Finite Group Databasebeta

← C3×#26387 #10644994

Subgroups — C3×#26387 #10644994

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#479C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2454C211yesmaximal, characteristic
4C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#2094C211yesmaximal, characteristic
6C3×#2083C211yesmaximal, characteristic, complemented
7C3×#2067C211yesmaximal, characteristic
8C3×#2059C211yesmaximal, characteristic, complemented
9C3×#1943C211yesmaximal, characteristic, complemented
10C3×#1917C211yesmaximal, characteristic, complemented
11C3×#1880C211yesmaximal, characteristic
12C3×#1880C211yesmaximal, characteristic
13C3×#1865C211yesmaximal, characteristic, complemented
14C3×#1863C211yesmaximal, characteristic, complemented
15C3×#1862C211yesmaximal, characteristic
16#26387C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C3×#479—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×#399—21no—
39C3×#396—21no—
40C3×#390C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#389—21no—
43C3×#389—21no—
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#389C2211yescharacteristic
47C3×#389C2211yescharacteristic
48C3×#385—21no—
49C3×#384—21no—
50C3×#384C2211yescharacteristic
51C3×#380—21no—
52C3×C4×C4⋊C4—21no—
53C3×C4×C4⋊C4C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#479C2×C311yescharacteristic, complemented
63C2×#440C2×C311yescharacteristic
64#2454C2×C311yescharacteristic
65Q8×C4⋊C4C2×C311yescharacteristic, complemented
66#2094C2×C311yescharacteristic
67#2083C2×C311yescharacteristic, complemented
68#2067C2×C311yescharacteristic
69#2059C2×C311yescharacteristic, complemented
70#1943C2×C311yescharacteristic, complemented
71#1917C2×C311yescharacteristic, complemented
72#1880C2×C311yescharacteristic
73#1880C2×C311yescharacteristic
74#1865C2×C311yescharacteristic, complemented
75#1863C2×C311yescharacteristic, complemented
76#1862C2×C311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16Q814yes—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#125—22no—
88C3×#125—41no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—41no—
94C3×C4×Q8—41no—
95C3×C4×D4—22no—
96C3×C4×D4—41no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
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⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4D411yescharacteristic
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C22⋊C4—21no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—21no—
121C2×C3×C42—21no—
122C2×C3×C42—21no—
123C2×C3×C42—21no—
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2311yescharacteristic
126C2×C3×C42C2311yescharacteristic
127C3×#107—22no—
128C3×#107—22no—
129C3×#107—22no—
130C3×#106—22no—
131C3×#106—41no—
132C3×C4⋊C8—22no—
133C3×#103—22no—
134C3×#103—22no—
135C3×#103D414yes—
136C3×#102—22no—
137C3×#102—22no—
138C3×#102—41no—
139C3×#97—22no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95D411yescharacteristic
145C3×#95D411yescharacteristic
146C22×SD16C22×C311yescharacteristic
147C2×#125C22×C311yescharacteristic
148C2×C4×Q8C22×C311yescharacteristic
149C2×C4×Q8C22×C311yescharacteristic
150C2×C4×Q8C22×C311yescharacteristic
151C2×C4×D4C22×C311yescharacteristic
152#479—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#399—21no—
168#396—21no—
169#390C22×C311yescharacteristic
170#390C22×C311yescharacteristic
171#389—21no—
172#389—21no—
173#389C22×C311yescharacteristic
174#389C22×C311yescharacteristic
175#389C22×C311yescharacteristic
176#389C22×C311yescharacteristic
177#385—21no—
178#384—21no—
179#384C22×C311yescharacteristic
180#380—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4C22×C311yescharacteristic
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×Q8C2×D412yes—
196C2×C3×Q8C2×Q812yes—
197C2×C3×Q8#4112yes—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4C2×Q812yes—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—81no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C3×SD16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8C2×D412yes—
233C2×C3×C8C2×Q812yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—81no—
253C3×C4⋊C4—81no—
254C3×C4⋊C4—81no—
255C3×C4⋊C4—81no—
256C3×C4⋊C4—81no—
257C3×C4⋊C4C2×D412yes—
258C3×C4⋊C4#4112yes—
259C3×C22⋊C4—42no—
260C3×C22⋊C4—42no—
261C3×C22⋊C4—81no—
262C3×C22⋊C4—81no—
263C3×C22⋊C4—81no—
264C3×C42—24no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—42no—
272C3×C42—42no—
273C22×Q8C23×C311yescharacteristic
274C22×D4C23×C311yescharacteristic
275C23×C4—21no—
276C2×SD16—22no—
277C2×SD16—22no—
278C2×SD16C3×Q814yes—
279C22×C8C23×C311yescharacteristic
280C22×C8C23×C311yescharacteristic
281C22×C8C23×C311yescharacteristic
282C22×C8C23×C311yescharacteristic
283#125—22no—
284#125—41no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—41no—
290C4×Q8—41no—
291C4×D4—22no—
292C4×D4—41no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
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⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C3×D411yescharacteristic
305C2×C4⋊C4C3×D411yescharacteristic
306C2×C4⋊C4C23×C311yescharacteristic
307C2×C4⋊C4C23×C311yescharacteristic
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C4⋊C4C23×C311yescharacteristic
311C2×C4⋊C4C23×C311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42C23×C311yescharacteristic
321C2×C42C23×C311yescharacteristic
322C2×C42C23×C311yescharacteristic
323#107—22no—
324#107—22no—
325#107—22no—
326#106—22no—
327#106—41no—
328C4⋊C8—22no—
329#103—22no—
330#103—22no—
331#103C3×D414yes—
332#102—22no—
333#102—22no—
334#102—41no—
335#97—22no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95C3×D411yescharacteristic
341#95C3×D411yescharacteristic
342C23×C3—22no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8C22⋊Q814yes—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—41no—
357C3×D4—41no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
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—21no—
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—84no—
397C2×C3×C4#13712yes—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×Q811yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C3×C8—22no—
405C3×C8—22no—
406C3×C8—41no—
407C3×C8—41no—
408C24—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×C3×D412yes—
413C2×Q8C2×C3×Q812yes—
414C2×Q8C3×#4112yes—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4C2×C3×Q812yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C24×C311yescharacteristic
438SD16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8C2×C3×D412yes—
450C2×C8C2×C3×Q812yes—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4C2×C3×D412yes—
475C4⋊C4C3×#4112yes—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3#52511yescharacteristic
498C22×C3D4211yescharacteristic
499C22×C3D4×Q811yescharacteristic
500C22×C3C2×#13711yescharacteristic
501C22×C3#57511yescharacteristic
502C22×C3#57611yescharacteristic
503C22×C3#57711yescharacteristic
504C3×C4—22no—
505C3×C4—24no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4C2×C22⋊Q811yescharacteristic
518C3×C4#53911yescharacteristic
519C3×C4#54711yescharacteristic
520C3×C4#55411yescharacteristic
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C22×C3×D411yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8C3×C22⋊Q814yes—
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
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—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
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—84no—
576C2×C4C3×#13712yes—
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C22×C3×Q811yescharacteristic
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C3×H(5,𝔽2)11yescharacteristic
582C2×C4C3×2−1+411yescharacteristic
583C8—22no—
584C8—22no—
585C8—41no—
586C8—41no—
587C2×C3—82no—
588C2×C3#244911yescharacteristic
589C2×C3#311711yescharacteristic
590C2×C3#311911yescharacteristic
591C2×C3#326811yescharacteristic
592C2×C3#327811yescharacteristic
593C2×C3#338511yescharacteristic
594C2×C3#338811yescharacteristic
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C3×#52511yescharacteristic
603C22C3×D4211yescharacteristic
604C22C3×D4×Q811yescharacteristic
605C22C2×C3×#13711yescharacteristic
606C22C3×#57511yescharacteristic
607C22C3×#57611yescharacteristic
608C22C3×#57711yescharacteristic
609C4—24no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4C2×C3×C22⋊Q811yescharacteristic
623C4C3×#53911yescharacteristic
624C4C3×#54711yescharacteristic
625C4C3×#55411yescharacteristic
626C3#2638711yescharacteristic, direct product factor, Sylow
627C2—82no—
628C2C3×#244911yescharacteristic
629C2C3×#311711yescharacteristic
630C2C3×#311911yescharacteristic
631C2C3×#326811yescharacteristic
632C2C3×#327811yescharacteristic
633C2C3×#338511yescharacteristic
634C2C3×#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
1. C2×C3×#479C2semidirect
4. C3×Q8×C4⋊C4C2semidirect
6. C3×#2083C2semidirect
8. C3×#2059C2semidirect
9. C3×#1943C2semidirect
10. C3×#1917C2semidirect
13. C3×#1865C2semidirect
14. C3×#1863C2semidirect
16. #26387C3direct (N × H)
62. C2×#479C2×C3semidirect
65. Q8×C4⋊C4C2×C3semidirect
67. #2083C2×C3semidirect
69. #2059C2×C3semidirect
70. #1943C2×C3semidirect
71. #1917C2×C3semidirect
74. #1865C2×C3semidirect
75. #1863C2×C3semidirect
626. C3#26387direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4792
C2×C3×#4402
C3×#24542
C3×Q8×C4⋊C42
C3×#20942
C3×#20832
C3×#20672
C3×#20592
C3×#19432
C3×#19172
C3×#18802
C3×#18652
C3×#18632
C3×#18622
#263873