Finite Group Databasebeta

← C3×#27622 #10646229

Subgroups — C3×#27622 #10646229

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2317C211yesmaximal, characteristic
2C3×#2314C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2085C211yesmaximal, characteristic, complemented
6C3×#2085C211yesmaximal, characteristic, complemented
7C3×#1921C211yesmaximal, characteristic, complemented
8C3×#1920C211yesmaximal, characteristic, complemented
9C3×#1919C211yesmaximal, characteristic, complemented
10C3×#1918C211yesmaximal, characteristic, complemented
11C3×#1854C211yesmaximal, characteristic
12C3×#1854C211yesmaximal, characteristic
13C3×#1824C211yesmaximal, characteristic
14C3×#1823C211yesmaximal, characteristic
15C3×#1818C211yesmaximal, characteristic
16#27622C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#388—21no—
31C3×#388—21no—
32C3×#387—21no—
33C3×#387—21no—
34C3×#385—21no—
35C3×#385—21no—
36C3×#384—21no—
37C3×#384C2211yescharacteristic
38C3×#382—21no—
39C3×#382C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×#340C2211yescharacteristic
45C3×#340C2211yescharacteristic
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2317C2×C311yescharacteristic
61#2314C2×C311yescharacteristic
62#2100C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2085C2×C311yescharacteristic, complemented
65#2085C2×C311yescharacteristic, complemented
66#1921C2×C311yescharacteristic, complemented
67#1920C2×C311yescharacteristic, complemented
68#1919C2×C311yescharacteristic, complemented
69#1918C2×C311yescharacteristic, complemented
70#1854C2×C311yescharacteristic
71#1854C2×C311yescharacteristic
72#1824C2×C311yescharacteristic
73#1823C2×C311yescharacteristic
74#1818C2×C311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#117—22no—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D411yescharacteristic
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×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×#107—22no—
116C3×#106—22no—
117C3×#106—22no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C2×#117C22×C311yescharacteristic
131C2×#117C22×C311yescharacteristic
132C22×C4⋊C4C22×C311yescharacteristic
133C2×#107C22×C311yescharacteristic
134C2×#107C22×C311yescharacteristic
135C2×#106C22×C311yescharacteristic
136C2×#106C22×C311yescharacteristic
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#389C22×C311yescharacteristic
140#389C22×C311yescharacteristic
141#389C22×C311yescharacteristic
142#389C22×C311yescharacteristic
143#388—21no—
144#388—21no—
145#387—21no—
146#387—21no—
147#385—21no—
148#385—21no—
149#384—21no—
150#384C22×C311yescharacteristic
151#382—21no—
152#382C22×C311yescharacteristic
153#380C22×C311yescharacteristic
154#380C22×C311yescharacteristic
155C4×C4⋊C4C22×C311yescharacteristic
156C4×C4⋊C4C22×C311yescharacteristic
157#340C22×C311yescharacteristic
158#340C22×C311yescharacteristic
159#340C22×C311yescharacteristic
160#340C22×C311yescharacteristic
161#340C22×C311yescharacteristic
162#340C22×C311yescharacteristic
163#340C22×C311yescharacteristic
164#340C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#336C22×C311yescharacteristic
170#336C22×C311yescharacteristic
171#336C22×C311yescharacteristic
172#336C22×C311yescharacteristic
173C24×C3C2×D411yescharacteristic
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—42no—
187C22×C3×C4—81no—
188C22×C3×C4—81no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4#4112yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
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—42no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4—81no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4#4112yes—
233C3×C4⋊C4#4112yes—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—81no—
238C3×C22⋊C4—81no—
239C3×C42—24no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—81no—
245C3×C42—81no—
246C23×C4—21no—
247C23×C4C23×C311yescharacteristic
248C22×C8C23×C311yescharacteristic
249C22×C8C23×C311yescharacteristic
250C22×C8C23×C311yescharacteristic
251C22×C8C23×C311yescharacteristic
252#117—22no—
253#117—22no—
254#117—22no—
255#117—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C3×D411yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4C3×D411yescharacteristic
279C2×C22⋊C4C3×D411yescharacteristic
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C23×C311yescharacteristic
284C2×C42C23×C311yescharacteristic
285#107—22no—
286#107—22no—
287#106—22no—
288#106—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3#12312yes—
308C23×C3C22×D411yescharacteristic
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
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—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
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—84no—
352C2×C3×C4#12312yes—
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×C3×C42−1+411yescharacteristic
359C3×C8—41no—
360C3×C8—41no—
361C3×C8—41no—
362C3×C8—41no—
363C24C2×C3×D411yescharacteristic
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C3×#4112yes—
385C22×C4C3×#4112yes—
386C22×C4C24×C311yescharacteristic
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4C3×#4112yes—
423C4⋊C4C3×#4112yes—
424C22⋊C4—24no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C42—24no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—81no—
435C42—81no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3C2×#12311yescharacteristic
444C22×C3#52511yescharacteristic
445C22×C3#53611yescharacteristic
446C22×C3#57611yescharacteristic
447C22×C3#57711yescharacteristic
448C22×C3#57711yescharacteristic
449C22×C3#57811yescharacteristic
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—81no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4#55211yescharacteristic
464C3×C4#55211yescharacteristic
465C3×C4#55211yescharacteristic
466C3×C4#55211yescharacteristic
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23C3×#12312yes—
474C23C22×C3×D411yescharacteristic
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
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—84no—
518C2×C4C3×#12312yes—
519C2×C4C2×C3×#4111yescharacteristic
520C2×C4C2×C3×#4111yescharacteristic
521C2×C4C2×C3×#4111yescharacteristic
522C2×C4C2×C3×#4111yescharacteristic
523C2×C4C3×2−1+411yescharacteristic
524C2×C4C3×2−1+411yescharacteristic
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C2×C3—42no—
530C2×C3#248411yescharacteristic
531C2×C3#309611yescharacteristic
532C2×C3#309711yescharacteristic
533C2×C3#312411yescharacteristic
534C2×C3#312511yescharacteristic
535C2×C3#324111yescharacteristic
536C2×C3#324211yescharacteristic
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22C2×C3×#12311yescharacteristic
545C22C3×#52511yescharacteristic
546C22C3×#53611yescharacteristic
547C22C3×#57611yescharacteristic
548C22C3×#57711yescharacteristic
549C22C3×#57711yescharacteristic
550C22C3×#57811yescharacteristic
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4C3×#55211yescharacteristic
565C4C3×#55211yescharacteristic
566C4C3×#55211yescharacteristic
567C4C3×#55211yescharacteristic
568C3#2762211yescharacteristic, direct product factor, Sylow
569C2—42no—
570C2C3×#248411yescharacteristic
571C2C3×#309611yescharacteristic
572C2C3×#309711yescharacteristic
573C2C3×#312411yescharacteristic
574C2C3×#312511yescharacteristic
575C2C3×#324111yescharacteristic
576C2C3×#324211yescharacteristic
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×#2100C2semidirect
4. C3×#2099C2semidirect
5. C3×#2085C2semidirect
6. C3×#2085C2semidirect
7. C3×#1921C2semidirect
8. C3×#1920C2semidirect
9. C3×#1919C2semidirect
10. C3×#1918C2semidirect
16. #27622C3direct (N × H)
62. #2100C2×C3semidirect
63. #2099C2×C3semidirect
64. #2085C2×C3semidirect
65. #2085C2×C3semidirect
66. #1921C2×C3semidirect
67. #1920C2×C3semidirect
68. #1919C2×C3semidirect
69. #1918C2×C3semidirect
568. C3#27622direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23172
C3×#23142
C3×#21002
C3×#20992
C3×#20852
C3×#19212
C3×#19202
C3×#19192
C3×#19182
C3×#18542
C3×#18242
C3×#18232
C3×#18182
#276223