Finite Group Databasebeta

← C3×#15101 #10633708

Subgroups — C3×#15101 #10633708

Order 768.

1226 conjugacy classes of subgroups.

The subgroup lattice has 570 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×#1444C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic
5C3×#1442C212yesmaximal
6C3×#1441C212yesmaximal
7C3×#1441C211yesmaximal, characteristic
8C3×#1441C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10C3×#1436C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15101C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×C3×C42C2212yes—
21C3×#387—24no—
22C3×C4×C4⋊C4—24no—
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
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#1444C2×C311yescharacteristic
41#1444C2×C311yescharacteristic
42#1442C2×C312yes—
43#1441C2×C311yescharacteristic
44#1441C2×C311yescharacteristic
45#1441C2×C312yes—
46#1436C2×C312yes—
47#1436C2×C312yes—
48#1434C2×C311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
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—22no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—42no—
74C2×C3×C4⋊C4—42no—
75C2×C3×C4⋊C4—42no—
76C2×C3×C42—22no—
77C2×C3×C42—22no—
78C2×C3×C42—24no—
79C2×C3×C42—42no—
80C2×C3×C42—42no—
81C2×C3×C42D414yes—
82C3×#95—24no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—24no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—42no—
93C3×#95—42no—
94C3×#95—42no—
95C3×#95—42no—
96C3×#95Q818yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C311yescharacteristic
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C311yescharacteristic
101C22×C4⋊C4C22×C312yes—
102C22×C4⋊C4C22×C312yes—
103C22×C42C22×C311yescharacteristic
104C22×C42C22×C312yes—
105#387—24no—
106C4×C4⋊C4—24no—
107C2×#95C22×C311yescharacteristic
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C311yescharacteristic
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C312yes—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C24×C3C2411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
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—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
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—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×Q814yes—
182C22×C3×C4C2×Q814yes—
183C22×C3×C4C2×Q814yes—
184C22×C3×C4#4112yes—
185C22×C3×C4#4114yes—
186C22×C3×C4#4114yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4112yes—
189C22×C3×C4#4112yes—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—48no—
194C3×C4⋊C4—48no—
195C3×C4⋊C4—44no—
196C3×C42—28no—
197C3×C42—48no—
198C23×C4C23×C311yescharacteristic
199C23×C4C23×C311yescharacteristic
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C312yes—
202C23×C4C23×C312yes—
203C23×C4C23×C311yescharacteristic
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C312yes—
206C23×C4C23×C312yes—
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C311yescharacteristic
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C3×D414yes—
231#95—24no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—24no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95C3×Q818yes—
246C23×C3C22×D411yescharacteristic
247C23×C3C22×Q812yes—
248C23×C3C2×#4111yescharacteristic
249C23×C3C2×#4111yescharacteristic
250C23×C3C2×#4112yes—
251C23×C3C2×#4112yes—
252C23×C3C2×#4111yescharacteristic
253C23×C3C2×#4111yescharacteristic
254C23×C3H(5,𝔽2)12yes—
255C23×C3H(5,𝔽2)11yescharacteristic
256C23×C32−1+411yescharacteristic
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
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—22no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
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—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4#12314yes—
323C2×C3×C4C4⋊Q814yes—
324C24C24×C311yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
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—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×Q814yes—
384C22×C4C2×C3×Q814yes—
385C22×C4C2×C3×Q814yes—
386C22×C4C3×#4112yes—
387C22×C4C3×#4114yes—
388C22×C4C3×#4114yes—
389C22×C4C3×#4112yes—
390C22×C4C3×#4112yes—
391C22×C4C3×#4112yes—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C42—28no—
399C42—48no—
400C22×C3C2×#12311yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#52511yescharacteristic
403C22×C3#52511yescharacteristic
404C22×C3#52912yes—
405C22×C3#52912yes—
406C22×C3C2×C4⋊Q811yescharacteristic
407C22×C3#53312yes—
408C22×C3#53312yes—
409C22×C3#53511yescharacteristic
410C22×C3#53711yescharacteristic
411C22×C3#53812yes—
412C22×C3#53912yes—
413C22×C3#55112yes—
414C22×C3#55211yescharacteristic
415C22×C3#55312yes—
416C22×C3#55311yescharacteristic
417C22×C3#55412yes—
418C22×C3#55412yes—
419C22×C3#55412yes—
420C22×C3#55511yescharacteristic
421C22×C3#55611yescharacteristic
422C22×C3#55712yes—
423C3×C4—24no—
424C3×C4—24no—
425C3×C4—44no—
426C3×C4—44no—
427C3×C4—42no—
428C3×C4—44no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—81no—
433C3×C4—81no—
434C23C22×C3×D411yescharacteristic
435C23C22×C3×Q812yes—
436C23C2×C3×#4111yescharacteristic
437C23C2×C3×#4111yescharacteristic
438C23C2×C3×#4112yes—
439C23C2×C3×#4112yes—
440C23C2×C3×#4111yescharacteristic
441C23C2×C3×#4111yescharacteristic
442C23C3×H(5,𝔽2)12yes—
443C23C3×H(5,𝔽2)11yescharacteristic
444C23C3×2−1+411yescharacteristic
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
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—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4C3×#12314yes—
511C2×C4C3×C4⋊Q814yes—
512C2×C3#239911yescharacteristic
513C2×C3#248511yescharacteristic
514C2×C3#250211yescharacteristic
515C2×C3#257412yes—
516C2×C3#258411yescharacteristic
517C2×C3#258512yes—
518C2×C3#258712yes—
519C2×C3#258812yes—
520C2×C3#259811yescharacteristic
521C2×C3#263611yescharacteristic
522C2×C3#285311yescharacteristic
523C22C2×C3×#12311yescharacteristic
524C22C3×#52511yescharacteristic
525C22C3×#52511yescharacteristic
526C22C3×#52511yescharacteristic
527C22C3×#52912yes—
528C22C3×#52912yes—
529C22C2×C3×C4⋊Q811yescharacteristic
530C22C3×#53312yes—
531C22C3×#53312yes—
532C22C3×#53511yescharacteristic
533C22C3×#53711yescharacteristic
534C22C3×#53812yes—
535C22C3×#53912yes—
536C22C3×#55112yes—
537C22C3×#55211yescharacteristic
538C22C3×#55311yescharacteristic
539C22C3×#55312yes—
540C22C3×#55412yes—
541C22C3×#55412yes—
542C22C3×#55412yes—
543C22C3×#55511yescharacteristic
544C22C3×#55611yescharacteristic
545C22C3×#55712yes—
546C4—24no—
547C4—24no—
548C4—44no—
549C4—42no—
550C4—42no—
551C4—44no—
552C4—44no—
553C4—42no—
554C4—44no—
555C4—81no—
556C4—81no—
557C3#1510111yescharacteristic, direct product factor, Sylow
558C2C3×#239911yescharacteristic
559C2C3×#248511yescharacteristic
560C2C3×#250211yescharacteristic
561C2C3×#257412yes—
562C2C3×#258411yescharacteristic
563C2C3×#258512yes—
564C2C3×#258712yes—
565C2C3×#258812yes—
566C2C3×#259811yescharacteristic
567C2C3×#263611yescharacteristic
568C2C3×#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. #15101C3direct (N × H)
557. C3#15101direct (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×#14442
C3×#14422
C3×#14412
C3×#14362
C3×#14342
#151013