Finite Group Databasebeta

← C3×#26396 #10645003

Subgroups — C3×#26396 #10645003

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#476C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2451C211yesmaximal, characteristic, complemented
4C3×#2349C211yesmaximal, characteristic
5C3×#2097C211yesmaximal, characteristic, complemented
6C3×#2085C211yesmaximal, characteristic, complemented
7C3×#2068C211yesmaximal, characteristic, complemented
8C3×#2057C211yesmaximal, characteristic, complemented
9C3×#1943C211yesmaximal, characteristic, complemented
10C3×#1917C211yesmaximal, characteristic, complemented
11C3×#1877C211yesmaximal, characteristic
12C3×#1877C211yesmaximal, characteristic
13C3×#1865C211yesmaximal, characteristic, complemented
14C3×#1862C211yesmaximal, characteristic
15C3×#1862C211yesmaximal, characteristic
16#26396C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C3×#476—24no—
24C3×#440—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2211yescharacteristic
37C2×C3×#97C2211yescharacteristic
38C3×#398—21no—
39C3×#395—21no—
40C3×#391—21no—
41C3×#391C2211yescharacteristic
42C3×#391C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#389C2211yescharacteristic
47C3×#387—21no—
48C3×#385—21no—
49C3×#385—21no—
50C3×#382C2211yescharacteristic
51C3×#380—21no—
52C3×C4×C4⋊C4C2211yescharacteristic
53C2×C3×#95—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#476C2×C311yescharacteristic
63C2×#440C2×C311yescharacteristic
64#2451C2×C311yescharacteristic, complemented
65#2349C2×C311yescharacteristic
66#2097C2×C311yescharacteristic, complemented
67#2085C2×C311yescharacteristic, complemented
68#2068C2×C311yescharacteristic, complemented
69#2057C2×C311yescharacteristic, complemented
70#1943C2×C311yescharacteristic, complemented
71#1917C2×C311yescharacteristic, complemented
72#1877C2×C311yescharacteristic
73#1877C2×C311yescharacteristic
74#1865C2×C311yescharacteristic, complemented
75#1862C2×C311yescharacteristic
76#1862C2×C311yescharacteristic
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16Q814yescomplemented
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4⋊Q8—22no—
91C3×C4⋊Q8—41no—
92C3×C4×Q8—22no—
93C3×C4×Q8—41no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—41no—
99C3×C4×D4—41no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4D411yescharacteristic
107C2×C3×C4⋊C4D411yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#107—22no—
123C3×#107—41no—
124C3×#106—22no—
125C3×#106—22no—
126C3×C4⋊C8—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#103—41no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#102D414yes—
133C3×#97—22no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95D411yescharacteristic
145C3×#95D411yescharacteristic
146C22×SD16C22×C311yescharacteristic
147C2×C4⋊Q8C22×C311yescharacteristic
148C2×C4×Q8C22×C311yescharacteristic
149C2×C4×D4C22×C311yescharacteristic
150C2×C4×D4C22×C311yescharacteristic
151C2×C4×D4C22×C311yescharacteristic
152#476—24no—
153#440—24no—
154C2×#107C22×C311yescharacteristic
155C2×#107C22×C311yescharacteristic
156C2×#107C22×C311yescharacteristic
157C2×#106C22×C311yescharacteristic
158C2×#106C22×C311yescharacteristic
159C2×C4⋊C8C22×C311yescharacteristic
160C2×#103C22×C311yescharacteristic
161C2×#103C22×C311yescharacteristic
162C2×#103C22×C311yescharacteristic
163C2×#102C22×C311yescharacteristic
164C2×#102C22×C311yescharacteristic
165C2×#102C22×C311yescharacteristic
166C2×#97C22×C311yescharacteristic
167#398—21no—
168#395—21no—
169#391—21no—
170#391C22×C311yescharacteristic
171#391C22×C311yescharacteristic
172#389C22×C311yescharacteristic
173#389C22×C311yescharacteristic
174#389C22×C311yescharacteristic
175#389C22×C311yescharacteristic
176#387—21no—
177#385—21no—
178#385—21no—
179#382C22×C311yescharacteristic
180#380—21no—
181C4×C4⋊C4C22×C311yescharacteristic
182C2×#95—21no—
183#340C22×C311yescharacteristic
184#340C22×C311yescharacteristic
185#340C22×C311yescharacteristic
186#340C22×C311yescharacteristic
187#336C22×C311yescharacteristic
188#336C22×C311yescharacteristic
189#336C22×C311yescharacteristic
190#336C22×C311yescharacteristic
191C24×C3—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—42no—
198C2×C3×Q8—42no—
199C2×C3×Q8—81no—
200C2×C3×Q8C2×Q812yes—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4C2×D412yes—
205C2×C3×D4C2×Q812yes—
206C2×C3×D4#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—81no—
220C22×C3×C4—81no—
221C22×C3×C4—81no—
222C22×C3×C4—81no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C3×SD16—28no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C2×D412yes—
242C2×C3×C8C2×Q812yes—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4—81no—
258C3×C4⋊C4—81no—
259C3×C4⋊C4—81no—
260C3×C4⋊C4—81no—
261C3×C4⋊C4C2×D412yes—
262C3×C4⋊C4#4112yes—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—42no—
265C3×C22⋊C4—42no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—42no—
273C22×Q8—21no—
274C22×Q8C23×C311yescharacteristic
275C22×D4C23×C311yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C2×SD16—22no—
280C2×SD16—22no—
281C2×SD16C3×Q814yescomplemented
282C22×C8C23×C311yescharacteristic
283C22×C8C23×C311yescharacteristic
284C22×C8C23×C311yescharacteristic
285C22×C8C23×C311yescharacteristic
286C4⋊Q8—22no—
287C4⋊Q8—41no—
288C4×Q8—22no—
289C4×Q8—41no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—41no—
295C4×D4—41no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C3×D411yescharacteristic
303C2×C4⋊C4C3×D411yescharacteristic
304C2×C4⋊C4C23×C311yescharacteristic
305C2×C4⋊C4C23×C311yescharacteristic
306C2×C4⋊C4C23×C311yescharacteristic
307C2×C4⋊C4C23×C311yescharacteristic
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C42—21no—
314C2×C42C23×C311yescharacteristic
315C2×C42C23×C311yescharacteristic
316C2×C42C23×C311yescharacteristic
317#107—22no—
318#107—22no—
319#107—41no—
320#106—22no—
321#106—22no—
322C4⋊C8—22no—
323#103—22no—
324#103—22no—
325#103—41no—
326#102—22no—
327#102—22no—
328#102C3×D414yes—
329#97—22no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95C3×D411yescharacteristic
341#95C3×D411yescharacteristic
342C23×C3—22no—
343C23×C3—22no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—84no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4C22⋊Q814yes—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—84no—
398C2×C3×C4—84no—
399C2×C3×C4—84no—
400C2×C3×C4#13712yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×Q811yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C3×C8—22no—
408C3×C8—22no—
409C3×C8—41no—
410C3×C8—41no—
411C24—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—81no—
420C2×Q8C2×C3×Q812yes—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×C3×D412yes—
425C2×D4C2×C3×Q812yes—
426C2×D4C3×#4112yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C2×C3×D411yescharacteristic
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C2×C3×D411yescharacteristic
449C22×C4C24×C311yescharacteristic
450SD16—28no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8C2×C3×D412yes—
462C2×C8C2×C3×Q812yes—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4C2×C3×D412yes—
482C4⋊C4C3×#4112yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—42no—
493C22×C3—24no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3#52511yescharacteristic
501C22×C3D4211yescharacteristic
502C22×C3D4×Q811yescharacteristic
503C22×C3C2×#13711yescharacteristic
504C22×C3#57511yescharacteristic
505C22×C3#57611yescharacteristic
506C22×C3#57711yescharacteristic
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4C2×C22⋊Q811yescharacteristic
521C3×C4#53911yescharacteristic
522C3×C4#54711yescharacteristic
523C3×C4#55411yescharacteristic
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×C3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—84no—
537D4—22no—
538D4—22no—
539D4—22no—
540D4C3×C22⋊Q814yes—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4C3×#13712yes—
583C2×C4C22×C3×D411yescharacteristic
584C2×C4C22×C3×Q811yescharacteristic
585C2×C4C2×C3×#4111yescharacteristic
586C2×C4C2×C3×#4111yescharacteristic
587C2×C4C3×H(5,𝔽2)11yescharacteristic
588C2×C4C3×2−1+411yescharacteristic
589C8—22no—
590C8—22no—
591C8—41no—
592C8—41no—
593C2×C3—44no—
594C2×C3#244911yescharacteristic
595C2×C3#311611yescharacteristic
596C2×C3#311811yescharacteristic
597C2×C3#326611yescharacteristic
598C2×C3#327311yescharacteristic
599C2×C3#338511yescharacteristic
600C2×C3#338811yescharacteristic
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22C3×#52511yescharacteristic
609C22C3×D4211yescharacteristic
610C22C3×D4×Q811yescharacteristic
611C22C2×C3×#13711yescharacteristic
612C22C3×#57511yescharacteristic
613C22C3×#57611yescharacteristic
614C22C3×#57711yescharacteristic
615C4—22no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4C2×C3×C22⋊Q811yescharacteristic
629C4C3×#53911yescharacteristic
630C4C3×#54711yescharacteristic
631C4C3×#55411yescharacteristic
632C3#2639611yescharacteristic, direct product factor, Sylow
633C2—44no—
634C2C3×#244911yescharacteristic
635C2C3×#311611yescharacteristic
636C2C3×#311811yescharacteristic
637C2C3×#326611yescharacteristic
638C2C3×#327311yescharacteristic
639C2C3×#338511yescharacteristic
640C2C3×#338811yescharacteristic
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
3. C3×#2451C2semidirect
5. C3×#2097C2semidirect
6. C3×#2085C2semidirect
7. C3×#2068C2semidirect
8. C3×#2057C2semidirect
9. C3×#1943C2semidirect
10. C3×#1917C2semidirect
13. C3×#1865C2semidirect
16. #26396C3direct (N × H)
64. #2451C2×C3semidirect
66. #2097C2×C3semidirect
67. #2085C2×C3semidirect
68. #2068C2×C3semidirect
69. #2057C2×C3semidirect
70. #1943C2×C3semidirect
71. #1917C2×C3semidirect
74. #1865C2×C3semidirect
85. C2×C3×SD16Q8semidirect
281. C2×SD16C3×Q8semidirect
632. C3#26396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×C3×#4402
C3×#24512
C3×#23492
C3×#20972
C3×#20852
C3×#20682
C3×#20572
C3×#19432
C3×#19172
C3×#18772
C3×#18652
C3×#18622
#263963