Finite Group Databasebeta

← C3×#27710 #10646317

Subgroups — C3×#27710 #10646317

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2500C211yesmaximal, characteristic
2C3×#2442C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2073C211yesmaximal, characteristic, complemented
6C3×#2025C211yesmaximal, characteristic, complemented
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1936C211yesmaximal, characteristic, complemented
9C3×#1934C211yesmaximal, characteristic, complemented
10C3×#1895C211yesmaximal, characteristic
11C3×#1874C211yesmaximal, characteristic
12C3×#1835C211yesmaximal, characteristic, complemented
13C3×#1825C211yesmaximal, characteristic, complemented
14C3×#1753C211yesmaximal, characteristic, complemented
15C3×#1736C211yesmaximal, characteristic, complemented
16#27710C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#400—21no—
30C3×#398—21no—
31C3×#394—21no—
32C3×#394—21no—
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#385—21no—
40C3×#384C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×#382C2211yescharacteristic
43C3×#380—21no—
44C3×#380C2211yescharacteristic
45C3×C4×C22⋊C4C2211yescharacteristic
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2500C2×C311yescharacteristic
61#2442C2×C311yescharacteristic, complemented
62#2101C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2073C2×C311yescharacteristic, complemented
65#2025C2×C311yescharacteristic, complemented
66#1992C2×C311yescharacteristic, complemented
67#1936C2×C311yescharacteristic, complemented
68#1934C2×C311yescharacteristic, complemented
69#1895C2×C311yescharacteristic
70#1874C2×C311yescharacteristic
71#1835C2×C311yescharacteristic, complemented
72#1825C2×C311yescharacteristic, complemented
73#1753C2×C311yescharacteristic, complemented
74#1736C2×C311yescharacteristic, complemented
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C2×C3×M16—22no—
80C2×C3×M16—22no—
81C2×C3×M16—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#125—22no—
87C3×#125—41no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#117—22no—
91C3×#117—22no—
92C2×C3×C4⋊C4—22no—
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—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—41no—
101C2×C3×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#102—22no—
121C3×#102—22no—
122C3×#102—41no—
123C3×#102—41no—
124C3×C22⋊C8—22no—
125C3×#97—41no—
126C3×C4×C8—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C22×M16C22×C311yescharacteristic
137C2×#125C22×C311yescharacteristic
138C2×#121C22×C311yescharacteristic
139C2×#117C22×C311yescharacteristic
140C22×C4⋊C4C22×C311yescharacteristic
141C2×#102C22×C311yescharacteristic
142C2×#102C22×C311yescharacteristic
143C2×#102C22×C311yescharacteristic
144C2×#102C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×#97C22×C311yescharacteristic
147C2×C4×C8C22×C311yescharacteristic
148#400—21no—
149#398—21no—
150#394—21no—
151#394—21no—
152#390C22×C311yescharacteristic
153#390C22×C311yescharacteristic
154#389C22×C311yescharacteristic
155#389C22×C311yescharacteristic
156#388—21no—
157#388—21no—
158#385—21no—
159#384C22×C311yescharacteristic
160#382C22×C311yescharacteristic
161#382C22×C311yescharacteristic
162#380—21no—
163#380C22×C311yescharacteristic
164C4×C22⋊C4C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×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×C3—21no—
180C24×C3C2×D411yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—42no—
188C2×C3×D4—81no—
189C2×C3×D4—81no—
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—42no—
202C22×C3×C4—42no—
203C22×C3×C4—81no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4#4112yes—
210C22×C3×C4#4112yes—
211C22×C3×C4C2411yescharacteristic
212C3×M16—44no—
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#4112yes—
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—42no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—81no—
254C3×C22⋊C4—81no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—42no—
260C22×D4—21no—
261C22×D4C23×C311yescharacteristic
262C23×C4—21no—
263C23×C4C23×C311yescharacteristic
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—41no—
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271#125—22no—
272#125—41no—
273#121—22no—
274#121—22no—
275#117—22no—
276#117—22no—
277C2×C4⋊C4—22no—
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—21no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4C3×D411yescharacteristic
301C2×C22⋊C4C3×D411yescharacteristic
302C2×C42—21no—
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305#102—22no—
306#102—22no—
307#102—41no—
308#102—41no—
309C22⋊C8—22no—
310#97—41no—
311C4×C8—22no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95C3×D411yescharacteristic
320#95C3×D411yescharacteristic
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3#12412yes—
335C23×C3C22×D411yescharacteristic
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—84no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
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—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
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—41no—
379C2×C3×C4—41no—
380C2×C3×C4—84no—
381C2×C3×C4#12412yes—
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C3×C8—22no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C24—21no—
393C24C2×C3×D411yescharacteristic
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—42no—
401C2×D4—81no—
402C2×D4—81no—
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—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—81no—
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C3×#4112yes—
423C22×C4C3×#4112yes—
424C22×C4C24×C311yescharacteristic
425M16—44no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8C3×#4112yes—
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—21no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4C3×#4112yes—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3C2×#12411yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57511yescharacteristic
492C22×C3#57611yescharacteristic
493C22×C3#57711yescharacteristic
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4—81no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4#52911yescharacteristic
507C3×C4#53911yescharacteristic
508C3×C4#55111yescharacteristic
509C3×C4#55211yescharacteristic
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C3×#12412yes—
524C23C22×C3×D411yescharacteristic
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—84no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
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—84no—
570C2×C4C3×#12412yes—
571C2×C4C2×C3×#4111yescharacteristic
572C2×C4C2×C3×#4111yescharacteristic
573C2×C4C2×C3×#4111yescharacteristic
574C2×C4C2×C3×#4111yescharacteristic
575C2×C4C3×H(5,𝔽2)11yescharacteristic
576C2×C4C3×2−1+411yescharacteristic
577C8—22no—
578C8—41no—
579C8—41no—
580C8—41no—
581C2×C3—42no—
582C2×C3—161no—
583C2×C3#248811yescharacteristic
584C2×C3#313611yescharacteristic
585C2×C3#314211yescharacteristic
586C2×C3#332211yescharacteristic
587C2×C3#332611yescharacteristic
588C2×C3#334711yescharacteristic
589C2×C3#334811yescharacteristic
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22C2×C3×#12411yescharacteristic
605C22C3×#54611yescharacteristic
606C22C3×#54711yescharacteristic
607C22C3×#57511yescharacteristic
608C22C3×#57511yescharacteristic
609C22C3×#57611yescharacteristic
610C22C3×#57711yescharacteristic
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4C3×#52911yescharacteristic
624C4C3×#53911yescharacteristic
625C4C3×#55111yescharacteristic
626C4C3×#55211yescharacteristic
627C3#2771011yescharacteristic, direct product factor, Sylow
628C2—42no—
629C2—161no—
630C2C3×#248811yescharacteristic
631C2C3×#313611yescharacteristic
632C2C3×#314211yescharacteristic
633C2C3×#332211yescharacteristic
634C2C3×#332611yescharacteristic
635C2C3×#334711yescharacteristic
636C2C3×#334811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2442C2semidirect
3. C3×#2101C2semidirect
4. C3×#2099C2semidirect
5. C3×#2073C2semidirect
6. C3×#2025C2semidirect
7. C3×#1992C2semidirect
8. C3×#1936C2semidirect
9. C3×#1934C2semidirect
12. C3×#1835C2semidirect
13. C3×#1825C2semidirect
14. C3×#1753C2semidirect
15. C3×#1736C2semidirect
16. #27710C3direct (N × H)
61. #2442C2×C3semidirect
62. #2101C2×C3semidirect
63. #2099C2×C3semidirect
64. #2073C2×C3semidirect
65. #2025C2×C3semidirect
66. #1992C2×C3semidirect
67. #1936C2×C3semidirect
68. #1934C2×C3semidirect
71. #1835C2×C3semidirect
72. #1825C2×C3semidirect
73. #1753C2×C3semidirect
74. #1736C2×C3semidirect
627. C3#27710direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25002
C3×#24422
C3×#21012
C3×#20992
C3×#20732
C3×#20252
C3×#19922
C3×#19362
C3×#19342
C3×#18952
C3×#18742
C3×#18352
C3×#18252
C3×#17532
C3×#17362
#277103