Finite Group Databasebeta

← C3×#25466 #10644073

Subgroups — C3×#25466 #10644073

Order 768.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#501C211yesmaximal, characteristic, complemented
2C2×C3×#478C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#2370C211yesmaximal, characteristic, complemented
5C3×#2085C211yesmaximal, characteristic, complemented
6C3×#2085C211yesmaximal, characteristic, complemented
7C3×#2067C211yesmaximal, characteristic
8C3×#2065C211yesmaximal, characteristic
9C3×#1988C211yesmaximal, characteristic, complemented
10C3×#1949C211yesmaximal, characteristic, complemented
11C3×#1949C211yesmaximal, characteristic, complemented
12C3×#1940C211yesmaximal, characteristic, complemented
13C3×#1926C211yesmaximal, characteristic
14C3×#1925C211yesmaximal, characteristic
15C3×#1762C211yesmaximal, characteristic
16#25466C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C3×#501—24no—
21C3×#478—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#97C2211yescharacteristic
36C3×#393—21no—
37C3×#393—21no—
38C3×#390C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#389—21no—
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#387—21no—
44C3×#387—21no—
45C3×#382C2211yescharacteristic
46C3×#382C2211yescharacteristic
47C3×#380—21no—
48C3×#380—21no—
49C3×C4×C4⋊C4—21no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×C4×C4⋊C4C2211yescharacteristic
52C3×C4×C4⋊C4C2211yescharacteristic
53C3×C4×C4⋊C4C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#501C2×C311yescharacteristic, complemented
63C2×#478C2×C311yescharacteristic
64#2563C2×C311yescharacteristic
65#2370C2×C311yescharacteristic, complemented
66#2085C2×C311yescharacteristic, complemented
67#2085C2×C311yescharacteristic, complemented
68#2067C2×C311yescharacteristic
69#2065C2×C311yescharacteristic
70#1988C2×C311yescharacteristic, complemented
71#1949C2×C311yescharacteristic, complemented
72#1949C2×C311yescharacteristic, complemented
73#1940C2×C311yescharacteristic, complemented
74#1926C2×C311yescharacteristic
75#1925C2×C311yescharacteristic
76#1762C2×C311yescharacteristic
77C22×C3×Q8—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C3×#125—22no—
87C3×#125D414yes—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C3×#107—22no—
116C3×#107—22no—
117C3×#107—22no—
118C3×#106—22no—
119C3×#106—22no—
120C3×#106—22no—
121C3×C4⋊C8—22no—
122C3×C4⋊C8—22no—
123C3×C4⋊C8—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#102—22no—
127C3×#102Q814yescomplemented
128C3×#97—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95D411yescharacteristic
139C3×#95D411yescharacteristic
140C2×C4⋊Q8C22×C311yescharacteristic
141C2×#125C22×C311yescharacteristic
142C2×C4×D4C22×C311yescharacteristic
143#501—24no—
144#478—24no—
145C2×#107C22×C311yescharacteristic
146C2×#107C22×C311yescharacteristic
147C2×#107C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×#106C22×C311yescharacteristic
150C2×#106C22×C311yescharacteristic
151C2×C4⋊C8C22×C311yescharacteristic
152C2×C4⋊C8C22×C311yescharacteristic
153C2×C4⋊C8C22×C311yescharacteristic
154C2×#102C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#102C22×C311yescharacteristic
158C2×#97C22×C311yescharacteristic
159#393—21no—
160#393—21no—
161#390C22×C311yescharacteristic
162#390C22×C311yescharacteristic
163#389—21no—
164#389C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#387—21no—
167#387—21no—
168#382C22×C311yescharacteristic
169#382C22×C311yescharacteristic
170#380—21no—
171#380—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4C22×C311yescharacteristic
174C4×C4⋊C4C22×C311yescharacteristic
175C4×C4⋊C4C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#336C22×C311yescharacteristic
184#336C22×C311yescharacteristic
185C24×C3—21no—
186C2×C3×Q8—42no—
187C2×C3×Q8—81no—
188C2×C3×Q8—81no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4C2×Q812yes—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8C2×Q812yes—
223C2×C3×C8#4112yes—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4C2×D412yes—
243C3×C4⋊C4C2×D412yes—
244C3×C4⋊C4C2×Q812yes—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C22⋊C4—81no—
249C3×C22⋊C4—81no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—42no—
255C3×C42—81no—
256C3×C42—81no—
257C3×C42C2×D412yes—
258C3×C42#4112yes—
259C22×Q8—21no—
260C22×D4C23×C311yescharacteristic
261C23×C4—21no—
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#125—22no—
269#125C3×D414yes—
270C4×D4—22no—
271C4×D4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C3×D411yescharacteristic
278C2×C4⋊C4C3×D411yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#107—22no—
298#107—22no—
299#107—22no—
300#106—22no—
301#106—22no—
302#106—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306#102—22no—
307#102—22no—
308#102—22no—
309#102C3×Q814yescomplemented
310#97—22no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95C3×D411yescharacteristic
321#95C3×D411yescharacteristic
322C23×C3—22no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C3×Q8—84no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—41no—
334C3×D4—41no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—84no—
373C2×C3×C4#12112yes—
374C2×C3×C4#12112yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×Q811yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C42−1+411yescharacteristic
380C2×C3×C42−1+411yescharacteristic
381C3×C8—22no—
382C3×C8—22no—
383C3×C8—41no—
384C3×C8—41no—
385C24—21no—
386C2×Q8—42no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4C2×C3×Q812yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C24×C311yescharacteristic
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C3×Q812yes—
423C2×C8C3×#4112yes—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—81no—
442C4⋊C4C2×C3×D412yes—
443C4⋊C4C2×C3×D412yes—
444C4⋊C4C2×C3×Q812yes—
445C22⋊C4—42no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—81no—
456C42—81no—
457C42C2×C3×D412yes—
458C42C3×#4112yes—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
465C22×C3—81no—
466C22×C3C2×#12111yescharacteristic
467C22×C3#54711yescharacteristic
468C22×C3D4×Q811yescharacteristic
469C22×C3#57511yescharacteristic
470C22×C3#57511yescharacteristic
471C22×C3#57611yescharacteristic
472C22×C3#57711yescharacteristic
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#53311yescharacteristic
487C3×C4#54111yescharacteristic
488C3×C4#54811yescharacteristic
489C3×C4D4×Q811yescharacteristic
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C3×D411yescharacteristic
498Q8—84no—
499D4—22no—
500D4—22no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4C3×#12112yes—
542C2×C4C3×#12112yes—
543C2×C4C22×C3×D411yescharacteristic
544C2×C4C22×C3×Q811yescharacteristic
545C2×C4C2×C3×#4111yescharacteristic
546C2×C4C2×C3×#4111yescharacteristic
547C2×C4C3×2−1+411yescharacteristic
548C2×C4C3×2−1+411yescharacteristic
549C8—22no—
550C8—22no—
551C8—41no—
552C8—41no—
553C2×C3—82no—
554C2×C3#242911yescharacteristic
555C2×C3#304011yescharacteristic
556C2×C3#304511yescharacteristic
557C2×C3#334711yescharacteristic
558C2×C3#334811yescharacteristic
559C2×C3#338811yescharacteristic
560C2×C3#338911yescharacteristic
561C22—42no—
562C22—42no—
563C22—42no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C2×C3×#12111yescharacteristic
569C22C3×#54711yescharacteristic
570C22C3×D4×Q811yescharacteristic
571C22C3×#57511yescharacteristic
572C22C3×#57511yescharacteristic
573C22C3×#57611yescharacteristic
574C22C3×#57711yescharacteristic
575C4—22no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4C3×#53311yescharacteristic
589C4C3×#54111yescharacteristic
590C4C3×#54811yescharacteristic
591C4C3×D4×Q811yescharacteristic
592C3#2546611yescharacteristic, direct product factor, Sylow
593C2—82no—
594C2C3×#242911yescharacteristic
595C2C3×#304011yescharacteristic
596C2C3×#304511yescharacteristic
597C2C3×#334711yescharacteristic
598C2C3×#334811yescharacteristic
599C2C3×#338811yescharacteristic
600C2C3×#338911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#501C2semidirect
4. C3×#2370C2semidirect
5. C3×#2085C2semidirect
6. C3×#2085C2semidirect
9. C3×#1988C2semidirect
10. C3×#1949C2semidirect
11. C3×#1949C2semidirect
12. C3×#1940C2semidirect
16. #25466C3direct (N × H)
62. C2×#501C2×C3semidirect
65. #2370C2×C3semidirect
66. #2085C2×C3semidirect
67. #2085C2×C3semidirect
70. #1988C2×C3semidirect
71. #1949C2×C3semidirect
72. #1949C2×C3semidirect
73. #1940C2×C3semidirect
127. C3×#102Q8semidirect
309. #102C3×Q8semidirect
592. C3#25466direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5012
C2×C3×#4782
C3×#25632
C3×#23702
C3×#20852
C3×#20672
C3×#20652
C3×#19882
C3×#19492
C3×#19402
C3×#19262
C3×#19252
C3×#17622
#254663