Finite Group Databasebeta

← C2×C3×#1521 #10636173

Subgroups — C2×C3×#1521 #10636173

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#431C211yesmaximal, characteristic, complemented
3C2×C3×#383C211yesmaximal, characteristic, complemented
4C3×#1521C218yesmaximal, direct product factor
5C2×C3×#328C212yesmaximal, complemented
6C2×C3×#327C212yesmaximal
7C2×#1521C311yesmaximal, characteristic, direct product factor, Sylow
8C3×#529—22no—
9C3×#529—22no—
10C3×#529C414yescomplemented
11C3×#529C2214yes—
12C2×C3×#126—21no—
13C2×C3×#125C411yescharacteristic, complemented
14C2×C3×#124C411yescharacteristic, complemented
15C2×C3×#123—21no—
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×Q8C411yescharacteristic, complemented
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C411yescharacteristic, complemented
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#431—22no—
24C3×#431C2214yescomplemented
25C2×C3×C4⋊C8—21no—
26C2×C3×C22⋊C8C2212yes—
27C2×C3×C4×C8—21no—
28C3×#383—22no—
29C3×#383C2214yescomplemented
30C2×C3×#95—21no—
31C2×C3×#95—21no—
32C2×C3×#95C2212yes—
33C3×#328C2218yescomplemented
34C3×#327C2218yes—
35C2×#529C2×C311yescharacteristic
36C2×#431C2×C311yescharacteristic, complemented
37C2×#383C2×C311yescharacteristic, complemented
38#1521C2×C318yesdirect product factor
39C2×#328C2×C312yescomplemented
40C2×#327C2×C312yes—
41C22×C3×Q8C2×C411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4C2×C411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4—22no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×C3×C8—22no—
50C3×#126—24no—
51C3×#126—24no—
52C3×#125—22no—
53C3×#125C2×C414yescomplemented
54C3×#124—22no—
55C3×#124C2×C414yescomplemented
56C3×#123—24no—
57C3×#123—24no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8C2×C412yes—
61C3×C22⋊Q8C2312yes—
62C3×#121—21no—
63C3×#121—21no—
64C3×#121C2×C412yes—
65C3×#121C2312yes—
66C3×C4×Q8—22no—
67C3×C4×Q8C2×C414yescomplemented
68C3×C4×D4—24no—
69C3×C4×D4—24no—
70C3×C4×D4—22no—
71C3×C4×D4C2×C414yescomplemented
72C3×#117—24no—
73C3×#117—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2×C411yescharacteristic
79C2×C3×C4⋊C4C2×C411yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—21no—
86C2×C3×C42C2×C411yescharacteristic
87C2×C3×C42C2×C411yescharacteristic
88C2×C3×C42C2×C412yes—
89C2×C3×C42D412yes—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D412yes—
93C2×C3×C42C2312yes—
94C3×C4⋊C8—24no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8C2314yes—
97C3×C4×C8—24no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—22no—
101C3×#95C2314yes—
102#529—22no—
103#529—22no—
104#529C3×C414yescomplemented
105#529C22×C314yes—
106C2×#126—21no—
107C2×#125C3×C411yescharacteristic, complemented
108C2×#124C3×C411yescharacteristic, complemented
109C2×#123—21no—
110C2×C22⋊Q8C22×C311yescharacteristic
111C2×#121C22×C311yescharacteristic
112C2×C4×Q8C3×C411yescharacteristic, complemented
113C2×C4×D4—21no—
114C2×C4×D4C3×C411yescharacteristic, complemented
115C2×#117—21no—
116C22×C42C22×C311yescharacteristic
117#431—22no—
118#431C22×C314yescomplemented
119C2×C4⋊C8—21no—
120C2×C22⋊C8C22×C312yes—
121C2×C4×C8—21no—
122#383—22no—
123#383C22×C314yescomplemented
124C2×#95—21no—
125C2×#95—21no—
126C2×#95C22×C312yes—
127#328C22×C318yescomplemented
128#327C22×C318yes—
129C24×C3—21no—
130C24×C3C2×D411yescharacteristic
131C2×C3×Q8—21no—
132C2×C3×Q8—42no—
133C2×C3×Q8C22×C412yes—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—44no—
139C2×C3×D4C22×C412yes—
140C22×C3×C4—24no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4—44no—
151C22×C3×C4—44no—
152C22×C3×C4C22⋊C412yes—
153C22×C3×C4C22⋊C411yescharacteristic
154C22×C3×C4C22⋊C411yescharacteristic
155C22×C3×C4C22⋊C411yescharacteristic
156C22×C3×C4C22⋊C411yescharacteristic
157C22×C3×C4C22⋊C412yes—
158C22×C3×C4C22×C411yescharacteristic
159C22×C3×C4C22×C411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×C3×C8—24no—
167C2×C3×C8—44no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4C22×C412yes—
176C3×C4⋊C4C22×C412yes—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—22no—
184C3×C42—24no—
185C3×C42C22⋊C412yes—
186C3×C42C22⋊C412yes—
187C3×C42C22⋊C412yes—
188C3×C42C22⋊C412yes—
189C3×C42C22×C412yes—
190C3×C42C22×C412yes—
191C3×C42C2×D412yes—
192C3×C42C2×D412yes—
193C22×Q8C2×C3×C411yescharacteristic
194C22×D4—21no—
195C22×D4C2×C3×C411yescharacteristic
196C23×C4—21no—
197C23×C4—22no—
198C23×C4C3×D411yescharacteristic
199C23×C4C3×D411yescharacteristic
200C23×C4C23×C311yescharacteristic
201C22×C8—22no—
202#126—24no—
203#126—24no—
204#125—22no—
205#125C2×C3×C414yescomplemented
206#124—22no—
207#124C2×C3×C414yescomplemented
208#123—24no—
209#123—24no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8C2×C3×C412yes—
213C22⋊Q8C23×C312yes—
214#121—21no—
215#121—21no—
216#121C2×C3×C412yes—
217#121C23×C312yes—
218C4×Q8—22no—
219C4×Q8C2×C3×C414yescomplemented
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4C2×C3×C414yescomplemented
224#117—24no—
225#117—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×C3×C411yescharacteristic
231C2×C4⋊C4C2×C3×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42C2×C3×C411yescharacteristic
239C2×C42C2×C3×C411yescharacteristic
240C2×C42C2×C3×C412yes—
241C2×C42C3×D412yes—
242C2×C42C3×D411yescharacteristic
243C2×C42C3×D411yescharacteristic
244C2×C42C3×D412yes—
245C2×C42C23×C312yes—
246C4⋊C8—24no—
247C22⋊C8—22no—
248C22⋊C8C23×C314yes—
249C4×C8—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95C23×C314yes—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3C2×C22⋊C411yescharacteristic
261C23×C3C2×C22⋊C411yescharacteristic
262C23×C3#13514yes—
263C23×C3C22×D411yescharacteristic
264C3×Q8—42no—
265C3×Q8—42no—
266C3×D4—42no—
267C3×D4—42no—
268C3×D4—44no—
269C3×D4—44no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—48no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—44no—
294C2×C3×C4—88no—
295C2×C3×C4C2×C22⋊C412yes—
296C2×C3×C4C2×C22⋊C411yescharacteristic
297C2×C3×C4C2×C22⋊C411yescharacteristic
298C2×C3×C4C2×C22⋊C412yes—
299C2×C3×C4C2×C22⋊C412yes—
300C2×C3×C4C2×C22⋊C412yes—
301C2×C3×C4C23×C411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C3×C8—44no—
304C24—21no—
305C24C2×C3×D411yescharacteristic
306C2×Q8—21no—
307C2×Q8—42no—
308C2×Q8C22×C3×C412yes—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4C22×C3×C412yes—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4C3×C22⋊C411yescharacteristic
328C22×C4C3×C22⋊C411yescharacteristic
329C22×C4C3×C22⋊C411yescharacteristic
330C22×C4C3×C22⋊C411yescharacteristic
331C22×C4C3×C22⋊C412yes—
332C22×C4C3×C22⋊C412yes—
333C22×C4C22×C3×C411yescharacteristic
334C22×C4C22×C3×C411yescharacteristic
335C22×C4C2×C3×D412yes—
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D412yes—
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C24×C311yescharacteristic
341C2×C8—24no—
342C2×C8—44no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4C22×C3×C412yes—
351C4⋊C4C22×C3×C412yes—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—22no—
359C42—24no—
360C42C3×C22⋊C412yes—
361C42C3×C22⋊C412yes—
362C42C3×C22⋊C412yes—
363C42C3×C22⋊C412yes—
364C42C22×C3×C412yes—
365C42C22×C3×C412yes—
366C42C2×C3×D412yes—
367C42C2×C3×D412yes—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—28no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—84no—
377C22×C3#41012yes—
378C22×C3#41612yes—
379C22×C3#41612yes—
380C22×C3#42112yes—
381C22×C3C22×C22⋊C411yescharacteristic
382C22×C3C2×#13512yes—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—84no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23C2×C3×C22⋊C411yescharacteristic
402C23C2×C3×C22⋊C411yescharacteristic
403C23C3×#13514yes—
404C23C22×C3×D411yescharacteristic
405Q8—42no—
406Q8—42no—
407D4—42no—
408D4—42no—
409D4—44no—
410D4—44no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4C2×C3×C22⋊C412yes—
437C2×C4C2×C3×C22⋊C411yescharacteristic
438C2×C4C2×C3×C22⋊C411yescharacteristic
439C2×C4C2×C3×C22⋊C412yes—
440C2×C4C2×C3×C22⋊C412yes—
441C2×C4C2×C3×C22⋊C412yes—
442C2×C4C23×C3×C411yescharacteristic
443C2×C4C22×C3×D411yescharacteristic
444C8—44no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—82no—
448C2×C3#152114yesdirect product factor
449C2×C3C2×#41011yescharacteristic
450C2×C3C2×#41611yescharacteristic
451C2×C3C2×#42111yescharacteristic
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—28no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—84no—
461C22C3×#41012yes—
462C22C3×#41612yes—
463C22C3×#41612yes—
464C22C3×#42112yes—
465C22C22×C3×C22⋊C411yescharacteristic
466C22C2×C3×#13512yes—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—84no—
479C3C2×#152111yescharacteristic, direct product factor, Sylow
480C2—22no—
481C2—22no—
482C2—82no—
483C2C3×#152114yesdirect product factor
484C2C2×C3×#41011yescharacteristic
485C2C2×C3×#41611yescharacteristic
486C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#431C2semidirect
3. C2×C3×#383C2semidirect
4. C3×#1521C2direct (N × H)
5. C2×C3×#328C2semidirect
7. C2×#1521C3direct (N × H)
10. C3×#529C4semidirect
13. C2×C3×#125C4semidirect
14. C2×C3×#124C4semidirect
18. C2×C3×C4×Q8C4semidirect
20. C2×C3×C4×D4C4semidirect
24. C3×#431C22semidirect
29. C3×#383C22semidirect
33. C3×#328C22semidirect
36. C2×#431C2×C3semidirect
37. C2×#383C2×C3semidirect
38. #1521C2×C3direct (N × H)
39. C2×#328C2×C3semidirect
53. C3×#125C2×C4semidirect
55. C3×#124C2×C4semidirect
67. C3×C4×Q8C2×C4semidirect
71. C3×C4×D4C2×C4semidirect
104. #529C3×C4semidirect
107. C2×#125C3×C4semidirect
108. C2×#124C3×C4semidirect
112. C2×C4×Q8C3×C4semidirect
114. C2×C4×D4C3×C4semidirect
118. #431C22×C3semidirect
123. #383C22×C3semidirect
127. #328C22×C3semidirect
205. #125C2×C3×C4semidirect
207. #124C2×C3×C4semidirect
219. C4×Q8C2×C3×C4semidirect
223. C4×D4C2×C3×C4semidirect
448. C2×C3#1521direct (N × H)
479. C3C2×#1521direct (N × H)
483. C2C3×#1521direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4312
C2×C3×#3832
C3×#15212
C2×C3×#3282
C2×C3×#3272
C2×#15213