Finite Group Databasebeta

← C3×#29698 #10648305

Subgroups — C3×#29698 #10648305

Order 768.

1076 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×#540C211yesmaximal, characteristic
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic, complemented
5C3×#2059C212yesmaximal, complemented
6C3×#2041C212yesmaximal, complemented
7C3×#2034C212yesmaximal, complemented
8C3×#2033C212yesmaximal, complemented
9C3×#1804C211yesmaximal, characteristic, complemented
10C3×#1804C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29698C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#540—24no—
14C3×#540—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×#123—21no—
20C2×C3×#121—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#476—24no—
28C3×#474—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#540C2×C311yescharacteristic
49C2×#476C2×C311yescharacteristic, complemented
50C2×#474C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic, complemented
52#2059C2×C312yescomplemented
53#2041C2×C312yescomplemented
54#2034C2×C312yescomplemented
55#2033C2×C312yescomplemented
56#1804C2×C311yescharacteristic, complemented
57#1804C2×C311yescharacteristic, complemented
58#1798C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—22no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#125—22no—
76C3×#125—22no—
77C3×#123—28no—
78C3×#123—28no—
79C3×#121—28no—
80C3×#121—28no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#121—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4D414yes—
88C3×C4×D4D414yes—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#106—24no—
112C3×C4⋊C8—22no—
113C3×C4⋊C8—22no—
114C3×#102—24no—
115C3×#102—24no—
116C3×#102—42no—
117C3×#102—42no—
118C3×C22⋊C8—42no—
119C3×#95—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—22no—
123#540—24no—
124#540—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#127C22×C311yescharacteristic
127C2×#125C22×C311yescharacteristic
128C2×#123—21no—
129C2×#123—21no—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121C22×C311yescharacteristic
133C2×#121C22×C311yescharacteristic
134C2×C4×D4C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137#476—24no—
138#474—24no—
139C2×#107C22×C312yes—
140C2×#106C22×C312yes—
141C2×C4⋊C8C22×C311yescharacteristic
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#102C22×C312yes—
144C2×#102C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#400—21no—
149#399—21no—
150#395—21no—
151#393—21no—
152#389C22×C312yes—
153#382C22×C311yescharacteristic
154#380C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3—21no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—42no—
162C2×C3×Q8—82no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—82no—
178C2×C3×D4—82no—
179C2×C3×D4—82no—
180C2×C3×D4—82no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D412yes—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—82no—
196C22×C3×C4—82no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—82no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—82no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234C22×Q8—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C23×C311yescharacteristic
240C22×D4C23×C311yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C23×C311yescharacteristic
244C22×C8C23×C312yes—
245C22×C8C23×C312yes—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248#127—22no—
249#127—22no—
250#125—22no—
251#125—22no—
252#123—28no—
253#123—28no—
254#121—28no—
255#121—28no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4C3×D414yes—
263C4×D4C3×D414yes—
264#117—22no—
265#117—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C3×D411yescharacteristic
272C2×C4⋊C4C23×C312yes—
273C2×C4⋊C4C23×C312yes—
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4C3×D411yescharacteristic
282C2×C22⋊C4C3×D411yescharacteristic
283C2×C42C23×C311yescharacteristic
284C2×C42C23×C311yescharacteristic
285#107—24no—
286#106—24no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289#102—24no—
290#102—24no—
291#102—42no—
292#102—42no—
293C22⋊C8—42no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3C22×D411yescharacteristic
314C3×Q8—84no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—41no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—84no—
324C3×D4—84no—
325C3×D4—84no—
326C3×D4—84no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—84no—
351C2×C3×C4—84no—
352C2×C3×C4#12012yes—
353C2×C3×C4#12012yes—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C2×C3×C42−1+411yescharacteristic
360C3×C8—42no—
361C3×C8—42no—
362C24—21no—
363C24—21no—
364C24C2×C3×D411yescharacteristic
365C2×Q8—42no—
366C2×Q8—82no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4C2×C3×D412yes—
386C2×D4C2×C3×D412yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C24×C311yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—82no—
425C4⋊C4C2×C3×D412yes—
426C4⋊C4C2×C3×D412yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—82no—
434C42—21no—
435C42—21no—
436C42C2×C3×D412yes—
437C42C2×C3×D412yes—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—42no—
449C22×C3—82no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3C2×#12011yescharacteristic
454C22×C3#53611yescharacteristic
455C22×C3#53711yescharacteristic
456C22×C3#57612yes—
457C22×C3#57712yes—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—42no—
463C3×C4—41no—
464C3×C4—42no—
465C3×C4—82no—
466C3×C4—82no—
467C3×C4#53611yescharacteristic
468C3×C4#53611yescharacteristic
469C3×C4#53711yescharacteristic
470C3×C4#53711yescharacteristic
471C23—22no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23C22×C3×D411yescharacteristic
487Q8—84no—
488D4—22no—
489D4—22no—
490D4—22no—
491D4—22no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—84no—
497D4—84no—
498D4—84no—
499D4—84no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4C3×#12012yes—
526C2×C4C3×#12012yes—
527C2×C4C22×C3×D411yescharacteristic
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C3×H(5,𝔽2)11yescharacteristic
530C2×C4C3×2−1+411yescharacteristic
531C2×C4C3×2−1+411yescharacteristic
532C2×C4C3×2−1+411yescharacteristic
533C8—42no—
534C8—42no—
535C2×C3—81no—
536C2×C3—82no—
537C2×C3—82no—
538C2×C3#261611yescharacteristic
539C2×C3#300811yescharacteristic
540C2×C3#300911yescharacteristic
541C2×C3#324112yes—
542C2×C3#325712yes—
543C22—41no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C2×C3×#12011yescharacteristic
559C22C3×#53611yescharacteristic
560C22C3×#53711yescharacteristic
561C22C3×#57612yes—
562C22C3×#57712yes—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—82no—
571C4—82no—
572C4C3×#53611yescharacteristic
573C4C3×#53611yescharacteristic
574C4C3×#53711yescharacteristic
575C4C3×#53711yescharacteristic
576C3#2969811yescharacteristic, direct product factor, Sylow
577C2—81no—
578C2—82no—
579C2—82no—
580C2C3×#261611yescharacteristic
581C2C3×#300811yescharacteristic
582C2C3×#300911yescharacteristic
583C2C3×#324112yes—
584C2C3×#325712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#476C2semidirect
3. C2×C3×#474C2semidirect
4. C3×#2628C2semidirect
5. C3×#2059C2semidirect
6. C3×#2041C2semidirect
7. C3×#2034C2semidirect
8. C3×#2033C2semidirect
9. C3×#1804C2semidirect
10. C3×#1804C2semidirect
11. C3×#1798C2semidirect
12. #29698C3direct (N × H)
49. C2×#476C2×C3semidirect
50. C2×#474C2×C3semidirect
51. #2628C2×C3semidirect
52. #2059C2×C3semidirect
53. #2041C2×C3semidirect
54. #2034C2×C3semidirect
55. #2033C2×C3semidirect
56. #1804C2×C3semidirect
57. #1804C2×C3semidirect
58. #1798C2×C3semidirect
576. C3#29698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4762
C2×C3×#4742
C3×#26282
C3×#20592
C3×#20412
C3×#20342
C3×#20332
C3×#18042
C3×#17982
#296983