Finite Group Databasebeta

← C3×#29203 #10647810

Subgroups — C3×#29203 #10647810

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2647C211yesmaximal, characteristic, complemented
2C3×#2326C211yesmaximal, characteristic
3C3×#2068C211yesmaximal, characteristic, complemented
4C3×#2067C211yesmaximal, characteristic
5C3×#2066C211yesmaximal, characteristic, complemented
6C3×#2065C211yesmaximal, characteristic
7C3×#1959C211yesmaximal, characteristic
8C3×#1958C211yesmaximal, characteristic
9C3×#1865C211yesmaximal, characteristic, complemented
10C3×#1864C211yesmaximal, characteristic
11C3×#1863C211yesmaximal, characteristic, complemented
12C3×#1862C211yesmaximal, characteristic
13C3×#1847C211yesmaximal, characteristic, complemented
14C3×#1834C211yesmaximal, characteristic, complemented
15C3×#1768C211yesmaximal, characteristic, complemented
16#29203C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
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×#102C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#97C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C3×#400—21no—
37C3×#400—21no—
38C3×#398—21no—
39C3×#398—21no—
40C3×#391C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#389C2211yescharacteristic
46C3×#388—21no—
47C3×#388—21no—
48C3×#385—21no—
49C3×#385—21no—
50C3×#384C2211yescharacteristic
51C3×#384C2211yescharacteristic
52C3×#382C2211yescharacteristic
53C3×#376C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2647C2×C311yescharacteristic, complemented
61#2326C2×C311yescharacteristic
62#2068C2×C311yescharacteristic, complemented
63#2067C2×C311yescharacteristic
64#2066C2×C311yescharacteristic, complemented
65#2065C2×C311yescharacteristic
66#1959C2×C311yescharacteristic
67#1958C2×C311yescharacteristic
68#1865C2×C311yescharacteristic, complemented
69#1864C2×C311yescharacteristic
70#1863C2×C311yescharacteristic, complemented
71#1862C2×C311yescharacteristic
72#1847C2×C311yescharacteristic, complemented
73#1834C2×C311yescharacteristic, complemented
74#1768C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#124—22no—
83C3×#124Q814yes—
84C3×C4×Q8—41no—
85C3×C4×Q8—41no—
86C3×C4×D4—41no—
87C3×C4×D4—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C42—21no—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#107—41no—
113C3×#107—41no—
114C3×#106—41no—
115C3×#106—41no—
116C3×C4⋊C8—22no—
117C3×C4⋊C8—22no—
118C3×#103—22no—
119C3×#103—22no—
120C3×#103—41no—
121C3×#103—41no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—41no—
125C3×#102—41no—
126C3×#97—22no—
127C3×C4×C8—22no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95D411yescharacteristic
137C3×#95D411yescharacteristic
138C3×#95D411yescharacteristic
139C3×#95D411yescharacteristic
140C2×#124C22×C311yescharacteristic
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×D4C22×C311yescharacteristic
143C2×#107C22×C311yescharacteristic
144C2×#107C22×C311yescharacteristic
145C2×#106C22×C311yescharacteristic
146C2×#106C22×C311yescharacteristic
147C2×C4⋊C8C22×C311yescharacteristic
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#102C22×C311yescharacteristic
154C2×#102C22×C311yescharacteristic
155C2×#102C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#97C22×C311yescharacteristic
158C2×C4×C8C22×C311yescharacteristic
159#400—21no—
160#400—21no—
161#398—21no—
162#398—21no—
163#391C22×C311yescharacteristic
164#390C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#389C22×C311yescharacteristic
167#389C22×C311yescharacteristic
168#389C22×C311yescharacteristic
169#388—21no—
170#388—21no—
171#385—21no—
172#385—21no—
173#384C22×C311yescharacteristic
174#384C22×C311yescharacteristic
175#382C22×C311yescharacteristic
176#376C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#336C22×C311yescharacteristic
182#336C22×C311yescharacteristic
183C24×C3—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×Q812yes—
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—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
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#4112yes—
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—42no—
238C3×C4⋊C4—81no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42C2×Q812yes—
257C3×C42#4112yes—
258C22×Q8C23×C311yescharacteristic
259C22×D4C23×C311yescharacteristic
260C23×C4—21no—
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265#124—22no—
266#124C3×Q814yes—
267C4×Q8—41no—
268C4×Q8—41no—
269C4×D4—41no—
270C4×D4—41no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#107—41no—
296#107—41no—
297#106—41no—
298#106—41no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301#103—22no—
302#103—22no—
303#103—41no—
304#103—41no—
305#102—22no—
306#102—22no—
307#102—41no—
308#102—41no—
309#97—22no—
310C4×C8—22no—
311#95—21no—
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
321#95C3×D411yescharacteristic
322#95C3×D411yescharacteristic
323C23×C3—22no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3C22×D411yescharacteristic
331C3×Q8—22no—
332C3×Q8—22no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—41no—
338C3×D4—41no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
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—84no—
377C2×C3×C4C22⋊Q812yes—
378C2×C3×C4C22⋊Q812yes—
379C2×C3×C4C22×Q811yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C3×C8—22no—
386C3×C8—22no—
387C3×C8—41no—
388C3×C8—41no—
389C24—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×C3×Q812yes—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C2×C3×Q812yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—42no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C24×C311yescharacteristic
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—
428C2×C8—21no—
429C2×C8C3×#4112yes—
430C2×C8C3×#4112yes—
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—21no—
443C4⋊C4—42no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—42no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—42no—
462C42C2×C3×Q812yes—
463C42C3×#4112yes—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×C22⋊Q811yescharacteristic
472C22×C3#53211yescharacteristic
473C22×C3#53511yescharacteristic
474C22×C3#57611yescharacteristic
475C22×C3#57711yescharacteristic
476C22×C3#57711yescharacteristic
477C22×C3#57811yescharacteristic
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4#55311yescharacteristic
492C3×C4#55311yescharacteristic
493C3×C4#55411yescharacteristic
494C3×C4#55411yescharacteristic
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C22×C3×D411yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—22no—
508D4—22no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
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×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4C3×C22⋊Q812yes—
550C2×C4C3×C22⋊Q812yes—
551C2×C4C22×C3×Q811yescharacteristic
552C2×C4C2×C3×#4111yescharacteristic
553C2×C4C2×C3×#4111yescharacteristic
554C2×C4C2×C3×#4111yescharacteristic
555C2×C4C3×H(5,𝔽2)11yescharacteristic
556C2×C4C3×H(5,𝔽2)11yescharacteristic
557C8—22no—
558C8—22no—
559C8—41no—
560C8—41no—
561C2×C3—82no—
562C2×C3#256611yescharacteristic
563C2×C3#308311yescharacteristic
564C2×C3#308411yescharacteristic
565C2×C3#317811yescharacteristic
566C2×C3#317911yescharacteristic
567C2×C3#322411yescharacteristic
568C2×C3#322511yescharacteristic
569C22—42no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22C2×C3×C22⋊Q811yescharacteristic
577C22C3×#53211yescharacteristic
578C22C3×#53511yescharacteristic
579C22C3×#57611yescharacteristic
580C22C3×#57711yescharacteristic
581C22C3×#57711yescharacteristic
582C22C3×#57811yescharacteristic
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4C3×#55311yescharacteristic
597C4C3×#55311yescharacteristic
598C4C3×#55411yescharacteristic
599C4C3×#55411yescharacteristic
600C3#2920311yescharacteristic, direct product factor, Sylow
601C2—82no—
602C2C3×#256611yescharacteristic
603C2C3×#308311yescharacteristic
604C2C3×#308411yescharacteristic
605C2C3×#317811yescharacteristic
606C2C3×#317911yescharacteristic
607C2C3×#322411yescharacteristic
608C2C3×#322511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2647C2semidirect
3. C3×#2068C2semidirect
5. C3×#2066C2semidirect
9. C3×#1865C2semidirect
11. C3×#1863C2semidirect
13. C3×#1847C2semidirect
14. C3×#1834C2semidirect
15. C3×#1768C2semidirect
16. #29203C3direct (N × H)
60. #2647C2×C3semidirect
62. #2068C2×C3semidirect
64. #2066C2×C3semidirect
68. #1865C2×C3semidirect
70. #1863C2×C3semidirect
72. #1847C2×C3semidirect
73. #1834C2×C3semidirect
74. #1768C2×C3semidirect
600. C3#29203direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26472
C3×#23262
C3×#20682
C3×#20672
C3×#20662
C3×#20652
C3×#19592
C3×#19582
C3×#18652
C3×#18642
C3×#18632
C3×#18622
C3×#18472
C3×#18342
C3×#17682
#292033