Finite Group Databasebeta

← C2×C3×#1511 #10636159

Subgroups — C2×C3×#1511 #10636159

Order 768.

1446 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#407C211yesmaximal, characteristic
3C2×C3×#381C211yesmaximal, characteristic
4C3×#1511C218yesmaximal, direct product factor
5C2×C3×#328C214yesmaximal, complemented
6C2×#1511C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×Q8C411yescharacteristic, complemented
8C24×C3×C4C2211yescharacteristic
9C22×C3×M16—21no—
10C2×C3×C22⋊Q8—21no—
11C2×C3×C22⋊Q8—22no—
12C2×C3×C22⋊Q8—21no—
13C2×C3×C22⋊Q8—22no—
14C2×C3×C22⋊Q8C412yescomplemented
15C2×C3×C22⋊Q8C414yescomplemented
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C22⋊Q8C2214yes—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C411yescharacteristic, complemented
20C22×C3×C22⋊C4—21no—
21C22×C3×C22⋊C4—21no—
22C3×#407—22no—
23C3×#407C2214yes—
24C2×C3×C22⋊C8—21no—
25C2×C3×C22⋊C8C2212yes—
26C3×#381—22no—
27C3×#381C2214yes—
28C2×C3×#95—21no—
29C2×C3×#95C2212yes—
30C3×#328C22116yescomplemented
31C22×C22⋊Q8C2×C311yescharacteristic
32C2×#407C2×C311yescharacteristic
33C2×#381C2×C311yescharacteristic
34#1511C2×C318yesdirect product factor
35C2×#328C2×C314yescomplemented
36C25×C3D411yescharacteristic
37C22×C3×Q8—22no—
38C22×C3×Q8—21no—
39C22×C3×Q8—44no—
40C22×C3×Q8C2×C412yes—
41C22×C3×Q8C2×C414yescomplemented
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C23×C3×C4—22no—
45C23×C3×C4—44no—
46C23×C3×C4C2×C411yescharacteristic
47C23×C3×C4C2×C412yes—
48C23×C3×C4C2×C411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D412yes—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D412yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C2×C3×M16—24no—
57C2×C3×M16—44no—
58C22×C3×C8—22no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—24no—
66C3×C22⋊Q8C2×C414yes—
67C3×C22⋊Q8C2×C418yescomplemented
68C3×C22⋊Q8C2314yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4C2×C412yes—
76C2×C3×C4⋊C4C2×C414yescomplemented
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—44no—
83C3×C22⋊C8—24no—
84C3×C22⋊C8—22no—
85C3×C22⋊C8C2314yes—
86C3×#95—24no—
87C3×#95—22no—
88C3×#95C2314yes—
89C23×Q8C3×C411yescharacteristic, complemented
90C24×C4C22×C311yescharacteristic
91C22×M16—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8C3×C412yescomplemented
97C2×C22⋊Q8C3×C414yescomplemented
98C2×C22⋊Q8C22×C312yes—
99C2×C22⋊Q8C22×C314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C3×C411yescharacteristic, complemented
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104#407—22no—
105#407C22×C314yes—
106C2×C22⋊C8—21no—
107C2×C22⋊C8C22×C312yes—
108#381—22no—
109#381C22×C314yes—
110C2×#95—21no—
111C2×#95C22×C312yes—
112#328C22×C3116yescomplemented
113C24×C3—24no—
114C24×C3—24no—
115C24×C3—42no—
116C24×C3C22⋊C411yescharacteristic
117C24×C3C22⋊C412yes—
118C24×C3C22⋊C411yescharacteristic
119C24×C3C2×D412yes—
120C24×C3C2×D411yescharacteristic
121C2×C3×Q8—22no—
122C2×C3×Q8—24no—
123C2×C3×Q8—48no—
124C2×C3×Q8—44no—
125C2×C3×Q8—44no—
126C2×C3×Q8—48no—
127C2×C3×Q8C22×C414yes—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—44no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—44no—
139C22×C3×C4—42no—
140C22×C3×C4—44no—
141C22×C3×C4—42no—
142C22×C3×C4—44no—
143C22×C3×C4—44no—
144C22×C3×C4—44no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4C22⋊C412yes—
149C22×C3×C4C22⋊C412yes—
150C22×C3×C4C22⋊C411yescharacteristic
151C22×C3×C4C22⋊C411yescharacteristic
152C22×C3×C4C22⋊C412yes—
153C22×C3×C4C22⋊C412yes—
154C22×C3×C4C22⋊C412yes—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C22×C411yescharacteristic
157C22×C3×C4C22×C411yescharacteristic
158C22×C3×C4C22×C412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C3×M16—48no—
167C2×C3×C8—24no—
168C2×C3×C8—44no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4C22×C414yes—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—28no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—48no—
179C25C3×D411yescharacteristic
180C22×Q8—21no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8C2×C3×C412yes—
184C22×Q8C2×C3×C414yescomplemented
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—44no—
189C23×C4C2×C3×C411yescharacteristic
190C23×C4C2×C3×C411yescharacteristic
191C23×C4C2×C3×C412yes—
192C23×C4C3×D411yescharacteristic
193C23×C4C3×D411yescharacteristic
194C23×C4C3×D412yes—
195C23×C4C3×D411yescharacteristic
196C23×C4C3×D412yes—
197C23×C4C23×C311yescharacteristic
198C23×C4C23×C312yes—
199C2×M16—24no—
200C2×M16—44no—
201C22×C8—22no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8—22no—
206C22⋊Q8—24no—
207C22⋊Q8—22no—
208C22⋊Q8—24no—
209C22⋊Q8C2×C3×C414yes—
210C22⋊Q8C2×C3×C418yescomplemented
211C22⋊Q8C23×C314yes—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4C2×C3×C412yes—
219C2×C4⋊C4C2×C3×C414yescomplemented
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—44no—
226C22⋊C8—24no—
227C22⋊C8—22no—
228C22⋊C8C23×C314yes—
229#95—24no—
230#95—22no—
231#95C23×C314yes—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—416no—
246C23×C3#9914yes—
247C23×C3C2×C22⋊C412yes—
248C23×C3C2×C22⋊C412yes—
249C23×C3C2×C22⋊C411yescharacteristic
250C23×C3C2×C22⋊C411yescharacteristic
251C23×C3#13714yes—
252C23×C3C22×D411yescharacteristic
253C3×Q8—44no—
254C3×Q8—44no—
255C3×Q8—48no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—48no—
277C2×C3×C4—48no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—88no—
284C2×C3×C4C2×C22⋊C411yescharacteristic
285C2×C3×C4C2×C22⋊C412yes—
286C2×C3×C4C2×C22⋊C411yescharacteristic
287C2×C3×C4C2×C22⋊C412yes—
288C2×C3×C4C23×C411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C3×C8—44no—
291C24—24no—
292C24—24no—
293C24—42no—
294C24C3×C22⋊C411yescharacteristic
295C24C3×C22⋊C412yes—
296C24C3×C22⋊C411yescharacteristic
297C24C2×C3×D412yes—
298C24C2×C3×D411yescharacteristic
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—48no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—48no—
305C2×Q8C22×C3×C414yes—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4C3×C22⋊C412yes—
327C22×C4C3×C22⋊C411yescharacteristic
328C22×C4C3×C22⋊C412yes—
329C22×C4C3×C22⋊C411yescharacteristic
330C22×C4C3×C22⋊C412yes—
331C22×C4C3×C22⋊C412yes—
332C22×C4C3×C22⋊C412yes—
333C22×C4C22×C3×C411yescharacteristic
334C22×C4C22×C3×C412yes—
335C22×C4C22×C3×C411yescharacteristic
336C22×C4C22×C3×C412yes—
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D412yes—
339C22×C4C2×C3×D412yes—
340C22×C4C2×C3×D411yescharacteristic
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C24×C311yescharacteristic
344M16—48no—
345C2×C8—24no—
346C2×C8—44no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4C22×C3×C414yes—
353C22⋊C4—24no—
354C22⋊C4—28no—
355C22⋊C4—24no—
356C22⋊C4—48no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—28no—
364C22×C3—28no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—48no—
369C22×C3—48no—
370C22×C3—44no—
371C22×C3C2×#9914yes—
372C22×C3C2×#9912yes—
373C22×C3#41912yes—
374C22×C3#41912yes—
375C22×C3#42112yes—
376C22×C3C22×C22⋊C411yescharacteristic
377C22×C3C2×#13712yes—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4—21no—
382C3×C4—22no—
383C3×C4—21no—
384C3×C4—44no—
385C3×C4—44no—
386C3×C4—84no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—416no—
401C23C3×#9914yes—
402C23C2×C3×C22⋊C412yes—
403C23C2×C3×C22⋊C412yes—
404C23C2×C3×C22⋊C411yescharacteristic
405C23C2×C3×C22⋊C411yescharacteristic
406C23C3×#13714yes—
407C23C22×C3×D411yescharacteristic
408Q8—44no—
409Q8—44no—
410Q8—48no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—88no—
439C2×C4C2×C3×C22⋊C411yescharacteristic
440C2×C4C2×C3×C22⋊C412yes—
441C2×C4C2×C3×C22⋊C411yescharacteristic
442C2×C4C2×C3×C22⋊C412yes—
443C2×C4C23×C3×C411yescharacteristic
444C2×C4C22×C3×D411yescharacteristic
445C8—44no—
446C2×C3—22no—
447C2×C3—24no—
448C2×C3—22no—
449C2×C3—42no—
450C2×C3#151114yesdirect product factor
451C2×C3C22×#9911yescharacteristic
452C2×C3C2×#41911yescharacteristic
453C2×C3C2×#42111yescharacteristic
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—28no—
461C22—28no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—48no—
466C22—48no—
467C22—44no—
468C22C2×C3×#9914yes—
469C22C2×C3×#9912yes—
470C22C3×#41912yes—
471C22C3×#41912yes—
472C22C3×#42112yes—
473C22C22×C3×C22⋊C411yescharacteristic
474C22C2×C3×#13712yes—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—21no—
479C4—22no—
480C4—21no—
481C4—44no—
482C4—44no—
483C4—84no—
484C3C2×#151111yescharacteristic, direct product factor, Sylow
485C2—22no—
486C2—24no—
487C2—22no—
488C2—42no—
489C2C3×#151114yesdirect product factor
490C2C22×C3×#9911yescharacteristic
491C2C2×C3×#41911yescharacteristic
492C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1511C2direct (N × H)
5. C2×C3×#328C2semidirect
6. C2×#1511C3direct (N × H)
7. C23×C3×Q8C4semidirect
14. C2×C3×C22⋊Q8C4semidirect
15. C2×C3×C22⋊Q8C4semidirect
19. C22×C3×C4⋊C4C4semidirect
30. C3×#328C22semidirect
34. #1511C2×C3direct (N × H)
35. C2×#328C2×C3semidirect
41. C22×C3×Q8C2×C4semidirect
67. C3×C22⋊Q8C2×C4semidirect
76. C2×C3×C4⋊C4C2×C4semidirect
89. C23×Q8C3×C4semidirect
96. C2×C22⋊Q8C3×C4semidirect
97. C2×C22⋊Q8C3×C4semidirect
101. C22×C4⋊C4C3×C4semidirect
112. #328C22×C3semidirect
184. C22×Q8C2×C3×C4semidirect
210. C22⋊Q8C2×C3×C4semidirect
219. C2×C4⋊C4C2×C3×C4semidirect
450. C2×C3#1511direct (N × H)
484. C3C2×#1511direct (N × H)
489. C2C3×#1511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#4072
C2×C3×#3812
C3×#15112
C2×C3×#3282
C2×#15113