Finite Group Databasebeta

← C3×#27642 #10646249

Subgroups — C3×#27642 #10646249

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic
2C3×#2314C211yesmaximal, characteristic
3C3×#2077C211yesmaximal, characteristic
4C3×#2077C211yesmaximal, characteristic
5C3×#2036C211yesmaximal, characteristic
6C3×#2035C211yesmaximal, characteristic
7C3×#1949C211yesmaximal, characteristic, complemented
8C3×#1948C211yesmaximal, characteristic, complemented
9C3×#1927C211yesmaximal, characteristic, complemented
10C3×#1927C211yesmaximal, characteristic, complemented
11C3×#1921C211yesmaximal, characteristic, complemented
12C3×#1919C211yesmaximal, characteristic, complemented
13C3×#1805C211yesmaximal, characteristic, complemented
14C3×#1803C211yesmaximal, characteristic, complemented
15C3×#1798C211yesmaximal, characteristic
16#27642C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
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×#389—21no—
34C3×#389C2211yescharacteristic
35C3×#388—21no—
36C3×#388—21no—
37C3×#388—21no—
38C3×#388—21no—
39C3×#387—21no—
40C3×#387—21no—
41C3×#387—21no—
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×#380C2211yescharacteristic
45C3×#380C2211yescharacteristic
46C3×#380C2211yescharacteristic
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×C4×C4⋊C4C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2323C2×C311yescharacteristic
61#2314C2×C311yescharacteristic
62#2077C2×C311yescharacteristic
63#2077C2×C311yescharacteristic
64#2036C2×C311yescharacteristic
65#2035C2×C311yescharacteristic
66#1949C2×C311yescharacteristic, complemented
67#1948C2×C311yescharacteristic, complemented
68#1927C2×C311yescharacteristic, complemented
69#1927C2×C311yescharacteristic, complemented
70#1921C2×C311yescharacteristic, complemented
71#1919C2×C311yescharacteristic, complemented
72#1805C2×C311yescharacteristic, complemented
73#1803C2×C311yescharacteristic, complemented
74#1798C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×#117—22no—
86C3×#117—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C3×#107—22no—
114C3×#106—22no—
115C3×#106—41no—
116C3×#106—41no—
117C3×C4⋊C8—22no—
118C3×C4⋊C8—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#103—41no—
122C3×#103—41no—
123C3×C22⋊C8—41no—
124C3×C22⋊C8—41no—
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
138C2×#117C22×C311yescharacteristic
139C2×#107C22×C311yescharacteristic
140C2×#106C22×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#389—21no—
153#389C22×C311yescharacteristic
154#388—21no—
155#388—21no—
156#388—21no—
157#388—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#384C22×C311yescharacteristic
162#382C22×C311yescharacteristic
163#380C22×C311yescharacteristic
164#380C22×C311yescharacteristic
165#380C22×C311yescharacteristic
166C4×C4⋊C4C22×C311yescharacteristic
167C4×C4⋊C4C22×C311yescharacteristic
168C4×C4⋊C4C22×C311yescharacteristic
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—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
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—42no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4#4112yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—81no—
243C3×C22⋊C4—81no—
244C3×C22⋊C4—81no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248C3×C42—81no—
249C3×C42—81no—
250C3×C42—81no—
251C3×C42#4112yes—
252C3×C42#4112yes—
253C22×Q8C23×C311yescharacteristic
254C23×C4C23×C311yescharacteristic
255C22×C8C23×C311yescharacteristic
256C22×C8C23×C311yescharacteristic
257C22×C8C23×C311yescharacteristic
258C22×C8C23×C311yescharacteristic
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263#117—22no—
264#117—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C3×D411yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C311yescharacteristic
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4C3×D411yescharacteristic
284C2×C22⋊C4C3×D411yescharacteristic
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C23×C311yescharacteristic
290C2×C42C23×C311yescharacteristic
291#107—22no—
292#106—22no—
293#106—41no—
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—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3C22×D411yescharacteristic
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—41no—
325C3×Q8—41no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
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#12312yes—
368C2×C3×C4#12312yes—
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C42−1+411yescharacteristic
375C3×C8—41no—
376C3×C8—41no—
377C3×C8—41no—
378C3×C8—41no—
379C24C2×C3×D411yescharacteristic
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8C3×#4112yes—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C24×C311yescharacteristic
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
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—42no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4C3×#4112yes—
439C22⋊C4—24no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—21no—
446C42—21no—
447C42—42no—
448C42—81no—
449C42—81no—
450C42—81no—
451C42C3×#4112yes—
452C42C3×#4112yes—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3C2×#12311yescharacteristic
461C22×C3#52511yescharacteristic
462C22×C3#53711yescharacteristic
463C22×C3#57711yescharacteristic
464C22×C3#57711yescharacteristic
465C22×C3#57711yescharacteristic
466C22×C3#57811yescharacteristic
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4#55211yescharacteristic
481C3×C4#55311yescharacteristic
482C3×C4#55511yescharacteristic
483C3×C4#55611yescharacteristic
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23C22×C3×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
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×C4C3×#12312yes—
538C2×C4C3×#12312yes—
539C2×C4C2×C3×#4111yescharacteristic
540C2×C4C2×C3×#4111yescharacteristic
541C2×C4C2×C3×#4111yescharacteristic
542C2×C4C2×C3×#4111yescharacteristic
543C2×C4C3×H(5,𝔽2)11yescharacteristic
544C2×C4C3×2−1+411yescharacteristic
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C2×C3—81no—
550C2×C3#248511yescharacteristic
551C2×C3#309711yescharacteristic
552C2×C3#309911yescharacteristic
553C2×C3#312111yescharacteristic
554C2×C3#312511yescharacteristic
555C2×C3#325811yescharacteristic
556C2×C3#326011yescharacteristic
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22C2×C3×#12311yescharacteristic
565C22C3×#52511yescharacteristic
566C22C3×#53711yescharacteristic
567C22C3×#57711yescharacteristic
568C22C3×#57711yescharacteristic
569C22C3×#57711yescharacteristic
570C22C3×#57811yescharacteristic
571C4—22no—
572C4—22no—
573C4—41no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4C3×#55211yescharacteristic
585C4C3×#55311yescharacteristic
586C4C3×#55511yescharacteristic
587C4C3×#55611yescharacteristic
588C3#2764211yescharacteristic, direct product factor, Sylow
589C2—81no—
590C2C3×#248511yescharacteristic
591C2C3×#309711yescharacteristic
592C2C3×#309911yescharacteristic
593C2C3×#312111yescharacteristic
594C2C3×#312511yescharacteristic
595C2C3×#325811yescharacteristic
596C2C3×#326011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
7. C3×#1949C2semidirect
8. C3×#1948C2semidirect
9. C3×#1927C2semidirect
10. C3×#1927C2semidirect
11. C3×#1921C2semidirect
12. C3×#1919C2semidirect
13. C3×#1805C2semidirect
14. C3×#1803C2semidirect
16. #27642C3direct (N × H)
66. #1949C2×C3semidirect
67. #1948C2×C3semidirect
68. #1927C2×C3semidirect
69. #1927C2×C3semidirect
70. #1921C2×C3semidirect
71. #1919C2×C3semidirect
72. #1805C2×C3semidirect
73. #1803C2×C3semidirect
588. C3#27642direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23232
C3×#23142
C3×#20772
C3×#20362
C3×#20352
C3×#19492
C3×#19482
C3×#19272
C3×#19212
C3×#19192
C3×#18052
C3×#18032
C3×#17982
#276423