Finite Group Databasebeta

← C3×#27635 #10646242

Subgroups — C3×#27635 #10646242

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C3×#2283C211yesmaximal, characteristic
3C3×#2077C211yesmaximal, characteristic
4C3×#2076C211yesmaximal, characteristic
5C3×#2036C211yesmaximal, characteristic
6C3×#2035C211yesmaximal, characteristic
7C3×#1949C211yesmaximal, characteristic, complemented
8C3×#1949C211yesmaximal, characteristic, complemented
9C3×#1928C211yesmaximal, characteristic, complemented
10C3×#1927C211yesmaximal, characteristic, complemented
11C3×#1919C211yesmaximal, characteristic, complemented
12C3×#1918C211yesmaximal, characteristic, complemented
13C3×#1805C211yesmaximal, characteristic, complemented
14C3×#1805C211yesmaximal, characteristic, complemented
15C3×#1797C211yesmaximal, characteristic
16#27635C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2211yescharacteristic
32C3×#391C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#387—21no—
35C3×#385—21no—
36C3×#383—21no—
37C3×#382C2211yescharacteristic
38C3×#382C2211yescharacteristic
39C3×#381—21no—
40C3×#380C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×#380C2211yescharacteristic
43C3×C4×C4⋊C4—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊C4C2211yescharacteristic
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C4×C22⋊C4—21no—
48C3×C4×C22⋊C4—21no—
49C3×#376—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60C4×C22⋊Q8C2×C311yescharacteristic
61#2283C2×C311yescharacteristic
62#2077C2×C311yescharacteristic
63#2076C2×C311yescharacteristic
64#2036C2×C311yescharacteristic
65#2035C2×C311yescharacteristic
66#1949C2×C311yescharacteristic, complemented
67#1949C2×C311yescharacteristic, complemented
68#1928C2×C311yescharacteristic, complemented
69#1927C2×C311yescharacteristic, complemented
70#1919C2×C311yescharacteristic, complemented
71#1918C2×C311yescharacteristic, complemented
72#1805C2×C311yescharacteristic, complemented
73#1805C2×C311yescharacteristic, complemented
74#1797C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4D411yescharacteristic
77C23×C3×C4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—22no—
102C2×C3×C42—22no—
103C2×C3×C42—21no—
104C2×C3×C42—21no—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#107—22no—
112C3×#107—41no—
113C3×#106—22no—
114C3×#106—41no—
115C3×C4⋊C8—22no—
116C3×C4⋊C8—22no—
117C3×#103—22no—
118C3×#103—22no—
119C3×#103—41no—
120C3×#103—41no—
121C3×C22⋊C8—41no—
122C3×C22⋊C8—41no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C2×C22⋊Q8C22×C311yescharacteristic
137C2×C4×Q8C22×C311yescharacteristic
138C22×C42C22×C311yescharacteristic
139C2×#107C22×C311yescharacteristic
140C2×#107C22×C311yescharacteristic
141C2×#106C22×C311yescharacteristic
142C2×#106C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×#103C22×C311yescharacteristic
148C2×#103C22×C311yescharacteristic
149C2×C22⋊C8C22×C311yescharacteristic
150C2×C22⋊C8C22×C311yescharacteristic
151#391C22×C311yescharacteristic
152#389C22×C311yescharacteristic
153#387—21no—
154#385—21no—
155#383—21no—
156#382C22×C311yescharacteristic
157#382C22×C311yescharacteristic
158#381—21no—
159#380C22×C311yescharacteristic
160#380C22×C311yescharacteristic
161#380C22×C311yescharacteristic
162C4×C4⋊C4—21no—
163C4×C4⋊C4C22×C311yescharacteristic
164C4×C4⋊C4C22×C311yescharacteristic
165C4×C4⋊C4C22×C311yescharacteristic
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168#376—21no—
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8#4112yes—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
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—21no—
203C22×C3×C4—21no—
204C22×C3×C4—41no—
205C22×C3×C4—41no—
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×C8—21no—
223C2×C3×C8—21no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—21no—
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—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4#4112yes—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250C3×C42—42no—
251C3×C42—42no—
252C3×C42—42no—
253C3×C42#4112yes—
254C3×C42#4112yes—
255C22×Q8C23×C311yescharacteristic
256C23×C4C3×D411yescharacteristic
257C23×C4C3×D411yescharacteristic
258C23×C4C23×C311yescharacteristic
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42C3×D411yescharacteristic
288C2×C42C3×D411yescharacteristic
289C2×C42C23×C311yescharacteristic
290C2×C42C23×C311yescharacteristic
291#107—22no—
292#107—41no—
293#106—22no—
294#106—41no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297#103—22no—
298#103—22no—
299#103—41no—
300#103—41no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3#13712yes—
323C23×C3C22×D411yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—41no—
327C3×Q8—41no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—44no—
343C2×C3×C4—44no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
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#12312yes—
371C2×C3×C4#12312yes—
372C2×C3×C4#13512yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C42−1+411yescharacteristic
379C3×C8—41no—
380C3×C8—41no—
381C3×C8—41no—
382C3×C8—41no—
383C24C2×C3×D411yescharacteristic
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C3×#4112yes—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C24×C311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C4⋊C4—24no—
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—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4C3×#4112yes—
445C22⋊C4—24no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42C3×#4112yes—
458C42C3×#4112yes—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3C2×#12311yescharacteristic
467C22×C3#52511yescharacteristic
468C22×C3#53711yescharacteristic
469C22×C3C2×#13511yescharacteristic
470C22×C3C2×#13711yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57511yescharacteristic
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4—81no—
485C3×C4—81no—
486C3×C4#55211yescharacteristic
487C3×C4#55311yescharacteristic
488C3×C4#55511yescharacteristic
489C3×C4#55611yescharacteristic
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C3×#13712yes—
497C23C22×C3×D411yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502C2×C4—21no—
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—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
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—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4C3×#12312yes—
545C2×C4C3×#12312yes—
546C2×C4C3×#13512yes—
547C2×C4C2×C3×#4111yescharacteristic
548C2×C4C2×C3×#4111yescharacteristic
549C2×C4C2×C3×#4111yescharacteristic
550C2×C4C2×C3×#4111yescharacteristic
551C2×C4C3×H(5,𝔽2)11yescharacteristic
552C2×C4C3×2−1+411yescharacteristic
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C2×C3—42no—
558C2×C3#248511yescharacteristic
559C2×C3#309111yescharacteristic
560C2×C3#309511yescharacteristic
561C2×C3#310211yescharacteristic
562C2×C3#311711yescharacteristic
563C2×C3#324811yescharacteristic
564C2×C3#325611yescharacteristic
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22C2×C3×#12311yescharacteristic
573C22C3×#52511yescharacteristic
574C22C3×#53711yescharacteristic
575C22C2×C3×#13511yescharacteristic
576C22C2×C3×#13711yescharacteristic
577C22C3×#57511yescharacteristic
578C22C3×#57511yescharacteristic
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4C3×#55211yescharacteristic
593C4C3×#55311yescharacteristic
594C4C3×#55511yescharacteristic
595C4C3×#55611yescharacteristic
596C3#2763511yescharacteristic, direct product factor, Sylow
597C2—42no—
598C2C3×#248511yescharacteristic
599C2C3×#309111yescharacteristic
600C2C3×#309511yescharacteristic
601C2C3×#310211yescharacteristic
602C2C3×#311711yescharacteristic
603C2C3×#324811yescharacteristic
604C2C3×#325611yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
7. C3×#1949C2semidirect
8. C3×#1949C2semidirect
9. C3×#1928C2semidirect
10. C3×#1927C2semidirect
11. C3×#1919C2semidirect
12. C3×#1918C2semidirect
13. C3×#1805C2semidirect
14. C3×#1805C2semidirect
16. #27635C3direct (N × H)
66. #1949C2×C3semidirect
67. #1949C2×C3semidirect
68. #1928C2×C3semidirect
69. #1927C2×C3semidirect
70. #1919C2×C3semidirect
71. #1918C2×C3semidirect
72. #1805C2×C3semidirect
73. #1805C2×C3semidirect
596. C3#27635direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C3×#22832
C3×#20772
C3×#20762
C3×#20362
C3×#20352
C3×#19492
C3×#19282
C3×#19272
C3×#19192
C3×#19182
C3×#18052
C3×#17972
#276353