Finite Group Databasebeta

← C3×#32918 #10651525

Subgroups — C3×#32918 #10651525

Order 768.

1372 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic, complemented
2C2×C3×#464C212yesmaximal, complemented
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C2×C3×#448C211yesmaximal, characteristic
5C3×#2620C211yesmaximal, characteristic
6C3×#1880C212yesmaximal, complemented
7C3×#1878C212yesmaximal, complemented
8C3×#1875C211yesmaximal, characteristic
9C3×#1873C211yesmaximal, characteristic, complemented
10C3×#1803C212yesmaximal, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#32918C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C3×#519—28no—
20C2×C3×C4×Q8—22no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2212yes—
23C2×C3×#117—22no—
24C2×C3×#117C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C3×#464—28no—
27C3×#451—24no—
28C3×#448—24no—
29C2×C3×C4⋊C8C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#396—22no—
37C3×#394—22no—
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#384C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#519C2×C311yescharacteristic, complemented
45C2×#464C2×C312yescomplemented
46C2×#451C2×C311yescharacteristic, complemented
47C2×#448C2×C311yescharacteristic
48#2620C2×C311yescharacteristic
49#1880C2×C312yescomplemented
50#1878C2×C312yescomplemented
51#1875C2×C311yescharacteristic
52#1873C2×C311yescharacteristic, complemented
53#1803C2×C312yescomplemented
54#1798C2×C311yescharacteristic, complemented
55C22×C3×Q8—24no—
56C22×C3×Q8—24no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8D414yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2312yes—
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C22×C3×C8C2312yes—
73C22×C3×C8C2312yes—
74C3×#124—24no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×C4×Q8—216no—
80C3×C4×Q8—24no—
81C3×C4×Q8—24no—
82C3×C4×Q8—24no—
83C3×C4×Q8D418yescomplemented
84C3×#117—216no—
85C3×#117—24no—
86C3×#117—24no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4D412yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4D412yes—
101C2×C3×C42—22no—
102C2×C3×C42—22no—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C3×C4⋊C8—24no—
107C3×#103—24no—
108C3×#103—24no—
109C3×#103—24no—
110C3×#102—24no—
111C3×C22⋊C8—22no—
112C3×C22⋊C8—22no—
113C3×#95—22no—
114C23×Q8C22×C311yescharacteristic
115C22×Q16C22×C312yes—
116C22×SD16C22×C312yes—
117C2×#124C22×C311yescharacteristic
118C2×C22⋊Q8C22×C311yescharacteristic
119C2×#121C22×C311yescharacteristic
120#519—28no—
121C2×C4×Q8—22no—
122C2×C4×Q8C22×C312yes—
123C2×C4×Q8C22×C312yes—
124C2×#117—22no—
125C2×#117C22×C311yescharacteristic
126C2×#117C22×C311yescharacteristic
127#464—28no—
128#451—24no—
129#448—24no—
130C2×C4⋊C8C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C312yes—
133C2×#103C22×C312yes—
134C2×#102C22×C312yes—
135C2×C22⋊C8C22×C311yescharacteristic
136C2×C22⋊C8C22×C311yescharacteristic
137#396—22no—
138#394—22no—
139#391C22×C311yescharacteristic
140#390C22×C311yescharacteristic
141#384C22×C311yescharacteristic
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145C24×C3—21no—
146C24×C3C2×D411yescharacteristic
147C2×C3×Q8—24no—
148C2×C3×Q8—216no—
149C2×C3×Q8—24no—
150C2×C3×Q8—216no—
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8C2×D414yes—
161C2×C3×Q8#4114yes—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—82no—
168C22×C3×C4—24no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C3×Q16—48no—
185C3×SD16—48no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4—82no—
202C3×C4⋊C4C2×D414yes—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—42no—
206C3×C22⋊C4—44no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—82no—
209C3×C42—28no—
210C3×C42—24no—
211C3×C42—28no—
212C3×C42—22no—
213C3×C42C2×D414yes—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8C3×D414yes—
219C22×Q8C23×C311yescharacteristic
220C22×Q8C23×C312yes—
221C22×D4—21no—
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C23×C4C23×C311yescharacteristic
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—24no—
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—24no—
231C22×C8C23×C312yes—
232C22×C8C23×C312yes—
233#124—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—22no—
237#121—22no—
238C4×Q8—216no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8C3×D418yescomplemented
243#117—216no—
244#117—24no—
245#117—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C3×D412yes—
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4C3×D412yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42C23×C311yescharacteristic
264C2×C42C23×C311yescharacteristic
265C4⋊C8—24no—
266#103—24no—
267#103—24no—
268#103—24no—
269#102—24no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272#95—22no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3#12112yes—
283C23×C3C22×D411yescharacteristic
284C3×Q8—216no—
285C3×Q8—24no—
286C3×Q8—24no—
287C3×Q8—24no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×Q8#12118yes—
291C3×D4—42no—
292C3×D4—42no—
293C3×D4—84no—
294C2×C3×C4—216no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4#12014yes—
318C2×C3×C4#12112yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D412yes—
321C2×C3×C4C2×#4112yes—
322C2×C3×C4C2×#4111yescharacteristic
323C3×C8—42no—
324C3×C8—42no—
325C24—21no—
326C24C2×C3×D411yescharacteristic
327C2×Q8—24no—
328C2×Q8—216no—
329C2×Q8—24no—
330C2×Q8—216no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8C2×C3×D414yes—
340C2×Q8C2×C3×D414yes—
341C2×Q8C3×#4114yes—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—42no—
347C2×D4—82no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D412yes—
362C22×C4C3×#4112yes—
363C22×C4C24×C311yescharacteristic
364Q16—48no—
365SD16—48no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—42no—
381C4⋊C4—82no—
382C4⋊C4C2×C3×D414yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—82no—
389C42—28no—
390C42—24no—
391C42—28no—
392C42—22no—
393C42C2×C3×D414yes—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—81no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3C2×#12011yescharacteristic
405C22×C3C2×#12111yescharacteristic
406C22×C3#52511yescharacteristic
407C22×C3#57712yes—
408C22×C3#57812yes—
409C3×C4—28no—
410C3×C4—22no—
411C3×C4—24no—
412C3×C4—42no—
413C3×C4—44no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4C2×#12112yes—
418C3×C4#52512yes—
419C23—22no—
420C23—22no—
421C23—21no—
422C23—21no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23C3×#12112yes—
429C23C22×C3×D411yescharacteristic
430Q8—216no—
431Q8—24no—
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8C3×#12118yes—
437D4—42no—
438D4—42no—
439D4—84no—
440C2×C4—216no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4C3×#12014yes—
464C2×C4C3×#12112yes—
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C22×C3×D412yes—
467C2×C4C2×C3×#4112yes—
468C2×C4C2×C3×#4111yescharacteristic
469C8—42no—
470C8—42no—
471C2×C3—42no—
472C2×C3—161no—
473C2×C3#284711yescharacteristic
474C2×C3#301312yes—
475C2×C3#304311yescharacteristic
476C2×C3#304411yescharacteristic
477C2×C3#312112yes—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—82no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22C2×C3×#12011yescharacteristic
489C22C2×C3×#12111yescharacteristic
490C22C3×#52511yescharacteristic
491C22C3×#57712yes—
492C22C3×#57812yes—
493C4—24no—
494C4—28no—
495C4—22no—
496C4—42no—
497C4—44no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4C2×C3×#12112yes—
502C4C3×#52512yes—
503C3#3291811yescharacteristic, direct product factor, Sylow
504C2—42no—
505C2—161no—
506C2C3×#284711yescharacteristic
507C2C3×#301312yes—
508C2C3×#304311yescharacteristic
509C2C3×#304411yescharacteristic
510C2C3×#312112yes—
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×#519C2semidirect
2. C2×C3×#464C2semidirect
3. C2×C3×#451C2semidirect
6. C3×#1880C2semidirect
7. C3×#1878C2semidirect
9. C3×#1873C2semidirect
10. C3×#1803C2semidirect
11. C3×#1798C2semidirect
12. #32918C3direct (N × H)
44. C2×#519C2×C3semidirect
45. C2×#464C2×C3semidirect
46. C2×#451C2×C3semidirect
49. #1880C2×C3semidirect
50. #1878C2×C3semidirect
52. #1873C2×C3semidirect
53. #1803C2×C3semidirect
54. #1798C2×C3semidirect
83. C3×C4×Q8D4semidirect
242. C4×Q8C3×D4semidirect
503. C3#32918direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×C3×#4642
C2×C3×#4512
C2×C3×#4482
C3×#26202
C3×#18802
C3×#18782
C3×#18752
C3×#18732
C3×#18032
C3×#17982
#329183