Finite Group Databasebeta

← C3×#14934 #10633541

Subgroups — C3×#14934 #10633541

Order 768.

2062 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#381C212yesmaximal
3C22×C3×#95C211yesmaximal, characteristic
4C3×#1446C212yesmaximal, complemented
5C3×#1442C214yesmaximal, complemented
6C3×#1442C212yesmaximal, complemented
7C3×C22⋊#95C212yesmaximal
8C3×C22⋊#95C211yesmaximal, characteristic
9#14934C311yesmaximal, 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×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C3×#381—28no—
21C2×C3×#95—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2214yes—
25C2×C3×#95C2214yes—
26C2×C3×#95C2214yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2214yes—
30C22×C22⋊Q8C2×C311yescharacteristic
31C2×#381C2×C312yes—
32C22×#95C2×C311yescharacteristic
33#1446C2×C312yescomplemented
34#1442C2×C312yescomplemented
35#1442C2×C314yescomplemented
36C22⋊#95C2×C311yescharacteristic
37C22⋊#95C2×C312yes—
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—24no—
47C23×C3×C4—22no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4—24no—
51C23×C3×C4—24no—
52C23×C3×C4—42no—
53C23×C3×C4—42no—
54C23×C3×C4D414yes—
55C23×C3×C4Q814yescomplemented
56C23×C3×C4C2314yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
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×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4D418yes—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—28no—
78C3×#95—28no—
79C3×#95—28no—
80C3×#95—24no—
81C3×#95—44no—
82C3×#95—42no—
83C23×Q8C22×C311yescharacteristic
84C24×C4C22×C311yescharacteristic
85C24×C4C22×C311yescharacteristic
86C24×C4C22×C311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—28no—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C22⋊C4C22×C312yes—
92C22×C22⋊C4C22×C312yes—
93#381—28no—
94C2×#95—24no—
95C2×#95C22×C314yes—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C314yes—
99C2×#95C22×C312yes—
100C2×#95C22×C314yes—
101C2×#95C22×C312yes—
102C2×#95C22×C314yes—
103C24×C3—22no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3C2×D414yes—
108C24×C3C2×D412yes—
109C24×C3C2×Q812yes—
110C24×C3#4112yes—
111C24×C3#4114yes—
112C24×C3C2411yescharacteristic
113C2×C3×Q8—28no—
114C2×C3×Q8—416no—
115C2×C3×Q8—44no—
116C2×C3×Q8—44no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—28no—
120C22×C3×C4—28no—
121C22×C3×C4—28no—
122C22×C3×C4—216no—
123C22×C3×C4—22no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
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—28no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—44no—
151C22×C3×C4—416no—
152C22×C3×C4—416no—
153C22×C3×C4—44no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4—48no—
157C22×C3×C4—44no—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×Q814yes—
163C22×C3×C4#4114yes—
164C22×C3×C4#4114yes—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—216no—
167C3×C22⋊C4—216no—
168C3×C22⋊C4—48no—
169C25C23×C311yescharacteristic
170C22×Q8—24no—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—42no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C3×D414yes—
186C23×C4C3×Q814yescomplemented
187C23×C4C23×C314yes—
188C23×C4C23×C312yes—
189C23×C4C23×C312yes—
190C23×C4C23×C311yescharacteristic
191C23×C4C23×C312yes—
192C23×C4C23×C312yes—
193C23×C4C23×C311yescharacteristic
194C22⋊Q8—232no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4C3×D418yes—
206#95—24no—
207#95—24no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—24no—
212#95—44no—
213#95—42no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—28no—
223C23×C3—28no—
224C23×C3—28no—
225C23×C3—28no—
226C23×C3#12014yes—
227C23×C3C22⋊Q814yes—
228C23×C3C22⋊Q818yes—
229C23×C3#12418yes—
230C23×C3#12414yes—
231C23×C3C22×D411yescharacteristic
232C23×C3C22×D412yes—
233C23×C3C22×Q811yescharacteristic
234C23×C3C2×#4111yescharacteristic
235C23×C3C2×#4112yes—
236C23×C3C2×#4112yes—
237C23×C3C2×#4112yes—
238C23×C3H(5,𝔽2)12yes—
239C23×C32−1+412yes—
240C3×Q8—416no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—28no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—28no—
253C2×C3×C4—28no—
254C2×C3×C4—28no—
255C2×C3×C4—416no—
256C2×C3×C4—416no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—44no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4—44no—
268C2×C3×C4—44no—
269C2×C3×C4—44no—
270C2×C3×C4—416no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4#12314yes—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24C2×C3×D414yes—
284C24C2×C3×D412yes—
285C24C2×C3×Q812yes—
286C24C3×#4112yes—
287C24C3×#4114yes—
288C24C24×C311yescharacteristic
289C2×Q8—28no—
290C2×Q8—416no—
291C2×Q8—44no—
292C2×Q8—44no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—216no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—44no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—48no—
333C22×C4—44no—
334C22×C4C2×C3×D414yes—
335C22×C4C2×C3×D414yes—
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×Q814yes—
339C22×C4C3×#4114yes—
340C22×C4C3×#4114yes—
341C4⋊C4—28no—
342C4⋊C4—216no—
343C22⋊C4—216no—
344C22⋊C4—48no—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—24no—
350C22×C3—216no—
351C22×C3—216no—
352C22×C3—24no—
353C22×C3#39318yes—
354C22×C3C2×#12011yescharacteristic
355C22×C3C2×C22⋊Q811yescharacteristic
356C22×C3C2×C22⋊Q812yes—
357C22×C3C2×#12311yescharacteristic
358C22×C3#52511yescharacteristic
359C22×C3C2×#12411yescharacteristic
360C22×C3C2×#12412yes—
361C22×C3#52912yes—
362C22×C3#53312yes—
363C22×C3#53411yescharacteristic
364C22×C3#53611yescharacteristic
365C22×C3#53912yes—
366C22×C3#54412yes—
367C22×C3#54614yes—
368C22×C3#54714yes—
369C22×C3#55114yes—
370C22×C3#55214yes—
371C3×C4—24no—
372C3×C4—44no—
373C3×C4—44no—
374C3×C4—42no—
375C3×C4—44no—
376C3×C4—44no—
377C3×C4—84no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—24no—
386C23—28no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23C3×#12014yes—
391C23C3×C22⋊Q814yes—
392C23C3×C22⋊Q818yes—
393C23C3×#12418yes—
394C23C3×#12414yes—
395C23C22×C3×D411yescharacteristic
396C23C22×C3×D412yes—
397C23C22×C3×Q811yescharacteristic
398C23C2×C3×#4111yescharacteristic
399C23C2×C3×#4112yes—
400C23C2×C3×#4112yes—
401C23C2×C3×#4112yes—
402C23C3×H(5,𝔽2)12yes—
403C23C3×2−1+412yes—
404Q8—416no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—28no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—416no—
420C2×C4—416no—
421C2×C4—416no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4C3×#12314yes—
443C2×C3—28no—
444C2×C3C2×#39311yescharacteristic
445C2×C3#240511yescharacteristic
446C2×C3#240611yescharacteristic
447C2×C3#243814yes—
448C2×C3#248814yes—
449C2×C3#250311yescharacteristic
450C2×C3#256111yescharacteristic
451C2×C3#262711yescharacteristic
452C2×C3#285511yescharacteristic
453C22—24no—
454C22—24no—
455C22—24no—
456C22—216no—
457C22—216no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22C3×#39318yes—
462C22C2×C3×#12011yescharacteristic
463C22C2×C3×C22⋊Q811yescharacteristic
464C22C2×C3×C22⋊Q812yes—
465C22C2×C3×#12311yescharacteristic
466C22C3×#52511yescharacteristic
467C22C2×C3×#12412yes—
468C22C2×C3×#12411yescharacteristic
469C22C3×#52912yes—
470C22C3×#53312yes—
471C22C3×#53411yescharacteristic
472C22C3×#53611yescharacteristic
473C22C3×#53912yes—
474C22C3×#54412yes—
475C22C3×#54614yes—
476C22C3×#54714yes—
477C22C3×#55114yes—
478C22C3×#55214yes—
479C4—24no—
480C4—44no—
481C4—42no—
482C4—44no—
483C4—44no—
484C4—44no—
485C4—84no—
486C3#1493411yescharacteristic, direct product factor, Sylow
487C2—28no—
488C2C2×C3×#39311yescharacteristic
489C2C3×#240511yescharacteristic
490C2C3×#240611yescharacteristic
491C2C3×#243814yes—
492C2C3×#248814yes—
493C2C3×#250311yescharacteristic
494C2C3×#256111yescharacteristic
495C2C3×#262711yescharacteristic
496C2C3×#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×#1442C2semidirect
6. C3×#1442C2semidirect
9. #14934C3direct (N × H)
33. #1446C2×C3semidirect
34. #1442C2×C3semidirect
35. #1442C2×C3semidirect
55. C23×C3×C4Q8semidirect
186. C23×C4C3×Q8semidirect
486. C3#14934direct (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×#3812
C22×C3×#952
C3×#14462
C3×#14422
C3×C22⋊#952
#149343