Finite Group Databasebeta

← C3×#15068 #10633675

Subgroups — C3×#15068 #10633675

Order 768.

1254 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×C3×#387C211yesmaximal, characteristic
3C3×#1444C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic
5C3×#1444C212yesmaximal
6C3×#1442C211yesmaximal, characteristic
7C3×#1442C211yesmaximal, characteristic
8C3×#1441C212yesmaximal
9C3×#1436C212yesmaximal
10C3×C4⋊#95C212yesmaximal
11C3×C4×#95C211yesmaximal, characteristic
12#15068C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C42C2212yes—
21C22×C3×C42C2211yescharacteristic
22C3×#389—24no—
23C3×#387—24no—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2211yescharacteristic
37C2×C3×#95C2211yescharacteristic
38C2×#389C2×C311yescharacteristic
39C2×#387C2×C311yescharacteristic
40#1444C2×C311yescharacteristic
41#1444C2×C311yescharacteristic
42#1444C2×C312yes—
43#1442C2×C311yescharacteristic
44#1442C2×C311yescharacteristic
45#1441C2×C312yes—
46#1436C2×C312yes—
47C4⋊#95C2×C312yes—
48C4×#95C2×C311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4—42no—
74C2×C3×C4⋊C4—42no—
75C2×C3×C4⋊C4—42no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4Q814yes—
78C2×C3×C42—24no—
79C2×C3×C42—24no—
80C2×C3×C42—42no—
81C2×C3×C42—42no—
82C2×C3×C42D414yes—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—42no—
93C3×#95—42no—
94C3×#95—42no—
95C3×#95Q814yes—
96C3×#95Q814yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C312yes—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C312yes—
101C22×C4⋊C4C22×C312yes—
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C42C22×C312yes—
105C22×C42C22×C311yescharacteristic
106#389—24no—
107#387—24no—
108C2×#95C22×C312yes—
109C2×#95C22×C311yescharacteristic
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C312yes—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C311yescharacteristic
121C2×#95C22×C311yescharacteristic
122C24×C3C2411yescharacteristic
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
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—21no—
144C22×C3×C4—21no—
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—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×Q814yes—
176C22×C3×C4C2×Q812yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4C2×Q812yes—
179C22×C3×C4C2×Q814yes—
180C22×C3×C4C2×Q812yes—
181C22×C3×C4C2×Q812yes—
182C22×C3×C4#4114yes—
183C22×C3×C4#4112yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4114yes—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—48no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—48no—
191C3×C4⋊C4—44no—
192C3×C4⋊C4—48no—
193C3×C42—44no—
194C3×C42—48no—
195C23×C4C23×C311yescharacteristic
196C23×C4C23×C312yes—
197C23×C4C23×C311yescharacteristic
198C23×C4C23×C311yescharacteristic
199C23×C4C23×C311yescharacteristic
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C312yes—
202C23×C4C23×C312yes—
203C23×C4C23×C311yescharacteristic
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C312yes—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C3×Q814yes—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C3×D414yes—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95C3×Q814yes—
242#95C3×Q814yes—
243C23×C3C22×D411yescharacteristic
244C23×C3C22×Q811yescharacteristic
245C23×C3C22×Q811yescharacteristic
246C23×C3C22×Q811yescharacteristic
247C23×C3C2×#4111yescharacteristic
248C23×C3C2×#4112yes—
249C23×C3C2×#4112yes—
250C23×C3C2×#4112yes—
251C23×C3H(5,𝔽2)12yes—
252C23×C3H(5,𝔽2)11yescharacteristic
253C23×C3H(5,𝔽2)11yescharacteristic
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4C22⋊Q814yes—
315C2×C3×C4#12518yes—
316C2×C3×C4C4⋊Q814yes—
317C24C24×C311yescharacteristic
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4C2×C3×D412yes—
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D412yes—
370C22×C4C2×C3×Q814yes—
371C22×C4C2×C3×Q812yes—
372C22×C4C2×C3×Q812yes—
373C22×C4C2×C3×Q812yes—
374C22×C4C2×C3×Q814yes—
375C22×C4C2×C3×Q812yes—
376C22×C4C2×C3×Q812yes—
377C22×C4C3×#4114yes—
378C22×C4C3×#4112yes—
379C22×C4C3×#4114yes—
380C22×C4C3×#4114yes—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C42—44no—
389C42—48no—
390C22×C3C2×C22⋊Q811yescharacteristic
391C22×C3#52511yescharacteristic
392C22×C3C2×#12512yes—
393C22×C3#52912yes—
394C22×C3C2×C4⋊Q811yescharacteristic
395C22×C3#53211yescharacteristic
396C22×C3#53211yescharacteristic
397C22×C3#53312yes—
398C22×C3#53312yes—
399C22×C3#53411yescharacteristic
400C22×C3#53511yescharacteristic
401C22×C3#53812yes—
402C22×C3#54012yes—
403C22×C3#55112yes—
404C22×C3#55112yes—
405C22×C3#55312yes—
406C22×C3#55412yes—
407C22×C3#55411yescharacteristic
408C22×C3#55411yescharacteristic
409C22×C3#55412yes—
410C22×C3#55412yes—
411C22×C3Q8211yescharacteristic
412C22×C3Q8211yescharacteristic
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—28no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—44no—
421C3×C4—81no—
422C3×C4—82no—
423C3×C4—81no—
424C23C22×C3×D411yescharacteristic
425C23C22×C3×Q811yescharacteristic
426C23C22×C3×Q811yescharacteristic
427C23C22×C3×Q811yescharacteristic
428C23C2×C3×#4111yescharacteristic
429C23C2×C3×#4112yes—
430C23C2×C3×#4112yes—
431C23C2×C3×#4112yes—
432C23C3×H(5,𝔽2)11yescharacteristic
433C23C3×H(5,𝔽2)11yescharacteristic
434C23C3×H(5,𝔽2)12yes—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4C3×C22⋊Q814yes—
496C2×C4C3×#12518yes—
497C2×C4C3×C4⋊Q814yes—
498C2×C3#239611yescharacteristic
499C2×C3#250512yes—
500C2×C3#250712yes—
501C2×C3#252212yes—
502C2×C3#256411yescharacteristic
503C2×C3#256711yescharacteristic
504C2×C3#258311yescharacteristic
505C2×C3#258612yes—
506C2×C3#260311yescharacteristic
507C2×C3#266111yescharacteristic
508C2×C3#285911yescharacteristic
509C22C2×C3×C22⋊Q811yescharacteristic
510C22C3×#52511yescharacteristic
511C22C2×C3×#12512yes—
512C22C3×#52912yes—
513C22C2×C3×C4⋊Q811yescharacteristic
514C22C3×#53211yescharacteristic
515C22C3×#53211yescharacteristic
516C22C3×#53312yes—
517C22C3×#53312yes—
518C22C3×#53411yescharacteristic
519C22C3×#53511yescharacteristic
520C22C3×#53812yes—
521C22C3×#54012yes—
522C22C3×#55112yes—
523C22C3×#55112yes—
524C22C3×#55312yes—
525C22C3×#55411yescharacteristic
526C22C3×#55412yes—
527C22C3×#55412yes—
528C22C3×#55411yescharacteristic
529C22C3×#55412yes—
530C22C3×Q8211yescharacteristic
531C22C3×Q8211yescharacteristic
532C4—28no—
533C4—24no—
534C4—24no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—44no—
540C4—82no—
541C4—81no—
542C4—81no—
543C3#1506811yescharacteristic, direct product factor, Sylow
544C2C3×#239611yescharacteristic
545C2C3×#250512yes—
546C2C3×#250712yes—
547C2C3×#252212yes—
548C2C3×#256411yescharacteristic
549C2C3×#256711yescharacteristic
550C2C3×#258311yescharacteristic
551C2C3×#258612yes—
552C2C3×#260311yescharacteristic
553C2C3×#266111yescharacteristic
554C2C3×#285911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #15068C3direct (N × H)
543. C3#15068direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×C3×#3872
C3×#14442
C3×#14422
C3×#14412
C3×#14362
C3×C4⋊#952
C3×C4×#952
#150683