Finite Group Databasebeta

← C3×#27698 #10646305

Subgroups — C3×#27698 #10646305

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2499C211yesmaximal, characteristic
2C3×#2439C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2099C211yesmaximal, characteristic, complemented
5C3×#2074C211yesmaximal, characteristic, complemented
6C3×#2024C211yesmaximal, characteristic, complemented
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1933C211yesmaximal, characteristic, complemented
9C3×#1933C211yesmaximal, characteristic, complemented
10C3×#1892C211yesmaximal, characteristic
11C3×#1873C211yesmaximal, characteristic
12C3×#1835C211yesmaximal, characteristic, complemented
13C3×#1825C211yesmaximal, characteristic
14C3×C8×C22⋊C4C211yesmaximal, characteristic
15C3×#1735C211yesmaximal, characteristic
16#27698C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#397—21no—
30C3×#397—21no—
31C3×#396—21no—
32C3×#393—21no—
33C3×#391—21no—
34C3×#391C2211yescharacteristic
35C3×#391C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#387—21no—
39C3×#387—21no—
40C3×#387—21no—
41C3×#384C2211yescharacteristic
42C3×#382C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×#380C2211yescharacteristic
45C3×C4×C22⋊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#2499C2×C311yescharacteristic
61#2439C2×C311yescharacteristic
62#2101C2×C311yescharacteristic, complemented
63#2099C2×C311yescharacteristic, complemented
64#2074C2×C311yescharacteristic, complemented
65#2024C2×C311yescharacteristic, complemented
66#1990C2×C311yescharacteristic, complemented
67#1933C2×C311yescharacteristic, complemented
68#1933C2×C311yescharacteristic, complemented
69#1892C2×C311yescharacteristic
70#1873C2×C311yescharacteristic
71#1835C2×C311yescharacteristic, complemented
72#1825C2×C311yescharacteristic
73C8×C22⋊C4C2×C311yescharacteristic
74#1735C2×C311yescharacteristic
75C22×C3×Q8—21no—
76C22×C3×Q8C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C22×C3×C8—22no—
79C22×C3×C8—22no—
80C22×C3×C8—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—41no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117—22no—
92C3×#117—41no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4D411yescharacteristic
111C2×C3×C22⋊C4D411yescharacteristic
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#103—22no—
117C3×#103—22no—
118C3×#103—41no—
119C3×#103—41no—
120C3×C22⋊C8—22no—
121C3×C4×C8—22no—
122C3×C4×C8—41no—
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×#95—21no—
133C3×#95—21no—
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136C23×C8C22×C311yescharacteristic
137C2×C4⋊Q8C22×C311yescharacteristic
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×#117C22×C311yescharacteristic
140C2×#117C22×C311yescharacteristic
141C2×#103C22×C311yescharacteristic
142C2×#103C22×C311yescharacteristic
143C2×#103C22×C311yescharacteristic
144C2×#103C22×C311yescharacteristic
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C4×C8C22×C311yescharacteristic
147C2×C4×C8C22×C311yescharacteristic
148#397—21no—
149#397—21no—
150#396—21no—
151#393—21no—
152#391—21no—
153#391C22×C311yescharacteristic
154#391C22×C311yescharacteristic
155#389C22×C311yescharacteristic
156#389C22×C311yescharacteristic
157#387—21no—
158#387—21no—
159#387—21no—
160#384C22×C311yescharacteristic
161#382C22×C311yescharacteristic
162#382C22×C311yescharacteristic
163#380C22×C311yescharacteristic
164C4×C22⋊C4C22×C311yescharacteristic
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#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#336C22×C311yescharacteristic
178#336C22×C311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—42no—
187C2×C3×Q8—42no—
188C2×C3×Q8—81no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4#4112yes—
207C22×C3×C4#4112yes—
208C22×C3×C4C2411yescharacteristic
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—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—44no—
221C2×C3×C8#4112yes—
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—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—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—42no—
253C3×C42—42no—
254C22×Q8—21no—
255C22×Q8C23×C311yescharacteristic
256C23×C4C23×C311yescharacteristic
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—41no—
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C4⋊Q8—22no—
265C4⋊Q8—41no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—41no—
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⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4C3×D411yescharacteristic
290C2×C22⋊C4C3×D411yescharacteristic
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#103—22no—
296#103—22no—
297#103—41no—
298#103—41no—
299C22⋊C8—22no—
300C4×C8—22no—
301C4×C8—41no—
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#95—21no—
311#95—21no—
312#95—21no—
313#95C3×D411yescharacteristic
314#95C3×D411yescharacteristic
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3#12412yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—41no—
327C3×Q8—84no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
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—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—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#12412yes—
369C2×C3×C4#13512yes—
370C2×C3×C4#13512yes—
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C42−1+411yescharacteristic
377C3×C8—22no—
378C3×C8—41no—
379C3×C8—41no—
380C3×C8—41no—
381C24C2×C3×D411yescharacteristic
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—81no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C3×#4112yes—
409C22×C4C3×#4112yes—
410C22×C4C24×C311yescharacteristic
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—44no—
423C2×C8C3×#4112yes—
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—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—81no—
443C4⋊C4C3×#4112yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—81no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—42no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3C2×#12411yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3#54711yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3C2×#13511yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57811yescharacteristic
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—42no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4#52911yescharacteristic
484C3×C4#53911yescharacteristic
485C3×C4#55111yescharacteristic
486C3×C4#55211yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23C3×#12412yes—
494C23C22×C3×D411yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
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—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×C4C3×#12412yes—
541C2×C4C3×#13512yes—
542C2×C4C3×#13512yes—
543C2×C4C2×C3×#4111yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C2×C4C2×C3×#4111yescharacteristic
546C2×C4C2×C3×#4111yescharacteristic
547C2×C4C3×H(5,𝔽2)11yescharacteristic
548C2×C4C3×2−1+411yescharacteristic
549C8—22no—
550C8—41no—
551C8—41no—
552C8—41no—
553C2×C3—42no—
554C2×C3#248811yescharacteristic
555C2×C3#313111yescharacteristic
556C2×C3#313711yescharacteristic
557C2×C3#329611yescharacteristic
558C2×C3#330011yescharacteristic
559C2×C3#334011yescharacteristic
560C2×C3#334211yescharacteristic
561C22—22no—
562C22—22no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22C2×C3×#12411yescharacteristic
569C22C3×#54611yescharacteristic
570C22C3×#54711yescharacteristic
571C22C2×C3×#13511yescharacteristic
572C22C2×C3×#13511yescharacteristic
573C22C3×#57711yescharacteristic
574C22C3×#57811yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4C3×#52911yescharacteristic
589C4C3×#53911yescharacteristic
590C4C3×#55111yescharacteristic
591C4C3×#55211yescharacteristic
592C3#2769811yescharacteristic, direct product factor, Sylow
593C2—42no—
594C2C3×#248811yescharacteristic
595C2C3×#313111yescharacteristic
596C2C3×#313711yescharacteristic
597C2C3×#329611yescharacteristic
598C2C3×#330011yescharacteristic
599C2C3×#334011yescharacteristic
600C2C3×#334211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2101C2semidirect
4. C3×#2099C2semidirect
5. C3×#2074C2semidirect
6. C3×#2024C2semidirect
7. C3×#1990C2semidirect
8. C3×#1933C2semidirect
9. C3×#1933C2semidirect
12. C3×#1835C2semidirect
16. #27698C3direct (N × H)
62. #2101C2×C3semidirect
63. #2099C2×C3semidirect
64. #2074C2×C3semidirect
65. #2024C2×C3semidirect
66. #1990C2×C3semidirect
67. #1933C2×C3semidirect
68. #1933C2×C3semidirect
71. #1835C2×C3semidirect
592. C3#27698direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24992
C3×#24392
C3×#21012
C3×#20992
C3×#20742
C3×#20242
C3×#19902
C3×#19332
C3×#18922
C3×#18732
C3×#18352
C3×#18252
C3×C8×C22⋊C42
C3×#17352
#276983