Finite Group Databasebeta

← C3×#29726 #10648333

Subgroups — C3×#29726 #10648333

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#452C212yesmaximal
2C3×#2621C211yesmaximal, characteristic, complemented
3C3×#2311C211yesmaximal, characteristic
4C3×#2074C211yesmaximal, characteristic, complemented
5C3×#2073C211yesmaximal, characteristic, complemented
6C3×#2056C212yesmaximal, complemented
7C3×#2055C212yesmaximal, complemented
8C3×#2016C211yesmaximal, characteristic, complemented
9C3×#2013C211yesmaximal, characteristic, complemented
10C3×#1872C212yesmaximal, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29726C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C3×#452—28no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×C22⋊C8C2212yes—
27C3×#398—22no—
28C3×#397—22no—
29C3×#391C2212yes—
30C3×#391C2211yescharacteristic
31C3×#391C2211yescharacteristic
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#386—22no—
35C3×#382C2211yescharacteristic
36C3×#380—22no—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40C3×#336C2211yescharacteristic
41C2×#452C2×C312yes—
42#2621C2×C311yescharacteristic, complemented
43#2311C2×C311yescharacteristic
44#2074C2×C311yescharacteristic, complemented
45#2073C2×C311yescharacteristic, complemented
46#2056C2×C312yescomplemented
47#2055C2×C312yescomplemented
48#2016C2×C311yescharacteristic, complemented
49#2013C2×C311yescharacteristic, complemented
50#1872C2×C312yescomplemented
51#1798C2×C311yescharacteristic, complemented
52C2×C3×#41—22no—
53C2×C3×#41—42no—
54C2×C3×#41—42no—
55C2×C3×#41D414yes—
56C22×C3×Q8C2312yes—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C2×C3×Q16—24no—
64C2×C3×Q16—42no—
65C2×C3×Q16—42no—
66C2×C3×SD16—24no—
67C2×C3×SD16—42no—
68C2×C3×SD16—42no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—42no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—24no—
74C3×#117—42no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C3×C4⋊C8—42no—
90C3×#103—24no—
91C3×#103—42no—
92C3×#103—42no—
93C3×#102—24no—
94C3×C22⋊C8—24no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—24no—
99C22×#41C22×C311yescharacteristic
100C22×Q16C22×C312yes—
101C22×SD16C22×C312yes—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C22⋊Q8C22×C312yes—
104C2×#117C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106#452—28no—
107C2×C4⋊C8C22×C312yes—
108C2×#103C22×C312yes—
109C2×#103C22×C312yes—
110C2×#103C22×C312yes—
111C2×#102C22×C312yes—
112C2×C22⋊C8C22×C312yes—
113#398—22no—
114#397—22no—
115#391C22×C312yes—
116#391C22×C311yescharacteristic
117#391C22×C311yescharacteristic
118#390C22×C311yescharacteristic
119#390C22×C311yescharacteristic
120#386—22no—
121#382C22×C311yescharacteristic
122#380—22no—
123#340C22×C312yes—
124#340C22×C312yes—
125#336C22×C311yescharacteristic
126#336C22×C311yescharacteristic
127C24×C3—21no—
128C24×C3C2×D411yescharacteristic
129C3×#41—48no—
130C3×#41—48no—
131C2×C3×Q8—21no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×D412yes—
141C2×C3×D4—24no—
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4C2×D412yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—82no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C3×Q16—48no—
167C3×SD16—48no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—44no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—44no—
184C3×C22⋊C4—82no—
185C3×C22⋊C4—82no—
186C3×C42—22no—
187C3×C42—22no—
188C2×#41—22no—
189C2×#41—42no—
190C2×#41—42no—
191C2×#41C3×D414yes—
192C22×Q8C23×C312yes—
193C22×Q8C23×C311yescharacteristic
194C22×D4—21no—
195C22×D4C23×C311yescharacteristic
196C23×C4—21no—
197C23×C4—21no—
198C23×C4C23×C311yescharacteristic
199C2×Q16—24no—
200C2×Q16—42no—
201C2×Q16—42no—
202C2×SD16—24no—
203C2×SD16—42no—
204C2×SD16—42no—
205C22×C8C23×C312yes—
206C22×C8C23×C312yes—
207C4⋊Q8—42no—
208C22⋊Q8—24no—
209C22⋊Q8—24no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C3×D412yes—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C312yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4C3×D412yes—
223C2×C42C23×C311yescharacteristic
224C2×C42C23×C311yescharacteristic
225C4⋊C8—42no—
226#103—24no—
227#103—42no—
228#103—42no—
229#102—24no—
230C22⋊C8—24no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—24no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3#12112yes—
245C23×C3C22×D411yescharacteristic
246C3×Q8—24no—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8—42no—
251C3×Q8—42no—
252C3×D4—24no—
253C3×D4—48no—
254C3×D4—42no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—48no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—84no—
280C2×C3×C4#12112yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4H(5,𝔽2)12yes—
284C2×C3×C4H(5,𝔽2)12yes—
285C3×C8—42no—
286C3×C8—42no—
287C24—21no—
288C24C2×C3×D411yescharacteristic
289#41—48no—
290#41—48no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8C2×C3×D412yes—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4C2×C3×D412yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—82no—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D412yes—
324C22×C4C3×#4112yes—
325C22×C4C24×C311yescharacteristic
326Q16—48no—
327SD16—48no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C42—22no—
347C42—22no—
348C22×C3—22no—
349C22×C3—24no—
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—44no—
355C22×C3—82no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3C2×#12111yescharacteristic
359C22×C3#53411yescharacteristic
360C22×C3#53511yescharacteristic
361C22×C3#57712yes—
362C22×C3#57812yes—
363C3×C4—22no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—82no—
368C3×C4—81no—
369C3×C4—81no—
370C3×C4—82no—
371C3×C4#53412yes—
372C3×C4#53812yes—
373C23—22no—
374C23—22no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23C3×#12112yes—
383C23C22×C3×D411yescharacteristic
384Q8—24no—
385Q8—42no—
386Q8—42no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390D4—24no—
391D4—48no—
392D4—42no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—48no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4C3×#12112yes—
419C2×C4C22×C3×D411yescharacteristic
420C2×C4C2×C3×#4111yescharacteristic
421C2×C4C3×H(5,𝔽2)12yes—
422C2×C4C3×H(5,𝔽2)12yes—
423C8—42no—
424C8—42no—
425C2×C3—42no—
426C2×C3—82no—
427C2×C3#261711yescharacteristic
428C2×C3#304612yes—
429C2×C3#320112yes—
430C2×C3#322211yescharacteristic
431C2×C3#322311yescharacteristic
432C22—22no—
433C22—24no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—42no—
438C22—44no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22C2×C3×#12111yescharacteristic
443C22C3×#53411yescharacteristic
444C22C3×#53511yescharacteristic
445C22C3×#57712yes—
446C22C3×#57812yes—
447C4—22no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—82no—
455C4C3×#53412yes—
456C4C3×#53812yes—
457C3#2972611yescharacteristic, direct product factor, Sylow
458C2—42no—
459C2—82no—
460C2C3×#261711yescharacteristic
461C2C3×#304612yes—
462C2C3×#320112yes—
463C2C3×#322211yescharacteristic
464C2C3×#322311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2621C2semidirect
4. C3×#2074C2semidirect
5. C3×#2073C2semidirect
6. C3×#2056C2semidirect
7. C3×#2055C2semidirect
8. C3×#2016C2semidirect
9. C3×#2013C2semidirect
10. C3×#1872C2semidirect
11. C3×#1798C2semidirect
12. #29726C3direct (N × H)
42. #2621C2×C3semidirect
44. #2074C2×C3semidirect
45. #2073C2×C3semidirect
46. #2056C2×C3semidirect
47. #2055C2×C3semidirect
48. #2016C2×C3semidirect
49. #2013C2×C3semidirect
50. #1872C2×C3semidirect
51. #1798C2×C3semidirect
457. C3#29726direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4522
C3×#26212
C3×#23112
C3×#20742
C3×#20732
C3×#20562
C3×#20552
C3×#20162
C3×#20132
C3×#18722
C3×#17982
#297263