Finite Group Databasebeta

← C3×#29544 #10648151

Subgroups — C3×#29544 #10648151

Order 768.

698 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2320C211yesmaximal, characteristic, complemented
2C3×#2319C211yesmaximal, characteristic
3C3×#2086C211yesmaximal, characteristic, complemented
4C3×#2068C211yesmaximal, characteristic, complemented
5C3×#2065C211yesmaximal, characteristic
6C3×#2055C211yesmaximal, characteristic
7C3×#1928C211yesmaximal, characteristic, complemented
8C3×#1925C211yesmaximal, characteristic
9C3×#1921C211yesmaximal, characteristic, complemented
10C3×#1920C211yesmaximal, characteristic, complemented
11C3×#1880C211yesmaximal, characteristic
12C3×#1877C211yesmaximal, characteristic
13C3×#1865C211yesmaximal, characteristic, complemented
14C3×#1862C211yesmaximal, characteristic
15C3×#1832C211yesmaximal, characteristic, complemented
16#29544C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C3×#391C2211yescharacteristic
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#388—21no—
40C3×#388—21no—
41C3×#387—21no—
42C3×#387—21no—
43C3×#383—21no—
44C3×#382—21no—
45C3×#381—21no—
46C3×#380—21no—
47C3×C4×C4⋊C4C2211yescharacteristic
48C3×C4×C4⋊C4C2211yescharacteristic
49C3×#376C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2320C2×C311yescharacteristic, complemented
61#2319C2×C311yescharacteristic
62#2086C2×C311yescharacteristic, complemented
63#2068C2×C311yescharacteristic, complemented
64#2065C2×C311yescharacteristic
65#2055C2×C311yescharacteristic
66#1928C2×C311yescharacteristic, complemented
67#1925C2×C311yescharacteristic
68#1921C2×C311yescharacteristic, complemented
69#1920C2×C311yescharacteristic, complemented
70#1880C2×C311yescharacteristic
71#1877C2×C311yescharacteristic
72#1865C2×C311yescharacteristic, complemented
73#1862C2×C311yescharacteristic
74#1832C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
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⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—21no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#107—22no—
112C3×#107—41no—
113C3×#106—22no—
114C3×C4⋊C8—41no—
115C3×C4⋊C8—41no—
116C3×#103—22no—
117C3×#103—22no—
118C3×#103—41no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—41no—
122C3×#95—21no—
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×#95D411yescharacteristic
133C3×#95D411yescharacteristic
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C22×SD16C22×C311yescharacteristic
137C2×C4×Q8C22×C311yescharacteristic
138C2×C4×D4C22×C311yescharacteristic
139C2×#107C22×C311yescharacteristic
140C2×#107C22×C311yescharacteristic
141C2×#106C22×C311yescharacteristic
142C2×C4⋊C8C22×C311yescharacteristic
143C2×C4⋊C8C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C311yescharacteristic
147C2×#102C22×C311yescharacteristic
148C2×#102C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150#391C22×C311yescharacteristic
151#391C22×C311yescharacteristic
152#390C22×C311yescharacteristic
153#390C22×C311yescharacteristic
154#389C22×C311yescharacteristic
155#389C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#389C22×C311yescharacteristic
158#388—21no—
159#388—21no—
160#387—21no—
161#387—21no—
162#383—21no—
163#382—21no—
164#381—21no—
165#380—21no—
166C4×C4⋊C4C22×C311yescharacteristic
167C4×C4⋊C4C22×C311yescharacteristic
168#376C22×C311yescharacteristic
169#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#336C22×C311yescharacteristic
176#336C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179C24×C3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8#4112yes—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—81no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C3×SD16—44no—
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—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—24no—
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—81no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4#4112yes—
238C3×C4⋊C4#4112yes—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—81no—
241C3×C22⋊C4—81no—
242C3×C22⋊C4—81no—
243C3×C22⋊C4—81no—
244C3×C42—24no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—81no—
252C3×C42—81no—
253C22×Q8C23×C311yescharacteristic
254C22×D4C23×C311yescharacteristic
255C23×C4—21no—
256C2×SD16—22no—
257C2×SD16—22no—
258C2×SD16—41no—
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C4×Q8—22no—
264C4×Q8—22no—
265C4×D4—22no—
266C4×D4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
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×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42C23×C311yescharacteristic
287C2×C42C23×C311yescharacteristic
288C2×C42C23×C311yescharacteristic
289#107—22no—
290#107—41no—
291#106—22no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294#103—22no—
295#103—22no—
296#103—41no—
297#102—22no—
298#102—22no—
299#102—41no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95C3×D411yescharacteristic
311#95C3×D411yescharacteristic
312#95C3×D411yescharacteristic
313#95C3×D411yescharacteristic
314C23×C3—22no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3C22×D411yescharacteristic
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—41no—
329C3×D4—41no—
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—22no—
337C2×C3×C4—21no—
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—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—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—84no—
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376C3×C8—41no—
377C3×C8—41no—
378C3×C8—41no—
379C3×C8—41no—
380C24—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8C3×#4112yes—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C3×#4112yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C24×C311yescharacteristic
410SD16—44no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C4⋊C4—24no—
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—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4C3×#4112yes—
439C4⋊C4C3×#4112yes—
440C22⋊C4—24no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—24no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—81no—
453C42—81no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3#52511yescharacteristic
462C22×C3#52511yescharacteristic
463C22×C3#53711yescharacteristic
464C22×C3#57611yescharacteristic
465C22×C3#57711yescharacteristic
466C22×C3#57711yescharacteristic
467C22×C3#57811yescharacteristic
468C3×C4—41no—
469C3×C4—42no—
470C3×C4—41no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4#55111yescharacteristic
482C3×C4#55211yescharacteristic
483C3×C4#55211yescharacteristic
484C3×C4#55511yescharacteristic
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C22×C3×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501C2×C4—21no—
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—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
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—84no—
541C2×C4C2×C3×#4111yescharacteristic
542C2×C4C2×C3×#4111yescharacteristic
543C2×C4C2×C3×#4111yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C2×C4C3×H(5,𝔽2)11yescharacteristic
546C2×C4C3×2−1+411yescharacteristic
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C2×C3—82no—
552C2×C3#260911yescharacteristic
553C2×C3#311811yescharacteristic
554C2×C3#312011yescharacteristic
555C2×C3#312411yescharacteristic
556C2×C3#312611yescharacteristic
557C2×C3#325711yescharacteristic
558C2×C3#325811yescharacteristic
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22C3×#52511yescharacteristic
567C22C3×#52511yescharacteristic
568C22C3×#53711yescharacteristic
569C22C3×#57611yescharacteristic
570C22C3×#57711yescharacteristic
571C22C3×#57711yescharacteristic
572C22C3×#57811yescharacteristic
573C4—41no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4C3×#55111yescharacteristic
587C4C3×#55211yescharacteristic
588C4C3×#55211yescharacteristic
589C4C3×#55511yescharacteristic
590C3#2954411yescharacteristic, direct product factor, Sylow
591C2—82no—
592C2C3×#260911yescharacteristic
593C2C3×#311811yescharacteristic
594C2C3×#312011yescharacteristic
595C2C3×#312411yescharacteristic
596C2C3×#312611yescharacteristic
597C2C3×#325711yescharacteristic
598C2C3×#325811yescharacteristic
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×#2320C2semidirect
3. C3×#2086C2semidirect
4. C3×#2068C2semidirect
7. C3×#1928C2semidirect
9. C3×#1921C2semidirect
10. C3×#1920C2semidirect
13. C3×#1865C2semidirect
15. C3×#1832C2semidirect
16. #29544C3direct (N × H)
60. #2320C2×C3semidirect
62. #2086C2×C3semidirect
63. #2068C2×C3semidirect
66. #1928C2×C3semidirect
68. #1921C2×C3semidirect
69. #1920C2×C3semidirect
72. #1865C2×C3semidirect
74. #1832C2×C3semidirect
590. C3#29544direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23202
C3×#23192
C3×#20862
C3×#20682
C3×#20652
C3×#20552
C3×#19282
C3×#19252
C3×#19212
C3×#19202
C3×#18802
C3×#18772
C3×#18652
C3×#18622
C3×#18322
#295443