Finite Group Databasebeta

← C3×#14166 #10632773

Subgroups — C3×#14166 #10632773

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic
2C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#376C211yesmaximal, characteristic, complemented
4C3×#1441C212yesmaximal, complemented
5C3×#1439C212yesmaximal
6C3×#1437C212yesmaximal
7C3×#1437C212yesmaximal
8C3×#1436C212yesmaximal, complemented
9C3×#1436C212yesmaximal, complemented
10#14166C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#126—24no—
12C2×C3×#126—24no—
13C2×C3×#126C418yescomplemented
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2214yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C42C2212yes—
23C22×C3×C42C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×C4×C4⋊C4—24no—
26C3×#376—24no—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2214yes—
31C2×C3×#95C2214yes—
32C2×C3×#95C2214yes—
33C22×#126C2×C311yescharacteristic
34C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
35C2×#376C2×C311yescharacteristic, complemented
36#1441C2×C312yescomplemented
37#1439C2×C312yes—
38#1437C2×C312yes—
39#1437C2×C312yes—
40#1436C2×C312yescomplemented
41#1436C2×C312yescomplemented
42C25×C3C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2314yes—
47C23×C3×C4C2314yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C3×#126—232no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—28no—
60C2×C3×C4⋊C4C2×C414yes—
61C2×C3×C4⋊C4C2×C414yes—
62C2×C3×C4⋊C4C2×C414yes—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—28no—
70C2×C3×C22⋊C4—28no—
71C2×C3×C22⋊C4—44no—
72C2×C3×C22⋊C4C2×C414yes—
73C2×C3×C22⋊C4C2×C414yes—
74C2×C3×C22⋊C4C2×C414yes—
75C2×C3×C42—24no—
76C2×C3×C42—28no—
77C2×C3×C42—24no—
78C2×C3×C42—44no—
79C2×C3×C42C2×C414yes—
80C2×C3×C42D414yes—
81C2×C3×C42D414yes—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—28no—
86C3×#95—28no—
87C3×#95—44no—
88C2×#126—24no—
89C2×#126—24no—
90C2×#126C3×C418yescomplemented
91C22×C4⋊C4C22×C312yes—
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C4⋊C4C22×C311yescharacteristic
94C22×C4⋊C4C22×C311yescharacteristic
95C22×C22⋊C4C22×C314yes—
96C22×C22⋊C4C22×C311yescharacteristic
97C22×C22⋊C4C22×C311yescharacteristic
98C22×C22⋊C4C22×C311yescharacteristic
99C22×C42C22×C312yes—
100C22×C42C22×C312yes—
101C22×C42C22×C311yescharacteristic
102C4×C4⋊C4—24no—
103#376—24no—
104C2×#95C22×C314yes—
105C2×#95C22×C314yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C314yes—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—24no—
113C24×C3—24no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3C22×C412yes—
117C24×C3C2411yescharacteristic
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—24no—
123C22×C3×C4—24no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4C22×C412yes—
145C22×C3×C4C22×C412yes—
146C22×C3×C4C22×C412yes—
147C22×C3×C4C22×C412yes—
148C22×C3×C4C22×C412yes—
149C22×C3×C4C22×C412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4#4118yes—
157C22×C3×C4#4118yes—
158C3×C4⋊C4—216no—
159C3×C4⋊C4—28no—
160C3×C4⋊C4—28no—
161C3×C4⋊C4—28no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—28no—
165C3×C22⋊C4—416no—
166C3×C42—216no—
167C3×C42—48no—
168C3×C42C22⋊C4116yescomplemented
169C25C23×C311yescharacteristic
170C23×C4C23×C311yescharacteristic
171C23×C4C23×C311yescharacteristic
172C23×C4C23×C311yescharacteristic
173C23×C4C23×C314yes—
174C23×C4C23×C314yes—
175C23×C4C23×C311yescharacteristic
176C23×C4C23×C311yescharacteristic
177C23×C4C23×C311yescharacteristic
178#126—232no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4C2×C3×C414yes—
188C2×C4⋊C4C2×C3×C414yes—
189C2×C4⋊C4C2×C3×C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4C2×C3×C414yes—
200C2×C22⋊C4C2×C3×C414yes—
201C2×C22⋊C4C2×C3×C414yes—
202C2×C42—24no—
203C2×C42—28no—
204C2×C42—24no—
205C2×C42—44no—
206C2×C42C2×C3×C414yes—
207C2×C42C3×D414yes—
208C2×C42C3×D414yes—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—28no—
213#95—28no—
214#95—44no—
215C23×C3—22no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—24no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—44no—
229C23×C3—44no—
230C23×C3C23×C411yescharacteristic
231C23×C3C22×D411yescharacteristic
232C23×C3C22×D411yescharacteristic
233C23×C3C2×#4114yes—
234C23×C3C2×#4114yes—
235C23×C3H(5,𝔽2)12yes—
236C23×C32−1+412yes—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—28no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—28no—
257C2×C3×C4—28no—
258C2×C3×C4—28no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—44no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4C2×C22⋊C414yes—
275C2×C3×C4C2×C22⋊C414yes—
276C2×C3×C4C2×C22⋊C414yes—
277C24—21no—
278C24—21no—
279C24—24no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24C22×C3×C412yes—
284C24C24×C311yescharacteristic
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4C22×C3×C412yes—
312C22×C4C22×C3×C412yes—
313C22×C4C22×C3×C412yes—
314C22×C4C22×C3×C412yes—
315C22×C4C22×C3×C412yes—
316C22×C4C22×C3×C412yes—
317C22×C4C2×C3×D412yes—
318C22×C4C2×C3×D412yes—
319C22×C4C2×C3×D412yes—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D412yes—
323C22×C4C3×#4118yes—
324C22×C4C3×#4118yes—
325C4⋊C4—216no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—416no—
333C42—216no—
334C42—48no—
335C42C3×C22⋊C4116yescomplemented
336C22×C3—42no—
337C22×C3—48no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—84no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3C22×C22⋊C411yescharacteristic
344C22×C3C4×#4112yes—
345C22×C3C4×#4112yes—
346C22×C3#52012yes—
347C22×C3#52512yes—
348C22×C3#52512yes—
349C22×C3#53212yes—
350C22×C3#53212yes—
351C22×C3#53511yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3#53712yes—
354C22×C3#55114yes—
355C22×C3#55214yes—
356C22×C3#55314yes—
357C22×C3#55614yes—
358C3×C4—24no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—42no—
362C3×C4—48no—
363C3×C4—48no—
364C3×C4—42no—
365C3×C4—42no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23C23×C3×C411yescharacteristic
382C23C22×C3×D411yescharacteristic
383C23C22×C3×D411yescharacteristic
384C23C2×C3×#4114yes—
385C23C2×C3×#4114yes—
386C23C3×H(5,𝔽2)12yes—
387C23C3×2−1+412yes—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4C2×C3×C22⋊C414yes—
426C2×C4C2×C3×C22⋊C414yes—
427C2×C4C2×C3×C22⋊C414yes—
428C2×C3—82no—
429C2×C3#229711yescharacteristic
430C2×C3#229711yescharacteristic
431C2×C3#231111yescharacteristic
432C2×C3#237314yes—
433C2×C3#260114yes—
434C2×C3#260312yes—
435C2×C3#260412yes—
436C22—48no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—84no—
441C22—82no—
442C22—82no—
443C22C22×C3×C22⋊C411yescharacteristic
444C22C3×C4×#4112yes—
445C22C3×C4×#4112yes—
446C22C3×#52012yes—
447C22C3×#52512yes—
448C22C3×#52512yes—
449C22C3×#53212yes—
450C22C3×#53212yes—
451C22C3×#53511yescharacteristic
452C22C3×#53611yescharacteristic
453C22C3×#53712yes—
454C22C3×#55114yes—
455C22C3×#55214yes—
456C22C3×#55314yes—
457C22C3×#55614yes—
458C4—24no—
459C4—24no—
460C4—24no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—48no—
465C4—48no—
466C3#1416611yescharacteristic, direct product factor, Sylow
467C2—82no—
468C2C3×#229711yescharacteristic
469C2C3×#229711yescharacteristic
470C2C3×#231111yescharacteristic
471C2C3×#237314yes—
472C2C3×#260114yes—
473C2C3×#260312yes—
474C2C3×#260412yes—
subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C4⋊C4C2semidirect
3. C2×C3×#376C2semidirect
4. C3×#1441C2semidirect
8. C3×#1436C2semidirect
9. C3×#1436C2semidirect
10. #14166C3direct (N × H)
13. C2×C3×#126C4semidirect
34. C2×C4×C4⋊C4C2×C3semidirect
35. C2×#376C2×C3semidirect
36. #1441C2×C3semidirect
40. #1436C2×C3semidirect
41. #1436C2×C3semidirect
90. C2×#126C3×C4semidirect
168. C3×C42C22⋊C4semidirect
335. C42C3×C22⋊C4semidirect
466. C3#14166direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C2×C3×C4×C4⋊C42
C2×C3×#3762
C3×#14412
C3×#14392
C3×#14372
C3×#14362
#141663