Finite Group Databasebeta

← C3×#29309 #10647916

Subgroups — C3×#29309 #10647916

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2629C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3C3×#2084C211yesmaximal, characteristic, complemented
4C3×#2083C211yesmaximal, characteristic, complemented
5C3×#2059C211yesmaximal, characteristic, complemented
6C3×#2059C211yesmaximal, characteristic, complemented
7C3×#1944C211yesmaximal, characteristic, complemented
8C3×#1943C211yesmaximal, characteristic, complemented
9C3×#1942C211yesmaximal, characteristic, complemented
10C3×#1941C211yesmaximal, characteristic, complemented
11C3×#1854C211yesmaximal, characteristic
12C3×#1854C211yesmaximal, characteristic
13C3×#1853C211yesmaximal, characteristic
14C3×#1852C211yesmaximal, characteristic
15C3×#1750C211yesmaximal, characteristic
16#29309C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C3×#398—21no—
34C3×#398—21no—
35C3×#396—21no—
36C3×#396—21no—
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#385—21no—
44C3×#385—21no—
45C3×#384—21no—
46C3×#384C2211yescharacteristic
47C3×#382—21no—
48C3×#380C2211yescharacteristic
49C3×#380C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2629C2×C311yescharacteristic
61#2317C2×C311yescharacteristic
62#2084C2×C311yescharacteristic, complemented
63#2083C2×C311yescharacteristic, complemented
64#2059C2×C311yescharacteristic, complemented
65#2059C2×C311yescharacteristic, complemented
66#1944C2×C311yescharacteristic, complemented
67#1943C2×C311yescharacteristic, complemented
68#1942C2×C311yescharacteristic, complemented
69#1941C2×C311yescharacteristic, complemented
70#1854C2×C311yescharacteristic
71#1854C2×C311yescharacteristic
72#1853C2×C311yescharacteristic
73#1852C2×C311yescharacteristic
74#1750C2×C311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#125—41no—
82C3×#125—41no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117—41no—
86C3×#117—41no—
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⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—41no—
99C2×C3×C4⋊C4—41no—
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4D411yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4D411yescharacteristic
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#107—22no—
119C3×#107—22no—
120C3×#107—41no—
121C3×#107—41no—
122C3×#106—22no—
123C3×#106—22no—
124C3×#106—41no—
125C3×#106—41no—
126C3×C22⋊C8Q814yes—
127C3×C22⋊C8Q814yes—
128C3×#97—41no—
129C3×C4×C8—41no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C2×#125C22×C311yescharacteristic
139C2×#117C22×C311yescharacteristic
140C2×#117C22×C311yescharacteristic
141C22×C4⋊C4C22×C311yescharacteristic
142C2×#107C22×C311yescharacteristic
143C2×#107C22×C311yescharacteristic
144C2×#107C22×C311yescharacteristic
145C2×#107C22×C311yescharacteristic
146C2×#106C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×#106C22×C311yescharacteristic
150C2×C22⋊C8C22×C311yescharacteristic
151C2×C22⋊C8C22×C311yescharacteristic
152C2×#97C22×C311yescharacteristic
153C2×C4×C8C22×C311yescharacteristic
154#398—21no—
155#398—21no—
156#396—21no—
157#396—21no—
158#389C22×C311yescharacteristic
159#389C22×C311yescharacteristic
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#385—21no—
165#385—21no—
166#384—21no—
167#384C22×C311yescharacteristic
168#382—21no—
169#380C22×C311yescharacteristic
170#380C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180#336C22×C311yescharacteristic
181C24×C3C2×D411yescharacteristic
182C22×C3×C4—21no—
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—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
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×Q812yes—
203C22×C3×C4C2×Q812yes—
204C22×C3×C4C2411yescharacteristic
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×C8C2×Q812yes—
214C2×C3×C8C2×Q812yes—
215C2×C3×C8C2×Q812yes—
216C2×C3×C8C2×Q812yes—
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—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—81no—
237C3×C4⋊C4—81no—
238C3×C4⋊C4—81no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4#4112yes—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C42—24no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C23×C4—21no—
257C23×C4C23×C311yescharacteristic
258C22×C8C23×C311yescharacteristic
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262#125—41no—
263#125—41no—
264#117—22no—
265#117—22no—
266#117—41no—
267#117—41no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
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⋊C4—21no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4C3×D411yescharacteristic
282C2×C4⋊C4C3×D411yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4C3×D411yescharacteristic
295C2×C22⋊C4C3×D411yescharacteristic
296C2×C42—21no—
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299#107—22no—
300#107—22no—
301#107—41no—
302#107—41no—
303#106—22no—
304#106—22no—
305#106—41no—
306#106—41no—
307C22⋊C8C3×Q814yes—
308C22⋊C8C3×Q814yes—
309#97—41no—
310C4×C8—41no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3C4⋊Q812yes—
326C23×C3C22×D411yescharacteristic
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
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—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
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—41no—
369C2×C3×C4—84no—
370C2×C3×C4C4⋊Q812yes—
371C2×C3×C4C22×Q811yescharacteristic
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C3×C8—22no—
378C3×C8—22no—
379C3×C8—22no—
380C3×C8—22no—
381C24C2×C3×D411yescharacteristic
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×Q812yes—
403C22×C4C2×C3×Q812yes—
404C22×C4C24×C311yescharacteristic
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C3×Q812yes—
414C2×C8C2×C3×Q812yes—
415C2×C8C2×C3×Q812yes—
416C2×C8C2×C3×Q812yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4C3×#4112yes—
445C4⋊C4C3×#4112yes—
446C22⋊C4—24no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C42—24no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3#52511yescharacteristic
464C22×C3C2×C4⋊Q811yescharacteristic
465C22×C3#53411yescharacteristic
466C22×C3#57611yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57811yescharacteristic
470C3×C4—22no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4#55411yescharacteristic
484C3×C4#55411yescharacteristic
485C3×C4#55411yescharacteristic
486C3×C4#55411yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23C3×C4⋊Q812yes—
494C23C22×C3×D411yescharacteristic
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
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—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
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—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—84no—
538C2×C4C3×C4⋊Q812yes—
539C2×C4C22×C3×Q811yescharacteristic
540C2×C4C22×C3×Q811yescharacteristic
541C2×C4C2×C3×#4111yescharacteristic
542C2×C4C2×C3×#4111yescharacteristic
543C2×C4C3×H(5,𝔽2)11yescharacteristic
544C2×C4C3×H(5,𝔽2)11yescharacteristic
545C8—22no—
546C8—22no—
547C8—22no—
548C8—22no—
549C2×C3—42no—
550C2×C3#258311yescharacteristic
551C2×C3#311911yescharacteristic
552C2×C3#312011yescharacteristic
553C2×C3#316311yescharacteristic
554C2×C3#316411yescharacteristic
555C2×C3#319811yescharacteristic
556C2×C3#320111yescharacteristic
557C22—22no—
558C22—22no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22C3×#52511yescharacteristic
565C22C2×C3×C4⋊Q811yescharacteristic
566C22C3×#53411yescharacteristic
567C22C3×#57611yescharacteristic
568C22C3×#57711yescharacteristic
569C22C3×#57711yescharacteristic
570C22C3×#57811yescharacteristic
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4C3×#55411yescharacteristic
585C4C3×#55411yescharacteristic
586C4C3×#55411yescharacteristic
587C4C3×#55411yescharacteristic
588C3#2930911yescharacteristic, direct product factor, Sylow
589C2—42no—
590C2C3×#258311yescharacteristic
591C2C3×#311911yescharacteristic
592C2C3×#312011yescharacteristic
593C2C3×#316311yescharacteristic
594C2C3×#316411yescharacteristic
595C2C3×#319811yescharacteristic
596C2C3×#320111yescharacteristic
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×#2084C2semidirect
4. C3×#2083C2semidirect
5. C3×#2059C2semidirect
6. C3×#2059C2semidirect
7. C3×#1944C2semidirect
8. C3×#1943C2semidirect
9. C3×#1942C2semidirect
10. C3×#1941C2semidirect
16. #29309C3direct (N × H)
62. #2084C2×C3semidirect
63. #2083C2×C3semidirect
64. #2059C2×C3semidirect
65. #2059C2×C3semidirect
66. #1944C2×C3semidirect
67. #1943C2×C3semidirect
68. #1942C2×C3semidirect
69. #1941C2×C3semidirect
588. C3#29309direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26292
C3×#23172
C3×#20842
C3×#20832
C3×#20592
C3×#19442
C3×#19432
C3×#19422
C3×#19412
C3×#18542
C3×#18532
C3×#18522
C3×#17502
#293093