Finite Group Databasebeta

← C3×#15108 #10633715

Subgroups — C3×#15108 #10633715

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 558 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×#1443C212yesmaximal
4C3×#1442C211yesmaximal, characteristic
5C3×#1442C211yesmaximal, characteristic
6C3×#1442C212yesmaximal
7C3×#1441C211yesmaximal, characteristic
8C3×#1441C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10C3×#1436C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15108C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C3×#389—24no—
22C3×C4×C4⋊C4—24no—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
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#1443C2×C312yes—
41#1442C2×C311yescharacteristic
42#1442C2×C312yes—
43#1442C2×C311yescharacteristic
44#1441C2×C311yescharacteristic
45#1441C2×C311yescharacteristic
46#1436C2×C312yes—
47#1436C2×C312yes—
48#1434C2×C311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
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—22no—
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—42no—
72C2×C3×C4⋊C4Q814yes—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42—42no—
78C2×C3×C42—42no—
79C2×C3×C42D414yes—
80C3×#95—22no—
81C3×#95—22no—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—24no—
86C3×#95—24no—
87C3×#95—24no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—42no—
93C3×#95—42no—
94C3×#95—42no—
95C22×C4⋊C4C22×C312yes—
96C22×C4⋊C4C22×C311yescharacteristic
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C4⋊C4C22×C311yescharacteristic
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C42C22×C311yescharacteristic
101C22×C42C22×C311yescharacteristic
102C22×C42C22×C311yescharacteristic
103#389—24no—
104C4×C4⋊C4—24no—
105C2×#95C22×C312yes—
106C2×#95C22×C311yescharacteristic
107C2×#95C22×C311yescharacteristic
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
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—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
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—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
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—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×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×Q812yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4C2×Q812yes—
179C22×C3×C4#4112yes—
180C22×C3×C4#4114yes—
181C22×C3×C4#4112yes—
182C22×C3×C4#4112yes—
183C22×C3×C4#4114yes—
184C22×C3×C4#4112yes—
185C22×C3×C4#4112yes—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—48no—
191C3×C42—28no—
192C3×C42—44no—
193C3×C42—44no—
194C23×C4C23×C312yes—
195C23×C4C23×C312yes—
196C23×C4C23×C312yes—
197C23×C4C23×C311yescharacteristic
198C23×C4C23×C311yescharacteristic
199C23×C4C23×C312yes—
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C311yescharacteristic
202C23×C4C23×C311yescharacteristic
203C23×C4C23×C311yescharacteristic
204C23×C4C23×C311yescharacteristic
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C3×Q814yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C3×D414yes—
225#95—22no—
226#95—22no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—22no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95—42no—
238#95—42no—
239#95—42no—
240C23×C3C22×D411yescharacteristic
241C23×C3C22×Q811yescharacteristic
242C23×C3C2×#4111yescharacteristic
243C23×C3C2×#4111yescharacteristic
244C23×C3C2×#4111yescharacteristic
245C23×C3C2×#4111yescharacteristic
246C23×C3C2×#4112yes—
247C23×C3C2×#4111yescharacteristic
248C23×C3C2×#4112yes—
249C23×C3H(5,𝔽2)12yes—
250C23×C32−1+412yes—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
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—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
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—42no—
302C2×C3×C4—42no—
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×C4C22⋊Q814yes—
314C2×C3×C4#12414yes—
315C2×C3×C4#12514yes—
316C24C24×C311yescharacteristic
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
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—22no—
348C22×C4—22no—
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—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×C4C2×C3×D412yes—
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×Q812yes—
373C22×C4C2×C3×Q812yes—
374C22×C4C2×C3×Q812yes—
375C22×C4C3×#4112yes—
376C22×C4C3×#4114yes—
377C22×C4C3×#4112yes—
378C22×C4C3×#4112yes—
379C22×C4C3×#4114yes—
380C22×C4C3×#4112yes—
381C22×C4C3×#4112yes—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C42—28no—
388C42—44no—
389C42—44no—
390C22×C3C2×C22⋊Q811yescharacteristic
391C22×C3#52511yescharacteristic
392C22×C3#52511yescharacteristic
393C22×C3#52511yescharacteristic
394C22×C3C2×#12411yescharacteristic
395C22×C3C2×#12511yescharacteristic
396C22×C3#52912yes—
397C22×C3#52911yescharacteristic
398C22×C3#52911yescharacteristic
399C22×C3#52912yes—
400C22×C3#53311yescharacteristic
401C22×C3#53411yescharacteristic
402C22×C3#53611yescharacteristic
403C22×C3#53912yes—
404C22×C3#54212yes—
405C22×C3#55212yes—
406C22×C3#55312yes—
407C22×C3#55312yes—
408C22×C3#55312yes—
409C22×C3#55412yes—
410C22×C3#55612yes—
411C22×C3#55612yes—
412C22×C3#55712yes—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—24no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—44no—
421C3×C4—82no—
422C3×C4—82no—
423C3×C4—82no—
424C23C22×C3×D411yescharacteristic
425C23C22×C3×Q811yescharacteristic
426C23C2×C3×#4111yescharacteristic
427C23C2×C3×#4111yescharacteristic
428C23C2×C3×#4111yescharacteristic
429C23C2×C3×#4111yescharacteristic
430C23C2×C3×#4112yes—
431C23C2×C3×#4111yescharacteristic
432C23C2×C3×#4112yes—
433C23C3×H(5,𝔽2)12yes—
434C23C3×2−1+412yes—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
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×C4—42no—
496C2×C4—42no—
497C2×C4C3×C22⋊Q814yes—
498C2×C4C3×#12414yes—
499C2×C4C3×#12514yes—
500C2×C3#239811yescharacteristic
501C2×C3#250912yes—
502C2×C3#252512yes—
503C2×C3#256811yescharacteristic
504C2×C3#257911yescharacteristic
505C2×C3#258512yes—
506C2×C3#259111yescharacteristic
507C2×C3#259412yes—
508C2×C3#259911yescharacteristic
509C2×C3#263811yescharacteristic
510C2×C3#285111yescharacteristic
511C22C2×C3×C22⋊Q811yescharacteristic
512C22C3×#52511yescharacteristic
513C22C3×#52511yescharacteristic
514C22C3×#52511yescharacteristic
515C22C2×C3×#12411yescharacteristic
516C22C2×C3×#12511yescharacteristic
517C22C3×#52912yes—
518C22C3×#52912yes—
519C22C3×#52911yescharacteristic
520C22C3×#52911yescharacteristic
521C22C3×#53311yescharacteristic
522C22C3×#53411yescharacteristic
523C22C3×#53611yescharacteristic
524C22C3×#53912yes—
525C22C3×#54212yes—
526C22C3×#55212yes—
527C22C3×#55312yes—
528C22C3×#55312yes—
529C22C3×#55312yes—
530C22C3×#55412yes—
531C22C3×#55612yes—
532C22C3×#55612yes—
533C22C3×#55712yes—
534C4—24no—
535C4—24no—
536C4—24no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—82no—
545C3#1510811yescharacteristic, direct product factor, Sylow
546C2C3×#239811yescharacteristic
547C2C3×#250912yes—
548C2C3×#252512yes—
549C2C3×#256811yescharacteristic
550C2C3×#257911yescharacteristic
551C2C3×#258512yes—
552C2C3×#259111yescharacteristic
553C2C3×#259412yes—
554C2C3×#259911yescharacteristic
555C2C3×#263811yescharacteristic
556C2C3×#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. #15108C3direct (N × H)
545. C3#15108direct (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×#14432
C3×#14422
C3×#14412
C3×#14362
C3×#14342
#151083