Finite Group Databasebeta

← C3×#32973 #10651580

Subgroups — C3×#32973 #10651580

Order 768.

1286 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#3080C212yesmaximal, complemented
3C3×#2566C211yesmaximal, characteristic, complemented
4C3×#1860C212yesmaximal, complemented
5C3×#1859C212yesmaximal, complemented
6C3×#1856C211yesmaximal, characteristic, complemented
7C3×#1856C211yesmaximal, characteristic, complemented
8C3×#1806C212yesmaximal, complemented
9C3×#1800C211yesmaximal, characteristic, complemented
10C3×#1792C211yesmaximal, characteristic
11C3×#1792C211yesmaximal, characteristic
12#32973C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×H(5,𝔽2)C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic, complemented
15C3×#520—22no—
16C3×#520C2212yes—
17C3×#518—22no—
18C3×#518—22no—
19C3×#518C2211yescharacteristic
20C3×C4×#41—22no—
21C3×C4×#41C2212yes—
22C2×C3×C4×D4—22no—
23C2×C3×C4×D4—22no—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#479—22no—
26C3×#479—22no—
27C3×#478—22no—
28C3×#478—22no—
29C3×#428C2212yescomplemented
30C3×#428C2212yescomplemented
31C3×#423C2212yescomplemented
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#417C2212yes—
35C3×#417C2212yes—
36C2×C3×#100C2211yescharacteristic
37C2×C3×#100C2211yescharacteristic
38C3×#396—22no—
39C3×#390C2211yescharacteristic
40C3×#390C2211yescharacteristic
41C3×#385—22no—
42C3×#384C2211yescharacteristic, complemented
43C3×#382C2211yescharacteristic, complemented
44C3×C4×C4⋊C4C2211yescharacteristic, complemented
45C3×#343C2212yescomplemented
46C3×#339C2212yescomplemented
47C3×#339C2212yescomplemented
48C4×H(5,𝔽2)C2×C311yescharacteristic
49#3080C2×C312yescomplemented
50#2566C2×C311yescharacteristic, complemented
51#1860C2×C312yescomplemented
52#1859C2×C312yescomplemented
53#1856C2×C311yescharacteristic, complemented
54#1856C2×C311yescharacteristic, complemented
55#1806C2×C312yescomplemented
56#1800C2×C311yescharacteristic, complemented
57#1792C2×C311yescharacteristic
58#1792C2×C311yescharacteristic
59C3×H(5,𝔽2)—24no—
60C3×H(5,𝔽2)—22no—
61C3×H(5,𝔽2)Q814yes—
62C2×C3×#41—22no—
63C2×C3×#41C2312yes—
64C22×C3×D4—22no—
65C22×C3×D4—22no—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4D412yes—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C3×#125—22no—
72C3×#125—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8D412yes—
75C3×C4×Q8D412yes—
76C3×C4×D4—24no—
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D412yes—
89C3×C4×D4D412yes—
90C3×#117—24no—
91C3×#117—22no—
92C3×#117—22no—
93C3×#117C2312yes—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—24no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4D412yes—
109C2×C3×C42—22no—
110C2×C3×C42—22no—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×#107—22no—
116C3×#106—22no—
117C3×#106—22no—
118C3×C4⋊C8—22no—
119C3×C4⋊C8—22no—
120C3×#104—24no—
121C3×#104—24no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#100—22no—
127C3×#100—22no—
128C3×#95—21no—
129C3×#95—22no—
130C2×H(5,𝔽2)C22×C311yescharacteristic
131C2×#125C22×C311yescharacteristic, complemented
132#520—22no—
133#520C22×C312yes—
134#518—22no—
135#518—22no—
136#518C22×C311yescharacteristic
137C4×#41—22no—
138C4×#41C22×C312yes—
139C2×C4×D4—22no—
140C2×C4×D4—22no—
141C2×C4×D4C22×C311yescharacteristic
142#479—22no—
143#479—22no—
144#478—22no—
145#478—22no—
146#428C22×C312yescomplemented
147#428C22×C312yescomplemented
148#423C22×C312yescomplemented
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#417C22×C312yes—
152#417C22×C312yes—
153C2×#100C22×C311yescharacteristic
154C2×#100C22×C311yescharacteristic
155#396—22no—
156#390C22×C311yescharacteristic
157#390C22×C311yescharacteristic
158#385—22no—
159#384C22×C311yescharacteristic, complemented
160#382C22×C311yescharacteristic, complemented
161C4×C4⋊C4C22×C311yescharacteristic, complemented
162#343C22×C312yescomplemented
163#339C22×C312yescomplemented
164#339C22×C312yescomplemented
165C24×C3—22no—
166C24×C3C2×D412yes—
167C3×#41—28no—
168C3×#41—28no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41C2×Q814yes—
172C3×#41#4114yes—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—24no—
175C2×C3×D4—28no—
176C2×C3×D4—28no—
177C2×C3×D4—28no—
178C2×C3×D4—22no—
179C2×C3×D4—28no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4C2×D412yes—
183C2×C3×D4C2×Q812yes—
184C2×C3×D4#4112yes—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—82no—
199C22×C3×C4—84no—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C3×M16—22no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4—41no—
220C3×C4⋊C4—82no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—44no—
230C3×C22⋊C4—82no—
231C3×C42—24no—
232C3×C42—22no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42C2×D412yes—
238C3×C42C2×D412yes—
239H(5,𝔽2)—24no—
240H(5,𝔽2)—22no—
241H(5,𝔽2)C3×Q814yes—
242C2×#41—22no—
243C2×#41C23×C312yes—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4C23×C311yescharacteristic
247C23×C4—22no—
248C23×C4C3×D412yes—
249C2×M16C23×C312yes—
250C2×M16C23×C312yes—
251#125—22no—
252#125—22no—
253C4×Q8—22no—
254C4×Q8C3×D412yes—
255C4×Q8C3×D412yes—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4C3×D412yes—
269C4×D4C3×D412yes—
270#117—24no—
271#117—22no—
272#117—22no—
273#117C23×C312yes—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4C3×D412yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42C23×C311yescharacteristic
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294#107—22no—
295#107—22no—
296#106—22no—
297#106—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300#104—24no—
301#104—24no—
302#103—22no—
303#103—22no—
304#102—22no—
305#102—22no—
306#100—22no—
307#100—22no—
308#95—21no—
309#95—22no—
310C23×C3—28no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—82no—
317C23×C3—82no—
318C23×C3C22⋊Q814yes—
319C23×C3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8C22⋊Q814yes—
322C3×D4—216no—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—22no—
327C3×D4—24no—
328C3×D4C22⋊Q814yes—
329C2×C3×C4—28no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—44no—
340C2×C3×C4—44no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—82no—
352C2×C3×C4—82no—
353C2×C3×C4—82no—
354C2×C3×C4—84no—
355C2×C3×C4—82no—
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4#12012yes—
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4C22×D412yes—
360C2×C3×C4C22×Q811yescharacteristic
361C2×C3×C4C2×#4112yes—
362C2×C3×C4C2×#4111yescharacteristic
363C3×C8—42no—
364C3×C8—42no—
365C24—22no—
366C24C2×C3×D412yes—
367#41—28no—
368#41—28no—
369#41—22no—
370#41—22no—
371#41C2×C3×Q814yes—
372#41C3×#4114yes—
373C2×Q8C2×C3×D412yes—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—28no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4C2×C3×D412yes—
383C2×D4C2×C3×Q812yes—
384C2×D4C3×#4112yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—82no—
399C22×C4—84no—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C24×C311yescharacteristic
404M16—22no—
405M16—22no—
406M16—22no—
407M16—22no—
408C2×C8—22no—
409C2×C8—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—82no—
421C4⋊C4C2×C3×D412yes—
422C4⋊C4C2×C3×D412yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42C2×C3×D412yes—
438C42C2×C3×D412yes—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—44no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—82no—
449C22×C3—88no—
450C22×C3C2×#12011yescharacteristic
451C22×C3C2×C22⋊Q811yescharacteristic
452C22×C3#52511yescharacteristic
453C3×C4—24no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4C2×C22⋊Q812yes—
465C3×C4#52512yes—
466C23—28no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—42no—
471C23—42no—
472C23—82no—
473C23—82no—
474C23C3×C22⋊Q814yes—
475C23C22×C3×D411yescharacteristic
476Q8—22no—
477Q8C3×C22⋊Q814yes—
478D4—216no—
479D4—24no—
480D4—24no—
481D4—24no—
482D4—22no—
483D4—24no—
484D4C3×C22⋊Q814yes—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4C3×#12011yescharacteristic
513C2×C4C3×#12012yes—
514C2×C4C3×#12011yescharacteristic
515C2×C4C22×C3×D412yes—
516C2×C4C22×C3×Q811yescharacteristic
517C2×C4C2×C3×#4112yes—
518C2×C4C2×C3×#4111yescharacteristic
519C8—42no—
520C8—42no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—44no—
524C2×C3—44no—
525C2×C3#284811yescharacteristic
526C2×C3#301811yescharacteristic
527C2×C3#301911yescharacteristic
528C22—21no—
529C22—21no—
530C22—24no—
531C22—24no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—88no—
539C22C2×C3×#12011yescharacteristic
540C22C2×C3×C22⋊Q811yescharacteristic
541C22C3×#52511yescharacteristic
542C4—22no—
543C4—21no—
544C4—21no—
545C4—24no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4C2×C3×C22⋊Q812yes—
554C4C3×#52512yes—
555C3#3297311yescharacteristic, direct product factor, Sylow
556C2—21no—
557C2—21no—
558C2—44no—
559C2—44no—
560C2C3×#284811yescharacteristic
561C2C3×#301811yescharacteristic
562C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3080C2semidirect
3. C3×#2566C2semidirect
4. C3×#1860C2semidirect
5. C3×#1859C2semidirect
6. C3×#1856C2semidirect
7. C3×#1856C2semidirect
8. C3×#1806C2semidirect
9. C3×#1800C2semidirect
12. #32973C3direct (N × H)
14. C2×C3×#125C22semidirect
29. C3×#428C22semidirect
30. C3×#428C22semidirect
31. C3×#423C22semidirect
42. C3×#384C22semidirect
43. C3×#382C22semidirect
44. C3×C4×C4⋊C4C22semidirect
45. C3×#343C22semidirect
46. C3×#339C22semidirect
47. C3×#339C22semidirect
49. #3080C2×C3semidirect
50. #2566C2×C3semidirect
51. #1860C2×C3semidirect
52. #1859C2×C3semidirect
53. #1856C2×C3semidirect
54. #1856C2×C3semidirect
55. #1806C2×C3semidirect
56. #1800C2×C3semidirect
131. C2×#125C22×C3semidirect
146. #428C22×C3semidirect
147. #428C22×C3semidirect
148. #423C22×C3semidirect
159. #384C22×C3semidirect
160. #382C22×C3semidirect
161. C4×C4⋊C4C22×C3semidirect
162. #343C22×C3semidirect
163. #339C22×C3semidirect
164. #339C22×C3semidirect
555. C3#32973direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C3×#30802
C3×#25662
C3×#18602
C3×#18592
C3×#18562
C3×#18062
C3×#18002
C3×#17922
#329733