Finite Group Databasebeta

← C3×#25504 #10644111

Subgroups — C3×#25504 #10644111

Order 768.

1108 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#470C212yesmaximal
3C3×#2855C211yesmaximal, characteristic
4C3×#2058C211yesmaximal, characteristic, complemented
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2024C211yesmaximal, characteristic, complemented
7C3×#2016C211yesmaximal, characteristic, complemented
8C3×#1894C212yesmaximal
9C3×#1878C212yesmaximal, complemented
10C3×#1865C212yesmaximal, complemented
11C3×#1728C211yesmaximal, characteristic
12#25504C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×M16C2212yes—
15C3×#533—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8C2212yes—
24C2×C3×#117—21no—
25C22×C3×C42C2211yescharacteristic
26C3×#470—28no—
27C2×C3×#106C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C3×#391C2211yescharacteristic
33C3×#391C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#387—21no—
37C3×#387—21no—
38C3×#385—21no—
39C3×#381—21no—
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#533C2×C311yescharacteristic
47C2×#470C2×C312yes—
48#2855C2×C311yescharacteristic
49#2058C2×C311yescharacteristic, complemented
50#2043C2×C311yescharacteristic, complemented
51#2024C2×C311yescharacteristic, complemented
52#2016C2×C311yescharacteristic, complemented
53#1894C2×C312yes—
54#1878C2×C312yescomplemented
55#1865C2×C312yescomplemented
56#1728C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2312yes—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×Q16—24no—
64C2×C3×Q16—24no—
65C2×C3×Q16—42no—
66C2×C3×M16—24no—
67C2×C3×M16—24no—
68C2×C3×M16—42no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×#125—28no—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—24no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8D418yes—
78C3×C4×Q8—28no—
79C3×C4×Q8—24no—
80C3×C4×Q8—24no—
81C3×#117—28no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—24no—
98C2×C3×C42—22no—
99C2×C3×C42D411yescharacteristic
100C2×C3×C42D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C3×#106—24no—
104C3×#103—24no—
105C3×#103—24no—
106C3×#103—24no—
107C3×#103—42no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C22×Q16C22×C312yes—
116C22×M16C22×C312yes—
117#533—28no—
118C2×C4⋊Q8C22×C311yescharacteristic
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×#125—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8C22×C311yescharacteristic
123C2×C22⋊Q8C22×C312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8C22×C312yes—
126C2×#117—21no—
127C22×C42C22×C311yescharacteristic
128#470—28no—
129C2×#106C22×C312yes—
130C2×#103C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C312yes—
133C2×#103C22×C312yes—
134#391C22×C311yescharacteristic
135#391C22×C311yescharacteristic
136#389C22×C311yescharacteristic
137#389C22×C311yescharacteristic
138#387—21no—
139#387—21no—
140#385—21no—
141#381—21no—
142#380C22×C311yescharacteristic
143#340C22×C312yes—
144#340C22×C312yes—
145#336C22×C311yescharacteristic
146#336C22×C312yes—
147#336C22×C311yescharacteristic
148C24×C3C2×D411yescharacteristic
149C2×C3×Q8—24no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—42no—
160C2×C3×Q8—42no—
161C2×C3×Q8C2×D414yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—41no—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C3×Q16—48no—
185C3×M16—48no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8#4114yes—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4C2×D414yes—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—42no—
209C3×C42—28no—
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—22no—
214C22×Q8—21no—
215C22×Q8C23×C311yescharacteristic
216C22×Q8C23×C312yes—
217C23×C4C3×D411yescharacteristic
218C23×C4C3×D411yescharacteristic
219C23×C4C23×C311yescharacteristic
220C2×Q16—24no—
221C2×Q16—24no—
222C2×Q16—42no—
223C2×M16—24no—
224C2×M16—24no—
225C2×M16—42no—
226C22×C8C23×C312yes—
227C22×C8C23×C312yes—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230#125—28no—
231C22⋊Q8—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8C3×D418yes—
235C4×Q8—28no—
236C4×Q8—24no—
237C4×Q8—24no—
238#117—28no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C23×C311yescharacteristic
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42C3×D411yescharacteristic
257C2×C42C3×D411yescharacteristic
258C2×C42C23×C311yescharacteristic
259C2×C42C23×C311yescharacteristic
260#106—24no—
261#103—24no—
262#103—24no—
263#103—24no—
264#103—42no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3#12012yes—
279C23×C3C22×D411yescharacteristic
280C3×Q8—24no—
281C3×Q8—24no—
282C3×Q8—48no—
283C3×Q8—42no—
284C3×Q8—42no—
285C3×Q8—42no—
286C3×Q8—42no—
287C2×C3×C4—28no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—21no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—44no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4#12012yes—
321C2×C3×C4#13712yes—
322C2×C3×C4#13712yes—
323C2×C3×C4#13712yes—
324C2×C3×C4C22×D412yes—
325C2×C3×C4C2×#4112yes—
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328C3×C8—24no—
329C3×C8—42no—
330C24C2×C3×D411yescharacteristic
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8C2×C3×D414yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D414yes—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C24×C311yescharacteristic
366Q16—48no—
367M16—48no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8C3×#4114yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4C2×C3×D414yes—
388C22⋊C4—28no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42—28no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#53211yescharacteristic
405C22×C3#53711yescharacteristic
406C22×C3C2×#13711yescharacteristic
407C22×C3C2×#13711yescharacteristic
408C22×C3C2×#13711yescharacteristic
409C22×C3#57511yescharacteristic
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—44no—
415C3×C4—41no—
416C3×C4—44no—
417C3×C4—81no—
418C3×C4—81no—
419C3×C4—81no—
420C3×C4—81no—
421C3×C4#54612yes—
422C3×C4#54712yes—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23C3×#12012yes—
430C23C22×C3×D411yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—48no—
434Q8—42no—
435Q8—42no—
436Q8—42no—
437Q8—42no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4C3×#12012yes—
472C2×C4C3×#13712yes—
473C2×C4C3×#13712yes—
474C2×C4C3×#13712yes—
475C2×C4C22×C3×D412yes—
476C2×C4C2×C3×#4112yes—
477C2×C4C3×H(5,𝔽2)11yescharacteristic
478C2×C4C3×2−1+411yescharacteristic
479C8—24no—
480C8—42no—
481C2×C3—42no—
482C2×C3#243011yescharacteristic
483C2×C3#300411yescharacteristic
484C2×C3#300711yescharacteristic
485C2×C3#317211yescharacteristic
486C2×C3#318511yescharacteristic
487C2×C3#325211yescharacteristic
488C2×C3#325511yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22C2×C3×#12011yescharacteristic
497C22C3×#53211yescharacteristic
498C22C3×#53711yescharacteristic
499C22C2×C3×#13711yescharacteristic
500C22C2×C3×#13711yescharacteristic
501C22C2×C3×#13711yescharacteristic
502C22C3×#57511yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—44no—
508C4—41no—
509C4—44no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4C3×#54612yes—
515C4C3×#54712yes—
516C3#2550411yescharacteristic, direct product factor, Sylow
517C2—42no—
518C2C3×#243011yescharacteristic
519C2C3×#300411yescharacteristic
520C2C3×#300711yescharacteristic
521C2C3×#317211yescharacteristic
522C2C3×#318511yescharacteristic
523C2C3×#325211yescharacteristic
524C2C3×#325511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2058C2semidirect
5. C3×#2043C2semidirect
6. C3×#2024C2semidirect
7. C3×#2016C2semidirect
9. C3×#1878C2semidirect
10. C3×#1865C2semidirect
12. #25504C3direct (N × H)
49. #2058C2×C3semidirect
50. #2043C2×C3semidirect
51. #2024C2×C3semidirect
52. #2016C2×C3semidirect
54. #1878C2×C3semidirect
55. #1865C2×C3semidirect
516. C3#25504direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4702
C3×#28552
C3×#20582
C3×#20432
C3×#20242
C3×#20162
C3×#18942
C3×#18782
C3×#18652
C3×#17282
#255043