Finite Group Databasebeta

← C2×C3×#1793 #10639221

Subgroups — C2×C3×#1793 #10639221

Order 768.

3682 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
1C22×C3×2−1+4C211yesmaximal, characteristic
2C22×C3×#104C212yesmaximal, complemented
3C2×C3×#417C212yesmaximal, complemented
4C22×C3×#101C211yesmaximal, characteristic, complemented
5C2×C3×#391C211yesmaximal, characteristic, complemented
6C3×#1793C218yesmaximal, direct product factor
7C2×#1793C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×2−1+4—28no—
9C2×C3×2−1+4—24no—
10C2×C3×2−1+4—24no—
11C2×C3×2−1+4—22no—
12C2×C3×2−1+4C418yescomplemented
13C2×C3×2−1+4C2214yes—
14C22×C3×#41—22no—
15C22×C3×#41—22no—
16C22×C3×#41C2212yes—
17C23×C3×Q8—22no—
18C23×C3×Q8C2211yescharacteristic
19C22×C3×M16C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C2×C3×#104—28no—
23C2×C3×#104C2218yescomplemented
24C3×#417—24no—
25C3×#417C2218yescomplemented
26C2×C3×#103—22no—
27C2×C3×#102—22no—
28C2×C3×#101—24no—
29C2×C3×#101C2214yescomplemented
30C3×#391—22no—
31C3×#391C2214yescomplemented
32C2×C3×#95—22no—
33C22×2−1+4C2×C311yescharacteristic
34C22×#104C2×C312yescomplemented
35C2×#417C2×C312yescomplemented
36C22×#101C2×C311yescharacteristic, complemented
37C2×#391C2×C311yescharacteristic, complemented
38#1793C2×C318yesdirect product factor
39C3×2−1+4—232no—
40C3×2−1+4—216no—
41C3×2−1+4—232no—
42C3×2−1+4—216no—
43C3×2−1+4—216no—
44C3×2−1+4—28no—
45C3×2−1+4C2×C4116yescomplemented
46C2×C3×#41—28no—
47C2×C3×#41—216no—
48C2×C3×#41—28no—
49C2×C3×#41—216no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—216no—
53C2×C3×#41—28no—
54C2×C3×#41—24no—
55C2×C3×#41—22no—
56C2×C3×#41—24no—
57C2×C3×#41—22no—
58C2×C3×#41C2×C418yes—
59C2×C3×#41D418yes—
60C2×C3×#41D414yes—
61C2×C3×#41C2314yes—
62C22×C3×Q8—22no—
63C22×C3×Q8—28no—
64C22×C3×Q8—216no—
65C22×C3×Q8—216no—
66C22×C3×Q8—28no—
67C22×C3×Q8—22no—
68C22×C3×Q8—24no—
69C22×C3×Q8—22no—
70C22×C3×Q8C2×C414yes—
71C22×C3×Q8D412yes—
72C22×C3×Q8D414yes—
73C22×C3×Q8D412yes—
74C22×C3×Q8C2312yes—
75C22×C3×D4—22no—
76C22×C3×D4—22no—
77C22×C3×D4D412yes—
78C23×C3×C4—24no—
79C23×C3×C4—22no—
80C23×C3×C4—21no—
81C23×C3×C4—21no—
82C23×C3×C4D412yes—
83C23×C3×C4C2311yescharacteristic
84C2×C3×M16—22no—
85C2×C3×M16—24no—
86C2×C3×M16—24no—
87C2×C3×M16C2314yes—
88C22×C3×C8—22no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C42—24no—
94C2×C3×C42—21no—
95C2×C3×C42C2312yes—
96C3×#104—216no—
97C3×#103—28no—
98C3×#102—28no—
99C3×#101—28no—
100C3×#95—28no—
101C2×2−1+4—28no—
102C2×2−1+4—24no—
103C2×2−1+4—24no—
104C2×2−1+4—22no—
105C2×2−1+4C3×C418yescomplemented
106C2×2−1+4C22×C314yes—
107C22×#41—22no—
108C22×#41—22no—
109C22×#41C22×C312yes—
110C23×Q8—22no—
111C23×Q8C22×C311yescharacteristic
112C22×M16C22×C312yes—
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C42C22×C311yescharacteristic
115C2×#104—28no—
116C2×#104C22×C318yescomplemented
117#417—24no—
118#417C22×C318yescomplemented
119C2×#103—22no—
120C2×#102—22no—
121C2×#101—24no—
122C2×#101C22×C314yescomplemented
123#391—22no—
124#391C22×C314yescomplemented
125C2×#95—22no—
126C24×C3—22no—
127C24×C3C2×D411yescharacteristic
128C3×#41—232no—
129C3×#41—264no—
130C3×#41—232no—
131C3×#41—264no—
132C3×#41—232no—
133C3×#41—232no—
134C3×#41—28no—
135C3×#41—28no—
136C3×#41—28no—
137C3×#41—28no—
138C3×#41—28no—
139C3×#41—24no—
140C3×#41—24no—
141C3×#41C22⋊C4116yes—
142C3×#41C22×C418yes—
143C3×#41C2×D418yes—
144C2×C3×Q8—28no—
145C2×C3×Q8—216no—
146C2×C3×Q8—232no—
147C2×C3×Q8—264no—
148C2×C3×Q8—232no—
149C2×C3×Q8—28no—
150C2×C3×Q8—28no—
151C2×C3×Q8—22no—
152C2×C3×Q8—24no—
153C2×C3×Q8—24no—
154C2×C3×Q8—28no—
155C2×C3×Q8—24no—
156C2×C3×Q8—28no—
157C2×C3×Q8—24no—
158C2×C3×Q8—24no—
159C2×C3×Q8C22⋊C418yes—
160C2×C3×Q8C22⋊C418yes—
161C2×C3×Q8C22×C414yes—
162C2×C3×Q8C2×D414yes—
163C2×C3×Q8C2×D414yes—
164C2×C3×D4—28no—
165C2×C3×D4—216no—
166C2×C3×D4—28no—
167C2×C3×D4—24no—
168C2×C3×D4—28no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4C22⋊C418yes—
173C2×C3×D4C2×D414yes—
174C22×C3×C4—216no—
175C22×C3×C4—232no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—28no—
181C22×C3×C4—24no—
182C22×C3×C4—28no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—44no—
190C22×C3×C4—41no—
191C22×C3×C4—41no—
192C22×C3×C4C22⋊C418yes—
193C22×C3×C4C22×C412yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D414yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2411yescharacteristic
201C3×M16—24no—
202C3×M16—24no—
203C3×M16—44no—
204C2×C3×C8—24no—
205C2×C3×C8—44no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—42no—
208C3×C42—24no—
209C3×C42—42no—
2102−1+4—216no—
2112−1+4—232no—
2122−1+4—232no—
2132−1+4—216no—
2142−1+4—216no—
2152−1+4—28no—
2162−1+4C2×C3×C4116yescomplemented
217C2×#41—216no—
218C2×#41—28no—
219C2×#41—216no—
220C2×#41—28no—
221C2×#41—216no—
222C2×#41—28no—
223C2×#41—216no—
224C2×#41—28no—
225C2×#41—24no—
226C2×#41—22no—
227C2×#41—24no—
228C2×#41—22no—
229C2×#41C2×C3×C418yes—
230C2×#41C3×D418yes—
231C2×#41C3×D414yes—
232C2×#41C23×C314yes—
233C22×Q8—22no—
234C22×Q8—216no—
235C22×Q8—28no—
236C22×Q8—216no—
237C22×Q8—28no—
238C22×Q8—22no—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8C2×C3×C414yes—
242C22×Q8C3×D412yes—
243C22×Q8C3×D414yes—
244C22×Q8C3×D412yes—
245C22×Q8C23×C312yes—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4C3×D412yes—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C3×D412yes—
254C23×C4C23×C311yescharacteristic
255C2×M16—22no—
256C2×M16—24no—
257C2×M16—24no—
258C2×M16C23×C314yes—
259C22×C8—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C23×C312yes—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42C23×C312yes—
267#104—216no—
268#103—28no—
269#102—28no—
270#101—28no—
271#95—28no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—28no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3C2×C22⋊C412yes—
279C23×C3#12012yes—
280C23×C3#12011yescharacteristic
281C23×C3#12011yescharacteristic
282C23×C3C22×D411yescharacteristic
283C3×Q8—232no—
284C3×Q8—28no—
285C3×Q8—24no—
286C3×Q8—28no—
287C3×Q8—28no—
288C3×Q8—216no—
289C3×Q8C2×C22⋊C418yes—
290C3×D4—232no—
291C3×D4—28no—
292C3×D4—28no—
293C3×D4—216no—
294C3×D4—24no—
295C3×D4—28no—
296C3×D4C2×C22⋊C418yes—
297C2×C3×C4—264no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—28no—
302C2×C3×C4—28no—
303C2×C3×C4—28no—
304C2×C3×C4—28no—
305C2×C3×C4—28no—
306C2×C3×C4—216no—
307C2×C3×C4—24no—
308C2×C3×C4—48no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4—82no—
314C2×C3×C4—82no—
315C2×C3×C4C2×C22⋊C418yes—
316C2×C3×C4C2×C22⋊C414yes—
317C2×C3×C4C2×C22⋊C414yes—
318C2×C3×C4C2×C22⋊C412yes—
319C2×C3×C4#12014yes—
320C2×C3×C4#12014yes—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4C23×C411yescharacteristic
324C2×C3×C4C22×D412yes—
325C2×C3×C4C22×D412yes—
326C2×C3×C4C22×D411yescharacteristic
327C3×C8—44no—
328C24—22no—
329C24C2×C3×D411yescharacteristic
330#41—264no—
331#41—232no—
332#41—264no—
333#41—232no—
334#41—232no—
335#41—232no—
336#41—28no—
337#41—28no—
338#41—28no—
339#41—28no—
340#41—28no—
341#41—24no—
342#41—24no—
343#41C3×C22⋊C4116yes—
344#41C22×C3×C418yes—
345#41C2×C3×D418yes—
346C2×Q8—216no—
347C2×Q8—28no—
348C2×Q8—264no—
349C2×Q8—232no—
350C2×Q8—232no—
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—24no—
354C2×Q8—28no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—28no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8C3×C22⋊C418yes—
362C2×Q8C3×C22⋊C418yes—
363C2×Q8C22×C3×C414yes—
364C2×Q8C2×C3×D414yes—
365C2×Q8C2×C3×D414yes—
366C2×D4—216no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4C3×C22⋊C418yes—
375C2×D4C2×C3×D414yes—
376C22×C4—232no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4C3×C22⋊C418yes—
395C22×C4C22×C3×C412yes—
396C22×C4C2×C3×D414yes—
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D414yes—
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D412yes—
402C22×C4C24×C311yescharacteristic
403M16—24no—
404M16—24no—
405M16—44no—
406C2×C8—24no—
407C2×C8—44no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C42—24no—
411C42—42no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—28no—
419C22×C3—416no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3#37912yes—
423C22×C3#37912yes—
424C22×C3#45412yes—
425C22×C3#45612yes—
426C22×C3C22×C22⋊C411yescharacteristic
427C22×C3C2×#12011yescharacteristic
428C22×C3C2×#12011yescharacteristic
429C3×C4—28no—
430C3×C4—28no—
431C3×C4—44no—
432C3×C4—82no—
433C3×C4#37914yes—
434C3×C4C22×C22⋊C412yes—
435C3×C4C2×#12012yes—
436C23—22no—
437C23—22no—
438C23—28no—
439C23—24no—
440C23—44no—
441C23—44no—
442C23C2×C3×C22⋊C412yes—
443C23C3×#12012yes—
444C23C3×#12011yescharacteristic
445C23C3×#12011yescharacteristic
446C23C22×C3×D411yescharacteristic
447Q8—232no—
448Q8—28no—
449Q8—216no—
450Q8—28no—
451Q8—24no—
452Q8—28no—
453Q8C2×C3×C22⋊C418yes—
454D4—232no—
455D4—216no—
456D4—28no—
457D4—28no—
458D4—28no—
459D4—24no—
460D4C2×C3×C22⋊C418yes—
461C2×C4—264no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—216no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4C2×C3×C22⋊C418yes—
480C2×C4C2×C3×C22⋊C412yes—
481C2×C4C2×C3×C22⋊C414yes—
482C2×C4C2×C3×C22⋊C414yes—
483C2×C4C3×#12012yes—
484C2×C4C3×#12012yes—
485C2×C4C3×#12014yes—
486C2×C4C3×#12014yes—
487C2×C4C23×C3×C411yescharacteristic
488C2×C4C22×C3×D412yes—
489C2×C4C22×C3×D412yes—
490C2×C4C22×C3×D411yescharacteristic
491C8—44no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—22no—
495C2×C3—48no—
496C2×C3#179314yesdirect product factor
497C2×C3C2×#37911yescharacteristic
498C2×C3C2×#45411yescharacteristic
499C2×C3C2×#45611yescharacteristic
500C22—24no—
501C22—24no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—28no—
507C22—416no—
508C22—44no—
509C22—44no—
510C22C3×#37912yes—
511C22C3×#37912yes—
512C22C3×#45412yes—
513C22C3×#45612yes—
514C22C22×C3×C22⋊C411yescharacteristic
515C22C2×C3×#12011yescharacteristic
516C22C2×C3×#12011yescharacteristic
517C4—28no—
518C4—28no—
519C4—44no—
520C4—82no—
521C4C3×#37914yes—
522C4C22×C3×C22⋊C412yes—
523C4C2×C3×#12012yes—
524C3C2×#179311yescharacteristic, direct product factor, Sylow
525C2—21no—
526C2—22no—
527C2—21no—
528C2—48no—
529C2C3×#179314yesdirect product factor
530C2C2×C3×#37911yescharacteristic
531C2C2×C3×#45411yescharacteristic
532C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×2−1+4C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#104C2semidirect
3. C2×C3×#417C2semidirect
4. C22×C3×#101C2semidirect
5. C2×C3×#391C2semidirect
6. C3×#1793C2direct (N × H)
7. C2×#1793C3direct (N × H)
12. C2×C3×2−1+4C4semidirect
23. C2×C3×#104C22semidirect
25. C3×#417C22semidirect
29. C2×C3×#101C22semidirect
31. C3×#391C22semidirect
34. C22×#104C2×C3semidirect
35. C2×#417C2×C3semidirect
36. C22×#101C2×C3semidirect
37. C2×#391C2×C3semidirect
38. #1793C2×C3direct (N × H)
45. C3×2−1+4C2×C4semidirect
105. C2×2−1+4C3×C4semidirect
116. C2×#104C22×C3semidirect
118. #417C22×C3semidirect
122. C2×#101C22×C3semidirect
124. #391C22×C3semidirect
216. 2−1+4C2×C3×C4semidirect
496. C2×C3#1793direct (N × H)
524. C3C2×#1793direct (N × H)
529. C2C3×#1793direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C22×C3×#1042
C2×C3×#4172
C22×C3×#1012
C2×C3×#3912
C3×#17932
C2×#17933