Finite Group Databasebeta

← C3×#29839 #10648446

Subgroups — C3×#29839 #10648446

Order 768.

802 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×#474C211yesmaximal, characteristic, complemented
2C2×C3×#463C211yesmaximal, characteristic
3C3×#2640C211yesmaximal, characteristic, complemented
4C3×#2319C211yesmaximal, characteristic, complemented
5C3×#2068C212yesmaximal, complemented
6C3×#2067C212yesmaximal, complemented
7C3×#2055C212yesmaximal, complemented
8C3×#2054C212yesmaximal
9C3×#1987C211yesmaximal, characteristic, complemented
10C3×#1983C211yesmaximal, characteristic, complemented
11C3×#1843C211yesmaximal, characteristic, complemented
12#29839C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C3×#474—24no—
19C3×#463—24no—
20C2×C3×#107C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2212yes—
30C3×#397—22no—
31C3×#393—21no—
32C3×#393—21no—
33C3×#391C2212yes—
34C3×#390C2212yes—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#388—22no—
38C3×#381—21no—
39C3×#380—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#474C2×C311yescharacteristic, complemented
45C2×#463C2×C311yescharacteristic
46#2640C2×C311yescharacteristic, complemented
47#2319C2×C311yescharacteristic, complemented
48#2068C2×C312yescomplemented
49#2067C2×C312yescomplemented
50#2055C2×C312yescomplemented
51#2054C2×C312yes—
52#1987C2×C311yescharacteristic, complemented
53#1983C2×C311yescharacteristic, complemented
54#1843C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×C3×SD16—22no—
61C2×C3×SD16—42no—
62C2×C3×D8—22no—
63C2×C3×D8—42no—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×#124—22no—
70C3×#124—22no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C3×#107—24no—
89C3×C4⋊C8—22no—
90C3×C4⋊C8—22no—
91C3×C4⋊C8—42no—
92C3×C4⋊C8—42no—
93C3×#103—22no—
94C3×#103—42no—
95C3×#102—24no—
96C3×#102—22no—
97C3×#102—42no—
98C3×#95—21no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×SD16C22×C311yescharacteristic
106C22×D8C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×#124C22×C311yescharacteristic
109C2×C4×D4C22×C311yescharacteristic
110#474—24no—
111#463—24no—
112C2×#107C22×C312yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C311yescharacteristic
118C2×#103C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C311yescharacteristic
121C2×#102C22×C312yes—
122#397—22no—
123#393—21no—
124#393—21no—
125#391C22×C312yes—
126#390C22×C312yes—
127#390C22×C312yes—
128#389C22×C312yes—
129#388—22no—
130#381—21no—
131#380—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133#376C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136C24×C3—21no—
137C24×C3—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—42no—
143C2×C3×Q8—82no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4C2×D412yes—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—82no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C3×SD16—44no—
164C3×D8—44no—
165C2×C3×C8—22no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—82no—
179C3×C4⋊C4C2×D412yes—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C22⋊C4—82no—
186C3×C22⋊C4—82no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—81no—
190C3×C42C2×D412yes—
191C3×C42#4112yes—
192C22×Q8—21no—
193C22×Q8C23×C311yescharacteristic
194C22×D4C23×C311yescharacteristic
195C22×D4C23×C311yescharacteristic
196C23×C4—21no—
197C2×SD16—22no—
198C2×SD16—42no—
199C2×D8—22no—
200C2×D8—42no—
201C22×C8C23×C311yescharacteristic
202C22×C8C23×C311yescharacteristic
203C22×C8C23×C312yes—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206#124—22no—
207#124—22no—
208C4×D4—22no—
209C4×D4C3×D414yes—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—21no—
223C2×C42C23×C312yes—
224C2×C42C23×C311yescharacteristic
225#107—24no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230#103—22no—
231#103—42no—
232#102—24no—
233#102—22no—
234#102—42no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95C3×D411yescharacteristic
240#95C3×D411yescharacteristic
241#95C3×D412yes—
242C23×C3—22no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3C22×D411yescharacteristic
253C3×Q8—41no—
254C3×Q8—41no—
255C3×Q8—42no—
256C3×Q8—84no—
257C3×D4—22no—
258C3×D4—22no—
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—42no—
262C3×D4—42no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—84no—
283C2×C3×C4#12112yes—
284C2×C3×C4#12112yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4H(5,𝔽2)12yes—
288C2×C3×C42−1+412yes—
289C3×C8—42no—
290C3×C8—41no—
291C3×C8—41no—
292C24—21no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—42no—
299C2×Q8—82no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4C2×C3×D412yes—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—82no—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C24×C311yescharacteristic
319SD16—44no—
320D8—44no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—82no—
335C4⋊C4C2×C3×D412yes—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—22no—
344C42—22no—
345C42—81no—
346C42C2×C3×D412yes—
347C42C3×#4112yes—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3C2×#12111yescharacteristic
359C22×C3#53712yes—
360C22×C3#57612yes—
361C22×C3#57711yescharacteristic
362C22×C3#57811yescharacteristic
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—81no—
369C3×C4—82no—
370C3×C4—82no—
371C3×C4#53511yescharacteristic
372C3×C4#53611yescharacteristic
373C3×C4#54212yes—
374C23—22no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23C22×C3×D411yescharacteristic
385Q8—41no—
386Q8—41no—
387Q8—42no—
388Q8—84no—
389D4—22no—
390D4—22no—
391D4—41no—
392D4—41no—
393D4—42no—
394D4—42no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—84no—
415C2×C4C3×#12112yes—
416C2×C4C3×#12112yes—
417C2×C4C22×C3×D411yescharacteristic
418C2×C4C2×C3×#4111yescharacteristic
419C2×C4C3×H(5,𝔽2)12yes—
420C2×C4C3×2−1+412yes—
421C8—41no—
422C8—41no—
423C8—42no—
424C2×C3—82no—
425C2×C3—161no—
426C2×C3#262011yescharacteristic
427C2×C3#304111yescharacteristic
428C2×C3#304611yescharacteristic
429C2×C3#325712yes—
430C2×C3#326212yes—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22—82no—
436C22—81no—
437C22—82no—
438C22—82no—
439C22—81no—
440C22—81no—
441C22C2×C3×#12111yescharacteristic
442C22C3×#53712yes—
443C22C3×#57612yes—
444C22C3×#57711yescharacteristic
445C22C3×#57811yescharacteristic
446C4—22no—
447C4—22no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—81no—
452C4—82no—
453C4—82no—
454C4C3×#53511yescharacteristic
455C4C3×#53611yescharacteristic
456C4C3×#54212yes—
457C3#2983911yescharacteristic, direct product factor, Sylow
458C2—82no—
459C2—161no—
460C2C3×#262011yescharacteristic
461C2C3×#304111yescharacteristic
462C2C3×#304611yescharacteristic
463C2C3×#325712yes—
464C2C3×#326212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#474C2semidirect
3. C3×#2640C2semidirect
4. C3×#2319C2semidirect
5. C3×#2068C2semidirect
6. C3×#2067C2semidirect
7. C3×#2055C2semidirect
9. C3×#1987C2semidirect
10. C3×#1983C2semidirect
11. C3×#1843C2semidirect
12. #29839C3direct (N × H)
44. C2×#474C2×C3semidirect
46. #2640C2×C3semidirect
47. #2319C2×C3semidirect
48. #2068C2×C3semidirect
49. #2067C2×C3semidirect
50. #2055C2×C3semidirect
52. #1987C2×C3semidirect
53. #1983C2×C3semidirect
54. #1843C2×C3semidirect
457. C3#29839direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×#4632
C3×#26402
C3×#23192
C3×#20682
C3×#20672
C3×#20552
C3×#20542
C3×#19872
C3×#19832
C3×#18432
#298393