Finite Group Databasebeta

← C3×#29716 #10648323

Subgroups — C3×#29716 #10648323

Order 768.

1156 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#449C212yesmaximal, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3C3×#2297C211yesmaximal, characteristic, complemented
4C3×#2072C211yesmaximal, characteristic, complemented
5C3×#2071C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2054C212yesmaximal, complemented
8C3×#2014C211yesmaximal, characteristic, complemented
9C3×#2011C211yesmaximal, characteristic, complemented
10C3×#1875C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29716C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2212yes—
20C2×C3×#117C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C3×#449—28no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#391C2211yescharacteristic
30C3×#391C2211yescharacteristic
31C3×#390—21no—
32C3×#390—21no—
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#390C2212yes—
36C3×#386—21no—
37C3×#386—21no—
38C3×#384C2211yescharacteristic
39C3×C4×C22⋊C4—21no—
40C3×C4×C22⋊C4—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C2×#449C2×C312yescomplemented
46#2856C2×C311yescharacteristic, complemented
47#2297C2×C311yescharacteristic, complemented
48#2072C2×C311yescharacteristic, complemented
49#2071C2×C311yescharacteristic, complemented
50#2055C2×C312yescomplemented
51#2054C2×C312yescomplemented
52#2014C2×C311yescharacteristic, complemented
53#2011C2×C311yescharacteristic, complemented
54#1875C2×C312yescomplemented
55#1797C2×C311yescharacteristic, complemented
56C2×C3×#41—22no—
57C2×C3×#41—42no—
58C2×C3×#41—42no—
59C2×C3×#41D414yescomplemented
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×SD16—42no—
74C2×C3×D8—24no—
75C2×C3×D8—42no—
76C2×C3×D8—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C3×#127—42no—
80C3×#121—24no—
81C3×#121—24no—
82C3×#121—44no—
83C3×#121—44no—
84C3×#117—42no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C42—21no—
99C2×C3×C42—42no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C3×C4⋊C8—42no—
105C3×#103—24no—
106C3×#102—24no—
107C3×#102—42no—
108C3×#102—42no—
109C3×C22⋊C8—24no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C22×#41C22×C311yescharacteristic
115C22×SD16C22×C312yes—
116C22×D8C22×C312yes—
117C2×#127C22×C311yescharacteristic
118C2×#121—21no—
119C2×#121—21no—
120C2×#121C22×C312yes—
121C2×#117C22×C311yescharacteristic
122C22×C42C22×C311yescharacteristic
123#449—28no—
124C2×C4⋊C8C22×C312yes—
125C2×#103C22×C312yes—
126C2×#102C22×C312yes—
127C2×#102C22×C312yes—
128C2×#102C22×C312yes—
129C2×C22⋊C8C22×C312yes—
130#391C22×C311yescharacteristic
131#391C22×C311yescharacteristic
132#390—21no—
133#390—21no—
134#390C22×C311yescharacteristic
135#390C22×C312yes—
136#390C22×C311yescharacteristic
137#386—21no—
138#386—21no—
139#384C22×C311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C311yescharacteristic
145#336C22×C311yescharacteristic
146C24×C3—21no—
147C24×C3—22no—
148C24×C3C2×D411yescharacteristic
149C3×#41—48no—
150C3×#41—48no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8C2×D412yes—
155C2×C3×D4—24no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4—44no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4C2×D412yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—41no—
188C22×C3×C4—41no—
189C22×C3×C4—41no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C3×SD16—48no—
201C3×D8—48no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—44no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—44no—
222C3×C42—22no—
223C3×C42—22no—
224C3×C42—41no—
225C3×C42—41no—
226C3×C42—42no—
227C2×#41—22no—
228C2×#41—42no—
229C2×#41—42no—
230C2×#41C3×D414yescomplemented
231C22×Q8C23×C311yescharacteristic
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C23×C312yes—
237C22×D4C23×C311yescharacteristic
238C23×C4—21no—
239C23×C4C3×D411yescharacteristic
240C23×C4C3×D411yescharacteristic
241C23×C4C23×C311yescharacteristic
242C2×SD16—24no—
243C2×SD16—42no—
244C2×SD16—42no—
245C2×D8—24no—
246C2×D8—42no—
247C2×D8—42no—
248C22×C8C23×C312yes—
249C22×C8C23×C312yes—
250#127—42no—
251#121—24no—
252#121—24no—
253#121—44no—
254#121—44no—
255#117—42no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C4⋊C4C23×C312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—21no—
270C2×C42—42no—
271C2×C42C3×D411yescharacteristic
272C2×C42C3×D411yescharacteristic
273C2×C42C23×C311yescharacteristic
274C2×C42C23×C311yescharacteristic
275C4⋊C8—42no—
276#103—24no—
277#102—24no—
278#102—42no—
279#102—42no—
280C22⋊C8—24no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3#12112yes—
306C23×C3C22×D411yescharacteristic
307C3×Q8—24no—
308C3×Q8—42no—
309C3×D4—24no—
310C3×D4—48no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—84no—
317C3×D4—84no—
318C3×D4—88no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4—48no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4#12112yes—
348C2×C3×C4#13512yes—
349C2×C3×C4AGL(1,ℤ/8)12yes—
350C2×C3×C4AGL(1,ℤ/8)12yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4H(5,𝔽2)12yes—
354C2×C3×C4H(5,𝔽2)12yes—
355C3×C8—42no—
356C3×C8—42no—
357C24—21no—
358C24—22no—
359C24C2×C3×D411yescharacteristic
360#41—48no—
361#41—48no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8C2×C3×D412yes—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4C2×C3×D412yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C3×#4112yes—
410C22×C4C24×C311yescharacteristic
411SD16—48no—
412D8—48no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C42—22no—
434C42—22no—
435C42—41no—
436C42—41no—
437C42—42no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3C2×#12111yescharacteristic
460C22×C3#53411yescharacteristic
461C22×C3#53511yescharacteristic
462C22×C3C2×#13511yescharacteristic
463C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
464C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
465C22×C3#57511yescharacteristic
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—41no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4—82no—
475C3×C4#53412yes—
476C3×C4#53812yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C3×#12112yes—
498C23C22×C3×D411yescharacteristic
499Q8—24no—
500Q8—42no—
501D4—24no—
502D4—48no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—84no—
509D4—84no—
510D4—88no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4C3×#12112yes—
540C2×C4C3×#13512yes—
541C2×C4C3×AGL(1,ℤ/8)12yes—
542C2×C4C3×AGL(1,ℤ/8)12yes—
543C2×C4C22×C3×D411yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C2×C4C3×H(5,𝔽2)12yes—
546C2×C4C3×H(5,𝔽2)12yes—
547C8—42no—
548C8—42no—
549C2×C3—42no—
550C2×C3—82no—
551C2×C3—162no—
552C2×C3#261711yescharacteristic
553C2×C3#303511yescharacteristic
554C2×C3#303811yescharacteristic
555C2×C3#318811yescharacteristic
556C2×C3#319411yescharacteristic
557C2×C3#320711yescharacteristic
558C2×C3#321411yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22C2×C3×#12111yescharacteristic
581C22C3×#53411yescharacteristic
582C22C3×#53511yescharacteristic
583C22C2×C3×#13511yescharacteristic
584C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
585C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
586C22C3×#57511yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—82no—
596C4C3×#53412yes—
597C4C3×#53812yes—
598C3#2971611yescharacteristic, direct product factor, Sylow
599C2—42no—
600C2—82no—
601C2—162no—
602C2C3×#261711yescharacteristic
603C2C3×#303511yescharacteristic
604C2C3×#303811yescharacteristic
605C2C3×#318811yescharacteristic
606C2C3×#319411yescharacteristic
607C2C3×#320711yescharacteristic
608C2C3×#321411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#449C2semidirect
2. C3×#2856C2semidirect
3. C3×#2297C2semidirect
4. C3×#2072C2semidirect
5. C3×#2071C2semidirect
6. C3×#2055C2semidirect
7. C3×#2054C2semidirect
8. C3×#2014C2semidirect
9. C3×#2011C2semidirect
10. C3×#1875C2semidirect
11. C3×#1797C2semidirect
12. #29716C3direct (N × H)
45. C2×#449C2×C3semidirect
46. #2856C2×C3semidirect
47. #2297C2×C3semidirect
48. #2072C2×C3semidirect
49. #2071C2×C3semidirect
50. #2055C2×C3semidirect
51. #2054C2×C3semidirect
52. #2014C2×C3semidirect
53. #2011C2×C3semidirect
54. #1875C2×C3semidirect
55. #1797C2×C3semidirect
59. C2×C3×#41D4semidirect
230. C2×#41C3×D4semidirect
598. C3#29716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4492
C3×#28562
C3×#22972
C3×#20722
C3×#20712
C3×#20552
C3×#20542
C3×#20142
C3×#20112
C3×#18752
C3×#17972
#297163