Finite Group Databasebeta

← C3×#15102 #10633709

Subgroups — C3×#15102 #10633709

Order 768.

1182 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
1C2×C3×#387C211yesmaximal, characteristic
2C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
3C3×#1445C212yesmaximal
4C3×#1444C211yesmaximal, characteristic
5C3×#1443C212yesmaximal
6C3×#1442C211yesmaximal, characteristic
7C3×#1441C211yesmaximal, characteristic
8C3×#1441C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10C3×#1436C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15102C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C22×C3×C42C2212yes—
20C3×#387—24no—
21C3×C4×C4⋊C4—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#387C2×C311yescharacteristic
39C2×C4×C4⋊C4C2×C311yescharacteristic
40#1445C2×C312yes—
41#1444C2×C311yescharacteristic
42#1443C2×C312yes—
43#1442C2×C311yescharacteristic
44#1441C2×C311yescharacteristic
45#1441C2×C311yescharacteristic
46#1436C2×C312yes—
47#1436C2×C312yes—
48#1434C2×C311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4Q814yes—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42—24no—
76C2×C3×C42—42no—
77C2×C3×C42—42no—
78C2×C3×C42D414yes—
79C3×#95—24no—
80C3×#95—22no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—24no—
85C3×#95—22no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—22no—
92C3×#95—42no—
93C3×#95—42no—
94C3×#95—42no—
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C4⋊C4C22×C311yescharacteristic
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C312yes—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C42C22×C311yescharacteristic
101C22×C42C22×C312yes—
102#387—24no—
103C4×C4⋊C4—24no—
104C2×#95C22×C312yes—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C311yescharacteristic
107C2×#95C22×C312yes—
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C312yes—
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C312yes—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C24×C3C2411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
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—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
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×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×Q812yes—
180C22×C3×C4C2×Q812yes—
181C22×C3×C4C2×Q812yes—
182C22×C3×C4#4114yes—
183C22×C3×C4#4114yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×C3×C4#4112yes—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—48no—
192C3×C4⋊C4—44no—
193C3×C42—28no—
194C3×C42—48no—
195C23×C4C23×C311yescharacteristic
196C23×C4C23×C311yescharacteristic
197C23×C4C23×C312yes—
198C23×C4C23×C311yescharacteristic
199C23×C4C23×C311yescharacteristic
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C312yes—
202C23×C4C23×C312yes—
203C23×C4C23×C312yes—
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C311yescharacteristic
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C3×Q814yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C3×D414yes—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—42no—
239#95—42no—
240#95—42no—
241C23×C3C22×D411yescharacteristic
242C23×C3C22×Q811yescharacteristic
243C23×C3C2×#4111yescharacteristic
244C23×C3C2×#4111yescharacteristic
245C23×C3C2×#4111yescharacteristic
246C23×C3C2×#4112yes—
247C23×C3C2×#4112yes—
248C23×C3C2×#4112yes—
249C23×C3H(5,𝔽2)11yescharacteristic
250C23×C3H(5,𝔽2)11yescharacteristic
251C23×C32−1+412yes—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
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—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
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—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4C22⋊Q814yes—
318C2×C3×C4#12314yes—
319C24C24×C311yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
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—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
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—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4C2×C3×D412yes—
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×Q812yes—
379C22×C4C2×C3×Q812yes—
380C22×C4C2×C3×Q812yes—
381C22×C4C3×#4114yes—
382C22×C4C3×#4114yes—
383C22×C4C3×#4114yes—
384C22×C4C3×#4112yes—
385C22×C4C3×#4112yes—
386C22×C4C3×#4112yes—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C42—28no—
393C42—48no—
394C22×C3C2×C22⋊Q811yescharacteristic
395C22×C3C2×#12311yescharacteristic
396C22×C3#52511yescharacteristic
397C22×C3#52511yescharacteristic
398C22×C3#52912yes—
399C22×C3#52912yes—
400C22×C3#52912yes—
401C22×C3#53211yescharacteristic
402C22×C3#53312yes—
403C22×C3#53511yescharacteristic
404C22×C3#53611yescharacteristic
405C22×C3#53912yes—
406C22×C3#53912yes—
407C22×C3#55112yes—
408C22×C3#55212yes—
409C22×C3#55311yescharacteristic
410C22×C3#55312yes—
411C22×C3#55311yescharacteristic
412C22×C3#55411yescharacteristic
413C22×C3#55411yescharacteristic
414C22×C3#55612yes—
415C22×C3#55612yes—
416C22×C3#55712yes—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—44no—
420C3×C4—44no—
421C3×C4—44no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—82no—
426C3×C4—81no—
427C3×C4—81no—
428C23C22×C3×D411yescharacteristic
429C23C22×C3×Q811yescharacteristic
430C23C2×C3×#4111yescharacteristic
431C23C2×C3×#4111yescharacteristic
432C23C2×C3×#4111yescharacteristic
433C23C2×C3×#4112yes—
434C23C2×C3×#4112yes—
435C23C2×C3×#4112yes—
436C23C3×H(5,𝔽2)11yescharacteristic
437C23C3×H(5,𝔽2)11yescharacteristic
438C23C3×2−1+412yes—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4C3×C22⋊Q814yes—
505C2×C4C3×#12314yes—
506C2×C3#239611yescharacteristic
507C2×C3#248711yescharacteristic
508C2×C3#249911yescharacteristic
509C2×C3#257312yes—
510C2×C3#258111yescharacteristic
511C2×C3#258512yes—
512C2×C3#258812yes—
513C2×C3#259412yes—
514C2×C3#260311yescharacteristic
515C2×C3#263311yescharacteristic
516C2×C3#285311yescharacteristic
517C22C2×C3×C22⋊Q811yescharacteristic
518C22C2×C3×#12311yescharacteristic
519C22C3×#52511yescharacteristic
520C22C3×#52511yescharacteristic
521C22C3×#52912yes—
522C22C3×#52912yes—
523C22C3×#52912yes—
524C22C3×#53211yescharacteristic
525C22C3×#53312yes—
526C22C3×#53511yescharacteristic
527C22C3×#53611yescharacteristic
528C22C3×#53912yes—
529C22C3×#53912yes—
530C22C3×#55112yes—
531C22C3×#55212yes—
532C22C3×#55311yescharacteristic
533C22C3×#55311yescharacteristic
534C22C3×#55312yes—
535C22C3×#55411yescharacteristic
536C22C3×#55411yescharacteristic
537C22C3×#55612yes—
538C22C3×#55612yes—
539C22C3×#55712yes—
540C4—24no—
541C4—24no—
542C4—44no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—44no—
548C4—82no—
549C4—81no—
550C4—81no—
551C3#1510211yescharacteristic, direct product factor, Sylow
552C2C3×#239611yescharacteristic
553C2C3×#248711yescharacteristic
554C2C3×#249911yescharacteristic
555C2C3×#257312yes—
556C2C3×#258111yescharacteristic
557C2C3×#258512yes—
558C2C3×#258812yes—
559C2C3×#259412yes—
560C2C3×#260311yescharacteristic
561C2C3×#263311yescharacteristic
562C2C3×#285311yescharacteristic
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. #15102C3direct (N × H)
551. C3#15102direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3872
C2×C3×C4×C4⋊C42
C3×#14452
C3×#14442
C3×#14432
C3×#14422
C3×#14412
C3×#14362
C3×#14342
#151023