Finite Group Databasebeta

← C3×#27690 #10646297

Subgroups — C3×#27690 #10646297

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2097C211yesmaximal, characteristic, complemented
6C3×#2095C211yesmaximal, characteristic, complemented
7C3×#1949C211yesmaximal, characteristic, complemented
8C3×#1947C211yesmaximal, characteristic, complemented
9C3×#1873C211yesmaximal, characteristic
10C3×#1872C211yesmaximal, characteristic
11C3×#1854C211yesmaximal, characteristic
12C3×#1852C211yesmaximal, characteristic
13C3×#1805C211yesmaximal, characteristic, complemented
14C3×#1803C211yesmaximal, characteristic, complemented
15C3×#1798C211yesmaximal, characteristic
16#27690C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#391C2211yescharacteristic
35C3×#389—21no—
36C3×#389—21no—
37C3×#389—21no—
38C3×#389C2211yescharacteristic
39C3×#388—21no—
40C3×#388—21no—
41C3×#387—21no—
42C3×#385—21no—
43C3×#385—21no—
44C3×#384C2211yescharacteristic
45C3×#384C2211yescharacteristic
46C3×#382C2211yescharacteristic
47C3×#382C2211yescharacteristic
48C3×#380C2211yescharacteristic
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#2317C2×C311yescharacteristic
62#2101C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2097C2×C311yescharacteristic, complemented
65#2095C2×C311yescharacteristic, complemented
66#1949C2×C311yescharacteristic, complemented
67#1947C2×C311yescharacteristic, complemented
68#1873C2×C311yescharacteristic
69#1872C2×C311yescharacteristic
70#1854C2×C311yescharacteristic
71#1852C2×C311yescharacteristic
72#1805C2×C311yescharacteristic, complemented
73#1803C2×C311yescharacteristic, complemented
74#1798C2×C311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8Q814yes—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×#117—22no—
87C3×#117—22no—
88C3×#117—41no—
89C3×#117—41no—
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⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—41no—
99C2×C3×C4⋊C4—41no—
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4D411yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4D411yescharacteristic
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C42—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#107—41no—
120C3×#107—41no—
121C3×#107—41no—
122C3×#106—41no—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×#103—41no—
126C3×#103—41no—
127C3×#103—41no—
128C3×#103—41no—
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×C4×Q8C22×C311yescharacteristic
142C2×#117C22×C311yescharacteristic
143C2×#117C22×C311yescharacteristic
144C22×C4⋊C4C22×C311yescharacteristic
145C2×#107C22×C311yescharacteristic
146C2×#107C22×C311yescharacteristic
147C2×#107C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×C4⋊C8C22×C311yescharacteristic
150C2×C4⋊C8C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#103C22×C311yescharacteristic
155C2×C22⋊C8C22×C311yescharacteristic
156C2×C22⋊C8C22×C311yescharacteristic
157#391C22×C311yescharacteristic
158#389—21no—
159#389—21no—
160#389—21no—
161#389C22×C311yescharacteristic
162#388—21no—
163#388—21no—
164#387—21no—
165#385—21no—
166#385—21no—
167#384C22×C311yescharacteristic
168#384C22×C311yescharacteristic
169#382C22×C311yescharacteristic
170#382C22×C311yescharacteristic
171#380C22×C311yescharacteristic
172C4×C4⋊C4C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183C24×C3C2×D411yescharacteristic
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
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×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—42no—
203C22×C3×C4—81no—
204C22×C3×C4—81no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×Q812yes—
211C22×C3×C4#4112yes—
212C22×C3×C4C2411yescharacteristic
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—
224C2×C3×C8—21no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
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—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4C2×Q812yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—42no—
257C3×C42—42no—
258C3×C42—81no—
259C3×C42#4112yes—
260C3×C42#4112yes—
261C22×Q8C23×C311yescharacteristic
262C23×C4—21no—
263C23×C4C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22⋊Q8—22no—
269C22⋊Q8C3×Q814yes—
270C4×Q8—22no—
271C4×Q8—22no—
272#117—22no—
273#117—22no—
274#117—41no—
275#117—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C3×D411yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4C3×D411yescharacteristic
299C2×C22⋊C4C3×D411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305#107—41no—
306#107—41no—
307#107—41no—
308#106—41no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311#103—41no—
312#103—41no—
313#103—41no—
314#103—41no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3C22⋊Q812yes—
333C23×C3C22×D411yescharacteristic
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—41no—
337C3×Q8—41no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
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—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—84no—
379C2×C3×C4C22⋊Q812yes—
380C2×C3×C4#12312yes—
381C2×C3×C4#12312yes—
382C2×C3×C4C22×Q811yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C24C2×C3×D411yescharacteristic
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C2×C3×Q812yes—
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—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×Q812yes—
420C22×C4C3×#4112yes—
421C22×C4C24×C311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4C2×C3×Q812yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—42no—
462C22⋊C4—81no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—42no—
467C42—81no—
468C42C3×#4112yes—
469C42C3×#4112yes—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×C22⋊Q811yescharacteristic
478C22×C3C2×#12311yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3#57611yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C22×C3#57811yescharacteristic
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#55311yescharacteristic
498C3×C4#55311yescharacteristic
499C3×C4#55411yescharacteristic
500C3×C4#55411yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C3×C22⋊Q812yes—
508C23C22×C3×D411yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—41no—
512Q8—41no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
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×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4C3×C22⋊Q812yes—
555C2×C4C3×#12312yes—
556C2×C4C3×#12312yes—
557C2×C4C22×C3×Q811yescharacteristic
558C2×C4C2×C3×#4111yescharacteristic
559C2×C4C2×C3×#4111yescharacteristic
560C2×C4C2×C3×#4111yescharacteristic
561C2×C4C3×H(5,𝔽2)11yescharacteristic
562C2×C4C3×H(5,𝔽2)11yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C2×C3—42no—
568C2×C3#248711yescharacteristic
569C2×C3#308211yescharacteristic
570C2×C3#308311yescharacteristic
571C2×C3#309711yescharacteristic
572C2×C3#309811yescharacteristic
573C2×C3#322311yescharacteristic
574C2×C3#322411yescharacteristic
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×C3×C22⋊Q811yescharacteristic
583C22C2×C3×#12311yescharacteristic
584C22C3×#53511yescharacteristic
585C22C3×#57611yescharacteristic
586C22C3×#57711yescharacteristic
587C22C3×#57811yescharacteristic
588C22C3×#57811yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4C3×#55311yescharacteristic
603C4C3×#55311yescharacteristic
604C4C3×#55411yescharacteristic
605C4C3×#55411yescharacteristic
606C3#2769011yescharacteristic, direct product factor, Sylow
607C2—42no—
608C2C3×#248711yescharacteristic
609C2C3×#308211yescharacteristic
610C2C3×#308311yescharacteristic
611C2C3×#309711yescharacteristic
612C2C3×#309811yescharacteristic
613C2C3×#322311yescharacteristic
614C2C3×#322411yescharacteristic
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
3. C3×#2101C2semidirect
4. C3×#2099C2semidirect
5. C3×#2097C2semidirect
6. C3×#2095C2semidirect
7. C3×#1949C2semidirect
8. C3×#1947C2semidirect
13. C3×#1805C2semidirect
14. C3×#1803C2semidirect
16. #27690C3direct (N × H)
62. #2101C2×C3semidirect
63. #2099C2×C3semidirect
64. #2097C2×C3semidirect
65. #2095C2×C3semidirect
66. #1949C2×C3semidirect
67. #1947C2×C3semidirect
72. #1805C2×C3semidirect
73. #1803C2×C3semidirect
606. C3#27690direct (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×#23172
C3×#21012
C3×#20992
C3×#20972
C3×#20952
C3×#19492
C3×#19472
C3×#18732
C3×#18722
C3×#18542
C3×#18522
C3×#18052
C3×#18032
C3×#17982
#276903