Finite Group Databasebeta

← C3×#15097 #10633704

Subgroups — C3×#15097 #10633704

Order 768.

1374 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#398C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic
4C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
5C3×#1446C211yesmaximal, characteristic
6C3×#1445C211yesmaximal, characteristic
7C3×#1442C212yesmaximal
8C3×#1441C212yesmaximal
9C3×#1436C212yesmaximal
10C3×#1436C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15097C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C3×#398—24no—
23C3×#395—24no—
24C3×#391—24no—
25C3×C4×C4⋊C4—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×#398C2×C311yescharacteristic
41C2×#395C2×C311yescharacteristic
42C2×#391C2×C311yescharacteristic
43C2×C4×C4⋊C4C2×C311yescharacteristic
44#1446C2×C311yescharacteristic
45#1445C2×C311yescharacteristic
46#1442C2×C312yes—
47#1441C2×C312yes—
48#1436C2×C312yes—
49#1436C2×C312yes—
50#1434C2×C311yescharacteristic
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—24no—
54C22×C3×Q8—22no—
55C22×C3×Q8—42no—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—42no—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C42—22no—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85C2×C3×C42—22no—
86C2×C3×C42—42no—
87C2×C3×C42—42no—
88C2×C3×C42D414yes—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—24no—
92C3×#95—24no—
93C3×#95—22no—
94C3×#95—24no—
95C3×#95—22no—
96C3×#95—24no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—42no—
101C3×#95—42no—
102C3×#95Q818yescomplemented
103C23×Q8C22×C311yescharacteristic
104C22×C4⋊C4C22×C312yes—
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C4⋊C4C22×C312yes—
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C42C22×C311yescharacteristic
110C22×C42C22×C311yescharacteristic
111C22×C42C22×C311yescharacteristic
112#398—24no—
113#395—24no—
114#391—24no—
115C4×C4⋊C4—24no—
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C312yes—
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C2×#95C22×C312yes—
130C24×C3C2411yescharacteristic
131C2×C3×Q8—28no—
132C2×C3×Q8—48no—
133C2×C3×Q8—48no—
134C2×C3×Q8—44no—
135C2×C3×Q8—44no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
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×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×Q814yes—
193C22×C3×C4C2×Q814yes—
194C22×C3×C4C2×Q814yes—
195C22×C3×C4#4112yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4112yes—
199C22×C3×C4#4112yes—
200C22×C3×C4#4112yes—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—48no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—48no—
206C3×C42—28no—
207C3×C42—44no—
208C3×C42—44no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—42no—
214C23×C4C23×C312yes—
215C23×C4C23×C311yescharacteristic
216C23×C4C23×C311yescharacteristic
217C23×C4C23×C312yes—
218C23×C4C23×C312yes—
219C23×C4C23×C312yes—
220C23×C4C23×C311yescharacteristic
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C311yescharacteristic
224C23×C4C23×C311yescharacteristic
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C3×D414yes—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C3×D414yes—
247#95—22no—
248#95—22no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—24no—
253#95—22no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—42no—
259#95—42no—
260#95C3×Q818yescomplemented
261C23×C3C22×D411yescharacteristic
262C23×C3C22×D411yescharacteristic
263C23×C3C22×Q812yes—
264C23×C3C2×#4111yescharacteristic
265C23×C3C2×#4111yescharacteristic
266C23×C3C2×#4111yescharacteristic
267C23×C3C2×#4111yescharacteristic
268C23×C3C2×#4111yescharacteristic
269C23×C3C2×#4112yes—
270C23×C3H(5,𝔽2)12yes—
271C23×C32−1+412yes—
272C3×Q8—416no—
273C2×C3×C4—22no—
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—22no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
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—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4#12114yes—
336C2×C3×C4#12414yes—
337C2×C3×C4C4⋊Q814yes—
338C24C24×C311yescharacteristic
339C2×Q8—28no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—44no—
343C2×Q8—44no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
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×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D412yes—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×Q814yes—
401C22×C4C2×C3×Q814yes—
402C22×C4C2×C3×Q814yes—
403C22×C4C3×#4112yes—
404C22×C4C3×#4112yes—
405C22×C4C3×#4114yes—
406C22×C4C3×#4112yes—
407C22×C4C3×#4112yes—
408C22×C4C3×#4112yes—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C42—28no—
415C42—44no—
416C42—44no—
417C22×C3C2×#12111yescharacteristic
418C22×C3#52511yescharacteristic
419C22×C3#52511yescharacteristic
420C22×C3#52511yescharacteristic
421C22×C3C2×#12411yescharacteristic
422C22×C3#52911yescharacteristic
423C22×C3#52911yescharacteristic
424C22×C3C2×C4⋊Q811yescharacteristic
425C22×C3#53211yescharacteristic
426C22×C3#53312yes—
427C22×C3#53312yes—
428C22×C3#53411yescharacteristic
429C22×C3#53611yescharacteristic
430C22×C3#53912yes—
431C22×C3#54212yes—
432C22×C3#54612yes—
433C22×C3D4×Q812yes—
434C22×C3#55112yes—
435C22×C3#55212yes—
436C22×C3#55412yes—
437C22×C3#55412yes—
438C22×C3#55612yes—
439C22×C3#55712yes—
440C3×C4—24no—
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—44no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—44no—
450C3×C4—82no—
451C23C22×C3×D411yescharacteristic
452C23C22×C3×D411yescharacteristic
453C23C22×C3×Q812yes—
454C23C2×C3×#4111yescharacteristic
455C23C2×C3×#4111yescharacteristic
456C23C2×C3×#4111yescharacteristic
457C23C2×C3×#4111yescharacteristic
458C23C2×C3×#4111yescharacteristic
459C23C2×C3×#4112yes—
460C23C3×H(5,𝔽2)12yes—
461C23C3×2−1+412yes—
462Q8—416no—
463C2×C4—22no—
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—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
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×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4C3×#12114yes—
526C2×C4C3×#12414yes—
527C2×C4C3×C4⋊Q814yes—
528C2×C3#239911yescharacteristic
529C2×C3#249212yes—
530C2×C3#252012yes—
531C2×C3#253411yescharacteristic
532C2×C3#254611yescharacteristic
533C2×C3#255612yes—
534C2×C3#258311yescharacteristic
535C2×C3#258812yes—
536C2×C3#259911yescharacteristic
537C2×C3#263811yescharacteristic
538C2×C3#285011yescharacteristic
539C22C2×C3×#12111yescharacteristic
540C22C3×#52511yescharacteristic
541C22C3×#52511yescharacteristic
542C22C3×#52511yescharacteristic
543C22C2×C3×#12411yescharacteristic
544C22C3×#52911yescharacteristic
545C22C3×#52911yescharacteristic
546C22C2×C3×C4⋊Q811yescharacteristic
547C22C3×#53211yescharacteristic
548C22C3×#53312yes—
549C22C3×#53312yes—
550C22C3×#53411yescharacteristic
551C22C3×#53611yescharacteristic
552C22C3×#53912yes—
553C22C3×#54212yes—
554C22C3×#54612yes—
555C22C3×D4×Q812yes—
556C22C3×#55112yes—
557C22C3×#55212yes—
558C22C3×#55412yes—
559C22C3×#55412yes—
560C22C3×#55612yes—
561C22C3×#55712yes—
562C4—24no—
563C4—24no—
564C4—24no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—44no—
569C4—42no—
570C4—44no—
571C4—42no—
572C4—82no—
573C3#1509711yescharacteristic, direct product factor, Sylow
574C2C3×#239911yescharacteristic
575C2C3×#249212yes—
576C2C3×#252012yes—
577C2C3×#253411yescharacteristic
578C2C3×#254611yescharacteristic
579C2C3×#255612yes—
580C2C3×#258311yescharacteristic
581C2C3×#258812yes—
582C2C3×#259911yescharacteristic
583C2C3×#263811yescharacteristic
584C2C3×#285011yescharacteristic
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. #15097C3direct (N × H)
102. C3×#95Q8semidirect
260. #95C3×Q8semidirect
573. C3#15097direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3952
C2×C3×#3912
C2×C3×C4×C4⋊C42
C3×#14462
C3×#14452
C3×#14422
C3×#14412
C3×#14362
C3×#14342
#150973