Finite Group Databasebeta

← C3×#10931 #10629538

Subgroups — C3×#10931 #10629538

Order 768.

1354 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#102C212yesmaximal
2C2×C3×#390C211yesmaximal, characteristic
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×#340C212yesmaximal, complemented
5C2×C3×#336C211yesmaximal, characteristic, complemented
6#10931C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4C2211yescharacteristic
8C23×C3×C8C2212yes—
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×C3×C4⋊C4C2212yes—
11C22×C3×C22⋊C4—21no—
12C22×C3×C22⋊C4—21no—
13C22×C3×C42—21no—
14C22×C3×C42C2211yescharacteristic
15C2×C3×#102—24no—
16C2×C3×#102—24no—
17C2×C3×#102C418yescomplemented
18C3×#390—22no—
19C3×#390C414yescomplemented
20C3×C4×C4⋊C4—24no—
21C2×C3×#95—21no—
22C3×#340—24no—
23C3×#336—22no—
24C22×#102C2×C312yes—
25C2×#390C2×C311yescharacteristic
26C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
27C2×#340C2×C312yescomplemented
28C2×#336C2×C311yescharacteristic, complemented
29C25×C3—21no—
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—21no—
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4C2×C412yes—
37C22×C3×D4C2×C414yes—
38C23×C3×C4—22no—
39C23×C3×C4—21no—
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×C3×C8—22no—
44C22×C3×C8—22no—
45C22×C3×C8—22no—
46C22×C3×C8—22no—
47C22×C3×C8—22no—
48C22×C3×C8—22no—
49C22×C3×C8C2×C414yes—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4C2×C412yes—
57C2×C3×C4⋊C4C2×C414yes—
58C2×C3×C4⋊C4D414yes—
59C2×C3×C4⋊C4D414yes—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—44no—
63C2×C3×C22⋊C4—42no—
64C2×C3×C22⋊C4—42no—
65C2×C3×C42—22no—
66C2×C3×C42—22no—
67C2×C3×C42—24no—
68C2×C3×C42—42no—
69C2×C3×C42—42no—
70C2×C3×C42C2×C412yes—
71C2×C3×C42D412yes—
72C3×#102—216no—
73C3×#95—24no—
74C23×D4C22×C311yescharacteristic
75C23×C8C22×C312yes—
76C22×C4⋊C4C22×C311yescharacteristic
77C22×C4⋊C4C22×C312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—21no—
81C22×C42C22×C311yescharacteristic
82C2×#102—24no—
83C2×#102—24no—
84C2×#102C3×C418yescomplemented
85#390—22no—
86#390C3×C414yescomplemented
87C4×C4⋊C4—24no—
88C2×#95—21no—
89#340—24no—
90#336—22no—
91C24×C3—22no—
92C24×C3—24no—
93C24×C3—41no—
94C24×C3—42no—
95C24×C3—41no—
96C24×C3—41no—
97C24×C3—41no—
98C24×C3—42no—
99C24×C3—41no—
100C24×C3—42no—
101C24×C3—41no—
102C24×C3C2×D411yescharacteristic
103C2×C3×D4—24no—
104C2×C3×D4—24no—
105C2×C3×D4—22no—
106C2×C3×D4—24no—
107C2×C3×D4—24no—
108C2×C3×D4—24no—
109C2×C3×D4—24no—
110C2×C3×D4—24no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—41no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C2×C3×D4—41no—
118C2×C3×D4C4214yescomplemented
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—24no—
124C22×C3×C4—21no—
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—42no—
139C22×C3×C4C22⋊C412yes—
140C22×C3×C4C22⋊C412yes—
141C22×C3×C4D812yes—
142C22×C3×C4SD1612yes—
143C22×C3×C4C22×C412yes—
144C22×C3×C4C22×C411yescharacteristic
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4#4112yes—
149C22×C3×C4#4112yes—
150C22×C3×C4#4112yes—
151C2×C3×C8—24no—
152C2×C3×C8—24no—
153C2×C3×C8—24no—
154C2×C3×C8—42no—
155C2×C3×C8—42no—
156C2×C3×C8—42no—
157C2×C3×C8—42no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4C22⋊C418yescomplemented
165C3×C22⋊C4—48no—
166C3×C22⋊C4—48no—
167C3×C42—24no—
168C3×C42—24no—
169C3×C42—48no—
170C25—21no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4C2×C3×C412yes—
178C22×D4C2×C3×C414yes—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4C3×D411yescharacteristic
182C23×C4C3×D411yescharacteristic
183C23×C4C23×C311yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×C3×C414yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C3×C412yes—
198C2×C4⋊C4C2×C3×C414yes—
199C2×C4⋊C4C3×D414yes—
200C2×C4⋊C4C3×D414yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C2×C3×C412yes—
212C2×C42C3×D412yes—
213#102—216no—
214#95—24no—
215C23×C3—24no—
216C23×C3—42no—
217C23×C3—44no—
218C23×C3—44no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—44no—
225C23×C3—44no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—81no—
229C23×C3—81no—
230C23×C3—82no—
231C23×C3—82no—
232C23×C3—82no—
233C23×C3—82no—
234C23×C3—82no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3—82no—
238C23×C3C2×C22⋊C411yescharacteristic
239C23×C3C4×D412yes—
240C23×C3#12011yescharacteristic
241C23×C3#12112yes—
242C23×C3#12311yescharacteristic
243C23×C3C2×D811yescharacteristic
244C23×C3C2×SD1611yescharacteristic
245C23×C3#13512yes—
246C23×C3AGL(1,ℤ/8)11yescharacteristic
247C23×C3AGL(1,ℤ/8)12yes—
248C23×C3#13711yescharacteristic
249C3×D4—28no—
250C3×D4—44no—
251C3×D4—44no—
252C3×D4—44no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—28no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—44no—
269C2×C3×C4—44no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—82no—
275C2×C3×C4—82no—
276C2×C3×C4#10214yescomplemented
277C2×C3×C4C2×C4211yescharacteristic
278C2×C3×C4C2×C22⋊C412yes—
279C2×C3×C4#11712yes—
280C2×C3×C4#11711yescharacteristic
281C2×C3×C4C4×D412yes—
282C2×C3×C4C4×D412yes—
283C2×C3×C4C4×D412yes—
284C2×C3×C4C4×D412yes—
285C2×C3×C4C4×D412yes—
286C2×C3×C4#12112yes—
287C2×C3×C4#12112yes—
288C2×C3×C4#12312yes—
289C2×C3×C4#12411yescharacteristic
290C2×C3×C4#12412yes—
291C2×C3×C4#12612yes—
292C2×C3×C4#12711yescharacteristic
293C3×C8—44no—
294C24—22no—
295C24—24no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—42no—
302C24—42no—
303C24—41no—
304C24—41no—
305C24C2×C3×D411yescharacteristic
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4C3×C4214yescomplemented
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4C3×C22⋊C412yes—
343C22×C4C3×C22⋊C412yes—
344C22×C4C3×D812yes—
345C22×C4C3×SD1612yes—
346C22×C4C22×C3×C412yes—
347C22×C4C22×C3×C411yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C3×#4112yes—
352C22×C4C3×#4112yes—
353C22×C4C3×#4112yes—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C3×C22⋊C418yescomplemented
368C22⋊C4—48no—
369C22⋊C4—48no—
370C42—24no—
371C42—24no—
372C42—48no—
373C22×C3—44no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—82no—
377C22×C3—82no—
378C22×C3—84no—
379C22×C3—84no—
380C22×C3—84no—
381C22×C3—84no—
382C22×C3—82no—
383C22×C3—82no—
384C22×C3C4×C22⋊C411yescharacteristic
385C22×C3#38611yescharacteristic
386C22×C3#38612yes—
387C22×C3#38812yes—
388C22×C3#39011yescharacteristic
389C22×C3C2×#10211yescharacteristic
390C22×C3#41611yescharacteristic
391C22×C3#41711yescharacteristic
392C22×C3#41811yescharacteristic
393C22×C3C4×D812yes—
394C22×C3C4×SD1612yes—
395C22×C3#44012yes—
396C22×C3#44212yes—
397C22×C3#44711yescharacteristic
398C22×C3#44811yescharacteristic
399C22×C3#44912yes—
400C22×C3#45912yes—
401C22×C3#46012yes—
402C22×C3#46312yes—
403C22×C3#46412yes—
404C22×C3#48011yescharacteristic
405C22×C3#48111yescharacteristic
406C22×C3#48212yes—
407C3×C4—24no—
408C3×C4—42no—
409C3×C4—48no—
410C3×C4—82no—
411C3×C4C4×C22⋊C412yes—
412C3×C4#38812yes—
413C3×C4#38812yes—
414C3×C4#39012yes—
415C23—24no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—81no—
429C23—81no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—81no—
436C23—81no—
437C23—82no—
438C23C2×C3×C22⋊C411yescharacteristic
439C23C3×C4×D412yes—
440C23C3×#12011yescharacteristic
441C23C3×#12112yes—
442C23C3×#12311yescharacteristic
443C23C2×C3×D811yescharacteristic
444C23C2×C3×SD1611yescharacteristic
445C23C3×#13512yes—
446C23C3×AGL(1,ℤ/8)11yescharacteristic
447C23C3×AGL(1,ℤ/8)12yes—
448C23C3×#13711yescharacteristic
449D4—28no—
450D4—44no—
451D4—44no—
452D4—44no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4C3×#10214yescomplemented
477C2×C4C2×C3×C4211yescharacteristic
478C2×C4C2×C3×C22⋊C412yes—
479C2×C4C3×#11712yes—
480C2×C4C3×#11711yescharacteristic
481C2×C4C3×C4×D412yes—
482C2×C4C3×C4×D412yes—
483C2×C4C3×C4×D412yes—
484C2×C4C3×C4×D412yes—
485C2×C4C3×C4×D412yes—
486C2×C4C3×#12112yes—
487C2×C4C3×#12112yes—
488C2×C4C3×#12312yes—
489C2×C4C3×#12411yescharacteristic
490C2×C4C3×#12412yes—
491C2×C4C3×#12612yes—
492C2×C4C3×#12711yescharacteristic
493C8—44no—
494C2×C3—84no—
495C2×C3#143711yescharacteristic
496C2×C3C4×#10211yescharacteristic
497C2×C3#176211yescharacteristic
498C2×C3#187411yescharacteristic
499C2×C3#187511yescharacteristic
500C2×C3#187912yes—
501C2×C3#188012yes—
502C2×C3#192512yes—
503C2×C3#192612yes—
504C2×C3#198211yescharacteristic
505C2×C3#198311yescharacteristic
506C22—44no—
507C22—44no—
508C22—44no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—82no—
516C22—82no—
517C22C3×C4×C22⋊C411yescharacteristic
518C22C3×#38611yescharacteristic
519C22C3×#38612yes—
520C22C3×#38812yes—
521C22C3×#39011yescharacteristic
522C22C2×C3×#10211yescharacteristic
523C22C3×#41611yescharacteristic
524C22C3×#41711yescharacteristic
525C22C3×#41811yescharacteristic
526C22C3×C4×D812yes—
527C22C3×C4×SD1612yes—
528C22C3×#44012yes—
529C22C3×#44212yes—
530C22C3×#44711yescharacteristic
531C22C3×#44811yescharacteristic
532C22C3×#44912yes—
533C22C3×#45912yes—
534C22C3×#46012yes—
535C22C3×#46312yes—
536C22C3×#46412yes—
537C22C3×#48011yescharacteristic
538C22C3×#48111yescharacteristic
539C22C3×#48212yes—
540C4—24no—
541C4—42no—
542C4—48no—
543C4—82no—
544C4C3×C4×C22⋊C412yes—
545C4C3×#38812yes—
546C4C3×#38812yes—
547C4C3×#39012yes—
548C3#1093111yescharacteristic, direct product factor, Sylow
549C2—84no—
550C2C3×#143711yescharacteristic
551C2C3×C4×#10211yescharacteristic
552C2C3×#176211yescharacteristic
553C2C3×#187411yescharacteristic
554C2C3×#187511yescharacteristic
555C2C3×#187912yes—
556C2C3×#188012yes—
557C2C3×#192512yes—
558C2C3×#192612yes—
559C2C3×#198211yescharacteristic
560C2C3×#198311yescharacteristic
subgroupG/H
center, socleC24×C3C2×D4
commutatorC2×C4C2×C3×C42
Frattini, cosocleC23×C4C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×C4⋊C4C2semidirect
4. C2×C3×#340C2semidirect
5. C2×C3×#336C2semidirect
6. #10931C3direct (N × H)
17. C2×C3×#102C4semidirect
19. C3×#390C4semidirect
26. C2×C4×C4⋊C4C2×C3semidirect
27. C2×#340C2×C3semidirect
28. C2×#336C2×C3semidirect
84. C2×#102C3×C4semidirect
86. #390C3×C4semidirect
118. C2×C3×D4C42semidirect
164. C3×C4⋊C4C22⋊C4semidirect
276. C2×C3×C4#102semidirect
321. C2×D4C3×C42semidirect
367. C4⋊C4C3×C22⋊C4semidirect
476. C2×C4C3×#102semidirect
548. C3#10931direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1022
C2×C3×#3902
C2×C3×C4×C4⋊C42
C2×C3×#3402
C2×C3×#3362
#109313