Finite Group Databasebeta

← C3×#34256 #10652863

Subgroups — C3×#34256 #10652863

Order 768.

5898 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C3×#518C211yesmaximal, characteristic, complemented
3C22×C3×C4×D4C211yesmaximal, characteristic, complemented
4C3×#2339C218yesmaximal, complemented
5C3×D4×C22⋊C4C218yesmaximal, complemented
6C3×#2311C212yesmaximal, complemented
7C3×#2308C214yesmaximal, complemented
8C3×#2297C212yesmaximal, complemented
9C2×C3×#386C214yesmaximal, complemented
10#34256C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×H(5,𝔽2)—28no—
12C2×C3×H(5,𝔽2)—28no—
13C2×C3×H(5,𝔽2)C4116yescomplemented
14C22×C3×#41C2212yes—
15C22×C3×#41C2214yes—
16C23×C3×D4C2214yes—
17C23×C3×D4C2214yes—
18C23×C3×D4C2211yescharacteristic
19C24×C3×C4C2212yescomplemented
20C3×#518—28no—
21C2×C3×C4×D4—28no—
22C2×C3×C4×D4C2218yescomplemented
23C2×C3×C4×D4C2218yescomplemented
24C2×C3×#117C2212yescomplemented
25C22×C3×C4⋊C4C2211yescharacteristic, complemented
26C22×C3×C22⋊C4C2212yescomplemented
27C22×C3×C22⋊C4C2214yescomplemented
28C22×C3×C22⋊C4C2218yescomplemented
29C22×C3×C42C2211yescharacteristic, complemented
30C3×#391C2214yescomplemented
31C3×#391C2214yescomplemented
32C3×#390C2214yescomplemented
33C3×#390C2214yescomplemented
34C3×#386C2214yescomplemented
35C3×#386C2218yescomplemented
36C3×#386C22116yescomplemented
37C3×#386C2214yescomplemented
38C3×#386C22116yescomplemented
39C3×#381C2218yescomplemented
40C3×#380C2218yescomplemented
41C3×#379C2218yescomplemented
42C3×C4×C22⋊C4C2218yescomplemented
43C2×C3×#95C2218yescomplemented
44C22×H(5,𝔽2)C2×C311yescharacteristic
45C2×#518C2×C311yescharacteristic, complemented
46C22×C4×D4C2×C311yescharacteristic, complemented
47#2339C2×C318yescomplemented
48D4×C22⋊C4C2×C318yescomplemented
49#2311C2×C312yescomplemented
50#2308C2×C314yescomplemented
51#2297C2×C312yescomplemented
52C2×#386C2×C314yescomplemented
53C25×C3C2314yes—
54C25×C3C2312yes—
55C3×H(5,𝔽2)—2128no—
56C2×C3×#41—216no—
57C2×C3×#41—216no—
58C2×C3×#41—28no—
59C2×C3×#41—28no—
60C2×C3×#41C2×C4116yescomplemented
61C2×C3×#41C2×C4132yescomplemented
62C22×C3×Q8C2312yes—
63C22×C3×D4—216no—
64C22×C3×D4—216no—
65C22×C3×D4—216no—
66C22×C3×D4—216no—
67C22×C3×D4C2×C418yes—
68C22×C3×D4C2×C4132yescomplemented
69C22×C3×D4C2×C4132yescomplemented
70C22×C3×D4D418yes—
71C22×C3×D4D418yes—
72C22×C3×D4C2314yes—
73C22×C3×D4C2318yes—
74C22×C3×D4C2312yes—
75C22×C3×D4C2314yes—
76C23×C3×C4—28no—
77C23×C3×C4—48no—
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2318yes—
80C23×C3×C4C2314yes—
81C23×C3×C4C2314yes—
82C23×C3×C4C2312yes—
83C23×C3×C4C2318yes—
84C23×C3×C4C2311yescharacteristic
85C23×C3×C4C2311yescharacteristic
86C3×C4×D4—232no—
87C3×C4×D4—232no—
88C3×#117—28no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4C2314yes—
91C2×C3×C4⋊C4C2314yes—
92C2×C3×C22⋊C4—28no—
93C2×C3×C22⋊C4—216no—
94C2×C3×C22⋊C4—416no—
95C2×C3×C22⋊C4C2318yes—
96C2×C3×C22⋊C4C2318yes—
97C2×C3×C22⋊C4C2318yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2318yes—
100C2×C3×C22⋊C4C23116yes—
101C2×C3×C42—24no—
102C2×C3×C42C2314yes—
103C2×C3×C42C2314yes—
104C3×#95C2318yes—
105C3×#95C23116yes—
106C3×#95C2318yes—
107C2×H(5,𝔽2)—28no—
108C2×H(5,𝔽2)—28no—
109C2×H(5,𝔽2)C3×C4116yescomplemented
110C22×#41C22×C312yes—
111C22×#41C22×C314yes—
112C23×D4C22×C314yes—
113C23×D4C22×C314yes—
114C23×D4C22×C311yescharacteristic
115C24×C4C22×C312yescomplemented
116#518—28no—
117C2×C4×D4—28no—
118C2×C4×D4C22×C318yescomplemented
119C2×C4×D4C22×C318yescomplemented
120C2×#117C22×C312yescomplemented
121C22×C4⋊C4C22×C311yescharacteristic, complemented
122C22×C22⋊C4C22×C312yescomplemented
123C22×C22⋊C4C22×C314yescomplemented
124C22×C22⋊C4C22×C318yescomplemented
125C22×C42C22×C311yescharacteristic, complemented
126#391C22×C314yescomplemented
127#391C22×C314yescomplemented
128#390C22×C314yescomplemented
129#390C22×C314yescomplemented
130#386C22×C314yescomplemented
131#386C22×C318yescomplemented
132#386C22×C3116yescomplemented
133#386C22×C3116yescomplemented
134#386C22×C314yescomplemented
135#381C22×C318yescomplemented
136#380C22×C318yescomplemented
137#379C22×C318yescomplemented
138C4×C22⋊C4C22×C318yescomplemented
139C2×#95C22×C318yescomplemented
140C24×C3—28no—
141C24×C3—28no—
142C24×C3—44no—
143C24×C3—44no—
144C24×C3—48no—
145C24×C3—48no—
146C24×C3C22×C418yes—
147C24×C3C22×C4116yescomplemented
148C24×C3C2×D418yes—
149C24×C3C2×D418yes—
150C24×C3C2414yes—
151C24×C3C2411yescharacteristic
152C24×C3C2414yes—
153C3×#41—2128no—
154C3×#41—264no—
155C2×C3×Q8—24no—
156C2×C3×Q8—24no—
157C2×C3×Q8C22×C418yescomplemented
158C2×C3×D4—2128no—
159C2×C3×D4—28no—
160C2×C3×D4—28no—
161C2×C3×D4—216no—
162C2×C3×D4—216no—
163C2×C3×D4—2128no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4C22⋊C4164yescomplemented
167C2×C3×D4C22×C4116yes—
168C2×C3×D4C22×C4116yes—
169C2×C3×D4C22×C4132yescomplemented
170C2×C3×D4C22×C418yescomplemented
171C2×C3×D4C2×D4116yes—
172C2×C3×D4C2×D4116yes—
173C22×C3×C4—22no—
174C22×C3×C4—28no—
175C22×C3×C4—28no—
176C22×C3×C4—28no—
177C22×C3×C4—28no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—24no—
181C22×C3×C4—216no—
182C22×C3×C4—216no—
183C22×C3×C4—216no—
184C22×C3×C4—48no—
185C22×C3×C4—44no—
186C22×C3×C4—42no—
187C22×C3×C4—48no—
188C22×C3×C4—48no—
189C22×C3×C4—432no—
190C22×C3×C4C22×C414yes—
191C22×C3×C4C22×C4116yescomplemented
192C22×C3×C4C22×C4116yescomplemented
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2418yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2414yes—
199C22×C3×C4C2412yes—
200C22×C3×C4C2412yes—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C22⋊C4—28no—
204C3×C22⋊C4—216no—
205C3×C22⋊C4—216no—
206C3×C22⋊C4—216no—
207C3×C22⋊C4—416no—
208C3×C22⋊C4—48no—
209C3×C42—28no—
210C3×C42—28no—
211C25C23×C314yes—
212C25C23×C312yes—
213H(5,𝔽2)—2128no—
214C2×#41—216no—
215C2×#41—216no—
216C2×#41—28no—
217C2×#41—28no—
218C2×#41C2×C3×C4116yescomplemented
219C2×#41C2×C3×C4132yescomplemented
220C22×Q8C23×C312yes—
221C22×D4—216no—
222C22×D4—216no—
223C22×D4—216no—
224C22×D4—216no—
225C22×D4C2×C3×C418yes—
226C22×D4C2×C3×C4132yescomplemented
227C22×D4C2×C3×C4132yescomplemented
228C22×D4C3×D418yes—
229C22×D4C3×D418yes—
230C22×D4C23×C314yes—
231C22×D4C23×C318yes—
232C22×D4C23×C312yes—
233C22×D4C23×C314yes—
234C23×C4—28no—
235C23×C4—48no—
236C23×C4C23×C311yescharacteristic
237C23×C4C23×C318yes—
238C23×C4C23×C314yes—
239C23×C4C23×C314yes—
240C23×C4C23×C318yes—
241C23×C4C23×C311yescharacteristic
242C23×C4C23×C312yes—
243C23×C4C23×C311yescharacteristic
244C4×D4—232no—
245C4×D4—232no—
246#117—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4C23×C314yes—
249C2×C4⋊C4C23×C314yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—416no—
253C2×C22⋊C4C23×C314yes—
254C2×C22⋊C4C23×C318yes—
255C2×C22⋊C4C23×C318yes—
256C2×C22⋊C4C23×C318yes—
257C2×C22⋊C4C23×C3116yes—
258C2×C22⋊C4C23×C318yes—
259C2×C42—24no—
260C2×C42C23×C314yes—
261C2×C42C23×C314yes—
262#95C23×C318yes—
263#95C23×C3116yes—
264#95C23×C318yes—
265C23×C3—22no—
266C23×C3—28no—
267C23×C3—22no—
268C23×C3—28no—
269C23×C3—24no—
270C23×C3—24no—
271C23×C3—28no—
272C23×C3—28no—
273C23×C3—232no—
274C23×C3—48no—
275C23×C3—416no—
276C23×C3—48no—
277C23×C3—416no—
278C23×C3—432no—
279C23×C3—88no—
280C23×C3—88no—
281C23×C3—816no—
282C23×C3C2×C22⋊C4132yescomplemented
283C23×C3C23×C418yes—
284C23×C3C23×C412yes—
285C23×C3C23×C418yes—
286C23×C3C22×D412yes—
287C23×C3C22×D418yes—
288C23×C3C22×D418yes—
289C23×C3C22×D412yes—
290C23×C3C2511yescharacteristic
291C3×Q8—216no—
292C3×D4—232no—
293C3×D4—264no—
294C3×D4—216no—
295C3×D4C2×C22⋊C4164yescomplemented
296C2×C3×C4—28no—
297C2×C3×C4—232no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—232no—
303C2×C3×C4—22no—
304C2×C3×C4—28no—
305C2×C3×C4—24no—
306C2×C3×C4—28no—
307C2×C3×C4—28no—
308C2×C3×C4—24no—
309C2×C3×C4—28no—
310C2×C3×C4—432no—
311C2×C3×C4—44no—
312C2×C3×C4—48no—
313C2×C3×C4—816no—
314C2×C3×C4C2×C22⋊C4116yescomplemented
315C2×C3×C4C23×C418yes—
316C2×C3×C4C23×C414yes—
317C2×C3×C4C22×D414yes—
318C2×C3×C4C22×D414yes—
319C2×C3×C4H(5,𝔽2)14yescomplemented
320C24—28no—
321C24—28no—
322C24—44no—
323C24—44no—
324C24—48no—
325C24—48no—
326C24C22×C3×C418yes—
327C24C22×C3×C4116yescomplemented
328C24C2×C3×D418yes—
329C24C2×C3×D418yes—
330C24C24×C314yes—
331C24C24×C311yescharacteristic
332C24C24×C314yes—
333#41—2128no—
334#41—264no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8C22×C3×C418yescomplemented
338C2×D4—2128no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—2128no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4C3×C22⋊C4164yescomplemented
347C2×D4C22×C3×C4116yes—
348C2×D4C22×C3×C4116yes—
349C2×D4C22×C3×C4132yescomplemented
350C2×D4C22×C3×C418yescomplemented
351C2×D4C2×C3×D4116yes—
352C2×D4C2×C3×D4116yes—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—432no—
370C22×C4C22×C3×C414yes—
371C22×C4C22×C3×C4116yescomplemented
372C22×C4C22×C3×C4116yescomplemented
373C22×C4C2×C3×D414yes—
374C22×C4C2×C3×D414yes—
375C22×C4C24×C312yes—
376C22×C4C24×C318yes—
377C22×C4C24×C314yes—
378C22×C4C24×C314yes—
379C22×C4C24×C312yes—
380C22×C4C24×C312yes—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—216no—
385C22⋊C4—216no—
386C22⋊C4—216no—
387C22⋊C4—416no—
388C22⋊C4—48no—
389C42—28no—
390C42—28no—
391C22×C3—24no—
392C22×C3—216no—
393C22×C3—22no—
394C22×C3—28no—
395C22×C3—22no—
396C22×C3—28no—
397C22×C3—216no—
398C22×C3—24no—
399C22×C3—28no—
400C22×C3—216no—
401C22×C3—432no—
402C22×C3—44no—
403C22×C3—44no—
404C22×C3—48no—
405C22×C3—832no—
406C22×C3C22×C22⋊C4116yes—
407C22×C3C22×C22⋊C414yes—
408C22×C3C24×C411yescharacteristic
409C22×C3C23×D411yescharacteristic
410C22×C3C23×D411yescharacteristic
411C22×C3C2×H(5,𝔽2)12yes—
412C22×C3#58512yes—
413C3×C4—28no—
414C3×C4—24no—
415C3×C4—44no—
416C3×C4—42no—
417C3×C4—48no—
418C3×C4C22×C22⋊C418yes—
419C23—22no—
420C23—28no—
421C23—22no—
422C23—28no—
423C23—24no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—232no—
428C23—48no—
429C23—416no—
430C23—48no—
431C23—416no—
432C23—432no—
433C23—88no—
434C23—88no—
435C23—816no—
436C23C2×C3×C22⋊C4132yescomplemented
437C23C23×C3×C418yes—
438C23C23×C3×C412yes—
439C23C23×C3×C418yes—
440C23C22×C3×D412yes—
441C23C22×C3×D418yes—
442C23C22×C3×D418yes—
443C23C22×C3×D412yes—
444C23C25×C311yescharacteristic
445Q8—216no—
446D4—232no—
447D4—264no—
448D4—216no—
449D4C2×C3×C22⋊C4164yescomplemented
450C2×C4—28no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—232no—
456C2×C4—232no—
457C2×C4—22no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—432no—
465C2×C4—48no—
466C2×C4—44no—
467C2×C4—816no—
468C2×C4C2×C3×C22⋊C4116yescomplemented
469C2×C4C23×C3×C418yes—
470C2×C4C23×C3×C414yes—
471C2×C4C22×C3×D414yes—
472C2×C4C22×C3×D414yes—
473C2×C4C3×H(5,𝔽2)14yescomplemented
474C2×C3—24no—
475C2×C3—216no—
476C2×C3—48no—
477C2×C3C23×C22⋊C411yescharacteristic
478C2×C3C4×H(5,𝔽2)12yes—
479C2×C3#348411yescharacteristic
480C2×C3#348511yescharacteristic
481C2×C3#349612yes—
482C22—24no—
483C22—216no—
484C22—22no—
485C22—28no—
486C22—22no—
487C22—28no—
488C22—216no—
489C22—24no—
490C22—28no—
491C22—216no—
492C22—432no—
493C22—44no—
494C22—44no—
495C22—48no—
496C22—832no—
497C22C22×C3×C22⋊C4116yes—
498C22C22×C3×C22⋊C414yes—
499C22C24×C3×C411yescharacteristic
500C22C23×C3×D411yescharacteristic
501C22C23×C3×D411yescharacteristic
502C22C2×C3×H(5,𝔽2)12yes—
503C22C3×#58512yes—
504C4—28no—
505C4—24no—
506C4—44no—
507C4—42no—
508C4—48no—
509C4C22×C3×C22⋊C418yes—
510C3#3425611yescharacteristic, direct product factor, Sylow
511C2—24no—
512C2—216no—
513C2—48no—
514C2C23×C3×C22⋊C411yescharacteristic
515C2C3×C4×H(5,𝔽2)12yes—
516C2C3×#348411yescharacteristic
517C2C3×#348511yescharacteristic
518C2C3×#349612yes—
subgroupG/H
center, socleC23×C3C25
commutatorC22C24×C3×C4
Frattini, cosocleC23C25×C3
prime radicalC22×C3×H(5,𝔽2)C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#518C2semidirect
3. C22×C3×C4×D4C2semidirect
4. C3×#2339C2semidirect
5. C3×D4×C22⋊C4C2semidirect
6. C3×#2311C2semidirect
7. C3×#2308C2semidirect
8. C3×#2297C2semidirect
9. C2×C3×#386C2semidirect
10. #34256C3direct (N × H)
13. C2×C3×H(5,𝔽2)C4semidirect
19. C24×C3×C4C22semidirect
22. C2×C3×C4×D4C22semidirect
23. C2×C3×C4×D4C22semidirect
24. C2×C3×#117C22semidirect
25. C22×C3×C4⋊C4C22semidirect
26. C22×C3×C22⋊C4C22semidirect
27. C22×C3×C22⋊C4C22semidirect
28. C22×C3×C22⋊C4C22semidirect
29. C22×C3×C42C22semidirect
30. C3×#391C22semidirect
31. C3×#391C22semidirect
32. C3×#390C22semidirect
33. C3×#390C22semidirect
34. C3×#386C22semidirect
35. C3×#386C22semidirect
36. C3×#386C22semidirect
37. C3×#386C22semidirect
38. C3×#386C22semidirect
39. C3×#381C22semidirect
40. C3×#380C22semidirect
41. C3×#379C22semidirect
42. C3×C4×C22⋊C4C22semidirect
43. C2×C3×#95C22semidirect
45. C2×#518C2×C3semidirect
46. C22×C4×D4C2×C3semidirect
47. #2339C2×C3semidirect
48. D4×C22⋊C4C2×C3semidirect
49. #2311C2×C3semidirect
50. #2308C2×C3semidirect
51. #2297C2×C3semidirect
52. C2×#386C2×C3semidirect
60. C2×C3×#41C2×C4semidirect
61. C2×C3×#41C2×C4semidirect
68. C22×C3×D4C2×C4semidirect
69. C22×C3×D4C2×C4semidirect
109. C2×H(5,𝔽2)C3×C4semidirect
115. C24×C4C22×C3semidirect
118. C2×C4×D4C22×C3semidirect
119. C2×C4×D4C22×C3semidirect
120. C2×#117C22×C3semidirect
121. C22×C4⋊C4C22×C3semidirect
122. C22×C22⋊C4C22×C3semidirect
123. C22×C22⋊C4C22×C3semidirect
124. C22×C22⋊C4C22×C3semidirect
125. C22×C42C22×C3semidirect
126. #391C22×C3semidirect
127. #391C22×C3semidirect
128. #390C22×C3semidirect
129. #390C22×C3semidirect
130. #386C22×C3semidirect
131. #386C22×C3semidirect
132. #386C22×C3semidirect
133. #386C22×C3semidirect
134. #386C22×C3semidirect
135. #381C22×C3semidirect
136. #380C22×C3semidirect
137. #379C22×C3semidirect
138. C4×C22⋊C4C22×C3semidirect
139. C2×#95C22×C3semidirect
147. C24×C3C22×C4semidirect
157. C2×C3×Q8C22×C4semidirect
166. C2×C3×D4C22⋊C4semidirect
169. C2×C3×D4C22×C4semidirect
170. C2×C3×D4C22×C4semidirect
191. C22×C3×C4C22×C4semidirect
192. C22×C3×C4C22×C4semidirect
218. C2×#41C2×C3×C4semidirect
219. C2×#41C2×C3×C4semidirect
226. C22×D4C2×C3×C4semidirect
227. C22×D4C2×C3×C4semidirect
282. C23×C3C2×C22⋊C4semidirect
295. C3×D4C2×C22⋊C4semidirect
314. C2×C3×C4C2×C22⋊C4semidirect
319. C2×C3×C4H(5,𝔽2)semidirect
327. C24C22×C3×C4semidirect
337. C2×Q8C22×C3×C4semidirect
346. C2×D4C3×C22⋊C4semidirect
349. C2×D4C22×C3×C4semidirect
350. C2×D4C22×C3×C4semidirect
371. C22×C4C22×C3×C4semidirect
372. C22×C4C22×C3×C4semidirect
436. C23C2×C3×C22⋊C4semidirect
449. D4C2×C3×C22⋊C4semidirect
468. C2×C4C2×C3×C22⋊C4semidirect
473. C2×C4C3×H(5,𝔽2)semidirect
510. C3#34256direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C2×C3×#5182
C22×C3×C4×D42
C3×#23392
C3×D4×C22⋊C42
C3×#23112
C3×#23082
C3×#22972
C2×C3×#3862
#342563