Finite Group Databasebeta

← C3×#27740 #10646347

Subgroups — C3×#27740 #10646347

Order 768.

662 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×#2570C211yesmaximal, characteristic, complemented
2C3×#2563C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2101C211yesmaximal, characteristic, complemented
5C3×#2073C211yesmaximal, characteristic
6C3×#2071C211yesmaximal, characteristic
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1988C211yesmaximal, characteristic, complemented
9C3×#1936C211yesmaximal, characteristic
10C3×#1934C211yesmaximal, characteristic
11C3×#1836C211yesmaximal, characteristic, complemented
12C3×#1835C211yesmaximal, characteristic, complemented
13C3×#1804C211yesmaximal, characteristic
14C3×#1771C211yesmaximal, characteristic, complemented
15C3×#1762C211yesmaximal, characteristic
16#27740C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#400—21no—
28C3×#398—21no—
29C3×#393—21no—
30C3×#393—21no—
31C3×#390C2211yescharacteristic
32C3×#390C2211yescharacteristic
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#387—21no—
36C3×#387—21no—
37C3×#384C2211yescharacteristic
38C3×#382C2211yescharacteristic
39C3×#382C2211yescharacteristic
40C3×#382C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×#380—21no—
43C3×#380—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊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#2570C2×C311yescharacteristic, complemented
61#2563C2×C311yescharacteristic
62#2101C2×C311yescharacteristic, complemented
63#2101C2×C311yescharacteristic, complemented
64#2073C2×C311yescharacteristic
65#2071C2×C311yescharacteristic
66#1993C2×C311yescharacteristic, complemented
67#1988C2×C311yescharacteristic, complemented
68#1936C2×C311yescharacteristic
69#1934C2×C311yescharacteristic
70#1836C2×C311yescharacteristic, complemented
71#1835C2×C311yescharacteristic, complemented
72#1804C2×C311yescharacteristic
73#1771C2×C311yescharacteristic, complemented
74#1762C2×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—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C3×C4⋊C8—22no—
110C3×#102—22no—
111C3×#102—22no—
112C3×#102—22no—
113C3×#102—22no—
114C3×#97—22no—
115C3×#97—22no—
116C3×C4×C8—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
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×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C2×C4⋊Q8C22×C311yescharacteristic
131C2×C4×D4C22×C311yescharacteristic
132C2×C4⋊C8C22×C311yescharacteristic
133C2×#102C22×C311yescharacteristic
134C2×#102C22×C311yescharacteristic
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C311yescharacteristic
137C2×#97C22×C311yescharacteristic
138C2×#97C22×C311yescharacteristic
139C2×C4×C8C22×C311yescharacteristic
140#400—21no—
141#398—21no—
142#393—21no—
143#393—21no—
144#390C22×C311yescharacteristic
145#390C22×C311yescharacteristic
146#389C22×C311yescharacteristic
147#389C22×C311yescharacteristic
148#387—21no—
149#387—21no—
150#384C22×C311yescharacteristic
151#382C22×C311yescharacteristic
152#382C22×C311yescharacteristic
153#382C22×C311yescharacteristic
154#382C22×C311yescharacteristic
155#380—21no—
156#380—21no—
157C4×C4⋊C4C22×C311yescharacteristic
158C4×C4⋊C4C22×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#340C22×C311yescharacteristic
170#340C22×C311yescharacteristic
171#336C22×C311yescharacteristic
172#336C22×C311yescharacteristic
173C24×C3—21no—
174C2×C3×Q8—42no—
175C2×C3×Q8—81no—
176C2×C3×Q8—81no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4#4112yes—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
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—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8#4112yes—
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—21no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—41no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4—81no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—81no—
235C3×C22⋊C4—81no—
236C3×C22⋊C4—81no—
237C3×C22⋊C4—81no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—81no—
243C3×C42#4112yes—
244C3×C42#4112yes—
245C22×Q8—21no—
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C22×C8C23×C311yescharacteristic
249C22×C8C23×C311yescharacteristic
250C22×C8C23×C311yescharacteristic
251C22×C8C23×C311yescharacteristic
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4×D4—22no—
255C4×D4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C3×D411yescharacteristic
262C2×C4⋊C4C3×D411yescharacteristic
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C311yescharacteristic
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42C23×C311yescharacteristic
277C2×C42C23×C311yescharacteristic
278C2×C42C23×C311yescharacteristic
279C4⋊C8—22no—
280#102—22no—
281#102—22no—
282#102—22no—
283#102—22no—
284#97—22no—
285#97—22no—
286C4×C8—22no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95C3×D411yescharacteristic
299#95C3×D411yescharacteristic
300C23×C3—22no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—84no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—41no—
312C3×D4—41no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
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—84no—
351C2×C3×C4#12412yes—
352C2×C3×C4#12412yes—
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—22no—
360C3×C8—41no—
361C3×C8—41no—
362C3×C8—41no—
363C24—21no—
364C2×Q8—42no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4C3×#4112yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C24×C311yescharacteristic
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—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8C3×#4112yes—
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—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C22⋊C4—42no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—81no—
433C42C3×#4112yes—
434C42C3×#4112yes—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3C2×#12411yescharacteristic
443C22×C3#54711yescharacteristic
444C22×C3#54811yescharacteristic
445C22×C3#57511yescharacteristic
446C22×C3#57511yescharacteristic
447C22×C3#57611yescharacteristic
448C22×C3#57711yescharacteristic
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—81no—
456C3×C4—81no—
457C3×C4—81no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4#52911yescharacteristic
463C3×C4#54111yescharacteristic
464C3×C4#55211yescharacteristic
465C3×C4#55211yescharacteristic
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23C22×C3×D411yescharacteristic
474Q8—84no—
475D4—22no—
476D4—22no—
477D4—41no—
478D4—41no—
479C2×C4—21no—
480C2×C4—21no—
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—21no—
494C2×C4—21no—
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—84no—
517C2×C4C3×#12412yes—
518C2×C4C3×#12412yes—
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—22no—
526C8—41no—
527C8—41no—
528C8—41no—
529C2×C3—82no—
530C2×C3#248911yescharacteristic
531C2×C3#313611yescharacteristic
532C2×C3#314011yescharacteristic
533C2×C3#334711yescharacteristic
534C2×C3#334811yescharacteristic
535C2×C3#337011yescharacteristic
536C2×C3#337611yescharacteristic
537C22—42no—
538C22—42no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C2×C3×#12411yescharacteristic
545C22C3×#54711yescharacteristic
546C22C3×#54811yescharacteristic
547C22C3×#57511yescharacteristic
548C22C3×#57511yescharacteristic
549C22C3×#57611yescharacteristic
550C22C3×#57711yescharacteristic
551C4—22no—
552C4—22no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4C3×#52911yescharacteristic
565C4C3×#54111yescharacteristic
566C4C3×#55211yescharacteristic
567C4C3×#55211yescharacteristic
568C3#2774011yescharacteristic, direct product factor, Sylow
569C2—82no—
570C2C3×#248911yescharacteristic
571C2C3×#313611yescharacteristic
572C2C3×#314011yescharacteristic
573C2C3×#334711yescharacteristic
574C2C3×#334811yescharacteristic
575C2C3×#337011yescharacteristic
576C2C3×#337611yescharacteristic
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×#2570C2semidirect
3. C3×#2101C2semidirect
4. C3×#2101C2semidirect
7. C3×#1993C2semidirect
8. C3×#1988C2semidirect
11. C3×#1836C2semidirect
12. C3×#1835C2semidirect
14. C3×#1771C2semidirect
16. #27740C3direct (N × H)
60. #2570C2×C3semidirect
62. #2101C2×C3semidirect
63. #2101C2×C3semidirect
66. #1993C2×C3semidirect
67. #1988C2×C3semidirect
70. #1836C2×C3semidirect
71. #1835C2×C3semidirect
73. #1771C2×C3semidirect
568. C3#27740direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25702
C3×#25632
C3×#21012
C3×#20732
C3×#20712
C3×#19932
C3×#19882
C3×#19362
C3×#19342
C3×#18362
C3×#18352
C3×#18042
C3×#17712
C3×#17622
#277403