Finite Group Databasebeta

← C3×#14194 #10632801

Subgroups — C3×#14194 #10632801

Order 768.

1586 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C211yesmaximal, characteristic
2C2×C3×#388C212yesmaximal
3C2×C3×C4×C4⋊C4C212yesmaximal, complemented
4C3×#1445C212yesmaximal, complemented
5C3×#1440C211yesmaximal, characteristic
6C3×#1437C212yesmaximal
7C3×#1437C211yesmaximal, characteristic
8C3×#1436C212yesmaximal, complemented
9C3×#1436C212yesmaximal, complemented
10#14194C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C42C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2212yes—
22C3×#399C418yescomplemented
23C3×#388—28no—
24C3×C4×C4⋊C4—28no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2211yescharacteristic
35C2×#399C2×C311yescharacteristic
36C2×#388C2×C312yes—
37C2×C4×C4⋊C4C2×C312yescomplemented
38#1445C2×C312yescomplemented
39#1440C2×C311yescharacteristic
40#1437C2×C311yescharacteristic
41#1437C2×C312yes—
42#1436C2×C312yescomplemented
43#1436C2×C312yescomplemented
44C25×C3C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4—41no—
59C2×C3×C4⋊C4C2×C418yes—
60C2×C3×C4⋊C4C2×C414yes—
61C2×C3×C4⋊C4D418yes—
62C2×C3×C22⋊C4—24no—
63C2×C3×C22⋊C4—24no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—41no—
70C2×C3×C22⋊C4—41no—
71C2×C3×C22⋊C4—41no—
72C2×C3×C22⋊C4—41no—
73C2×C3×C22⋊C4C2×C418yes—
74C2×C3×C22⋊C4C2×C414yes—
75C2×C3×C42—24no—
76C2×C3×C42—24no—
77C2×C3×C42—24no—
78C2×C3×C42—24no—
79C2×C3×C42—24no—
80C2×C3×C42—24no—
81C2×C3×C42—41no—
82C2×C3×C42—42no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—24no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—42no—
91C3×#95—41no—
92C3×#95C2×C414yes—
93C22×C4⋊C4C22×C312yes—
94C22×C4⋊C4C22×C312yes—
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C22⋊C4C22×C312yes—
97C22×C22⋊C4C22×C311yescharacteristic
98C22×C22⋊C4C22×C311yescharacteristic
99C22×C22⋊C4C22×C312yes—
100C22×C22⋊C4C22×C311yescharacteristic
101C22×C42C22×C312yes—
102C22×C42C22×C312yes—
103C22×C42C22×C311yescharacteristic
104#399C3×C418yescomplemented
105#388—28no—
106C4×C4⋊C4—28no—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C311yescharacteristic
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—22no—
121C24×C3—22no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—22no—
125C24×C3C22×C412yes—
126C24×C3C2411yescharacteristic
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C22×C414yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C22×C414yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D414yes—
174C22×C3×C4C2×D414yes—
175C22×C3×C4#4114yes—
176C22×C3×C4#4114yes—
177C22×C3×C4#4114yes—
178C22×C3×C4#4114yes—
179C3×C4⋊C4—216no—
180C3×C4⋊C4—216no—
181C3×C4⋊C4—44no—
182C3×C22⋊C4—216no—
183C3×C22⋊C4—44no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—48no—
187C3×C42—216no—
188C3×C42—44no—
189C3×C42—48no—
190C25C23×C311yescharacteristic
191C23×C4C23×C312yes—
192C23×C4C23×C312yes—
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C311yescharacteristic
195C23×C4C23×C312yes—
196C23×C4C23×C312yes—
197C23×C4C23×C312yes—
198C23×C4C23×C312yes—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×C3×C418yes—
206C2×C4⋊C4C2×C3×C414yes—
207C2×C4⋊C4C3×D418yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4C2×C3×C418yes—
220C2×C22⋊C4C2×C3×C414yes—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—41no—
228C2×C42—42no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—42no—
237#95—41no—
238#95C2×C3×C414yes—
239C23×C3—24no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3C23×C411yescharacteristic
261C23×C3C22×D412yes—
262C23×C3C2×#4112yes—
263C23×C3C2×#4112yes—
264C23×C3C2×#4112yes—
265C23×C3C2×#4112yes—
266C23×C3H(5,𝔽2)11yescharacteristic
267C23×C3H(5,𝔽2)11yescharacteristic
268C23×C32−1+412yes—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4C4×D418yes—
320C2×C3×C4#12318yes—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24C22×C3×C412yes—
330C24C24×C311yescharacteristic
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4C22×C3×C412yes—
373C22×C4C22×C3×C414yes—
374C22×C4C22×C3×C412yes—
375C22×C4C22×C3×C414yes—
376C22×C4C2×C3×D414yes—
377C22×C4C2×C3×D414yes—
378C22×C4C2×C3×D414yes—
379C22×C4C3×#4114yes—
380C22×C4C3×#4114yes—
381C22×C4C3×#4114yes—
382C22×C4C3×#4114yes—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—44no—
386C22⋊C4—216no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—48no—
391C42—216no—
392C42—44no—
393C42—48no—
394C22×C3—42no—
395C22×C3—44no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3C2×C4×D412yes—
404C22×C3C4×#4111yescharacteristic
405C22×C3C4×#4112yes—
406C22×C3#51811yescharacteristic
407C22×C3#51911yescharacteristic
408C22×C3C2×#12312yes—
409C22×C3#52512yes—
410C22×C3#52912yes—
411C22×C3#52912yes—
412C22×C3#53912yes—
413C22×C3#54212yes—
414C22×C3D4211yescharacteristic
415C22×C3#54712yes—
416C22×C3#54812yes—
417C22×C3#55012yes—
418C22×C3#55111yescharacteristic
419C22×C3#55212yes—
420C22×C3#55312yes—
421C22×C3#55612yes—
422C22×C3#55911yescharacteristic
423C22×C3#55911yescharacteristic
424C3×C4—28no—
425C3×C4—28no—
426C3×C4—44no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—44no—
430C3×C4—44no—
431C3×C4—82no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23C23×C3×C411yescharacteristic
454C23C22×C3×D412yes—
455C23C2×C3×#4112yes—
456C23C2×C3×#4112yes—
457C23C2×C3×#4112yes—
458C23C2×C3×#4112yes—
459C23C3×H(5,𝔽2)11yescharacteristic
460C23C3×H(5,𝔽2)11yescharacteristic
461C23C3×2−1+412yes—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4C3×C4×D418yes—
513C2×C4C3×#12318yes—
514C2×C3—82no—
515C2×C3C4×#12312yes—
516C2×C3#233211yescharacteristic
517C2×C3#236212yes—
518C2×C3#236611yescharacteristic
519C2×C3#237411yescharacteristic
520C2×C3#244312yes—
521C2×C3#244812yes—
522C2×C3#247912yes—
523C2×C3#259612yes—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C2×C3×C4×D412yes—
534C22C3×C4×#4112yes—
535C22C3×C4×#4111yescharacteristic
536C22C3×#51811yescharacteristic
537C22C3×#51911yescharacteristic
538C22C2×C3×#12312yes—
539C22C3×#52512yes—
540C22C3×#52912yes—
541C22C3×#52912yes—
542C22C3×#53912yes—
543C22C3×#54212yes—
544C22C3×D4211yescharacteristic
545C22C3×#54712yes—
546C22C3×#54812yes—
547C22C3×#55012yes—
548C22C3×#55111yescharacteristic
549C22C3×#55212yes—
550C22C3×#55312yes—
551C22C3×#55612yes—
552C22C3×#55911yescharacteristic
553C22C3×#55911yescharacteristic
554C4—28no—
555C4—28no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—44no—
561C4—82no—
562C3#1419411yescharacteristic, direct product factor, Sylow
563C2—82no—
564C2C3×C4×#12312yes—
565C2C3×#233211yescharacteristic
566C2C3×#236212yes—
567C2C3×#236611yescharacteristic
568C2C3×#237411yescharacteristic
569C2C3×#244312yes—
570C2C3×#244812yes—
571C2C3×#247912yes—
572C2C3×#259612yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×C4⋊C4C2semidirect
4. C3×#1445C2semidirect
8. C3×#1436C2semidirect
9. C3×#1436C2semidirect
10. #14194C3direct (N × H)
22. C3×#399C4semidirect
37. C2×C4×C4⋊C4C2×C3semidirect
38. #1445C2×C3semidirect
42. #1436C2×C3semidirect
43. #1436C2×C3semidirect
104. #399C3×C4semidirect
562. C3#14194direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3882
C2×C3×C4×C4⋊C42
C3×#14452
C3×#14402
C3×#14372
C3×#14362
#141943