Finite Group Databasebeta

← C3×#26028 #10644635

Subgroups — C3×#26028 #10644635

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2550C211yesmaximal, characteristic, complemented
2C3×#2357C211yesmaximal, characteristic
3C3×#2067C211yesmaximal, characteristic
4C3×#2065C211yesmaximal, characteristic
5C3×#2025C211yesmaximal, characteristic
6C3×#2023C211yesmaximal, characteristic
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1988C211yesmaximal, characteristic, complemented
9C3×#1949C211yesmaximal, characteristic, complemented
10C3×#1939C211yesmaximal, characteristic, complemented
11C3×#1921C211yesmaximal, characteristic, complemented
12C3×#1920C211yesmaximal, characteristic, complemented
13C3×#1835C211yesmaximal, characteristic, complemented
14C3×#1804C211yesmaximal, characteristic
15C3×C4×#102C211yesmaximal, characteristic
16#26028C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C3×#398—21no—
31C3×#395—21no—
32C3×#391—21no—
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389—21no—
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#388—21no—
39C3×#388—21no—
40C3×#382C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×#381—21no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊C4C2211yescharacteristic
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×C4×C22⋊C4—21no—
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#2550C2×C311yescharacteristic, complemented
61#2357C2×C311yescharacteristic
62#2067C2×C311yescharacteristic
63#2065C2×C311yescharacteristic
64#2025C2×C311yescharacteristic
65#2023C2×C311yescharacteristic
66#1989C2×C311yescharacteristic, complemented
67#1988C2×C311yescharacteristic, complemented
68#1949C2×C311yescharacteristic, complemented
69#1939C2×C311yescharacteristic, complemented
70#1921C2×C311yescharacteristic, complemented
71#1920C2×C311yescharacteristic, complemented
72#1835C2×C311yescharacteristic, complemented
73#1804C2×C311yescharacteristic
74C4×#102C2×C311yescharacteristic
75C22×C3×Q8—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—41no—
84C3×#125—22no—
85C3×#125—41no—
86C3×C4×D4—22no—
87C3×C4×D4Q814yescomplemented
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⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
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×C4⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C42—21no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C3×#107—22no—
114C3×#106—22no—
115C3×C4⋊C8—22no—
116C3×C4⋊C8—41no—
117C3×C4⋊C8—41no—
118C3×#102—22no—
119C3×#102—22no—
120C3×#102—41no—
121C3×#102—41no—
122C3×C4×C8—22no—
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
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×#125C22×C311yescharacteristic
136C2×C4×D4C22×C311yescharacteristic
137C2×#107C22×C311yescharacteristic
138C2×#106C22×C311yescharacteristic
139C2×C4⋊C8C22×C311yescharacteristic
140C2×C4⋊C8C22×C311yescharacteristic
141C2×C4⋊C8C22×C311yescharacteristic
142C2×#102C22×C311yescharacteristic
143C2×#102C22×C311yescharacteristic
144C2×#102C22×C311yescharacteristic
145C2×#102C22×C311yescharacteristic
146C2×C4×C8C22×C311yescharacteristic
147#398—21no—
148#395—21no—
149#391—21no—
150#390C22×C311yescharacteristic
151#390C22×C311yescharacteristic
152#389—21no—
153#389C22×C311yescharacteristic
154#389C22×C311yescharacteristic
155#388—21no—
156#388—21no—
157#382C22×C311yescharacteristic
158#382C22×C311yescharacteristic
159#381—21no—
160C4×C4⋊C4C22×C311yescharacteristic
161C4×C4⋊C4C22×C311yescharacteristic
162C4×C4⋊C4C22×C311yescharacteristic
163C4×C4⋊C4C22×C311yescharacteristic
164C4×C22⋊C4—21no—
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#336C22×C311yescharacteristic
176#336C22×C311yescharacteristic
177C24×C3—21no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×Q8—81no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4C2×Q812yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—81no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C2×C3×C8—21no—
205C2×C3×C8—21no—
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—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
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—41no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—81no—
234C3×C4⋊C4C2×Q812yes—
235C3×C4⋊C4#4112yes—
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—42no—
239C3×C22⋊C4—42no—
240C3×C22⋊C4—81no—
241C3×C22⋊C4—81no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—81no—
247C3×C42—81no—
248C3×C42—81no—
249C3×C42C2×Q812yes—
250C3×C42#4112yes—
251C22×Q8—21no—
252C22×D4C23×C311yescharacteristic
253C23×C4—21no—
254C22×C8C23×C311yescharacteristic
255C22×C8C23×C311yescharacteristic
256C22×C8C23×C311yescharacteristic
257C22×C8C23×C311yescharacteristic
258C4⋊Q8—22no—
259C4⋊Q8—41no—
260#125—22no—
261#125—41no—
262C4×D4—22no—
263C4×D4C3×Q814yescomplemented
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C3×D411yescharacteristic
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
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#106—22no—
291C4⋊C8—22no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294#102—22no—
295#102—22no—
296#102—41no—
297#102—41no—
298C4×C8—22no—
299#95—21no—
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#95C3×D411yescharacteristic
309#95C3×D411yescharacteristic
310C23×C3—22no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3C22×D411yescharacteristic
318C3×Q8—84no—
319C3×D4—22no—
320C3×D4—22no—
321C3×D4—41no—
322C3×D4—41no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
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—84no—
360C2×C3×C4C22⋊Q812yes—
361C2×C3×C4C22⋊Q812yes—
362C2×C3×C4#13512yes—
363C2×C3×C4C22×Q811yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×C3×C42−1+411yescharacteristic
369C3×C8—41no—
370C3×C8—41no—
371C3×C8—41no—
372C3×C8—41no—
373C24—21no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—81no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4C2×C3×Q812yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C24×C311yescharacteristic
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4C2×C3×Q812yes—
431C4⋊C4C3×#4112yes—
432C4⋊C4C3×#4112yes—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—81no—
443C42—81no—
444C42—81no—
445C42C2×C3×Q812yes—
446C42C3×#4112yes—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3C2×C22⋊Q811yescharacteristic
455C22×C3#54611yescharacteristic
456C22×C3#54811yescharacteristic
457C22×C3C2×#13511yescharacteristic
458C22×C3#57511yescharacteristic
459C22×C3#57611yescharacteristic
460C22×C3#57711yescharacteristic
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4—42no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—81no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4—81no—
474C3×C4#53311yescharacteristic
475C3×C4#54411yescharacteristic
476C3×C4#55511yescharacteristic
477C3×C4#55611yescharacteristic
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22×C3×D411yescharacteristic
486Q8—84no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
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—84no—
528C2×C4C3×C22⋊Q812yes—
529C2×C4C3×C22⋊Q812yes—
530C2×C4C3×#13512yes—
531C2×C4C22×C3×Q811yescharacteristic
532C2×C4C2×C3×#4111yescharacteristic
533C2×C4C2×C3×#4111yescharacteristic
534C2×C4C2×C3×#4111yescharacteristic
535C2×C4C3×H(5,𝔽2)11yescharacteristic
536C2×C4C3×2−1+411yescharacteristic
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C2×C3—82no—
542C2×C3#243911yescharacteristic
543C2×C3#307611yescharacteristic
544C2×C3#308111yescharacteristic
545C2×C3#329711yescharacteristic
546C2×C3#332611yescharacteristic
547C2×C3#336511yescharacteristic
548C2×C3#337011yescharacteristic
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22C2×C3×C22⋊Q811yescharacteristic
557C22C3×#54611yescharacteristic
558C22C3×#54811yescharacteristic
559C22C2×C3×#13511yescharacteristic
560C22C3×#57511yescharacteristic
561C22C3×#57611yescharacteristic
562C22C3×#57711yescharacteristic
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C3×#53311yescharacteristic
577C4C3×#54411yescharacteristic
578C4C3×#55511yescharacteristic
579C4C3×#55611yescharacteristic
580C3#2602811yescharacteristic, direct product factor, Sylow
581C2—82no—
582C2C3×#243911yescharacteristic
583C2C3×#307611yescharacteristic
584C2C3×#308111yescharacteristic
585C2C3×#329711yescharacteristic
586C2C3×#332611yescharacteristic
587C2C3×#336511yescharacteristic
588C2C3×#337011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2550C2semidirect
7. C3×#1989C2semidirect
8. C3×#1988C2semidirect
9. C3×#1949C2semidirect
10. C3×#1939C2semidirect
11. C3×#1921C2semidirect
12. C3×#1920C2semidirect
13. C3×#1835C2semidirect
16. #26028C3direct (N × H)
60. #2550C2×C3semidirect
66. #1989C2×C3semidirect
67. #1988C2×C3semidirect
68. #1949C2×C3semidirect
69. #1939C2×C3semidirect
70. #1921C2×C3semidirect
71. #1920C2×C3semidirect
72. #1835C2×C3semidirect
87. C3×C4×D4Q8semidirect
263. C4×D4C3×Q8semidirect
580. C3#26028direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25502
C3×#23572
C3×#20672
C3×#20652
C3×#20252
C3×#20232
C3×#19892
C3×#19882
C3×#19492
C3×#19392
C3×#19212
C3×#19202
C3×#18352
C3×#18042
C3×C4×#1022
#260283