Finite Group Databasebeta

← C3×#15071 #10633678

Subgroups — C3×#15071 #10633678

Order 768.

1478 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic
2C2×C3×#387C212yesmaximal
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic
5C3×#1444C212yesmaximal
6C3×#1442C212yesmaximal
7C3×#1442C211yesmaximal, characteristic
8C3×#1441C211yesmaximal, characteristic
9C3×#1441C211yesmaximal, characteristic
10C3×#1436C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15071C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125—28no—
14C2×C3×#125—24no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C22×C3×C42C2211yescharacteristic
25C3×#387—28no—
26C3×C4×C4⋊C4—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C22×#125C2×C311yescharacteristic
41C2×#387C2×C312yes—
42C2×C4×C4⋊C4C2×C311yescharacteristic
43#1444C2×C311yescharacteristic
44#1444C2×C312yes—
45#1442C2×C311yescharacteristic
46#1442C2×C312yes—
47#1441C2×C311yescharacteristic
48#1441C2×C311yescharacteristic
49#1436C2×C312yes—
50#1434C2×C311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C3×#125—232no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D418yes—
81C2×C3×C4⋊C4Q814yes—
82C2×C3×C42—22no—
83C2×C3×C42—24no—
84C2×C3×C42—22no—
85C2×C3×C42—22no—
86C2×C3×C42—22no—
87C2×C3×C42—24no—
88C2×C3×C42D414yes—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—22no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—24no—
101C3×#95—22no—
102C2×#125—28no—
103C2×#125—24no—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C4⋊C4C22×C312yes—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C312yes—
109C22×C4⋊C4C22×C312yes—
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C42C22×C311yescharacteristic
112C22×C42C22×C311yescharacteristic
113C22×C42C22×C311yescharacteristic
114#387—28no—
115C4×C4⋊C4—24no—
116C2×#95C22×C312yes—
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C312yes—
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C312yes—
129C24×C3C2411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×Q812yes—
187C22×C3×C4C2×Q812yes—
188C22×C3×C4C2×Q812yes—
189C22×C3×C4#4112yes—
190C22×C3×C4#4114yes—
191C22×C3×C4#4112yes—
192C22×C3×C4#4112yes—
193C22×C3×C4#4112yes—
194C22×C3×C4#4112yes—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—216no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—216no—
200C3×C4⋊C4—216no—
201C3×C4⋊C4—28no—
202C3×C42—28no—
203C3×C42—28no—
204C3×C42—44no—
205C23×C4C23×C312yes—
206C23×C4C23×C312yes—
207C23×C4C23×C312yes—
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C311yescharacteristic
211C23×C4C23×C312yes—
212C23×C4C23×C311yescharacteristic
213C23×C4C23×C311yescharacteristic
214C23×C4C23×C311yescharacteristic
215C23×C4C23×C311yescharacteristic
216#125—232no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C3×D418yes—
235C2×C4⋊C4C3×Q814yes—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42C3×D414yes—
243#95—22no—
244#95—22no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256C23×C3C22×D411yescharacteristic
257C23×C3C22×D412yes—
258C23×C3C22×Q811yescharacteristic
259C23×C3C2×#4111yescharacteristic
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4111yescharacteristic
262C23×C3C2×#4111yescharacteristic
263C23×C3C2×#4111yescharacteristic
264C23×C3C2×#4112yes—
265C23×C32−1+412yes—
266C23×C32−1+412yes—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4#12014yes—
330C2×C3×C4C22⋊Q814yes—
331C2×C3×C4#12514yescomplemented
332C24C24×C311yescharacteristic
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4C2×C3×D414yes—
384C22×C4C2×C3×D412yes—
385C22×C4C2×C3×D414yes—
386C22×C4C2×C3×D414yes—
387C22×C4C2×C3×D412yes—
388C22×C4C2×C3×D412yes—
389C22×C4C2×C3×Q812yes—
390C22×C4C2×C3×Q812yes—
391C22×C4C2×C3×Q812yes—
392C22×C4C3×#4112yes—
393C22×C4C3×#4114yes—
394C22×C4C3×#4112yes—
395C22×C4C3×#4112yes—
396C22×C4C3×#4112yes—
397C22×C4C3×#4112yes—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—216no—
401C4⋊C4—28no—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—28no—
405C42—28no—
406C42—28no—
407C42—44no—
408C22×C3C2×#12011yescharacteristic
409C22×C3C2×C22⋊Q811yescharacteristic
410C22×C3#52511yescharacteristic
411C22×C3#52511yescharacteristic
412C22×C3#52512yes—
413C22×C3C2×#12511yescharacteristic
414C22×C3#52911yescharacteristic
415C22×C3#52911yescharacteristic
416C22×C3#53211yescharacteristic
417C22×C3#53212yes—
418C22×C3#53311yescharacteristic
419C22×C3#53611yescharacteristic
420C22×C3#53611yescharacteristic
421C22×C3#54112yes—
422C22×C3#54112yes—
423C22×C3#54712yes—
424C22×C3#54712yes—
425C22×C3#54812yes—
426C22×C3D4×Q812yes—
427C22×C3#55212yes—
428C22×C3#55212yes—
429C22×C3#55212yes—
430C22×C3#55712yes—
431C3×C4—24no—
432C3×C4—24no—
433C3×C4—24no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—44no—
437C3×C4—42no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—44no—
441C3×C4—82no—
442C23C22×C3×D411yescharacteristic
443C23C22×C3×D412yes—
444C23C22×C3×Q811yescharacteristic
445C23C2×C3×#4111yescharacteristic
446C23C2×C3×#4111yescharacteristic
447C23C2×C3×#4111yescharacteristic
448C23C2×C3×#4111yescharacteristic
449C23C2×C3×#4111yescharacteristic
450C23C2×C3×#4112yes—
451C23C3×2−1+412yes—
452C23C3×2−1+412yes—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4C3×#12014yes—
516C2×C4C3×C22⋊Q814yes—
517C2×C4C3×#12514yescomplemented
518C2×C3#238811yescharacteristic
519C2×C3#240611yescharacteristic
520C2×C3#240911yescharacteristic
521C2×C3#246212yes—
522C2×C3#247512yes—
523C2×C3#250612yes—
524C2×C3#251912yes—
525C2×C3#256511yescharacteristic
526C2×C3#261111yescharacteristic
527C2×C3#263411yescharacteristic
528C2×C3#285111yescharacteristic
529C22C2×C3×#12011yescharacteristic
530C22C2×C3×C22⋊Q811yescharacteristic
531C22C3×#52511yescharacteristic
532C22C3×#52512yes—
533C22C3×#52511yescharacteristic
534C22C2×C3×#12511yescharacteristic
535C22C3×#52911yescharacteristic
536C22C3×#52911yescharacteristic
537C22C3×#53212yes—
538C22C3×#53211yescharacteristic
539C22C3×#53311yescharacteristic
540C22C3×#53611yescharacteristic
541C22C3×#53611yescharacteristic
542C22C3×#54112yes—
543C22C3×#54112yes—
544C22C3×#54712yes—
545C22C3×#54712yes—
546C22C3×#54812yes—
547C22C3×D4×Q812yes—
548C22C3×#55212yes—
549C22C3×#55212yes—
550C22C3×#55212yes—
551C22C3×#55712yes—
552C4—24no—
553C4—24no—
554C4—24no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—44no—
559C4—42no—
560C4—44no—
561C4—42no—
562C4—82no—
563C3#1507111yescharacteristic, direct product factor, Sylow
564C2C3×#238811yescharacteristic
565C2C3×#240611yescharacteristic
566C2C3×#240911yescharacteristic
567C2C3×#246212yes—
568C2C3×#247512yes—
569C2C3×#250612yes—
570C2C3×#251912yes—
571C2C3×#256511yescharacteristic
572C2C3×#261111yescharacteristic
573C2C3×#263411yescharacteristic
574C2C3×#285111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #15071C3direct (N × H)
331. C2×C3×C4#125semidirect
517. C2×C4C3×#125semidirect
563. C3#15071direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×#3872
C2×C3×C4×C4⋊C42
C3×#14442
C3×#14422
C3×#14412
C3×#14362
C3×#14342
#150713