Finite Group Databasebeta

← C3×#29629 #10648236

Subgroups — C3×#29629 #10648236

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#452C212yesmaximal, complemented
3C3×#2855C211yesmaximal, characteristic, complemented
4C3×#2043C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C211yesmaximal, characteristic, complemented
7C3×#2016C211yesmaximal, characteristic, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#2014C211yesmaximal, characteristic, complemented
10C3×#1817C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#29629C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—21no—
22C2×C3×#121—21no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4—21no—
25C22×C3×C42C2211yescharacteristic
26C3×#452—28no—
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C3×#391C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#387—21no—
39C3×#387—21no—
40C3×#385—21no—
41C3×#381—21no—
42C3×#380C2211yescharacteristic
43C3×#336C2211yescharacteristic
44C3×#336C2212yes—
45C3×#336C2211yescharacteristic
46C2×#532C2×C311yescharacteristic
47C2×#452C2×C312yescomplemented
48#2855C2×C311yescharacteristic, complemented
49#2043C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2041C2×C311yescharacteristic, complemented
52#2016C2×C311yescharacteristic, complemented
53#2015C2×C312yescomplemented
54#2014C2×C311yescharacteristic, complemented
55#1817C2×C311yescharacteristic, complemented
56#1788C2×C312yescomplemented
57C2×C3×#41—24no—
58C2×C3×#41—42no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41—42no—
62C2×C3×#41D418yes—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2312yes—
65C22×C3×D4—22no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4C2312yes—
69C23×C3×C4—22no—
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D411yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C2×C3×Q16—24no—
74C2×C3×Q16—42no—
75C2×C3×Q16—42no—
76C2×C3×SD16—24no—
77C2×C3×SD16—42no—
78C2×C3×SD16—42no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C3×C4⋊Q8—24no—
82C3×#127—24no—
83C3×#124—24no—
84C3×#124—24no—
85C3×C22⋊Q8—24no—
86C3×#121—28no—
87C3×#121—28no—
88C3×C4×D4—28no—
89C3×C4×D4—28no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C42—24no—
103C2×C3×C42D411yescharacteristic
104C2×C3×C42D411yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C3×#103—24no—
108C3×#103—42no—
109C3×#103—42no—
110C3×#102—24no—
111C3×#102—42no—
112C3×#102—42no—
113C3×C22⋊C8—24no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C22×#41C22×C312yes—
122C22×Q16C22×C312yes—
123C22×SD16C22×C312yes—
124#532—28no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#127C22×C311yescharacteristic
127C2×#124C22×C312yes—
128C2×C22⋊Q8C22×C311yescharacteristic
129C2×#121—21no—
130C2×#121—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C22×C42C22×C311yescharacteristic
134#452—28no—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×#103C22×C312yes—
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×#102C22×C312yes—
141C2×C22⋊C8C22×C312yes—
142#391C22×C311yescharacteristic
143#391C22×C311yescharacteristic
144#389C22×C311yescharacteristic
145#389C22×C311yescharacteristic
146#387—21no—
147#387—21no—
148#385—21no—
149#381—21no—
150#380C22×C311yescharacteristic
151#336C22×C311yescharacteristic
152#336C22×C312yes—
153#336C22×C311yescharacteristic
154C24×C3—22no—
155C24×C3C2×D411yescharacteristic
156C3×#41—48no—
157C3×#41—48no—
158C3×#41—48no—
159C3×#41—48no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8C2×D414yes—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—28no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C2×C3×D4—44no—
181C2×C3×D4—44no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4C2×D414yes—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—28no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—41no—
199C22×C3×C4—41no—
200C22×C3×C4—44no—
201C22×C3×C4—44no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C3×Q16—48no—
210C3×SD16—48no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C22⋊C4—28no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42—22no—
234C2×#41—24no—
235C2×#41—42no—
236C2×#41—42no—
237C2×#41—42no—
238C2×#41—42no—
239C2×#41C3×D418yes—
240C22×Q8C23×C311yescharacteristic
241C22×Q8C23×C312yes—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C23×C312yes—
246C23×C4—22no—
247C23×C4C3×D411yescharacteristic
248C23×C4C3×D411yescharacteristic
249C23×C4C23×C311yescharacteristic
250C2×Q16—24no—
251C2×Q16—42no—
252C2×Q16—42no—
253C2×SD16—24no—
254C2×SD16—42no—
255C2×SD16—42no—
256C22×C8C23×C312yes—
257C22×C8C23×C312yes—
258C4⋊Q8—24no—
259#127—24no—
260#124—24no—
261#124—24no—
262C22⋊Q8—24no—
263#121—28no—
264#121—28no—
265C4×D4—28no—
266C4×D4—28no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C42—24no—
280C2×C42C3×D411yescharacteristic
281C2×C42C3×D411yescharacteristic
282C2×C42C23×C311yescharacteristic
283C2×C42C23×C311yescharacteristic
284#103—24no—
285#103—42no—
286#103—42no—
287#102—24no—
288#102—42no—
289#102—42no—
290C22⋊C8—24no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—24no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3#12012yes—
312C23×C3C22×D411yescharacteristic
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—42no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—42no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—48no—
322C3×D4—48no—
323C3×D4—416no—
324C3×D4—42no—
325C3×D4—42no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—24no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—44no—
342C2×C3×C4—44no—
343C2×C3×C4—416no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4#12012yes—
359C2×C3×C4#13512yes—
360C2×C3×C4#13712yes—
361C2×C3×C4C22×D412yes—
362C2×C3×C4H(5,𝔽2)12yes—
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C3×C8—42no—
366C3×C8—42no—
367C24—22no—
368C24C2×C3×D411yescharacteristic
369#41—48no—
370#41—48no—
371#41—48no—
372#41—48no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8C2×C3×D414yes—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C2×C3×D414yes—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D414yes—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C24×C311yescharacteristic
422Q16—48no—
423SD16—48no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C22×C3—22no—
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—44no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3C2×#12011yescharacteristic
462C22×C3#53511yescharacteristic
463C22×C3#53511yescharacteristic
464C22×C3C2×#13511yescharacteristic
465C22×C3C2×#13711yescharacteristic
466C22×C3#57511yescharacteristic
467C22×C3#57511yescharacteristic
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—44no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4#53412yes—
479C3×C4#53412yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23C3×#12012yes—
494C23C22×C3×D411yescharacteristic
495Q8—24no—
496Q8—24no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501D4—24no—
502D4—24no—
503D4—48no—
504D4—48no—
505D4—416no—
506D4—42no—
507D4—42no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4C3×#12012yes—
541C2×C4C3×#13512yes—
542C2×C4C3×#13712yes—
543C2×C4C22×C3×D412yes—
544C2×C4C3×H(5,𝔽2)12yes—
545C2×C4C3×H(5,𝔽2)11yescharacteristic
546C2×C4C3×H(5,𝔽2)11yescharacteristic
547C8—42no—
548C8—42no—
549C2×C3—42no—
550C2×C3—84no—
551C2×C3#261311yescharacteristic
552C2×C3#300211yescharacteristic
553C2×C3#300711yescharacteristic
554C2×C3#321011yescharacteristic
555C2×C3#321211yescharacteristic
556C2×C3#321611yescharacteristic
557C2×C3#321911yescharacteristic
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C2×C3×#12011yescharacteristic
573C22C3×#53511yescharacteristic
574C22C3×#53511yescharacteristic
575C22C2×C3×#13511yescharacteristic
576C22C2×C3×#13711yescharacteristic
577C22C3×#57511yescharacteristic
578C22C3×#57511yescharacteristic
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—44no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4C3×#53412yes—
590C4C3×#53412yes—
591C3#2962911yescharacteristic, direct product factor, Sylow
592C2—42no—
593C2—84no—
594C2C3×#261311yescharacteristic
595C2C3×#300211yescharacteristic
596C2C3×#300711yescharacteristic
597C2C3×#321011yescharacteristic
598C2C3×#321211yescharacteristic
599C2C3×#321611yescharacteristic
600C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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
2. C2×C3×#452C2semidirect
3. C3×#2855C2semidirect
4. C3×#2043C2semidirect
5. C3×#2042C2semidirect
6. C3×#2041C2semidirect
7. C3×#2016C2semidirect
8. C3×#2015C2semidirect
9. C3×#2014C2semidirect
10. C3×#1817C2semidirect
11. C3×#1788C2semidirect
12. #29629C3direct (N × H)
47. C2×#452C2×C3semidirect
48. #2855C2×C3semidirect
49. #2043C2×C3semidirect
50. #2042C2×C3semidirect
51. #2041C2×C3semidirect
52. #2016C2×C3semidirect
53. #2015C2×C3semidirect
54. #2014C2×C3semidirect
55. #1817C2×C3semidirect
56. #1788C2×C3semidirect
591. C3#29629direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4522
C3×#28552
C3×#20432
C3×#20422
C3×#20412
C3×#20162
C3×#20152
C3×#20142
C3×#18172
C3×#17882
#296293