Finite Group Databasebeta

← C3×#23577 #10642184

Subgroups — C3×#23577 #10642184

Order 768.

1144 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#441C212yesmaximal, complemented
3C3×C4×C22⋊Q8C211yesmaximal, characteristic
4C3×#2058C211yesmaximal, characteristic, complemented
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2035C212yesmaximal
7C3×#2024C211yesmaximal, characteristic, complemented
8C3×#2016C211yesmaximal, characteristic, complemented
9C3×#2001C212yesmaximal
10C3×#1763C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic
12#23577C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C3×#533—28no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2212yes—
24C2×C3×#117—21no—
25C22×C3×C42C2211yescharacteristic
26C3×#441—28no—
27C2×C3×#106C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C2×C3×#97C2212yes—
34C3×#391C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#387—21no—
37C3×#385—21no—
38C3×#380C2211yescharacteristic
39C3×C4×C4⋊C4—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#533C2×C311yescharacteristic
47C2×#441C2×C312yescomplemented
48C4×C22⋊Q8C2×C311yescharacteristic
49#2058C2×C311yescharacteristic, complemented
50#2043C2×C311yescharacteristic, complemented
51#2035C2×C312yes—
52#2024C2×C311yescharacteristic, complemented
53#2016C2×C311yescharacteristic, complemented
54#2001C2×C312yes—
55#1763C2×C312yescomplemented
56#1749C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C2×C3×Q16—24no—
64C2×C3×Q16—24no—
65C2×C3×Q16—24no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8—24no—
70C3×#125—28no—
71C3×C22⋊Q8—28no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—22no—
76C3×C4×Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8D418yes—
80C3×#117—28no—
81C2×C3×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—22no—
97C2×C3×C42—24no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42D411yescharacteristic
101C2×C3×C42D411yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C3×#106—24no—
105C3×#103—24no—
106C3×#103—24no—
107C3×#103—24no—
108C3×#103—24no—
109C3×C22⋊C8—24no—
110C3×#97—24no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C22×Q16C22×C312yes—
117#533—28no—
118C2×C4⋊Q8C22×C311yescharacteristic
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×#125—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8C22×C312yes—
123C2×C22⋊Q8C22×C311yescharacteristic
124C2×C4×Q8—21no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×Q8C22×C312yes—
127C2×#117—21no—
128C22×C42C22×C311yescharacteristic
129#441—28no—
130C2×#106C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C312yes—
133C2×#103C22×C312yes—
134C2×#103C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×#97C22×C312yes—
137#391C22×C311yescharacteristic
138#389C22×C311yescharacteristic
139#387—21no—
140#385—21no—
141#380C22×C311yescharacteristic
142C4×C4⋊C4—21no—
143C4×C4⋊C4C22×C311yescharacteristic
144C4×C22⋊C4—21no—
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C311yescharacteristic
148#336C22×C311yescharacteristic
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—42no—
160C2×C3×Q8—42no—
161C2×C3×Q8C2×D414yes—
162C2×C3×Q8#4112yes—
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—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—41no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C3×Q16—216no—
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—24no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4C2×D414yes—
205C3×C4⋊C4#4112yes—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—42no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—28no—
212C3×C42—22no—
213C3×C42—42no—
214C3×C42—42no—
215C3×C42C2×D414yes—
216C22×Q8—21no—
217C22×Q8C23×C312yes—
218C22×Q8C23×C311yescharacteristic
219C23×C4C3×D411yescharacteristic
220C23×C4C3×D411yescharacteristic
221C23×C4C23×C311yescharacteristic
222C2×Q16—24no—
223C2×Q16—24no—
224C2×Q16—24no—
225C22×C8C23×C312yes—
226C22×C8C23×C312yes—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229#125—28no—
230C22⋊Q8—28no—
231C22⋊Q8—24no—
232C22⋊Q8—22no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C4×Q8—28no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8C3×D418yes—
239#117—28no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C3×D411yescharacteristic
260C2×C42C3×D411yescharacteristic
261C2×C42C23×C311yescharacteristic
262C2×C42C23×C311yescharacteristic
263#106—24no—
264#103—24no—
265#103—24no—
266#103—24no—
267#103—24no—
268C22⋊C8—24no—
269#97—24no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3#13712yes—
282C23×C3C22×D411yescharacteristic
283C3×Q8—24no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—48no—
287C3×Q8—42no—
288C3×Q8—42no—
289C3×Q8—42no—
290C2×C3×C4—28no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—44no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12012yes—
325C2×C3×C4#13712yes—
326C2×C3×C4#13712yes—
327C2×C3×C4C22×D412yes—
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4112yes—
330C2×C3×C4C2×#4111yescharacteristic
331C3×C8—24no—
332C3×C8—42no—
333C24C2×C3×D411yescharacteristic
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8C2×C3×D414yes—
346C2×Q8C3×#4112yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C24×C311yescharacteristic
369Q16—216no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8C3×#4114yes—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4C2×C3×D414yes—
389C4⋊C4C3×#4112yes—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—42no—
393C42—22no—
394C42—22no—
395C42—28no—
396C42—22no—
397C42—42no—
398C42—42no—
399C42C2×C3×D414yes—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3C2×#12011yescharacteristic
408C22×C3#52511yescharacteristic
409C22×C3#53211yescharacteristic
410C22×C3C2×#13711yescharacteristic
411C22×C3C2×#13711yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57511yescharacteristic
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—41no—
420C3×C4—44no—
421C3×C4—42no—
422C3×C4—44no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4#52512yes—
426C3×C4#53212yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C3×#13712yes—
434C23C22×C3×D411yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—24no—
438Q8—48no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442C2×C4—28no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4C3×#12012yes—
476C2×C4C3×#12012yes—
477C2×C4C3×#13712yes—
478C2×C4C3×#13712yes—
479C2×C4C22×C3×D412yes—
480C2×C4C2×C3×#4111yescharacteristic
481C2×C4C2×C3×#4112yes—
482C2×C4C2×C3×#4111yescharacteristic
483C8—24no—
484C8—42no—
485C2×C3—42no—
486C2×C3#238811yescharacteristic
487C2×C3#300411yescharacteristic
488C2×C3#300711yescharacteristic
489C2×C3#311311yescharacteristic
490C2×C3#311611yescharacteristic
491C2×C3#318111yescharacteristic
492C2×C3#318511yescharacteristic
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22C2×C3×#12011yescharacteristic
501C22C3×#52511yescharacteristic
502C22C3×#53211yescharacteristic
503C22C2×C3×#13711yescharacteristic
504C22C2×C3×#13711yescharacteristic
505C22C2×C3×#13711yescharacteristic
506C22C3×#57511yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—41no—
512C4—44no—
513C4—42no—
514C4—44no—
515C4—42no—
516C4—81no—
517C4—81no—
518C4C3×#52512yes—
519C4C3×#53212yes—
520C3#2357711yescharacteristic, direct product factor, Sylow
521C2—42no—
522C2C3×#238811yescharacteristic
523C2C3×#300411yescharacteristic
524C2C3×#300711yescharacteristic
525C2C3×#311311yescharacteristic
526C2C3×#311611yescharacteristic
527C2C3×#318111yescharacteristic
528C2C3×#318511yescharacteristic
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
2. C2×C3×#441C2semidirect
4. C3×#2058C2semidirect
5. C3×#2043C2semidirect
7. C3×#2024C2semidirect
8. C3×#2016C2semidirect
10. C3×#1763C2semidirect
12. #23577C3direct (N × H)
47. C2×#441C2×C3semidirect
49. #2058C2×C3semidirect
50. #2043C2×C3semidirect
52. #2024C2×C3semidirect
53. #2016C2×C3semidirect
55. #1763C2×C3semidirect
520. C3#23577direct (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×#4412
C3×C4×C22⋊Q82
C3×#20582
C3×#20432
C3×#20352
C3×#20242
C3×#20162
C3×#20012
C3×#17632
C3×#17492
#235773