Finite Group Databasebeta

← C2×C3×#3100 #10679134

Subgroups — C2×C3×#3100 #10679134

Order 768.

2850 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C22×C3×C4×D4C211yesmaximal, characteristic, complemented
3C3×#3100C2116yesmaximal, direct product factor
4C2×C3×#480C212yesmaximal, complemented
5C2×C3×#474C212yesmaximal, complemented
6C2×C3×#459C212yesmaximal, complemented
7C2×C3×#447C212yesmaximal, complemented
8C2×C3×C4×D8C214yesmaximal, complemented
9C2×C3×#431C211yesmaximal, characteristic, complemented
10C2×#3100C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C23×C3×D4C2211yescharacteristic
13C24×C3×C4—21no—
14C22×C3×D8C2214yes—
15C3×#532—24no—
16C3×#532C2218yescomplemented
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C2×C3×#127C2211yescharacteristic, complemented
19C2×C3×#124—21no—
20C2×C3×#121—21no—
21C2×C3×#121C2212yes—
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4—24no—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2212yes—
27C2×C3×C4×D4C2212yes—
28C2×C3×C4×D4C2218yescomplemented
29C2×C3×C4×D4C2212yescomplemented
30C22×C3×C4⋊C4C2211yescharacteristic
31C22×C3×C22⋊C4—21no—
32C22×C3×C42C2211yescharacteristic
33C3×#480C22116yescomplemented
34C3×#474C22116yescomplemented
35C3×#459C22116yescomplemented
36C3×#447C22116yescomplemented
37C3×C4×D8C22132yescomplemented
38C3×#431C2218yescomplemented
39C2×C3×#107C2214yescomplemented
40C2×C3×C4⋊C8C2212yescomplemented
41C2×C3×#102C2214yes—
42C2×C3×#102C2214yescomplemented
43C2×C3×C22⋊C8C2212yes—
44C2×C3×C4×C8C2212yescomplemented
45C2×#532C2×C311yescharacteristic, complemented
46C22×C4×D4C2×C311yescharacteristic, complemented
47#3100C2×C3116yesdirect product factor
48C2×#480C2×C312yescomplemented
49C2×#474C2×C312yescomplemented
50C2×#459C2×C312yescomplemented
51C2×#447C2×C312yescomplemented
52C2×C4×D8C2×C314yescomplemented
53C2×#431C2×C311yescharacteristic, complemented
54C25×C3—21no—
55C2×C3×#41—28no—
56C2×C3×#41—48no—
57C22×C3×Q8—21no—
58C22×C3×D4—28no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2314yes—
66C22×C3×D4C2312yes—
67C23×C3×C4—21no—
68C23×C3×C4—22no—
69C23×C3×C4—28no—
70C23×C3×C4—22no—
71C23×C3×C4—22no—
72C23×C3×C4—48no—
73C23×C3×C4D411yescharacteristic
74C23×C3×C4D411yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C2×C3×D8—216no—
77C2×C3×D8C23116yes—
78C22×C3×C8C2314yes—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8C2314yescomplemented
81C3×#127—22no—
82C3×#127C2314yescomplemented
83C3×#124—28no—
84C3×#121—28no—
85C3×#121—24no—
86C3×#121C2318yes—
87C3×C4×D4—24no—
88C3×C4×D4—24no—
89C3×C4×D4—28no—
90C3×C4×D4—28no—
91C3×C4×D4—28no—
92C3×C4×D4—24no—
93C3×C4×D4C2318yes—
94C3×C4×D4C2318yes—
95C3×C4×D4C2318yescomplemented
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—28no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D414yes—
109C2×C3×C42D411yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2314yes—
112C2×C3×C42C2311yescharacteristic
113C3×#107C23116yescomplemented
114C3×C4⋊C8C2318yescomplemented
115C3×#102C23116yes—
116C3×#102C23116yescomplemented
117C3×C22⋊C8C2318yes—
118C3×C4×C8C2318yescomplemented
119C22×#41—21no—
120C23×D4C22×C311yescharacteristic
121C24×C4—21no—
122C22×D8C22×C314yes—
123#532—24no—
124#532C22×C318yescomplemented
125C2×C4⋊Q8C22×C311yescharacteristic, complemented
126C2×#127C22×C311yescharacteristic, complemented
127C2×#124—21no—
128C2×#121—21no—
129C2×#121C22×C312yes—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—24no—
133C2×C4×D4—21no—
134C2×C4×D4C22×C312yes—
135C2×C4×D4C22×C312yes—
136C2×C4×D4C22×C318yescomplemented
137C2×C4×D4C22×C312yescomplemented
138C22×C4⋊C4C22×C311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C42C22×C311yescharacteristic
141#480C22×C3116yescomplemented
142#474C22×C3116yescomplemented
143#459C22×C3116yescomplemented
144#447C22×C3116yescomplemented
145C4×D8C22×C3132yescomplemented
146#431C22×C318yescomplemented
147C2×#107C22×C314yescomplemented
148C2×C4⋊C8C22×C312yescomplemented
149C2×#102C22×C314yes—
150C2×#102C22×C314yescomplemented
151C2×C22⋊C8C22×C312yes—
152C2×C4×C8C22×C312yescomplemented
153C24×C3—28no—
154C24×C3—24no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—44no—
158C24×C3—44no—
159C24×C3C2×D411yescharacteristic
160C3×#41—432no—
161C2×C3×Q8—24no—
162C2×C3×Q8—44no—
163C2×C3×D4—28no—
164C2×C3×D4—216no—
165C2×C3×D4—216no—
166C2×C3×D4—28no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—24no—
173C2×C3×D4—28no—
174C2×C3×D4—44no—
175C2×C3×D4—48no—
176C2×C3×D4—44no—
177C2×C3×D4#4118yescomplemented
178C2×C3×D4C2414yes—
179C2×C3×D4C2414yes—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—28no—
185C22×C3×C4—28no—
186C22×C3×C4—28no—
187C22×C3×C4—42no—
188C22×C3×C4—44no—
189C22×C3×C4—48no—
190C22×C3×C4—48no—
191C22×C3×C4—48no—
192C22×C3×C4—432no—
193C22×C3×C4D818yescomplemented
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2412yes—
203C22×C3×C4C2411yescharacteristic
204C3×D8—232no—
205C2×C3×C8—28no—
206C2×C3×C8C2418yes—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4C2414yes—
212C3×C4⋊C4C2414yes—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—28no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—28no—
217C3×C42C2×D414yes—
218C3×C42C2×D412yes—
219C3×C42C2×D414yes—
220C3×C42C2×D412yes—
221C3×C42C2412yes—
222C3×C42C2412yes—
223C25—21no—
224C2×#41—28no—
225C2×#41—48no—
226C22×Q8—21no—
227C22×D4—28no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C23×C314yes—
234C22×D4C23×C312yes—
235C22×D4C23×C312yes—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—28no—
240C23×C4—22no—
241C23×C4—48no—
242C23×C4C3×D411yescharacteristic
243C23×C4C3×D411yescharacteristic
244C23×C4C23×C311yescharacteristic
245C2×D8—216no—
246C2×D8C23×C3116yes—
247C22×C8C23×C314yes—
248C4⋊Q8—22no—
249C4⋊Q8C23×C314yescomplemented
250#127—22no—
251#127C23×C314yescomplemented
252#124—28no—
253#121—28no—
254#121—24no—
255#121C23×C318yes—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—24no—
262C4×D4C23×C318yes—
263C4×D4C23×C318yes—
264C4×D4C23×C318yescomplemented
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C314yes—
270C2×C4⋊C4C23×C312yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C42C3×D411yescharacteristic
277C2×C42C3×D411yescharacteristic
278C2×C42C3×D414yes—
279C2×C42C23×C311yescharacteristic
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C314yes—
282#107C23×C3116yescomplemented
283C4⋊C8C23×C318yescomplemented
284#102C23×C3116yes—
285#102C23×C3116yescomplemented
286C22⋊C8C23×C318yes—
287C4×C8C23×C318yescomplemented
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—216no—
291C23×C3—28no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—432no—
295C23×C3—48no—
296C23×C3—48no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3C2×D814yes—
303C23×C3C22×D412yes—
304C23×C3C22×D411yescharacteristic
305C3×Q8—48no—
306C3×D4—216no—
307C3×D4—28no—
308C3×D4—24no—
309C3×D4—28no—
310C3×D4—48no—
311C3×D4—416no—
312C3×D4—44no—
313C3×D4—48no—
314C3×D4C2×#4118yescomplemented
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—44no—
323C2×C3×C4—48no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—416no—
327C2×C3×C4—416no—
328C2×C3×C4—416no—
329C2×C3×C4C2×D814yes—
330C2×C3×C4C2×D814yes—
331C2×C3×C4C2×D8116yescomplemented
332C2×C3×C4C22×D412yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C2×#4114yes—
339C2×C3×C4C2511yescharacteristic
340C3×C8—28no—
341C24—28no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—44no—
346C24—44no—
347C24C2×C3×D411yescharacteristic
348#41—432no—
349C2×Q8—24no—
350C2×Q8—44no—
351C2×D4—28no—
352C2×D4—216no—
353C2×D4—216no—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—28no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4C3×#4118yescomplemented
366C2×D4C24×C314yes—
367C2×D4C24×C314yes—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—432no—
381C22×C4C3×D818yescomplemented
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D414yes—
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D414yes—
390C22×C4C24×C312yes—
391C22×C4C24×C311yescharacteristic
392D8—232no—
393C2×C8—28no—
394C2×C8C24×C318yes—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4C24×C314yes—
400C4⋊C4C24×C314yes—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C42C2×C3×D414yes—
406C42C2×C3×D412yes—
407C42C2×C3×D414yes—
408C42C2×C3×D412yes—
409C42C24×C312yes—
410C42C24×C312yes—
411C22×C3—28no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—28no—
415C22×C3—432no—
416C22×C3—416no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—88no—
423C22×C3#52512yes—
424C22×C3C22×D814yes—
425C22×C3C22×D812yes—
426C22×C3#57512yes—
427C22×C3C23×D411yescharacteristic
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—44no—
432C3×C4—44no—
433C3×C4#52512yes—
434C3×C4C22×D814yes—
435C3×C4C22×D814yes—
436C3×C4C22×#4112yes—
437C23—22no—
438C23—22no—
439C23—216no—
440C23—24no—
441C23—28no—
442C23—24no—
443C23—48no—
444C23—432no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23C2×C3×D814yes—
452C23C22×C3×D412yes—
453C23C22×C3×D411yescharacteristic
454Q8—48no—
455D4—216no—
456D4—24no—
457D4—28no—
458D4—28no—
459D4—48no—
460D4—416no—
461D4—48no—
462D4—44no—
463D4C2×C3×#4118yescomplemented
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—44no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—416no—
476C2×C4—416no—
477C2×C4—416no—
478C2×C4C2×C3×D814yes—
479C2×C4C2×C3×D814yes—
480C2×C4C2×C3×D8116yescomplemented
481C2×C4C22×C3×D412yes—
482C2×C4C22×C3×D412yes—
483C2×C4C22×C3×D412yes—
484C2×C4C22×C3×D412yes—
485C2×C4C22×C3×D412yes—
486C2×C4C22×C3×D411yescharacteristic
487C2×C4C2×C3×#4114yes—
488C2×C4C25×C311yescharacteristic
489C8—28no—
490C2×C3—24no—
491C2×C3—48no—
492C2×C3—84no—
493C2×C3#310014yesdirect product factor
494C2×C3C2×#52511yescharacteristic
495C2×C3C23×D811yescharacteristic
496C2×C3C2×#57511yescharacteristic
497C22—22no—
498C22—22no—
499C22—28no—
500C22—28no—
501C22—432no—
502C22—44no—
503C22—44no—
504C22—416no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—88no—
509C22C3×#52512yes—
510C22C22×C3×D814yes—
511C22C22×C3×D812yes—
512C22C3×#57512yes—
513C22C23×C3×D411yescharacteristic
514C4—22no—
515C4—22no—
516C4—22no—
517C4—44no—
518C4—44no—
519C4C3×#52512yes—
520C4C22×C3×D814yes—
521C4C22×C3×D814yes—
522C4C22×C3×#4112yes—
523C3C2×#310011yescharacteristic, direct product factor, Sylow
524C2—24no—
525C2—48no—
526C2—84no—
527C2C3×#310014yesdirect product factor
528C2C2×C3×#52511yescharacteristic
529C2C23×C3×D811yescharacteristic
530C2C2×C3×#57511yescharacteristic
subgroupG/H
centerC22×C3×C4C2×D4
commutator, Frattini, cosocleC2×C4C25×C3
socleC23×C3C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C22×C3×C4×D4C2semidirect
3. C3×#3100C2direct (N × H)
4. C2×C3×#480C2semidirect
5. C2×C3×#474C2semidirect
6. C2×C3×#459C2semidirect
7. C2×C3×#447C2semidirect
8. C2×C3×C4×D8C2semidirect
9. C2×C3×#431C2semidirect
10. C2×#3100C3direct (N × H)
16. C3×#532C22semidirect
17. C2×C3×C4⋊Q8C22semidirect
18. C2×C3×#127C22semidirect
28. C2×C3×C4×D4C22semidirect
29. C2×C3×C4×D4C22semidirect
33. C3×#480C22semidirect
34. C3×#474C22semidirect
35. C3×#459C22semidirect
36. C3×#447C22semidirect
37. C3×C4×D8C22semidirect
38. C3×#431C22semidirect
39. C2×C3×#107C22semidirect
40. C2×C3×C4⋊C8C22semidirect
42. C2×C3×#102C22semidirect
44. C2×C3×C4×C8C22semidirect
45. C2×#532C2×C3semidirect
46. C22×C4×D4C2×C3semidirect
47. #3100C2×C3direct (N × H)
48. C2×#480C2×C3semidirect
49. C2×#474C2×C3semidirect
50. C2×#459C2×C3semidirect
51. C2×#447C2×C3semidirect
52. C2×C4×D8C2×C3semidirect
53. C2×#431C2×C3semidirect
80. C3×C4⋊Q8C23semidirect
82. C3×#127C23semidirect
95. C3×C4×D4C23semidirect
113. C3×#107C23semidirect
114. C3×C4⋊C8C23semidirect
116. C3×#102C23semidirect
118. C3×C4×C8C23semidirect
124. #532C22×C3semidirect
125. C2×C4⋊Q8C22×C3semidirect
126. C2×#127C22×C3semidirect
136. C2×C4×D4C22×C3semidirect
137. C2×C4×D4C22×C3semidirect
141. #480C22×C3semidirect
142. #474C22×C3semidirect
143. #459C22×C3semidirect
144. #447C22×C3semidirect
145. C4×D8C22×C3semidirect
146. #431C22×C3semidirect
147. C2×#107C22×C3semidirect
148. C2×C4⋊C8C22×C3semidirect
150. C2×#102C22×C3semidirect
152. C2×C4×C8C22×C3semidirect
177. C2×C3×D4#41semidirect
193. C22×C3×C4D8semidirect
249. C4⋊Q8C23×C3semidirect
251. #127C23×C3semidirect
264. C4×D4C23×C3semidirect
282. #107C23×C3semidirect
283. C4⋊C8C23×C3semidirect
285. #102C23×C3semidirect
287. C4×C8C23×C3semidirect
314. C3×D4C2×#41semidirect
331. C2×C3×C4C2×D8semidirect
365. C2×D4C3×#41semidirect
381. C22×C4C3×D8semidirect
463. D4C2×C3×#41semidirect
480. C2×C4C2×C3×D8semidirect
493. C2×C3#3100direct (N × H)
523. C3C2×#3100direct (N × H)
527. C2C3×#3100direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C22×C3×C4×D42
C3×#31002
C2×C3×#4802
C2×C3×#4742
C2×C3×#4592
C2×C3×#4472
C2×C3×C4×D82
C2×C3×#4312
C2×#31003