Finite Group Databasebeta

← C3×#27679 #10646286

Subgroups — C3×#27679 #10646286

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C3×#2285C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2092C212yesmaximal, complemented
5C3×#1948C211yesmaximal, characteristic, complemented
6C3×#1947C211yesmaximal, characteristic, complemented
7C3×#1875C212yesmaximal
8C3×#1854C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic, complemented
10C3×#1802C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#27679C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#117C2212yes—
17C22×C3×C42C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C22⋊C8C2212yes—
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#388—21no—
28C3×#386—21no—
29C3×#384C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C3×#382C2212yes—
32C3×#380—21no—
33C3×#380—21no—
34C3×#380C2211yescharacteristic
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C4×#121C2×C311yescharacteristic
45#2285C2×C311yescharacteristic, complemented
46#2101C2×C312yescomplemented
47#2092C2×C312yescomplemented
48#1948C2×C311yescharacteristic, complemented
49#1947C2×C311yescharacteristic, complemented
50#1875C2×C312yes—
51#1854C2×C312yescomplemented
52#1802C2×C311yescharacteristic, complemented
53#1802C2×C311yescharacteristic, complemented
54#1797C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×#121—22no—
64C3×#121Q814yes—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—44no—
68C3×#117—42no—
69C3×#117—42no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—21no—
89C2×C3×C42—22no—
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×#107—42no—
95C3×#106—42no—
96C3×C4⋊C8—22no—
97C3×C4⋊C8—22no—
98C3×#102—42no—
99C3×#102—42no—
100C3×C22⋊C8—24no—
101C3×#95—21no—
102C3×#95—21no—
103C3×#95—22no—
104C3×#95—22no—
105C3×#95—21no—
106C2×#121C22×C311yescharacteristic
107C2×C4×D4—21no—
108C2×C4×D4C22×C311yescharacteristic
109C2×#117C22×C312yes—
110C22×C42C22×C311yescharacteristic
111C2×#107C22×C312yes—
112C2×#106C22×C312yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×C22⋊C8C22×C312yes—
118#390C22×C311yescharacteristic
119#389C22×C311yescharacteristic
120#388—21no—
121#386—21no—
122#384C22×C311yescharacteristic
123#384C22×C311yescharacteristic
124#382C22×C312yes—
125#380—21no—
126#380—21no—
127#380C22×C311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C22×C311yescharacteristic
131C4×C22⋊C4—21no—
132#340C22×C312yes—
133#340C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136#336C22×C312yes—
137C24×C3—21no—
138C24×C3C2×D411yescharacteristic
139C2×C3×D4—24no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4C2×Q812yes—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—41no—
162C22×C3×C4—41no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×Q812yes—
172C22×C3×C4#4112yes—
173C22×C3×C4C2411yescharacteristic
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4C2×Q812yes—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—42no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—21no—
204C3×C42—21no—
205C3×C42—44no—
206C3×C42—42no—
207C3×C42#4112yes—
208C3×C42#4112yes—
209C22×D4—21no—
210C22×D4C23×C311yescharacteristic
211C23×C4—21no—
212C23×C4C3×D411yescharacteristic
213C23×C4C3×D411yescharacteristic
214C23×C4C23×C311yescharacteristic
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217#121—22no—
218#121C3×Q814yes—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—44no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C312yes—
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42C3×D411yescharacteristic
245C2×C42C3×D411yescharacteristic
246C2×C42C23×C311yescharacteristic
247C2×C42C23×C311yescharacteristic
248#107—42no—
249#106—42no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252#102—42no—
253#102—42no—
254C22⋊C8—24no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3C22⋊Q812yes—
274C23×C3C22×D411yescharacteristic
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—48no—
278C3×D4—41no—
279C3×D4—41no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—44no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—44no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—84no—
312C2×C3×C4C22⋊Q812yes—
313C2×C3×C4#12312yes—
314C2×C3×C4#12312yes—
315C2×C3×C4#13512yes—
316C2×C3×C4AGL(1,ℤ/8)12yes—
317C2×C3×C4C22×Q811yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4H(5,𝔽2)11yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C3×C8—42no—
324C3×C8—42no—
325C24—21no—
326C24C2×C3×D411yescharacteristic
327C2×D4—24no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4C2×C3×Q812yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C2×C3×Q812yes—
360C22×C4C3×#4112yes—
361C22×C4C24×C311yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4C2×C3×Q812yes—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C42—22no—
390C42—22no—
391C42—21no—
392C42—21no—
393C42—44no—
394C42—42no—
395C42C3×#4112yes—
396C42C3×#4112yes—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3C2×C22⋊Q811yescharacteristic
412C22×C3C2×#12311yescharacteristic
413C22×C3#53511yescharacteristic
414C22×C3C2×#13511yescharacteristic
415C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C3#57511yescharacteristic
417C22×C3#57511yescharacteristic
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—81no—
424C3×C4—82no—
425C3×C4—82no—
426C3×C4—81no—
427C3×C4#48214yes—
428C3×C4#55311yescharacteristic
429C3×C4#55311yescharacteristic
430C3×C4#55411yescharacteristic
431C3×C4#55411yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C3×C22⋊Q812yes—
446C23C22×C3×D411yescharacteristic
447D4—22no—
448D4—22no—
449D4—48no—
450D4—41no—
451D4—41no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—84no—
484C2×C4C3×C22⋊Q812yes—
485C2×C4C3×#12312yes—
486C2×C4C3×#12312yes—
487C2×C4C3×#13512yes—
488C2×C4C3×AGL(1,ℤ/8)12yes—
489C2×C4C22×C3×Q811yescharacteristic
490C2×C4C2×C3×#4111yescharacteristic
491C2×C4C2×C3×#4111yescharacteristic
492C2×C4C2×C3×#4111yescharacteristic
493C2×C4C3×H(5,𝔽2)11yescharacteristic
494C2×C4C3×H(5,𝔽2)11yescharacteristic
495C8—42no—
496C8—42no—
497C2×C3—42no—
498C2×C3—82no—
499C2×C3#248711yescharacteristic
500C2×C3#307611yescharacteristic
501C2×C3#307711yescharacteristic
502C2×C3C2×#48211yescharacteristic
503C2×C3#309311yescharacteristic
504C2×C3#321211yescharacteristic
505C2×C3#321411yescharacteristic
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22C2×C3×C22⋊Q811yescharacteristic
521C22C2×C3×#12311yescharacteristic
522C22C3×#53511yescharacteristic
523C22C2×C3×#13511yescharacteristic
524C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
525C22C3×#57511yescharacteristic
526C22C3×#57511yescharacteristic
527C4—22no—
528C4—22no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—82no—
533C4—81no—
534C4—82no—
535C4—81no—
536C4C3×#48214yes—
537C4C3×#55311yescharacteristic
538C4C3×#55311yescharacteristic
539C4C3×#55411yescharacteristic
540C4C3×#55411yescharacteristic
541C3#2767911yescharacteristic, direct product factor, Sylow
542C2—42no—
543C2—82no—
544C2C3×#248711yescharacteristic
545C2C3×#307611yescharacteristic
546C2C3×#307711yescharacteristic
547C2C2×C3×#48211yescharacteristic
548C2C3×#309311yescharacteristic
549C2C3×#321211yescharacteristic
550C2C3×#321411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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. C3×#2285C2semidirect
3. C3×#2101C2semidirect
4. C3×#2092C2semidirect
5. C3×#1948C2semidirect
6. C3×#1947C2semidirect
8. C3×#1854C2semidirect
9. C3×#1802C2semidirect
10. C3×#1802C2semidirect
11. C3×#1797C2semidirect
12. #27679C3direct (N × H)
45. #2285C2×C3semidirect
46. #2101C2×C3semidirect
47. #2092C2×C3semidirect
48. #1948C2×C3semidirect
49. #1947C2×C3semidirect
51. #1854C2×C3semidirect
52. #1802C2×C3semidirect
53. #1802C2×C3semidirect
54. #1797C2×C3semidirect
541. C3#27679direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C3×#22852
C3×#21012
C3×#20922
C3×#19482
C3×#19472
C3×#18752
C3×#18542
C3×#18022
C3×#17972
#276793