Finite Group Databasebeta

← C3×#23590 #10642197

Subgroups — C3×#23590 #10642197

Order 768.

1074 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
1C3×#3461C211yesmaximal, characteristic
2C3×C4×#137C212yesmaximal, complemented
3C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4C3×#2060C211yesmaximal, characteristic, complemented
5C3×#2039C211yesmaximal, characteristic
6C3×#2038C212yesmaximal, complemented
7C3×#2028C211yesmaximal, characteristic, complemented
8C3×#2007C211yesmaximal, characteristic
9C3×#2005C212yesmaximal, complemented
10C3×C4×#104C212yesmaximal, complemented
11C3×C4×#101C211yesmaximal, characteristic, complemented
12#23590C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#544C2211yescharacteristic
15C3×#542—22no—
16C3×#541—22no—
17C3×#539—22no—
18C3×#537C2211yescharacteristic
19C3×#536C2212yes—
20C3×#533—22no—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C3×#529—22no—
23C3×#529—22no—
24C3×C4×#41C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C3×#441—22no—
28C3×#440—22no—
29C3×C4×Q16—22no—
30C3×C4×SD16—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#419C2212yes—
35C3×#417C2212yes—
36C2×C3×#101C2212yes—
37C3×C4×M16C2212yes—
38C3×#391C2211yescharacteristic
39C3×#391C2211yescharacteristic
40C3×#389—22no—
41C3×#384C2211yescharacteristic
42C3×C4×C4⋊C4—22no—
43C3×C43C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3461C2×C311yescharacteristic
49C4×#137C2×C312yescomplemented
50C4×C4⋊Q8C2×C311yescharacteristic, complemented
51#2060C2×C311yescharacteristic, complemented
52#2039C2×C311yescharacteristic
53#2038C2×C312yescomplemented
54#2028C2×C311yescharacteristic, complemented
55#2007C2×C311yescharacteristic
56#2005C2×C312yescomplemented
57C4×#104C2×C312yescomplemented
58C4×#101C2×C311yescharacteristic, complemented
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C3×#137—24no—
63C3×#137—24no—
64C2×C3×Q16—22no—
65C2×C3×SD16—22no—
66C2×C3×M16C2312yes—
67C2×C3×M16C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×#126—24no—
73C3×#126—24no—
74C3×#125—22no—
75C3×#125—24no—
76C3×#125—22no—
77C3×#124—22no—
78C3×#124—22no—
79C3×#123—24no—
80C3×#123—24no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8D412yes—
94C3×C4×Q8D412yes—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4D412yes—
99C3×C4×D4D412yes—
100C3×#117—24no—
101C3×#117—22no—
102C3×#117C2312yes—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4D412yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C42—21no—
110C2×C3×C42—22no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42D412yes—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#106—22no—
119C3×#104—24no—
120C3×#104—24no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#102—22no—
125C3×#101—24no—
126C3×#97—22no—
127C3×C4×C8—22no—
128C3×#95—22no—
129C3×#95—22no—
130C2×#137C22×C312yes—
131#544C22×C311yescharacteristic
132#542—22no—
133#541—22no—
134#539—22no—
135#537C22×C311yescharacteristic
136#536C22×C312yes—
137#533—22no—
138C2×C4⋊Q8C22×C311yescharacteristic
139#529—22no—
140#529—22no—
141C4×#41C22×C312yes—
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×Q8C22×C311yescharacteristic
144#441—22no—
145#440—22no—
146C4×Q16—22no—
147C4×SD16—22no—
148#428C22×C312yes—
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#419C22×C312yes—
152#417C22×C312yes—
153C2×#101C22×C312yes—
154C4×M16C22×C312yes—
155#391C22×C311yescharacteristic
156#391C22×C311yescharacteristic
157#389—22no—
158#384C22×C311yescharacteristic
159C4×C4⋊C4—22no—
160C43C22×C311yescharacteristic
161#343C22×C311yescharacteristic
162#343C22×C311yescharacteristic
163#339C22×C312yes—
164#339C22×C312yes—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8#4112yes—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C3×Q16—28no—
195C3×SD16—28no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16#4114yes—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—44no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C42—24no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—21no—
230C3×C42—22no—
231C3×C42—42no—
232C3×C42—41no—
233C3×C42—41no—
234C3×C42C2×D412yes—
235C3×C42C2×D412yes—
236C3×C42#4112yes—
237C2×#41C23×C312yes—
238C22×Q8C23×C311yescharacteristic
239C22×Q8C23×C311yescharacteristic
240#137—24no—
241#137—24no—
242C2×Q16—22no—
243C2×SD16—22no—
244C2×M16C23×C312yes—
245C2×M16C23×C312yes—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250#126—24no—
251#126—24no—
252#125—22no—
253#125—24no—
254#125—22no—
255#124—22no—
256#124—22no—
257#123—24no—
258#123—24no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264#121—22no—
265#121—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8C3×D412yes—
272C4×Q8C3×D412yes—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4C3×D412yes—
277C4×D4C3×D412yes—
278#117—24no—
279#117—22no—
280#117C23×C312yes—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4C3×D412yes—
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C3×D412yes—
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#107—22no—
296#106—22no—
297#104—24no—
298#104—24no—
299#103—22no—
300#103—22no—
301#103—22no—
302#102—22no—
303#101—24no—
304#97—22no—
305C4×C8—22no—
306#95—22no—
307#95—22no—
308C23×C3—22no—
309C23×C3C22×D411yescharacteristic
310C3×Q8—24no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8—82no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—82no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—44no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—82no—
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12012yes—
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4C22×D412yes—
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4112yes—
353C2×C3×C4C2×#4111yescharacteristic
354C3×C8—24no—
355C3×C8—42no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8C2×C3×D412yes—
369C2×Q8C3×#4112yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4C2×C3×D412yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C24×C311yescharacteristic
385Q16—28no—
386SD16—28no—
387M16—22no—
388M16—22no—
389M16—22no—
390M16C3×#4114yes—
391C2×C8—22no—
392C2×C8—22no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4C2×C3×D412yes—
407C4⋊C4C2×C3×D412yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C42—24no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—21no—
421C42—22no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42C2×C3×D412yes—
426C42C2×C3×D412yes—
427C42C3×#4112yes—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3C2×#12011yescharacteristic
434C22×C3#52511yescharacteristic
435C22×C3#53211yescharacteristic
436C3×C4—24no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—81no—
447C3×C4#45612yes—
448C3×C4#52512yes—
449C3×C4#53212yes—
450C23—22no—
451C23C22×C3×D411yescharacteristic
452Q8—24no—
453Q8—22no—
454Q8—22no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—82no—
459D4—22no—
460D4—22no—
461D4—42no—
462D4—42no—
463D4—82no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—82no—
489C2×C4C3×#12011yescharacteristic
490C2×C4C3×#12012yes—
491C2×C4C3×#12011yescharacteristic
492C2×C4C22×C3×D412yes—
493C2×C4C2×C3×#4111yescharacteristic
494C2×C4C2×C3×#4112yes—
495C2×C4C2×C3×#4111yescharacteristic
496C8—24no—
497C8—42no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—82no—
501C2×C3#238811yescharacteristic
502C2×C3C2×#45611yescharacteristic
503C2×C3#301811yescharacteristic
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22C2×C3×#12011yescharacteristic
510C22C3×#52511yescharacteristic
511C22C3×#53211yescharacteristic
512C4—24no—
513C4—21no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4C3×#45612yes—
524C4C3×#52512yes—
525C4C3×#53212yes—
526C3#2359011yescharacteristic, direct product factor, Sylow
527C2—21no—
528C2—21no—
529C2—82no—
530C2C3×#238811yescharacteristic
531C2C2×C3×#45611yescharacteristic
532C2C3×#301811yescharacteristic
subgroupG/H
centerC2×C3×C4#120
commutator, Frattini, cosocleC22×C4C24×C3
socleC22×C3C2×#120
prime radicalC3×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C4×#137C2semidirect
3. C3×C4×C4⋊Q8C2semidirect
4. C3×#2060C2semidirect
6. C3×#2038C2semidirect
7. C3×#2028C2semidirect
9. C3×#2005C2semidirect
10. C3×C4×#104C2semidirect
11. C3×C4×#101C2semidirect
12. #23590C3direct (N × H)
49. C4×#137C2×C3semidirect
50. C4×C4⋊Q8C2×C3semidirect
51. #2060C2×C3semidirect
53. #2038C2×C3semidirect
54. #2028C2×C3semidirect
56. #2005C2×C3semidirect
57. C4×#104C2×C3semidirect
58. C4×#101C2×C3semidirect
526. C3#23590direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×C4×#1372
C3×C4×C4⋊Q82
C3×#20602
C3×#20392
C3×#20382
C3×#20282
C3×#20072
C3×#20052
C3×C4×#1042
C3×C4×#1012
#235903