Finite Group Databasebeta

← C3×#72138 #11784790

Subgroups — C3×#72138 #11784790

Order 1152.

1681 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C211yesmaximal, characteristic
2C2×C3×#837C214yesmaximal, complemented
3C3×C4×#217C211yesmaximal, characteristic, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5C3×#1789—31nomaximal
6#72138C311yesmaximal, characteristic, direct product factor
7C2×C32×#41—28no—
8C2×C32×#41—21no—
9C2×C32×#41—21no—
10C2×C32×#41C412yescomplemented
11C2×C32×#41C2212yes—
12C22×C32×Q8C411yescharacteristic, complemented
13C22×C32×D4—21no—
14C22×C32×D4C411yescharacteristic, complemented
15C23×C32×C4—21no—
16C23×C32×C4C2211yescharacteristic
17C2×C3×#217—21no—
18C2×C3×C4×Dic3C2212yescomplemented
19C2×C3×#208C2212yescomplemented
20C3×#837—28no—
21C3×C3⋊#95—21no—
22C3×#830—21no—
23C3×#830—21no—
24C22×C3×#41—21no—
25C22×C3×#41D311yescharacteristic
26C22×C3×#41C2×C311yescharacteristic
27C2×C3×#104—34no—
28C3×#407—31no—
29C3×C4×C22⋊C4—31no—
30C2×#837C2×C314yescomplemented
31C4×#217C2×C311yescharacteristic, complemented
32#5207C2×C311yescharacteristic, complemented
33C24×C32—21no—
34C24×C32D411yescharacteristic
35C32×#41—28no—
36C32×#41—28no—
37C32×#41—24no—
38C32×#41—24no—
39C32×#41—24no—
40C32×#41D418yes—
41C2×C32×Q8—24no—
42C2×C32×Q8—21no—
43C2×C32×Q8C2×C412yes—
44C2×C32×D4—28no—
45C2×C32×D4—22no—
46C2×C32×D4—22no—
47C2×C32×D4—24no—
48C2×C32×D4—21no—
49C2×C32×D4C2×C412yes—
50C22×C32×C4—28no—
51C22×C32×C4—21no—
52C22×C32×C4—22no—
53C22×C32×C4—22no—
54C22×C32×C4—24no—
55C22×C32×C4C2×C411yescharacteristic
56C22×C32×C4C2×C411yescharacteristic
57C22×C32×C4D411yescharacteristic
58C22×C32×C4D411yescharacteristic
59C22×C32×C4D411yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—22no—
62C3×#217—24no—
63C3×C4×Dic3—24no—
64C3×#208—24no—
65C2×C3×C3⋊C8—22no—
66#1789—31noSylow
67C2×C3×#41—216no—
68C2×C3×#41—22no—
69C2×C3×#41—22no—
70C2×C3×#41—28no—
71C2×C3×#41—28no—
72C2×C3×#41—21no—
73C2×C3×#41—21no—
74C2×C3×#41—21no—
75C2×C3×#41—21no—
76C2×C3×#41—22no—
77C2×C3×#41—22no—
78C2×C3×#41Dic312yescomplemented
79C2×C3×#41C3×C412yescomplemented
80C2×C3×#41C2×D312yes—
81C2×C3×#41C22×C312yes—
82C22×C3×Q8—21no—
83C22×C3×Q8Dic311yescharacteristic, complemented
84C22×C3×Q8C3×C411yescharacteristic, complemented
85C22×C3×D4—22no—
86C22×C3×D4—21no—
87C22×C3×D4—21no—
88C22×C3×D4—21no—
89C22×C3×D4Dic311yescharacteristic, complemented
90C22×C3×D4C3×C411yescharacteristic, complemented
91C23×C3×C4—22no—
92C23×C3×C4—21no—
93C23×C3×C4—21no—
94C23×C3×C4—21no—
95C23×C3×C4C2×D311yescharacteristic
96C23×C3×C4C22×C311yescharacteristic
97C2×C3×M16—32no—
98C2×C3×C22⋊C4—61no—
99C2×C3×C42—32no—
100C2×#217—21no—
101C2×C4×Dic3C22×C312yescomplemented
102C2×#208C22×C312yescomplemented
103C3×#104—68no—
104C3×C22⋊C8—61no—
105C3×C22⋊C8—61no—
106C3×#95—61no—
107#837—28no—
108C3⋊#95—21no—
109#830—21no—
110#830—21no—
111C23×C32—24no—
112C23×C32—22no—
113C23×C32—22no—
114C23×C32—22no—
115C23×C32—41no—
116C23×C32—42no—
117C23×C32—42no—
118C23×C32C22⋊C411yescharacteristic
119C23×C32C22⋊C411yescharacteristic
120C23×C32C2×D411yescharacteristic
121C32×Q8—24no—
122C32×Q8—22no—
123C32×Q8C22⋊C414yescomplemented
124C32×D4—28no—
125C32×D4—24no—
126C32×D4—28no—
127C32×D4—22no—
128C32×D4C22⋊C414yescomplemented
129C2×C32×C4—24no—
130C2×C32×C4—28no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—28no—
137C2×C32×C4—22no—
138C2×C32×C4C22⋊C411yescharacteristic
139C2×C32×C4C22⋊C411yescharacteristic
140C2×C32×C4C22×C411yescharacteristic
141C2×C32×C4C2×D414yes—
142C2×C32×C4C2×D411yescharacteristic
143C2×C3×Dic3—24no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—42no—
146C3×C3⋊C8—42no—
147C22×#41C3×D311yescharacteristic
148C2×#104—34no—
149#407—31no—
150C4×C22⋊C4—31no—
151C24×C3—22no—
152C24×C3—21no—
153C24×C3—21no—
154C24×C3—21no—
155C24×C3#6711yescharacteristic
156C24×C3C3×D411yescharacteristic
157C3×#41—216no—
158C3×#41—216no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—28no—
162C3×#41—28no—
163C3×#41—28no—
164C3×#41—28no—
165C3×#41—28no—
166C3×#41—24no—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—24no—
172C3×#41—28no—
173C3×#41#6718yes—
174C3×#41C3×D418yes—
175C2×C3×Q8—28no—
176C2×C3×Q8—22no—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8C2×Dic312yes—
183C2×C3×Q8C2×C3×C412yes—
184C2×C3×D4—216no—
185C2×C3×D4—28no—
186C2×C3×D4—28no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—28no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4C2×Dic312yes—
201C2×C3×D4C2×C3×C412yes—
202C22×C3×C4—22no—
203C22×C3×C4—216no—
204C22×C3×C4—28no—
205C22×C3×C4—28no—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—28no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—24no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—62no—
224C22×C3×C4C2×Dic311yescharacteristic
225C22×C3×C4C2×Dic311yescharacteristic
226C22×C3×C4#6711yescharacteristic
227C22×C3×C4#6711yescharacteristic
228C22×C3×C4#6711yescharacteristic
229C22×C3×C4C2×C3×C411yescharacteristic
230C22×C3×C4C2×C3×C411yescharacteristic
231C22×C3×C4C3×D411yescharacteristic
232C22×C3×C4C3×D411yescharacteristic
233C22×C3×C4C3×D411yescharacteristic
234C22×C3×C4C22×D311yescharacteristic
235C22×C3×C4C23×C311yescharacteristic
236C22×Dic3—22no—
237C3×M16—64no—
238C2×C3×C8—62no—
239C3×C22⋊C4—64no—
240C3×C42—64no—
241#217—24no—
242C4×Dic3—24no—
243#208—24no—
244C2×C3⋊C8—22no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—24no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—44no—
254C22×C32—48no—
255C22×C32—42no—
256C22×C32—42no—
257C22×C32#10412yescomplemented
258C22×C32C2×C22⋊C411yescharacteristic
259C22×C32#12012yes—
260C32×C4—21no—
261C32×C4—24no—
262C32×C4—21no—
263C32×C4C2×C22⋊C412yes—
264C32×C4#12012yes—
265C3×Dic3—44no—
266C2×#41—28no—
267C2×#41—21no—
268C2×#41—21no—
269C2×#41C3×Dic312yescomplemented
270C2×#41C2×C3×D312yes—
271C22×Q8C3×Dic311yescharacteristic, complemented
272C22×D4—21no—
273C22×D4C3×Dic311yescharacteristic, complemented
274C23×C4—21no—
275C23×C4C2×C3×D311yescharacteristic
276C2×M16—32no—
277C2×C22⋊C4—61no—
278C2×C42—32no—
279#104—68no—
280C22⋊C8—61no—
281C22⋊C8—61no—
282#95—61no—
283C23×C3—28no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—44no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—44no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3#21711yescharacteristic
308C23×C3#21711yescharacteristic
309C23×C3C3×C22⋊C411yescharacteristic
310C23×C3C3×C22⋊C411yescharacteristic
311C23×C3C2×#6711yescharacteristic
312C23×C3C2×C3×D411yescharacteristic
313C3×Q8—28no—
314C3×Q8—24no—
315C3×Q8—24no—
316C3×Q8—24no—
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—24no—
320C3×Q8#21714yescomplemented
321C3×Q8C3×C22⋊C414yescomplemented
322C3×D4—216no—
323C3×D4—28no—
324C3×D4—28no—
325C3×D4—24no—
326C3×D4—28no—
327C3×D4—24no—
328C3×D4—216no—
329C3×D4—24no—
330C3×D4—28no—
331C3×D4—28no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—24no—
335C3×D4#21714yescomplemented
336C3×D4C3×C22⋊C414yescomplemented
337C2×C3×C4—24no—
338C2×C3×C4—28no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—216no—
344C2×C3×C4—28no—
345C2×C3×C4—28no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—216no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—28no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—28no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—24no—
369C2×C3×C4—64no—
370C2×C3×C4—122no—
371C2×C3×C4—122no—
372C2×C3×C4#21711yescharacteristic
373C2×C3×C4#21711yescharacteristic
374C2×C3×C4C3×C22⋊C411yescharacteristic
375C2×C3×C4C3×C22⋊C411yescharacteristic
376C2×C3×C4C22×Dic311yescharacteristic
377C2×C3×C4C2×#6714yes—
378C2×C3×C4C2×#6711yescharacteristic
379C2×C3×C4C22×C3×C411yescharacteristic
380C2×C3×C4C2×C3×D414yes—
381C2×C3×C4C2×C3×D411yescharacteristic
382C2×Dic3—24no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C3×C8—122no—
386C3⋊C8—42no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—41no—
390C2×C32—44no—
391C2×C32#37911yescharacteristic
392C2×C32C2×#10411yescharacteristic
393C2×C32#42111yescharacteristic
394C24—21no—
395C24C3×#6711yescharacteristic
396#41—28no—
397#41—28no—
398#41—24no—
399#41—24no—
400#41—24no—
401#41C3×#6718yes—
402C2×Q8—24no—
403C2×Q8—21no—
404C2×Q8C2×C3×Dic312yes—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4C2×C3×Dic312yes—
411C22×C4—28no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—62no—
417C22×C4C2×C3×Dic311yescharacteristic
418C22×C4C2×C3×Dic311yescharacteristic
419C22×C4C3×#6711yescharacteristic
420C22×C4C3×#6711yescharacteristic
421C22×C4C3×#6711yescharacteristic
422C22×C4C22×C3×D311yescharacteristic
423M16—64no—
424C2×C8—62no—
425C22⋊C4—64no—
426C42—64no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—28no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—24no—
443C22×C3—21no—
444C22×C3—24no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—48no—
457C22×C3—416no—
458C22×C3—48no—
459C22×C3—48no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3#83712yescomplemented
467C22×C3C3×#10412yescomplemented
468C22×C3C2×#21711yescharacteristic
469C22×C3#95312yes—
470C22×C3C2×C3×C22⋊C411yescharacteristic
471C22×C3C3×#12012yes—
472C3×C4—22no—
473C3×C4—28no—
474C3×C4—22no—
475C3×C4—21no—
476C3×C4—24no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—24no—
480C3×C4—21no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—124no—
484C3×C4C2×#21712yes—
485C3×C4#95312yes—
486C3×C4C2×C3×C22⋊C412yes—
487C3×C4C3×#12012yes—
488Dic3—44no—
489C32#178911yescharacteristic, complemented, Sylow
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23C3×#21711yescharacteristic
498C23C3×#21711yescharacteristic
499C23C2×C3×#6711yescharacteristic
500Q8—24no—
501Q8—22no—
502Q8C3×#21714yescomplemented
503D4—28no—
504D4—24no—
505D4—28no—
506D4—22no—
507D4C3×#21714yescomplemented
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—64no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4C3×#21711yescharacteristic
521C2×C4C3×#21711yescharacteristic
522C2×C4C22×C3×Dic311yescharacteristic
523C2×C4C2×C3×#6714yes—
524C2×C4C2×C3×#6711yescharacteristic
525C8—122no—
526C2×C3—24no—
527C2×C3—22no—
528C2×C3—22no—
529C2×C3—22no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—42no—
536C2×C3—48no—
537C2×C3—41no—
538C2×C3—44no—
539C2×C3—41no—
540C2×C3—44no—
541C2×C3C2×#83711yescharacteristic
542C2×C3#523611yescharacteristic
543C2×C3#524711yescharacteristic
544C2×C3C3×#37911yescharacteristic
545C2×C3C2×C3×#10411yescharacteristic
546C2×C3C3×#42111yescharacteristic
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—24no—
555C22—44no—
556C22—48no—
557C22—42no—
558C22—42no—
559C22C3×#83712yescomplemented
560C22C2×C3×#21711yescharacteristic
561C22C3×#95312yes—
562C4—21no—
563C4—24no—
564C4—21no—
565C4—124no—
566C4C2×C3×#21712yes—
567C4C3×#95312yes—
568C3—21no—
569C3#7213811yescharacteristic, direct product factor
570C3C3×#178911yescharacteristic, complemented
571C2—22no—
572C2—22no—
573C2—41no—
574C2—44no—
575C2C2×C3×#83711yescharacteristic
576C2C3×#523611yescharacteristic
577C2C3×#524711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC22×C32×#41C2
socleC22×C32C2×C22⋊C4
cosocleC22×C3×C4C23×C3
hypercenterC22×C3×#41D3
hypocenterC3C3×#1789
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#837C2semidirect
3. C3×C4×#217C2semidirect
4. C3×#5207C2semidirect
6. #72138C3direct (N × H)
10. C2×C32×#41C4semidirect
12. C22×C32×Q8C4semidirect
14. C22×C32×D4C4semidirect
18. C2×C3×C4×Dic3C22semidirect
19. C2×C3×#208C22semidirect
30. C2×#837C2×C3semidirect
31. C4×#217C2×C3semidirect
32. #5207C2×C3semidirect
78. C2×C3×#41Dic3semidirect
79. C2×C3×#41C3×C4semidirect
83. C22×C3×Q8Dic3semidirect
84. C22×C3×Q8C3×C4semidirect
89. C22×C3×D4Dic3semidirect
90. C22×C3×D4C3×C4semidirect
101. C2×C4×Dic3C22×C3semidirect
102. C2×#208C22×C3semidirect
123. C32×Q8C22⋊C4semidirect
128. C32×D4C22⋊C4semidirect
257. C22×C32#104semidirect
269. C2×#41C3×Dic3semidirect
271. C22×Q8C3×Dic3semidirect
273. C22×D4C3×Dic3semidirect
320. C3×Q8#217semidirect
321. C3×Q8C3×C22⋊C4semidirect
335. C3×D4#217semidirect
336. C3×D4C3×C22⋊C4semidirect
466. C22×C3#837semidirect
467. C22×C3C3×#104semidirect
489. C32#1789semidirect
502. Q8C3×#217semidirect
507. D4C3×#217semidirect
559. C22C3×#837semidirect
569. C3#72138direct (N × H)
570. C3C3×#1789semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#412
C2×C3×#8372
C3×C4×#2172
C3×#52072
C3×#17893
#721383