Finite Group Databasebeta

← C3×#29676 #10648283

Subgroups — C3×#29676 #10648283

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×C3×#464C212yesmaximal, complemented
3C3×#2620C211yesmaximal, characteristic
4C3×#2026C211yesmaximal, characteristic, complemented
5C3×#2025C211yesmaximal, characteristic, complemented
6C3×#2016C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#2001C212yesmaximal, complemented
9C3×#2000C212yesmaximal
10C3×#1803C212yesmaximal, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29676C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C3×#541—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—22no—
18C2×C3×#125—22no—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C2×C3×#117C2211yescharacteristic
26C3×#464—28no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C3×#396—22no—
34C3×#394—22no—
35C3×#391C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#384C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C2×#541C2×C311yescharacteristic, complemented
43C2×#464C2×C312yescomplemented
44#2620C2×C311yescharacteristic
45#2026C2×C311yescharacteristic, complemented
46#2025C2×C311yescharacteristic, complemented
47#2016C2×C311yescharacteristic, complemented
48#2013C2×C311yescharacteristic, complemented
49#2001C2×C312yescomplemented
50#2000C2×C312yes—
51#1803C2×C312yescomplemented
52#1798C2×C311yescharacteristic, complemented
53C22×C3×Q8C2311yescharacteristic
54C22×C3×Q8C2312yes—
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C2×C3×Q16—24no—
59C2×C3×Q16—42no—
60C2×C3×Q16—42no—
61C2×C3×SD16—24no—
62C2×C3×SD16—42no—
63C2×C3×SD16—42no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—24no—
67C3×#126—216no—
68C3×#125—216no—
69C3×#125—24no—
70C3×#124—24no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—42no—
74C3×#121—42no—
75C3×C4×Q8—24no—
76C3×C4×Q8D418yescomplemented
77C3×#117—24no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4D412yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4D412yes—
94C2×C3×C42—22no—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—24no—
98C3×#103—24no—
99C3×#103—42no—
100C3×#103—42no—
101C3×#102—24no—
102C3×C22⋊C8—42no—
103C3×#95—22no—
104C22×Q16C22×C312yes—
105C22×SD16C22×C312yes—
106#541—28no—
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×#126—22no—
109C2×#125—22no—
110C2×#125C22×C311yescharacteristic
111C2×#124C22×C311yescharacteristic
112C2×C22⋊Q8C22×C312yes—
113C2×C22⋊Q8C22×C311yescharacteristic
114C2×#121C22×C311yescharacteristic
115C2×C4×Q8C22×C312yes—
116C2×#117C22×C311yescharacteristic
117#464—28no—
118C2×C4⋊C8C22×C312yes—
119C2×#103C22×C312yes—
120C2×#103C22×C312yes—
121C2×#103C22×C312yes—
122C2×#102C22×C312yes—
123C2×C22⋊C8C22×C312yes—
124#396—22no—
125#394—22no—
126#391C22×C311yescharacteristic
127#390C22×C311yescharacteristic
128#384C22×C311yescharacteristic
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C311yescharacteristic
132#336C22×C311yescharacteristic
133C24×C3—21no—
134C24×C3C2×D411yescharacteristic
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—21no—
144C2×C3×Q8C2×D414yes—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—42no—
150C2×C3×D4—82no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2411yescharacteristic
164C3×Q16—48no—
165C3×SD16—48no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—28no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—82no—
182C3×C4⋊C4C2×D414yes—
183C3×C22⋊C4—28no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—82no—
189C3×C42—28no—
190C3×C42—22no—
191C3×C42C2×D414yes—
192C22×Q8C23×C311yescharacteristic
193C22×Q8C23×C312yes—
194C22×D4—21no—
195C22×D4C23×C311yescharacteristic
196C23×C4C23×C311yescharacteristic
197C2×Q16—24no—
198C2×Q16—42no—
199C2×Q16—42no—
200C2×SD16—24no—
201C2×SD16—42no—
202C2×SD16—42no—
203C22×C8C23×C312yes—
204C22×C8C23×C312yes—
205C4⋊Q8—24no—
206#126—216no—
207#125—216no—
208#125—24no—
209#124—24no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—42no—
213#121—42no—
214C4×Q8—24no—
215C4×Q8C3×D418yescomplemented
216#117—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C3×D412yes—
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C312yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C3×D412yes—
233C2×C42—22no—
234C2×C42C23×C311yescharacteristic
235C2×C42C23×C311yescharacteristic
236C4⋊C8—24no—
237#103—24no—
238#103—42no—
239#103—42no—
240#102—24no—
241C22⋊C8—42no—
242#95—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3C22×D411yescharacteristic
254C3×Q8—24no—
255C3×Q8—24no—
256C3×Q8—42no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—84no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4#12014yes—
287C2×C3×C4C22×D412yes—
288C2×C3×C4H(5,𝔽2)11yescharacteristic
289C2×C3×C4H(5,𝔽2)12yes—
290C2×C3×C42−1+411yescharacteristic
291C3×C8—42no—
292C3×C8—42no—
293C24—21no—
294C24C2×C3×D411yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—21no—
302C2×Q8—22no—
303C2×Q8—21no—
304C2×Q8C2×C3×D414yes—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—82no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D412yes—
323C22×C4C24×C311yescharacteristic
324Q16—48no—
325SD16—48no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—82no—
342C4⋊C4C2×C3×D414yes—
343C22⋊C4—28no—
344C22⋊C4—24no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C42—28no—
350C42—22no—
351C42C2×C3×D414yes—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3C2×#12011yescharacteristic
363C22×C3#53511yescharacteristic
364C22×C3#53711yescharacteristic
365C22×C3#57712yes—
366C22×C3#57812yes—
367C3×C4—24no—
368C3×C4—41no—
369C3×C4—44no—
370C3×C4—42no—
371C3×C4—44no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4—81no—
375C3×C4#53512yes—
376C3×C4#53712yes—
377C23—22no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C22×C3×D411yescharacteristic
388Q8—24no—
389Q8—24no—
390Q8—42no—
391Q8—42no—
392Q8—42no—
393Q8—42no—
394D4—42no—
395D4—42no—
396D4—84no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4C3×#12014yes—
421C2×C4C22×C3×D412yes—
422C2×C4C3×H(5,𝔽2)11yescharacteristic
423C2×C4C3×H(5,𝔽2)12yes—
424C2×C4C3×2−1+411yescharacteristic
425C8—42no—
426C8—42no—
427C2×C3—81no—
428C2×C3—161no—
429C2×C3#261511yescharacteristic
430C2×C3#301312yes—
431C2×C3#322211yescharacteristic
432C2×C3#322311yescharacteristic
433C2×C3#325812yes—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—41no—
439C22—81no—
440C22—81no—
441C22—82no—
442C22—82no—
443C22—81no—
444C22C2×C3×#12011yescharacteristic
445C22C3×#53511yescharacteristic
446C22C3×#53711yescharacteristic
447C22C3×#57712yes—
448C22C3×#57812yes—
449C4—24no—
450C4—42no—
451C4—44no—
452C4—41no—
453C4—44no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4C3×#53512yes—
458C4C3×#53712yes—
459C3#2967611yescharacteristic, direct product factor, Sylow
460C2—81no—
461C2—161no—
462C2C3×#261511yescharacteristic
463C2C3×#301312yes—
464C2C3×#322211yescharacteristic
465C2C3×#322311yescharacteristic
466C2C3×#325812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
1. C2×C3×#541C2semidirect
2. C2×C3×#464C2semidirect
4. C3×#2026C2semidirect
5. C3×#2025C2semidirect
6. C3×#2016C2semidirect
7. C3×#2013C2semidirect
8. C3×#2001C2semidirect
10. C3×#1803C2semidirect
11. C3×#1798C2semidirect
12. #29676C3direct (N × H)
42. C2×#541C2×C3semidirect
43. C2×#464C2×C3semidirect
45. #2026C2×C3semidirect
46. #2025C2×C3semidirect
47. #2016C2×C3semidirect
48. #2013C2×C3semidirect
49. #2001C2×C3semidirect
51. #1803C2×C3semidirect
52. #1798C2×C3semidirect
76. C3×C4×Q8D4semidirect
215. C4×Q8C3×D4semidirect
459. C3#29676direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4642
C3×#26202
C3×#20262
C3×#20252
C3×#20162
C3×#20132
C3×#20012
C3×#20002
C3×#18032
C3×#17982
#296763