Finite Group Databasebeta

← C3×#15098 #10633705

Subgroups — C3×#15098 #10633705

Order 768.

1270 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×#389C211yesmaximal, characteristic
2C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
3C3×#1444C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic
5C3×#1444C212yesmaximal
6C3×#1441C212yesmaximal
7C3×#1441C211yesmaximal, characteristic
8C3×#1441C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10C3×#1436C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15098C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C3×#389—24no—
24C3×C4×C4⋊C4—24no—
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×#389C2×C311yescharacteristic
39C2×C4×C4⋊C4C2×C311yescharacteristic
40#1444C2×C311yescharacteristic
41#1444C2×C312yes—
42#1444C2×C311yescharacteristic
43#1441C2×C311yescharacteristic
44#1441C2×C312yes—
45#1441C2×C311yescharacteristic
46#1436C2×C312yes—
47#1436C2×C312yes—
48#1434C2×C311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
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—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
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⋊C4—42no—
78C2×C3×C4⋊C4Q814yes—
79C2×C3×C42—22no—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42—42no—
84C2×C3×C42—42no—
85C2×C3×C42D414yes—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—24no—
93C3×#95—24no—
94C3×#95—42no—
95C3×#95—42no—
96C3×#95—42no—
97C3×#95—42no—
98C3×#95Q818yes—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C311yescharacteristic
101C22×C4⋊C4C22×C312yes—
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C312yes—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C4⋊C4C22×C312yes—
106C22×C42C22×C311yescharacteristic
107C22×C42C22×C311yescharacteristic
108C22×C42C22×C311yescharacteristic
109#389—24no—
110C4×C4⋊C4—24no—
111C2×#95C22×C311yescharacteristic
112C2×#95C22×C311yescharacteristic
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C24×C3C2411yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
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—21no—
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—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×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×Q814yes—
179C22×C3×C4C2×Q814yes—
180C22×C3×C4C2×Q812yes—
181C22×C3×C4C2×Q812yes—
182C22×C3×C4C2×Q812yes—
183C22×C3×C4C2×Q814yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
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—44no—
195C3×C4⋊C4—48no—
196C3×C4⋊C4—48no—
197C3×C42—28no—
198C3×C42—44no—
199C3×C42—44no—
200C23×C4C23×C312yes—
201C23×C4C23×C311yescharacteristic
202C23×C4C23×C312yes—
203C23×C4C23×C312yes—
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C312yes—
206C23×C4C23×C311yescharacteristic
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C311yescharacteristic
210C23×C4C23×C311yescharacteristic
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C3×Q814yes—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42C3×D414yes—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95—24no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95C3×Q818yes—
250C23×C3C22×D411yescharacteristic
251C23×C3C22×Q811yescharacteristic
252C23×C3C22×Q812yes—
253C23×C3C2×#4111yescharacteristic
254C23×C3C2×#4111yescharacteristic
255C23×C3C2×#4111yescharacteristic
256C23×C3C2×#4111yescharacteristic
257C23×C3C2×#4111yescharacteristic
258C23×C3C2×#4112yes—
259C23×C3H(5,𝔽2)12yes—
260C23×C3H(5,𝔽2)12yes—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
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—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
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×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4C22⋊Q814yes—
324C2×C3×C4#12514yes—
325C2×C3×C4C4⋊Q814yes—
326C24C24×C311yescharacteristic
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
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—21no—
348C22×C4—21no—
349C22×C4—21no—
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—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—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D412yes—
379C22×C4C2×C3×D412yes—
380C22×C4C2×C3×Q814yes—
381C22×C4C2×C3×Q814yes—
382C22×C4C2×C3×Q812yes—
383C22×C4C2×C3×Q812yes—
384C22×C4C2×C3×Q812yes—
385C22×C4C2×C3×Q814yes—
386C22×C4C3×#4114yes—
387C22×C4C3×#4112yes—
388C22×C4C3×#4112yes—
389C22×C4C3×#4112yes—
390C22×C4C3×#4112yes—
391C22×C4C3×#4112yes—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C42—28no—
400C42—44no—
401C42—44no—
402C22×C3C2×C22⋊Q811yescharacteristic
403C22×C3#52511yescharacteristic
404C22×C3#52511yescharacteristic
405C22×C3#52511yescharacteristic
406C22×C3C2×#12511yescharacteristic
407C22×C3#52911yescharacteristic
408C22×C3#52911yescharacteristic
409C22×C3C2×C4⋊Q811yescharacteristic
410C22×C3#53311yescharacteristic
411C22×C3#53312yes—
412C22×C3#53312yes—
413C22×C3#53411yescharacteristic
414C22×C3#53511yescharacteristic
415C22×C3#53812yes—
416C22×C3#54012yes—
417C22×C3#55112yes—
418C22×C3#55312yes—
419C22×C3#55312yes—
420C22×C3#55412yes—
421C22×C3#55412yes—
422C22×C3#55412yes—
423C22×C3#55412yes—
424C22×C3Q8212yes—
425C3×C4—24no—
426C3×C4—24no—
427C3×C4—24no—
428C3×C4—42no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—44no—
435C3×C4—82no—
436C23C22×C3×D411yescharacteristic
437C23C22×C3×Q811yescharacteristic
438C23C22×C3×Q812yes—
439C23C2×C3×#4111yescharacteristic
440C23C2×C3×#4111yescharacteristic
441C23C2×C3×#4111yescharacteristic
442C23C2×C3×#4111yescharacteristic
443C23C2×C3×#4111yescharacteristic
444C23C2×C3×#4112yes—
445C23C3×H(5,𝔽2)12yes—
446C23C3×H(5,𝔽2)12yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
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—24no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4C3×C22⋊Q814yes—
510C2×C4C3×#12514yes—
511C2×C4C3×C4⋊Q814yes—
512C2×C3#239911yescharacteristic
513C2×C3#250712yes—
514C2×C3#252212yes—
515C2×C3#256711yescharacteristic
516C2×C3#257911yescharacteristic
517C2×C3#258311yescharacteristic
518C2×C3#258612yes—
519C2×C3#258712yes—
520C2×C3#259811yescharacteristic
521C2×C3#263511yescharacteristic
522C2×C3#285111yescharacteristic
523C22C2×C3×C22⋊Q811yescharacteristic
524C22C3×#52511yescharacteristic
525C22C3×#52511yescharacteristic
526C22C3×#52511yescharacteristic
527C22C2×C3×#12511yescharacteristic
528C22C3×#52911yescharacteristic
529C22C3×#52911yescharacteristic
530C22C2×C3×C4⋊Q811yescharacteristic
531C22C3×#53312yes—
532C22C3×#53312yes—
533C22C3×#53311yescharacteristic
534C22C3×#53411yescharacteristic
535C22C3×#53511yescharacteristic
536C22C3×#53812yes—
537C22C3×#54012yes—
538C22C3×#55112yes—
539C22C3×#55312yes—
540C22C3×#55312yes—
541C22C3×#55412yes—
542C22C3×#55412yes—
543C22C3×#55412yes—
544C22C3×#55412yes—
545C22C3×Q8212yes—
546C4—24no—
547C4—24no—
548C4—24no—
549C4—42no—
550C4—42no—
551C4—44no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—44no—
556C4—82no—
557C3#1509811yescharacteristic, direct product factor, Sylow
558C2C3×#239911yescharacteristic
559C2C3×#250712yes—
560C2C3×#252212yes—
561C2C3×#256711yescharacteristic
562C2C3×#257911yescharacteristic
563C2C3×#258311yescharacteristic
564C2C3×#258612yes—
565C2C3×#258712yes—
566C2C3×#259811yescharacteristic
567C2C3×#263511yescharacteristic
568C2C3×#285111yescharacteristic
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. #15098C3direct (N × H)
557. C3#15098direct (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×C4×C4⋊C42
C3×#14442
C3×#14412
C3×#14362
C3×#14342
#150983