Finite Group Databasebeta

← C3×#29745 #10648352

Subgroups — C3×#29745 #10648352

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#469C211yesmaximal, characteristic, complemented
2C2×C3×#408C211yesmaximal, characteristic, complemented
3C3×#2615C212yesmaximal, complemented
4C3×#2077C212yesmaximal, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1999C212yesmaximal, complemented
8C3×#1893C212yesmaximal, complemented
9C3×#1809C211yesmaximal, characteristic, complemented
10#29745C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×D8C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×C3×#121C2212yes—
17C2×C3×#121C2212yes—
18C3×#469—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C3×#408—24no—
26C2×C3×C22⋊C8C2212yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#393—22no—
29C3×#392—22no—
30C3×#392—22no—
31C3×#392—22no—
32C3×#380C2212yes—
33C3×#380C2212yes—
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C2×#469C2×C311yescharacteristic, complemented
39C2×#408C2×C311yescharacteristic, complemented
40#2615C2×C312yescomplemented
41#2077C2×C312yescomplemented
42#2034C2×C312yescomplemented
43#2033C2×C312yescomplemented
44#1999C2×C312yescomplemented
45#1893C2×C312yescomplemented
46#1809C2×C311yescharacteristic, complemented
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—41no—
50C2×C3×#41—41no—
51C2×C3×#41—41no—
52C2×C3×#41—41no—
53C22×C3×Q8D411yescharacteristic
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—22no—
57C22×C3×D4D412yes—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C2×C3×D8—22no—
63C2×C3×D8—41no—
64C2×C3×D8—41no—
65C2×C3×M16—41no—
66C2×C3×M16—41no—
67C2×C3×M16D414yescomplemented
68C22×C3×C8—22no—
69C22×C3×C8—41no—
70C22×C3×C8—41no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#121—24no—
75C3×#121—24no—
76C3×#121—42no—
77C3×#121—42no—
78C3×#121—42no—
79C3×#121—42no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C3×#107—41no—
90C3×#106—22no—
91C3×#106—41no—
92C3×#102—24no—
93C3×#102—42no—
94C3×#102—42no—
95C3×C22⋊C8—24no—
96C3×C22⋊C8—24no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C22×#41C22×C311yescharacteristic
101C22×D8C22×C311yescharacteristic
102C22×M16C22×C311yescharacteristic
103C23×C8C22×C311yescharacteristic
104C2×#121C22×C312yes—
105C2×#121C22×C312yes—
106C2×#121C22×C312yes—
107#469—24no—
108C2×#107C22×C311yescharacteristic
109C2×#106C22×C311yescharacteristic
110C2×#106C22×C311yescharacteristic
111C2×#102C22×C312yes—
112C2×#102C22×C312yes—
113C2×#102C22×C312yes—
114#408—24no—
115C2×C22⋊C8C22×C312yes—
116C2×C22⋊C8C22×C312yes—
117#393—22no—
118#392—22no—
119#392—22no—
120#392—22no—
121#380C22×C312yes—
122#380C22×C312yes—
123#380C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127C24×C3—22no—
128C24×C3C2×D412yes—
129C24×C3C2×D411yescharacteristic
130C3×#41—44no—
131C3×#41—44no—
132C3×#41—44no—
133C3×#41—44no—
134C2×C3×Q8—22no—
135C2×C3×Q8—41no—
136C2×C3×Q8—41no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—44no—
150C2×C3×D4—44no—
151C2×C3×D4—44no—
152C2×C3×D4—82no—
153C2×C3×D4—82no—
154C2×C3×D4—82no—
155C2×C3×D4—82no—
156C2×C3×D4—82no—
157C2×C3×D4—82no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C3×D8—44no—
177C3×M16—44no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—44no—
186C2×C3×C8C2×D412yes—
187C2×C3×C8C2×D412yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—82no—
198C3×C22⋊C4—82no—
199C3×C22⋊C4—82no—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41—41no—
206C22×Q8C3×D411yescharacteristic
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C3×D412yes—
211C22×D4C3×D411yescharacteristic
212C22×D4C23×C312yes—
213C23×C4C23×C312yes—
214C23×C4C23×C311yescharacteristic
215C2×D8—22no—
216C2×D8—41no—
217C2×D8—41no—
218C2×M16—41no—
219C2×M16—41no—
220C2×M16C3×D414yescomplemented
221C22×C8—22no—
222C22×C8—41no—
223C22×C8—41no—
224C22×C8C23×C312yes—
225C22×C8C23×C311yescharacteristic
226C22×C8C23×C311yescharacteristic
227#121—24no—
228#121—24no—
229#121—42no—
230#121—42no—
231#121—42no—
232#121—42no—
233C2×C4⋊C4C23×C312yes—
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C312yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242#107—41no—
243#106—22no—
244#106—41no—
245#102—24no—
246#102—42no—
247#102—42no—
248C22⋊C8—24no—
249C22⋊C8—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3#12112yes—
272C23×C3C22×D411yescharacteristic
273C3×Q8—44no—
274C3×D4—44no—
275C3×D4—48no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—88no—
281C3×D4—88no—
282C3×D4—88no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4#12112yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312C3×C8—22no—
313C3×C8—22no—
314C3×C8—42no—
315C24—22no—
316C24C2×C3×D412yes—
317C24C2×C3×D411yescharacteristic
318#41—44no—
319#41—44no—
320#41—44no—
321#41—44no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D412yes—
362C22×C4C3×#4112yes—
363C22×C4C24×C311yescharacteristic
364D8—44no—
365M16—44no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—44no—
374C2×C8C2×C3×D412yes—
375C2×C8C2×C3×D412yes—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22×C3—22no—
389C22×C3—24no—
390C22×C3—42no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3C2×#12111yescharacteristic
408C22×C3#53411yescharacteristic
409C22×C3#53411yescharacteristic
410C22×C3#57612yes—
411C22×C3#57712yes—
412C3×C4—22no—
413C3×C4—42no—
414C3×C4—82no—
415C3×C4—82no—
416C3×C4—82no—
417C3×C4#53411yescharacteristic
418C3×C4#53511yescharacteristic
419C3×C4#53811yescharacteristic
420C3×C4#53811yescharacteristic
421C23—22no—
422C23—22no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23C3×#12112yes—
440C23C22×C3×D411yescharacteristic
441Q8—44no—
442D4—44no—
443D4—48no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—88no—
449D4—88no—
450D4—88no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4C3×#12112yes—
474C2×C4C22×C3×D411yescharacteristic
475C2×C4C2×C3×#4111yescharacteristic
476C2×C4C3×H(5,𝔽2)11yescharacteristic
477C2×C4C3×H(5,𝔽2)11yescharacteristic
478C2×C4C3×H(5,𝔽2)11yescharacteristic
479C2×C4C3×H(5,𝔽2)11yescharacteristic
480C8—22no—
481C8—22no—
482C8—42no—
483C2×C3—42no—
484C2×C3—82no—
485C2×C3—162no—
486C2×C3#261711yescharacteristic
487C2×C3#306212yes—
488C2×C3#319812yes—
489C2×C3#319911yescharacteristic
490C2×C3#320411yescharacteristic
491C22—22no—
492C22—24no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22C2×C3×#12111yescharacteristic
511C22C3×#53411yescharacteristic
512C22C3×#53411yescharacteristic
513C22C3×#57612yes—
514C22C3×#57712yes—
515C4—22no—
516C4—42no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4C3×#53411yescharacteristic
521C4C3×#53511yescharacteristic
522C4C3×#53811yescharacteristic
523C4C3×#53811yescharacteristic
524C3#2974511yescharacteristic, direct product factor, Sylow
525C2—42no—
526C2—82no—
527C2—162no—
528C2C3×#261711yescharacteristic
529C2C3×#306212yes—
530C2C3×#319812yes—
531C2C3×#319911yescharacteristic
532C2C3×#320411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#469C2semidirect
2. C2×C3×#408C2semidirect
3. C3×#2615C2semidirect
4. C3×#2077C2semidirect
5. C3×#2034C2semidirect
6. C3×#2033C2semidirect
7. C3×#1999C2semidirect
8. C3×#1893C2semidirect
9. C3×#1809C2semidirect
10. #29745C3direct (N × H)
38. C2×#469C2×C3semidirect
39. C2×#408C2×C3semidirect
40. #2615C2×C3semidirect
41. #2077C2×C3semidirect
42. #2034C2×C3semidirect
43. #2033C2×C3semidirect
44. #1999C2×C3semidirect
45. #1893C2×C3semidirect
46. #1809C2×C3semidirect
67. C2×C3×M16D4semidirect
220. C2×M16C3×D4semidirect
524. C3#29745direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4692
C2×C3×#4082
C3×#26152
C3×#20772
C3×#20342
C3×#20332
C3×#19992
C3×#18932
C3×#18092
#297453