Finite Group Databasebeta

← C3×#29695 #10648302

Subgroups — C3×#29695 #10648302

Order 768.

1024 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#479C212yesmaximal, complemented
3C3×#2851C211yesmaximal, characteristic
4C3×#2058C211yesmaximal, characteristic, complemented
5C3×#2057C211yesmaximal, characteristic, complemented
6C3×#2043C211yesmaximal, characteristic, complemented
7C3×#2043C211yesmaximal, characteristic, complemented
8C3×#2036C212yesmaximal
9C3×#2035C212yesmaximal
10C3×#1805C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic
12#29695C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×#117—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#479—28no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2212yes—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C3×#389C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#387—21no—
36C3×#387—21no—
37C3×#385—21no—
38C3×#385—21no—
39C3×#382C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#533C2×C311yescharacteristic
47C2×#479C2×C312yescomplemented
48#2851C2×C311yescharacteristic
49#2058C2×C311yescharacteristic, complemented
50#2057C2×C311yescharacteristic, complemented
51#2043C2×C311yescharacteristic, complemented
52#2043C2×C311yescharacteristic, complemented
53#2036C2×C312yes—
54#2035C2×C312yes—
55#1805C2×C312yescomplemented
56#1797C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—24no—
65C3×C4⋊Q8—24no—
66C3×#125—28no—
67C3×#125—24no—
68C3×C22⋊Q8—28no—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—24no—
71C3×C4×Q8—28no—
72C3×C4×Q8—24no—
73C3×C4×Q8D418yes—
74C3×#117—28no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—22no—
93C2×C3×C42—24no—
94C2×C3×C42D411yescharacteristic
95C2×C3×C42D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×#107—24no—
99C3×#106—24no—
100C3×C4⋊C8—24no—
101C3×#103—24no—
102C3×#103—24no—
103C3×#103—42no—
104C3×#103—42no—
105C3×C22⋊C8—42no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112#533—28no—
113C2×C4⋊Q8C22×C311yescharacteristic
114C2×C4⋊Q8C22×C311yescharacteristic
115C2×#125—21no—
116C2×#125C22×C311yescharacteristic
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C22×C312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C22×C312yes—
121C2×#117—21no—
122C22×C42C22×C311yescharacteristic
123#479—28no—
124C2×#107C22×C312yes—
125C2×#106C22×C312yes—
126C2×C4⋊C8C22×C312yes—
127C2×#103C22×C312yes—
128C2×#103C22×C312yes—
129C2×#103C22×C312yes—
130C2×#103C22×C312yes—
131C2×C22⋊C8C22×C312yes—
132#389C22×C311yescharacteristic
133#389C22×C311yescharacteristic
134#387—21no—
135#387—21no—
136#385—21no—
137#385—21no—
138#382C22×C311yescharacteristic
139#380C22×C311yescharacteristic
140#380C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#336C22×C311yescharacteristic
144#336C22×C311yescharacteristic
145C24×C3C2×D411yescharacteristic
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—42no—
153C2×C3×Q8—42no—
154C2×C3×Q8C2×D414yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4C2×D414yes—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—42no—
202C3×C22⋊C4—42no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—28no—
206C3×C42—22no—
207C3×C42C2×D414yes—
208C22×Q8—21no—
209C22×Q8C23×C312yes—
210C23×C4C3×D411yescharacteristic
211C23×C4C3×D411yescharacteristic
212C23×C4C23×C311yescharacteristic
213C22×C8C23×C312yes—
214C22×C8C23×C312yes—
215C4⋊Q8—24no—
216C4⋊Q8—24no—
217#125—28no—
218#125—24no—
219C22⋊Q8—28no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C4×Q8—28no—
223C4×Q8—24no—
224C4×Q8C3×D418yes—
225#117—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C312yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42C3×D411yescharacteristic
246C2×C42C3×D411yescharacteristic
247C2×C42C23×C311yescharacteristic
248C2×C42C23×C311yescharacteristic
249#107—24no—
250#106—24no—
251C4⋊C8—24no—
252#103—24no—
253#103—24no—
254#103—42no—
255#103—42no—
256C22⋊C8—42no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3#13712yes—
270C23×C3C22×D411yescharacteristic
271C3×Q8—24no—
272C3×Q8—24no—
273C3×Q8—48no—
274C3×Q8—42no—
275C3×Q8—42no—
276C2×C3×C4—28no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4#13512yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316C2×C3×C42−1+412yes—
317C3×C8—42no—
318C3×C8—42no—
319C24C2×C3×D411yescharacteristic
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8C2×C3×D414yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C24×C311yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4C2×C3×D414yes—
374C22⋊C4—28no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C42—22no—
378C42—22no—
379C42—28no—
380C42—22no—
381C42C2×C3×D414yes—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3C2×#12011yescharacteristic
390C22×C3#53611yescharacteristic
391C22×C3#53711yescharacteristic
392C22×C3C2×#13511yescharacteristic
393C22×C3C2×#13711yescharacteristic
394C22×C3C2×#13711yescharacteristic
395C22×C3#57511yescharacteristic
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—41no—
401C3×C4—44no—
402C3×C4—44no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4—81no—
407C3×C4#53612yes—
408C3×C4#53712yes—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23C3×#13712yes—
416C23C22×C3×D411yescharacteristic
417Q8—24no—
418Q8—24no—
419Q8—48no—
420Q8—42no—
421Q8—42no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4C3×#12012yes—
456C2×C4C3×#12012yes—
457C2×C4C3×#13512yes—
458C2×C4C3×#13712yes—
459C2×C4C22×C3×D412yes—
460C2×C4C3×H(5,𝔽2)11yescharacteristic
461C2×C4C3×2−1+411yescharacteristic
462C2×C4C3×2−1+412yes—
463C8—42no—
464C8—42no—
465C2×C3—42no—
466C2×C3#261611yescharacteristic
467C2×C3#300211yescharacteristic
468C2×C3#300411yescharacteristic
469C2×C3#323111yescharacteristic
470C2×C3#323611yescharacteristic
471C2×C3#324811yescharacteristic
472C2×C3#325211yescharacteristic
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22C2×C3×#12011yescharacteristic
481C22C3×#53611yescharacteristic
482C22C3×#53711yescharacteristic
483C22C2×C3×#13511yescharacteristic
484C22C2×C3×#13711yescharacteristic
485C22C2×C3×#13711yescharacteristic
486C22C3×#57511yescharacteristic
487C4—22no—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—41no—
492C4—44no—
493C4—44no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4C3×#53612yes—
499C4C3×#53712yes—
500C3#2969511yescharacteristic, direct product factor, Sylow
501C2—42no—
502C2C3×#261611yescharacteristic
503C2C3×#300211yescharacteristic
504C2C3×#300411yescharacteristic
505C2C3×#323111yescharacteristic
506C2C3×#323611yescharacteristic
507C2C3×#324811yescharacteristic
508C2C3×#325211yescharacteristic
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×#479C2semidirect
4. C3×#2058C2semidirect
5. C3×#2057C2semidirect
6. C3×#2043C2semidirect
7. C3×#2043C2semidirect
10. C3×#1805C2semidirect
12. #29695C3direct (N × H)
47. C2×#479C2×C3semidirect
49. #2058C2×C3semidirect
50. #2057C2×C3semidirect
51. #2043C2×C3semidirect
52. #2043C2×C3semidirect
55. #1805C2×C3semidirect
500. C3#29695direct (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×#4792
C3×#28512
C3×#20582
C3×#20572
C3×#20432
C3×#20362
C3×#20352
C3×#18052
C3×#17972
#296953