Finite Group Databasebeta

← C3×#14914 #10633521

Subgroups — C3×#14914 #10633521

Order 768.

2150 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#385C212yesmaximal
3C22×C3×#95C211yesmaximal, characteristic
4C3×#1446C212yesmaximal, complemented
5C3×#1444C212yesmaximal, complemented
6C3×#1441C214yesmaximal, complemented
7C3×C22⋊#95C211yesmaximal, characteristic
8C3×C22⋊#95C212yesmaximal
9#14914C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C24×C3×C4C2211yescharacteristic
14C2×C3×C22⋊Q8—24no—
15C2×C3×C22⋊Q8—28no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2214yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C3×#385—28no—
22C2×C3×#95—24no—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2214yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2214yes—
29C2×C3×#95C2214yes—
30C22×C22⋊Q8C2×C311yescharacteristic
31C2×#385C2×C312yes—
32C22×#95C2×C311yescharacteristic
33#1446C2×C312yescomplemented
34#1444C2×C312yescomplemented
35#1441C2×C314yescomplemented
36C22⋊#95C2×C312yes—
37C22⋊#95C2×C311yescharacteristic
38C25×C3C2311yescharacteristic
39C22×C3×Q8—24no—
40C22×C3×Q8—22no—
41C22×C3×Q8—24no—
42C22×C3×Q8—42no—
43C23×C3×C4—22no—
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—22no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—42no—
51C23×C3×C4—42no—
52C23×C3×C4—42no—
53C23×C3×C4—42no—
54C23×C3×C4D414yes—
55C23×C3×C4Q814yescomplemented
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2314yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C3×C22⋊Q8—232no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—28no—
70C2×C3×C4⋊C4—28no—
71C2×C3×C4⋊C4—44no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4D418yes—
77C2×C3×C22⋊C4Q818yescomplemented
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—28no—
81C3×#95—24no—
82C3×#95—42no—
83C3×#95—44no—
84C3×#95—44no—
85C23×Q8C22×C311yescharacteristic
86C24×C4C22×C311yescharacteristic
87C24×C4C22×C311yescharacteristic
88C24×C4C22×C311yescharacteristic
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C22×C4⋊C4C22×C312yes—
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C4⋊C4C22×C314yes—
94C22×C22⋊C4C22×C312yes—
95C22×C22⋊C4C22×C312yes—
96#385—28no—
97C2×#95—24no—
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C314yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C314yes—
104C2×#95C22×C314yes—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3C2×D414yes—
110C24×C3C2×D412yes—
111C24×C3C2×Q814yes—
112C24×C3C2×Q812yes—
113C24×C3#4112yes—
114C24×C3C2411yescharacteristic
115C2×C3×Q8—28no—
116C2×C3×Q8—416no—
117C2×C3×Q8—44no—
118C2×C3×Q8—44no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—28no—
122C22×C3×C4—28no—
123C22×C3×C4—28no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—28no—
150C22×C3×C4—44no—
151C22×C3×C4—48no—
152C22×C3×C4—416no—
153C22×C3×C4—416no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—48no—
158C22×C3×C4—44no—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×Q818yes—
164C22×C3×C4C2×Q814yes—
165C22×C3×C4#4114yes—
166C22×C3×C4#4114yes—
167C3×C4⋊C4—28no—
168C3×C4⋊C4—216no—
169C3×C4⋊C4—416no—
170C3×C22⋊C4—216no—
171C3×C22⋊C4—216no—
172C25C23×C311yescharacteristic
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—42no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C3×D414yes—
189C23×C4C3×Q814yescomplemented
190C23×C4C23×C312yes—
191C23×C4C23×C312yes—
192C23×C4C23×C311yescharacteristic
193C23×C4C23×C314yes—
194C23×C4C23×C312yes—
195C23×C4C23×C312yes—
196C23×C4C23×C311yescharacteristic
197C22⋊Q8—232no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—44no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4C3×D418yes—
211C2×C22⋊C4C3×Q818yescomplemented
212#95—24no—
213#95—24no—
214#95—28no—
215#95—24no—
216#95—42no—
217#95—44no—
218#95—44no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—24no—
222C23×C3—24no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—28no—
228C23×C3—28no—
229C23×C3—28no—
230C23×C3—28no—
231C23×C3#12014yes—
232C23×C3C22⋊Q818yes—
233C23×C3C22⋊Q814yes—
234C23×C3C4⋊Q814yes—
235C23×C3C4⋊Q818yes—
236C23×C3C22×D411yescharacteristic
237C23×C3C22×D412yes—
238C23×C3C22×Q811yescharacteristic
239C23×C3C22×Q812yes—
240C23×C3C2×#4111yescharacteristic
241C23×C3C2×#4112yes—
242C23×C3C2×#4112yes—
243C23×C3H(5,𝔽2)12yes—
244C23×C3H(5,𝔽2)12yes—
245C3×Q8—416no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—28no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—28no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—28no—
258C2×C3×C4—28no—
259C2×C3×C4—28no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—28no—
263C2×C3×C4—416no—
264C2×C3×C4—416no—
265C2×C3×C4—416no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—44no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4#12314yes—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C2×C3×D414yes—
289C24C2×C3×D412yes—
290C24C2×C3×Q814yes—
291C24C2×C3×Q812yes—
292C24C3×#4112yes—
293C24C24×C311yescharacteristic
294C2×Q8—28no—
295C2×Q8—416no—
296C2×Q8—44no—
297C2×Q8—44no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—44no—
330C22×C4—48no—
331C22×C4—416no—
332C22×C4—416no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4C2×C3×D414yes—
339C22×C4C2×C3×D412yes—
340C22×C4C2×C3×D414yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×Q818yes—
343C22×C4C2×C3×Q814yes—
344C22×C4C3×#4114yes—
345C22×C4C3×#4114yes—
346C4⋊C4—28no—
347C4⋊C4—216no—
348C4⋊C4—416no—
349C22⋊C4—216no—
350C22⋊C4—216no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—216no—
357C22×C3—24no—
358C22×C3—216no—
359C22×C3#39518yes—
360C22×C3C2×#12011yescharacteristic
361C22×C3C2×C22⋊Q811yescharacteristic
362C22×C3C2×C22⋊Q812yes—
363C22×C3C2×#12311yescharacteristic
364C22×C3#52511yescharacteristic
365C22×C3#52912yes—
366C22×C3C2×C4⋊Q811yescharacteristic
367C22×C3C2×C4⋊Q812yes—
368C22×C3#53312yes—
369C22×C3#53411yescharacteristic
370C22×C3#53511yescharacteristic
371C22×C3#53812yes—
372C22×C3#54312yes—
373C22×C3#54614yes—
374C22×C3#54614yes—
375C22×C3#55414yes—
376C22×C3#55414yes—
377C3×C4—24no—
378C3×C4—44no—
379C3×C4—44no—
380C3×C4—44no—
381C3×C4—42no—
382C3×C4—44no—
383C3×C4—48no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—28no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23C3×#12014yes—
397C23C3×C22⋊Q818yes—
398C23C3×C22⋊Q814yes—
399C23C3×C4⋊Q814yes—
400C23C3×C4⋊Q818yes—
401C23C22×C3×D411yescharacteristic
402C23C22×C3×D412yes—
403C23C22×C3×Q811yescharacteristic
404C23C22×C3×Q812yes—
405C23C2×C3×#4111yescharacteristic
406C23C2×C3×#4112yes—
407C23C2×C3×#4112yes—
408C23C3×H(5,𝔽2)12yes—
409C23C3×H(5,𝔽2)12yes—
410Q8—416no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—416no—
429C2×C4—416no—
430C2×C4—416no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4C3×#12314yes—
449C2×C3—28no—
450C2×C3C2×#39511yescharacteristic
451C2×C3#240411yescharacteristic
452C2×C3#240511yescharacteristic
453C2×C3#245014yes—
454C2×C3#250211yescharacteristic
455C2×C3#254914yes—
456C2×C3#258311yescharacteristic
457C2×C3#262611yescharacteristic
458C2×C3#285511yescharacteristic
459C22—24no—
460C22—24no—
461C22—24no—
462C22—216no—
463C22—24no—
464C22—216no—
465C22—24no—
466C22—24no—
467C22C3×#39518yes—
468C22C2×C3×#12011yescharacteristic
469C22C2×C3×C22⋊Q811yescharacteristic
470C22C2×C3×C22⋊Q812yes—
471C22C2×C3×#12311yescharacteristic
472C22C3×#52511yescharacteristic
473C22C3×#52912yes—
474C22C2×C3×C4⋊Q811yescharacteristic
475C22C2×C3×C4⋊Q812yes—
476C22C3×#53312yes—
477C22C3×#53411yescharacteristic
478C22C3×#53511yescharacteristic
479C22C3×#53812yes—
480C22C3×#54312yes—
481C22C3×#54614yes—
482C22C3×#54614yes—
483C22C3×#55414yes—
484C22C3×#55414yes—
485C4—24no—
486C4—44no—
487C4—44no—
488C4—44no—
489C4—42no—
490C4—44no—
491C4—48no—
492C3#1491411yescharacteristic, direct product factor, Sylow
493C2—28no—
494C2C2×C3×#39511yescharacteristic
495C2C3×#240411yescharacteristic
496C2C3×#240511yescharacteristic
497C2C3×#245014yes—
498C2C3×#250211yescharacteristic
499C2C3×#254914yes—
500C2C3×#258311yescharacteristic
501C2C3×#262611yescharacteristic
502C2C3×#285511yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1446C2semidirect
5. C3×#1444C2semidirect
6. C3×#1441C2semidirect
9. #14914C3direct (N × H)
33. #1446C2×C3semidirect
34. #1444C2×C3semidirect
35. #1441C2×C3semidirect
55. C23×C3×C4Q8semidirect
77. C2×C3×C22⋊C4Q8semidirect
189. C23×C4C3×Q8semidirect
211. C2×C22⋊C4C3×Q8semidirect
492. C3#14914direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#3852
C22×C3×#952
C3×#14462
C3×#14442
C3×#14412
C3×C22⋊#952
#149143