Finite Group Databasebeta

← C3×#35315 #10653922

Subgroups — C3×#35315 #10653922

Order 768.

1602 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C3×#2855C212yesmaximal
3C3×#2851C211yesmaximal, characteristic
4C3×#2647C211yesmaximal, characteristic, complemented
5C3×#2645C212yesmaximal, complemented
6C3×#2642C211yesmaximal, characteristic, complemented
7C3×#2533C214yesmaximal, complemented
8C3×#2528C214yesmaximal, complemented
9C3×#2505C214yesmaximal
10C3×#2493C214yesmaximal
11C3×#2330C211yesmaximal, characteristic, complemented
12C3×#2327C211yesmaximal, characteristic, complemented
13C3×#2320C212yesmaximal, complemented
14C3×C4×C22⋊Q8C212yesmaximal
15C2×C3×#376C211yesmaximal, characteristic
16#35315C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C3×#401C2214yes—
25C3×#400C2214yes—
26C3×#400C2214yes—
27C3×#400C2214yes—
28C3×#400C2214yes—
29C3×#400C2214yes—
30C3×#400C2214yes—
31C3×#398C2214yes—
32C3×#391C2212yes—
33C3×#391C2212yes—
34C3×#391C2212yes—
35C3×#391C2212yes—
36C3×#389C2212yes—
37C3×#389C2212yes—
38C3×#389C2212yes—
39C3×#389C2212yes—
40C3×#389C2214yes—
41C3×#389C2214yes—
42C3×#387C2214yes—
43C3×#387C2214yes—
44C3×#387C2214yes—
45C3×#387C2214yes—
46C3×#387C2214yes—
47C3×#387C2214yes—
48C3×#387C2214yes—
49C3×#387C2214yes—
50C3×#387C2214yes—
51C3×#385C2214yes—
52C3×#385C2214yes—
53C3×#385C2214yes—
54C3×#384C2211yescharacteristic
55C3×#383C2214yes—
56C3×#382C2211yescharacteristic
57C3×#382C2214yes—
58C3×#381C2214yes—
59C3×#380C2214yes—
60C3×#380C2212yes—
61C3×C4×C4⋊C4C2214yes—
62C3×C4×C4⋊C4C2212yes—
63C3×C4×C22⋊C4C2214yes—
64C3×#376C2211yescharacteristic
65C3×#376C2211yescharacteristic
66C3×#376C2212yes—
67C3×#376C2212yes—
68C3×#376C2211yescharacteristic
69C3×#376C2211yescharacteristic
70C2×C3×#95C2212yes—
71C2×C3×#95C2212yes—
72#2857C2×C311yescharacteristic
73#2855C2×C312yes—
74#2851C2×C311yescharacteristic
75#2647C2×C311yescharacteristic, complemented
76#2645C2×C312yescomplemented
77#2642C2×C311yescharacteristic, complemented
78#2533C2×C314yescomplemented
79#2528C2×C314yescomplemented
80#2505C2×C314yes—
81#2493C2×C314yes—
82#2330C2×C311yescharacteristic, complemented
83#2327C2×C311yescharacteristic, complemented
84#2320C2×C312yescomplemented
85C4×C22⋊Q8C2×C312yes—
86C2×#376C2×C311yescharacteristic
87C22×C3×Q8C2312yes—
88C23×C3×C4C2311yescharacteristic
89C23×C3×C4C2312yes—
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C3×C4⋊Q8—24no—
94C3×#125—24no—
95C3×C22⋊Q8—28no—
96C3×C4×Q8—28no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2314yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2314yes—
111C2×C3×C42—24no—
112C2×C3×C42—24no—
113C2×C3×C42—24no—
114C2×C3×C42C2312yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2314yes—
123C2×C3×C42C2314yes—
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2311yescharacteristic
126C3×#95C2312yes—
127C3×#95C2312yes—
128C3×#95C2312yes—
129C3×#95C2312yes—
130C3×#95C2312yes—
131C3×#95C2312yes—
132C3×#95C2312yes—
133C3×#95C2312yes—
134C3×#95C2314yes—
135C3×#95C2314yes—
136C3×#95C2314yes—
137C3×#95C2314yes—
138C3×#95C2314yes—
139C3×#95C2314yes—
140C3×#95C2314yes—
141C3×#95C2314yes—
142C3×#95C2314yes—
143C3×#95C2314yes—
144C3×#95C2314yes—
145C3×#95C2314yes—
146C3×#95C2314yes—
147C3×#95C2314yes—
148C3×#95C2314yes—
149C2×C4⋊Q8C22×C311yescharacteristic
150C2×#125C22×C311yescharacteristic
151C2×C22⋊Q8C22×C312yes—
152C2×C4×Q8C22×C312yes—
153C22×C42C22×C311yescharacteristic
154C22×C42C22×C311yescharacteristic
155C22×C42C22×C311yescharacteristic
156#401C22×C314yes—
157#400C22×C314yes—
158#400C22×C314yes—
159#400C22×C314yes—
160#400C22×C314yes—
161#400C22×C314yes—
162#400C22×C314yes—
163#398C22×C314yes—
164#391C22×C312yes—
165#391C22×C312yes—
166#391C22×C312yes—
167#391C22×C312yes—
168#389C22×C312yes—
169#389C22×C312yes—
170#389C22×C312yes—
171#389C22×C314yes—
172#389C22×C314yes—
173#389C22×C312yes—
174#387C22×C314yes—
175#387C22×C314yes—
176#387C22×C314yes—
177#387C22×C314yes—
178#387C22×C314yes—
179#387C22×C314yes—
180#387C22×C314yes—
181#387C22×C314yes—
182#387C22×C314yes—
183#385C22×C314yes—
184#385C22×C314yes—
185#385C22×C314yes—
186#384C22×C311yescharacteristic
187#383C22×C314yes—
188#382C22×C314yes—
189#382C22×C311yescharacteristic
190#381C22×C314yes—
191#380C22×C312yes—
192#380C22×C314yes—
193C4×C4⋊C4C22×C312yes—
194C4×C4⋊C4C22×C314yes—
195C4×C22⋊C4C22×C314yes—
196#376C22×C311yescharacteristic
197#376C22×C311yescharacteristic
198#376C22×C311yescharacteristic
199#376C22×C311yescharacteristic
200#376C22×C312yes—
201#376C22×C312yes—
202C2×#95C22×C312yes—
203C2×#95C22×C312yes—
204C24×C3C2411yescharacteristic
205C2×C3×Q8—24no—
206C2×C3×Q8—24no—
207C2×C3×Q8—24no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—24no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—24no—
216C22×C3×C4—24no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—42no—
220C22×C3×C4#4114yes—
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2412yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2414yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2414yes—
230C22×C3×C4C2414yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2412yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C3×C4⋊C4—24no—
236C3×C4⋊C4—28no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—28no—
239C3×C4⋊C4—28no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—28no—
242C3×C4⋊C4—44no—
243C3×C4⋊C4—44no—
244C3×C4⋊C4—44no—
245C3×C4⋊C4—44no—
246C3×C4⋊C4—44no—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—28no—
249C3×C42—24no—
250C3×C42—24no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—22no—
254C3×C42—22no—
255C3×C42—22no—
256C3×C42—22no—
257C3×C42—28no—
258C3×C42—44no—
259C3×C42#4114yes—
260C3×C42#4114yes—
261C22×Q8C23×C312yes—
262C23×C4C23×C311yescharacteristic
263C23×C4C23×C312yes—
264C23×C4C23×C312yes—
265C23×C4C23×C311yescharacteristic
266C23×C4C23×C311yescharacteristic
267C4⋊Q8—24no—
268#125—24no—
269C22⋊Q8—28no—
270C4×Q8—28no—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C314yes—
273C2×C4⋊C4C23×C312yes—
274C2×C4⋊C4C23×C314yes—
275C2×C4⋊C4C23×C312yes—
276C2×C4⋊C4C23×C314yes—
277C2×C4⋊C4C23×C314yes—
278C2×C4⋊C4C23×C314yes—
279C2×C4⋊C4C23×C314yes—
280C2×C4⋊C4C23×C314yes—
281C2×C4⋊C4C23×C314yes—
282C2×C4⋊C4C23×C314yes—
283C2×C22⋊C4C23×C314yes—
284C2×C22⋊C4C23×C314yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42C23×C314yes—
289C2×C42C23×C312yes—
290C2×C42C23×C311yescharacteristic
291C2×C42C23×C311yescharacteristic
292C2×C42C23×C312yes—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C314yes—
298C2×C42C23×C311yescharacteristic
299C2×C42C23×C311yescharacteristic
300#95C23×C312yes—
301#95C23×C312yes—
302#95C23×C312yes—
303#95C23×C312yes—
304#95C23×C312yes—
305#95C23×C312yes—
306#95C23×C312yes—
307#95C23×C312yes—
308#95C23×C314yes—
309#95C23×C314yes—
310#95C23×C314yes—
311#95C23×C314yes—
312#95C23×C314yes—
313#95C23×C314yes—
314#95C23×C314yes—
315#95C23×C314yes—
316#95C23×C314yes—
317#95C23×C314yes—
318#95C23×C314yes—
319#95C23×C314yes—
320#95C23×C314yes—
321#95C23×C314yes—
322#95C23×C314yes—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—22no—
326C23×C3C2×#4112yes—
327C23×C32−1+414yes—
328C23×C3C2511yescharacteristic
329C3×Q8—48no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—28no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—48no—
364C2×C3×C4—48no—
365C2×C3×C4—44no—
366C2×C3×C4—44no—
367C2×C3×C4C2×#4114yes—
368C2×C3×C4C2×#4112yes—
369C2×C3×C4C2×#4114yes—
370C2×C3×C4C2×#4112yes—
371C2×C3×C4C2×#4112yes—
372C2×C3×C4C2×#4112yes—
373C24C24×C311yescharacteristic
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4C3×#4114yes—
390C22×C4C24×C311yescharacteristic
391C22×C4C24×C311yescharacteristic
392C22×C4C24×C312yes—
393C22×C4C24×C312yes—
394C22×C4C24×C311yescharacteristic
395C22×C4C24×C311yescharacteristic
396C22×C4C24×C314yes—
397C22×C4C24×C314yes—
398C22×C4C24×C314yes—
399C22×C4C24×C314yes—
400C22×C4C24×C312yes—
401C22×C4C24×C312yes—
402C22×C4C24×C311yescharacteristic
403C22×C4C24×C311yescharacteristic
404C4⋊C4—24no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—28no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C42—24no—
419C42—24no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—28no—
427C42—44no—
428C42C3×#4114yes—
429C42C3×#4114yes—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3#54114yes—
435C22×C3C22×#4111yescharacteristic
436C22×C3C22×#4111yescharacteristic
437C22×C3C22×#4111yescharacteristic
438C22×C3C2×2−1+412yes—
439C22×C3#58512yes—
440C3×C4—22no—
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—44no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—44no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—44no—
450C3×C4—44no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4#52914yes—
454C23—21no—
455C23—21no—
456C23—22no—
457C23C2×C3×#4112yes—
458C23C3×2−1+414yes—
459C23C25×C311yescharacteristic
460Q8—48no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—28no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4C2×C3×#4114yes—
499C2×C4C2×C3×#4112yes—
500C2×C4C2×C3×#4114yes—
501C2×C4C2×C3×#4112yes—
502C2×C4C2×C3×#4112yes—
503C2×C4C2×C3×#4112yes—
504C2×C3—24no—
505C2×C3C2×#52911yescharacteristic
506C2×C3C2×#54111yescharacteristic
507C2×C3#345511yescharacteristic
508C2×C3#349512yes—
509C2×C3#349512yes—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22C3×#54114yes—
515C22C22×C3×#4111yescharacteristic
516C22C22×C3×#4111yescharacteristic
517C22C22×C3×#4111yescharacteristic
518C22C2×C3×2−1+412yes—
519C22C3×#58512yes—
520C4—22no—
521C4—24no—
522C4—24no—
523C4—44no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—44no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—44no—
533C4C3×#52914yes—
534C3#3531511yescharacteristic, direct product factor, Sylow
535C2—24no—
536C2C2×C3×#52911yescharacteristic
537C2C2×C3×#54111yescharacteristic
538C2C3×#345511yescharacteristic
539C2C3×#349512yes—
540C2C3×#349512yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC24×C3C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2647C2semidirect
5. C3×#2645C2semidirect
6. C3×#2642C2semidirect
7. C3×#2533C2semidirect
8. C3×#2528C2semidirect
11. C3×#2330C2semidirect
12. C3×#2327C2semidirect
13. C3×#2320C2semidirect
16. #35315C3direct (N × H)
75. #2647C2×C3semidirect
76. #2645C2×C3semidirect
77. #2642C2×C3semidirect
78. #2533C2×C3semidirect
79. #2528C2×C3semidirect
82. #2330C2×C3semidirect
83. #2327C2×C3semidirect
84. #2320C2×C3semidirect
534. C3#35315direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28572
C3×#28552
C3×#28512
C3×#26472
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C3×#25282
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C3×#23302
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#353153