yescharacteristic
458C2C3×#201711yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3246C2semidirect
2. C3×#3077C2semidirect
4. C3×#1954C2semidirect
5. C3×#1493C2semidirect
6. C3×#1492C2semidirect
7. C3×#1292C2semidirect
8. #10256C3direct (N × H)
34. #3246C2×C3semidirect
35. #3077C2×C3semidirect
37. #1954C2×C3semidirect
38. #1493C2×C3semidirect
39. #1492C2×C3semidirect
40. #1292C2×C3semidirect
135. C2×C3×D4SD16semidirect
298. C2×D4C3×SD16semidirect
451. C3#10256direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32462
C3×#30772
C3×#30362
C3×#19542
C3×#14932
C3×#14922
C3×#12922
#102563