Finite Group Databasebeta

← C3×#10242 #10628849

Subgroups — C3×#10242 #10628849

Order 768.

658 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3205C211yesmaximal, characteristic
2C3×#3037C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic, complemented
4C3×#1545C211yesmaximal, characteristic, complemented
5C3×#1502C211yesmaximal, characteristic, complemented
6C3×#1479C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10242C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×C3×#124—21no—
13C2×C3×C4×Q8C2211yescharacteristic
14C3×#493—21no—
15C3×#486—21no—
16C3×#464—22no—
17C3×#462—21no—
18C3×#460—21no—
19C3×#459—22no—
20C3×#447—22no—
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#423C2211yescharacteristic
24C3×#417—21no—
25C3×#394—22no—
26C3×#390—22no—
27C3×#390—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#331—22no—
30C3×#327—22no—
31C3×Q8⋊C8—22no—
32#3205C2×C311yescharacteristic
33#3037C2×C311yescharacteristic, complemented
34#2548C2×C311yescharacteristic, complemented
35#1545C2×C311yescharacteristic, complemented
36#1502C2×C311yescharacteristic, complemented
37#1479C2×C311yescharacteristic, complemented
38#1303C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4D412yes—
44C3×#137—42no—
45C3×#137—42no—
46C2×C3×Q16—22no—
47C2×C3×SD16—22no—
48C2×C3×D8—24no—
49C2×C3×M16—21no—
50C3×C4⋊Q8D411yescharacteristic
51C3×#127—21no—
52C3×#127—21no—
53C3×#127—21no—
54C3×#127—21no—
55C3×#127D411yescharacteristic
56C3×#124—21no—
57C3×#124—42no—
58C3×#124—42no—
59C3×#121—21no—
60C3×#121—21no—
61C3×C4×Q8—21no—
62C3×C4×Q8—21no—
63C3×C4×Q8—21no—
64C3×C4×Q8—21no—
65C3×C4×D4—21no—
66C3×C4×D4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yes—
78C3×C4⋊C8D412yes—
79C3×#103—22no—
80C3×#102—22no—
81C3×#102—24no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×C4×C8—21no—
85C3×C4×C8—21no—
86C3×#95D412yes—
87C2×#137—21no—
88#532C22×C311yescharacteristic
89C2×#127C22×C311yescharacteristic
90C2×#124—21no—
91C2×C4×Q8C22×C311yescharacteristic
92#493—21no—
93#486—21no—
94#464—22no—
95#462—21no—
96#460—21no—
97#459—22no—
98#447—22no—
99#431C22×C311yescharacteristic
100#431C22×C311yescharacteristic
101#423C22×C311yescharacteristic
102#417—21no—
103#394—22no—
104#390—22no—
105#390—21no—
106C4×C4⋊C4C22×C311yescharacteristic
107#331—22no—
108#327—22no—
109Q8⋊C8—22no—
110C24×C3—22no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8D812yescomplemented
122C2×C3×D4—21no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—22no—
127C2×C3×D4—44no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—82no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C3×Q16—44no—
146C3×SD16—44no—
147C3×D8—84no—
148C3×M16—41no—
149C3×M16—41no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—41no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C22⋊C4—48no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—84no—
164C3×C42—41no—
165C3×C42—41no—
166C3×C42—41no—
167C3×C42—41no—
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42#4111yescharacteristic
172C2×#41—21no—
173C22×Q8C3×D411yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C3×D412yes—
177#137—42no—
178#137—42no—
179C2×Q16—22no—
180C2×SD16—22no—
181C2×D8—24no—
182C2×M16—21no—
183C4⋊Q8C3×D411yescharacteristic
184#127—21no—
185#127—21no—
186#127—21no—
187#127—21no—
188#127C3×D411yescharacteristic
189#124—21no—
190#124—42no—
191#124—42no—
192#121—21no—
193#121—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×D4—21no—
199C4×D4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C3×D411yescharacteristic
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C23×C311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8C3×D412yes—
211C4⋊C8C3×D412yes—
212#103—22no—
213#102—22no—
214#102—24no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219#95C3×D412yes—
220C23×C3—44no—
221C23×C3—44no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3#12011yescharacteristic
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—42no—
236C3×D4—44no—
237C3×D4—81no—
238C3×D4—82no—
239C3×D4—82no—
240C3×D4—82no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—81no—
259C2×C3×C4—81no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12711yescharacteristic
266C2×C3×C4C2×D811yescharacteristic
267C2×C3×C4#13511yescharacteristic
268C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
269C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
270C3×C8—42no—
271C3×C8—42no—
272C3×C8—81no—
273C24—22no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8C3×D812yescomplemented
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—44no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C2×C3×D411yescharacteristic
307C22×C4C2×C3×D411yescharacteristic
308Q16—44no—
309SD16—44no—
310D8—84no—
311M16—41no—
312M16—41no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—48no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—84no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42C2×C3×D411yescharacteristic
332C42C2×C3×D411yescharacteristic
333C42C2×C3×D411yescharacteristic
334C42C3×#4111yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—84no—
338C22×C3—84no—
339C22×C3—81no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—81no—
343C22×C3—82no—
344C22×C3—81no—
345C22×C3#39211yescharacteristic
346C22×C3#44711yescharacteristic
347C22×C3#44911yescharacteristic
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4—82no—
353C3×C4—81no—
354C3×C4—81no—
355C3×C4#44711yescharacteristic
356C3×C4#44911yescharacteristic
357C3×C4#45512yes—
358C3×C4#46611yescharacteristic
359C3×C4#46911yescharacteristic
360C23—44no—
361C23—44no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23C3×#12011yescharacteristic
367Q8—22no—
368Q8—22no—
369Q8—41no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373D4—41no—
374D4—41no—
375D4—42no—
376D4—44no—
377D4—82no—
378D4—81no—
379D4—82no—
380D4—82no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4C3×#12011yescharacteristic
401C2×C4C3×#12011yescharacteristic
402C2×C4C3×#12111yescharacteristic
403C2×C4C3×#12111yescharacteristic
404C2×C4C3×#12111yescharacteristic
405C2×C4C3×#12711yescharacteristic
406C2×C4C2×C3×D811yescharacteristic
407C2×C4C3×#13511yescharacteristic
408C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
409C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
410C8—42no—
411C8—42no—
412C8—81no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—161no—
416C2×C3—162no—
417C2×C3#199911yescharacteristic
418C2×C3#200411yescharacteristic
419C2×C3#200811yescharacteristic
420C22—21no—
421C22—21no—
422C22—84no—
423C22—84no—
424C22—81no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—82no—
429C22—81no—
430C22C3×#39211yescharacteristic
431C22C3×#44711yescharacteristic
432C22C3×#44911yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4—81no—
440C4C3×#44711yescharacteristic
441C4C3×#44911yescharacteristic
442C4C3×#45512yes—
443C4C3×#46611yescharacteristic
444C4C3×#46911yescharacteristic
445C3#1024211yescharacteristic, direct product factor, Sylow
446C2—21no—
447C2—21no—
448C2—162no—
449C2—161no—
450C2C3×#199911yescharacteristic
451C2C3×#200411yescharacteristic
452C2C3×#200811yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#3205C2
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
2. C3×#3037C2semidirect
3. C3×#2548C2semidirect
4. C3×#1545C2semidirect
5. C3×#1502C2semidirect
6. C3×#1479C2semidirect
7. C3×#1303C2semidirect
8. #10242C3direct (N × H)
33. #3037C2×C3semidirect
34. #2548C2×C3semidirect
35. #1545C2×C3semidirect
36. #1502C2×C3semidirect
37. #1479C2×C3semidirect
38. #1303C2×C3semidirect
121. C2×C3×Q8D8semidirect
284. C2×Q8C3×D8semidirect
445. C3#10242direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32052
C3×#30372
C3×#25482
C3×#15452
C3×#15022
C3×#14792
C3×#13032
#102423