Finite Group Databasebeta

← C3×#30429 #10649036

Subgroups — C3×#30429 #10649036

Order 768.

920 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3060C212yesmaximal, complemented
2C3×#3012C212yesmaximal, complemented
3C3×#2094C211yesmaximal, characteristic, complemented
4C3×#2079C212yesmaximal, complemented
5C3×#2039C212yesmaximal, complemented
6C3×#2006C211yesmaximal, characteristic, complemented
7C3×#1907C212yesmaximal, complemented
8C3×#1876C212yesmaximal, complemented
9C3×#1792C211yesmaximal, characteristic, complemented
10#30429C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×H(5,𝔽2)C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C2×C3×#135C2212yes—
14C2×C3×#131C2212yescomplemented
15C3×#537C2212yes—
16C3×#537C2211yescharacteristic, complemented
17C3×#470—22no—
18C3×#469—22no—
19C3×#465—22no—
20C3×#465—22no—
21C3×#452—22no—
22C3×#450—22no—
23C3×#449—22no—
24C3×#447—22no—
25C2×C3×#106C2212yescomplemented
26C2×C3×#104C2212yes—
27C2×C3×#104C2212yescomplemented
28C3×#417C2212yes—
29C3×#417C2212yes—
30C3×#417C2212yescomplemented
31C2×C3×#102C2212yes—
32C2×C3×#100C2211yescharacteristic
33C3×#408C2212yes—
34C3×#390C2211yescharacteristic
35C3×#384C2211yescharacteristic, complemented
36C3×#343C2211yescharacteristic, complemented
37C3×#343C2212yescomplemented
38C3×#341C2212yescomplemented
39C3×#336C2211yescharacteristic, complemented
40#3060C2×C312yescomplemented
41#3012C2×C312yescomplemented
42#2094C2×C311yescharacteristic, complemented
43#2079C2×C312yescomplemented
44#2039C2×C312yescomplemented
45#2006C2×C311yescharacteristic, complemented
46#1907C2×C312yescomplemented
47#1876C2×C312yescomplemented
48#1792C2×C311yescharacteristic, complemented
49C3×H(5,𝔽2)—22no—
50C3×H(5,𝔽2)—42no—
51C3×H(5,𝔽2)—41no—
52C2×C3×#41—22no—
53C2×C3×#41C2312yes—
54C2×C3×#41C2312yes—
55C22×C3×D4—22no—
56C22×C3×D4—41no—
57C22×C3×D4C2311yescharacteristic
58C3×AGL(1,ℤ/8)—24no—
59C3×AGL(1,ℤ/8)—42no—
60C3×#135—42no—
61C3×#135—42no—
62C2×C3×Q16D412yes—
63C2×C3×SD16—22no—
64C2×C3×SD16D412yes—
65C2×C3×SD16D412yes—
66C2×C3×D8—22no—
67C2×C3×D8D412yes—
68C3×#131—42no—
69C3×#131—42no—
70C2×C3×M16—22no—
71C2×C3×M16C2312yes—
72C2×C3×M16C2312yes—
73C22×C3×C8—22no—
74C22×C3×C8C2312yes—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—42no—
82C3×#121—41no—
83C2×C3×C4⋊C4—41no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—41no—
87C2×C3×C42C2311yescharacteristic
88C3×#106—22no—
89C3×#106—22no—
90C3×#104—42no—
91C3×#104—42no—
92C3×#103—22no—
93C3×#103—22no—
94C3×#103—22no—
95C3×#102—22no—
96C3×#102—22no—
97C3×#102—22no—
98C3×#102—22no—
99C3×#102—22no—
100C3×#100—41no—
101C3×C22⋊C8—22no—
102C3×C22⋊C8—22no—
103C2×H(5,𝔽2)C22×C311yescharacteristic
104C2×AGL(1,ℤ/8)C22×C312yes—
105C2×#135C22×C312yes—
106C2×#131C22×C312yescomplemented
107#537C22×C312yes—
108#537C22×C311yescharacteristic, complemented
109#470—22no—
110#469—22no—
111#465—22no—
112#465—22no—
113#452—22no—
114#450—22no—
115#449—22no—
116#447—22no—
117C2×#106C22×C312yescomplemented
118C2×#104C22×C312yes—
119C2×#104C22×C312yescomplemented
120#417C22×C312yes—
121#417C22×C312yes—
122#417C22×C312yescomplemented
123C2×#102C22×C312yes—
124C2×#100C22×C311yescharacteristic
125#408C22×C312yes—
126#390C22×C311yescharacteristic
127#384C22×C311yescharacteristic, complemented
128#343C22×C311yescharacteristic, complemented
129#343C22×C312yescomplemented
130#341C22×C312yescomplemented
131#336C22×C311yescharacteristic, complemented
132C24×C3—21no—
133C24×C3—41no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—22no—
142C3×#41—44no—
143C3×#41—44no—
144C2×C3×Q8C2×D412yes—
145C2×C3×Q8C2×D412yes—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—42no—
152C2×C3×D4—44no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—44no—
156C2×C3×D4—41no—
157C2×C3×D4—42no—
158C2×C3×D4C2×D412yes—
159C2×C3×D4C2×D412yes—
160C2×C3×D4#4112yes—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—41no—
167C22×C3×C4C2411yescharacteristic
168C3×Q16—42no—
169C3×SD16—42no—
170C3×SD16—42no—
171C3×SD16—44no—
172C3×D8—42no—
173C3×D8—44no—
174C3×M16—22no—
175C3×M16—22no—
176C3×M16—22no—
177C3×M16—22no—
178C3×M16—44no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—44no—
184C2×C3×C8C2×D412yes—
185C2×C3×C8C2×D412yes—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—82no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—82no—
196C3×C42—22no—
197H(5,𝔽2)—22no—
198H(5,𝔽2)—42no—
199H(5,𝔽2)—41no—
200C2×#41—22no—
201C2×#41C23×C312yes—
202C2×#41C23×C312yes—
203C22×D4—22no—
204C22×D4—41no—
205C22×D4C23×C311yescharacteristic
206AGL(1,ℤ/8)—24no—
207AGL(1,ℤ/8)—42no—
208#135—42no—
209#135—42no—
210C2×Q16C3×D412yes—
211C2×SD16—22no—
212C2×SD16C3×D412yes—
213C2×SD16C3×D412yes—
214C2×D8—22no—
215C2×D8C3×D412yes—
216#131—42no—
217#131—42no—
218C2×M16—22no—
219C2×M16C23×C312yes—
220C2×M16C23×C312yes—
221C22×C8—22no—
222C22×C8C23×C312yes—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#121—42no—
230#121—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C23×C312yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C22⋊C4—41no—
235C2×C42C23×C311yescharacteristic
236#106—22no—
237#106—22no—
238#104—42no—
239#104—42no—
240#103—22no—
241#103—22no—
242#103—22no—
243#102—22no—
244#102—22no—
245#102—22no—
246#102—22no—
247#102—22no—
248#100—41no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—44no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—81no—
258C23×C3—81no—
259C23×C3—161no—
260C23×C3—161no—
261C23×C3H(5,𝔽2)11yescharacteristic
262C3×Q8—22no—
263C3×Q8—22no—
264C3×Q8—22no—
265C3×Q8—22no—
266C3×D4—22no—
267C3×D4—22no—
268C3×D4—22no—
269C3×D4—24no—
270C3×D4—22no—
271C3×D4—48no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—162no—
278C3×D4—161no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—41no—
283C2×C3×C4—44no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—81no—
292C2×C3×C4—82no—
293C2×C3×C4C22×D412yes—
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4H(5,𝔽2)11yescharacteristic
296C2×C3×C4H(5,𝔽2)12yes—
297C3×C8—22no—
298C3×C8—22no—
299C3×C8—42no—
300C3×C8—42no—
301C24—21no—
302C24—41no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—44no—
312#41—44no—
313C2×Q8C2×C3×D412yes—
314C2×Q8C2×C3×D412yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4C2×C3×D412yes—
328C2×D4C2×C3×D412yes—
329C2×D4C3×#4112yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4C24×C311yescharacteristic
337Q16—42no—
338SD16—42no—
339SD16—42no—
340SD16—44no—
341D8—42no—
342D8—44no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—44no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—44no—
353C2×C8C2×C3×D412yes—
354C2×C8C2×C3×D412yes—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—82no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—82no—
365C42—22no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—161no—
379C22×C3—164no—
380C22×C3#53811yescharacteristic
381C22×C3D4211yescharacteristic
382C22×C3#56011yescharacteristic
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—82no—
387C3×C4—81no—
388C3×C4#53412yes—
389C3×C4#54612yes—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—41no—
395C23—41no—
396C23—81no—
397C23—81no—
398C23—161no—
399C23—161no—
400C23C3×H(5,𝔽2)11yescharacteristic
401Q8—22no—
402Q8—22no—
403Q8—22no—
404Q8—22no—
405D4—22no—
406D4—22no—
407D4—22no—
408D4—24no—
409D4—22no—
410D4—48no—
411D4—42no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—162no—
417D4—161no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4C22×C3×D412yes—
433C2×C4C2×C3×#4111yescharacteristic
434C2×C4C3×H(5,𝔽2)11yescharacteristic
435C2×C4C3×H(5,𝔽2)12yes—
436C8—22no—
437C8—22no—
438C8—42no—
439C8—42no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—82no—
443C2×C3—82no—
444C2×C3—82no—
445C2×C3#266811yescharacteristic
446C2×C3#328811yescharacteristic
447C2×C3#329111yescharacteristic
448C22—21no—
449C22—21no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—82no—
458C22—81no—
459C22—81no—
460C22—161no—
461C22—164no—
462C22C3×#53811yescharacteristic
463C22C3×D4211yescharacteristic
464C22C3×#56011yescharacteristic
465C4—42no—
466C4—42no—
467C4—42no—
468C4—82no—
469C4—81no—
470C4C3×#53412yes—
471C4C3×#54612yes—
472C3#3042911yescharacteristic, direct product factor, Sylow
473C2—21no—
474C2—21no—
475C2—82no—
476C2—82no—
477C2—82no—
478C2C3×#266811yescharacteristic
479C2C3×#328811yescharacteristic
480C2C3×#329111yescharacteristic
subgroupG/H
center, socleC22×C3D42
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3060C2semidirect
2. C3×#3012C2semidirect
3. C3×#2094C2semidirect
4. C3×#2079C2semidirect
5. C3×#2039C2semidirect
6. C3×#2006C2semidirect
7. C3×#1907C2semidirect
8. C3×#1876C2semidirect
9. C3×#1792C2semidirect
10. #30429C3direct (N × H)
14. C2×C3×#131C22semidirect
16. C3×#537C22semidirect
25. C2×C3×#106C22semidirect
27. C2×C3×#104C22semidirect
30. C3×#417C22semidirect
35. C3×#384C22semidirect
36. C3×#343C22semidirect
37. C3×#343C22semidirect
38. C3×#341C22semidirect
39. C3×#336C22semidirect
40. #3060C2×C3semidirect
41. #3012C2×C3semidirect
42. #2094C2×C3semidirect
43. #2079C2×C3semidirect
44. #2039C2×C3semidirect
45. #2006C2×C3semidirect
46. #1907C2×C3semidirect
47. #1876C2×C3semidirect
48. #1792C2×C3semidirect
106. C2×#131C22×C3semidirect
108. #537C22×C3semidirect
117. C2×#106C22×C3semidirect
119. C2×#104C22×C3semidirect
122. #417C22×C3semidirect
127. #384C22×C3semidirect
128. #343C22×C3semidirect
129. #343C22×C3semidirect
130. #341C22×C3semidirect
131. #336C22×C3semidirect
472. C3#30429direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30602
C3×#30122
C3×#20942
C3×#20792
C3×#20392
C3×#20062
C3×#19072
C3×#18762
C3×#17922
#304293