Finite Group Databasebeta

← C3×#14946 #10633553

Subgroups — C3×#14946 #10633553

Order 768.

2162 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#386C212yesmaximal
2C2×C3×#386C211yesmaximal, characteristic
3C2×C3×#385C212yesmaximal, complemented
4C22×C3×#95C211yesmaximal, characteristic, complemented
5C3×#1445C214yesmaximal, complemented
6C3×#1439C214yesmaximal, complemented
7C3×C22⋊#95C211yesmaximal, characteristic, complemented
8#14946C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C24×C3×C4C2212yes—
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2214yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2214yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C3×#386—24no—
18C3×#386—28no—
19C3×#385—28no—
20C2×C3×#95—24no—
21C2×C3×#95C2214yes—
22C2×C3×#95C2214yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2214yes—
25C2×C3×#95C2214yes—
26C2×#386C2×C311yescharacteristic
27C2×#386C2×C312yes—
28C2×#385C2×C312yescomplemented
29C22×#95C2×C311yescharacteristic, complemented
30#1445C2×C314yescomplemented
31#1439C2×C314yescomplemented
32C22⋊#95C2×C311yescharacteristic, complemented
33C25×C3C2311yescharacteristic
34C25×C3C2311yescharacteristic
35C22×C3×D4—24no—
36C22×C3×D4—22no—
37C22×C3×D4—22no—
38C22×C3×D4—24no—
39C22×C3×D4Q814yes—
40C23×C3×C4—24no—
41C23×C3×C4—24no—
42C23×C3×C4—28no—
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—44no—
46C23×C3×C4—44no—
47C23×C3×C4D414yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2314yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2314yes—
53C2×C3×C4⋊C4—28no—
54C2×C3×C4⋊C4—28no—
55C2×C3×C4⋊C4—44no—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—28no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—28no—
63C2×C3×C22⋊C4—44no—
64C2×C3×C22⋊C4—44no—
65C2×C3×C22⋊C4D418yes—
66C3×#95—28no—
67C3×#95—28no—
68C3×#95—28no—
69C3×#95—24no—
70C3×#95—44no—
71C23×D4C22×C311yescharacteristic
72C24×C4C22×C312yes—
73C24×C4C22×C311yescharacteristic
74C22×C4⋊C4C22×C314yes—
75C22×C22⋊C4C22×C311yescharacteristic
76C22×C22⋊C4C22×C314yes—
77C22×C22⋊C4C22×C312yes—
78C22×C22⋊C4C22×C312yes—
79#386—24no—
80#386—28no—
81#385—28no—
82C2×#95—24no—
83C2×#95C22×C314yes—
84C2×#95C22×C314yes—
85C2×#95C22×C312yes—
86C2×#95C22×C314yes—
87C2×#95C22×C314yes—
88C24×C3—24no—
89C24×C3—24no—
90C24×C3—21no—
91C24×C3—22no—
92C24×C3—24no—
93C24×C3—22no—
94C24×C3—21no—
95C24×C3—24no—
96C24×C3C2×D414yes—
97C24×C3C2×D412yes—
98C24×C3C2×Q812yes—
99C24×C3C2×Q812yes—
100C24×C3#4112yes—
101C24×C3#4114yes—
102C24×C3C2411yescharacteristic
103C2×C3×D4—28no—
104C2×C3×D4—216no—
105C2×C3×D4—48no—
106C2×C3×D4—44no—
107C2×C3×D4—44no—
108C22×C3×C4—216no—
109C22×C3×C4—28no—
110C22×C3×C4—24no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—24no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—48no—
135C22×C3×C4—416no—
136C22×C3×C4—416no—
137C22×C3×C4—48no—
138C22×C3×C4—416no—
139C22×C3×C4—48no—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D418yes—
142C22×C3×C4C2×Q812yes—
143C22×C3×C4#4118yes—
144C3×C4⋊C4—416no—
145C3×C22⋊C4—216no—
146C3×C22⋊C4—416no—
147C3×C22⋊C4—48no—
148C3×C22⋊C4—44no—
149C25C23×C311yescharacteristic
150C25C23×C311yescharacteristic
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4C3×Q814yes—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—28no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4C3×D414yes—
164C23×C4C23×C312yes—
165C23×C4C23×C314yes—
166C23×C4C23×C312yes—
167C23×C4C23×C311yescharacteristic
168C23×C4C23×C314yes—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—44no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4C3×D418yes—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—44no—
187C23×C3—24no—
188C23×C3—28no—
189C23×C3—24no—
190C23×C3—28no—
191C23×C3—28no—
192C23×C3—28no—
193C23×C3—28no—
194C23×C3—28no—
195C23×C3—22no—
196C23×C3—24no—
197C23×C3—28no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3—42no—
201C23×C3—44no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3—44no—
208C23×C3C22⋊Q814yes—
209C23×C3C22⋊Q818yes—
210C23×C3#12318yes—
211C23×C3#12314yes—
212C23×C3#12714yes—
213C23×C3C22×D411yescharacteristic
214C23×C3C22×D412yes—
215C23×C3C22×Q811yescharacteristic
216C23×C3C2×#4112yes—
217C23×C3C2×#4111yescharacteristic
218C23×C3C2×#4114yes—
219C23×C3H(5,𝔽2)14yes—
220C3×D4—416no—
221C2×C3×C4—22no—
222C2×C3×C4—24no—
223C2×C3×C4—28no—
224C2×C3×C4—24no—
225C2×C3×C4—28no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—28no—
229C2×C3×C4—28no—
230C2×C3×C4—28no—
231C2×C3×C4—28no—
232C2×C3×C4—416no—
233C2×C3×C4—432no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—44no—
237C2×C3×C4—44no—
238C2×C3×C4—41no—
239C2×C3×C4—44no—
240C2×C3×C4—44no—
241C2×C3×C4—48no—
242C2×C3×C4—44no—
243C2×C3×C4—44no—
244C2×C3×C4—44no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—44no—
251C24—24no—
252C24—24no—
253C24—21no—
254C24—22no—
255C24—24no—
256C24—22no—
257C24—21no—
258C24—24no—
259C24C2×C3×D414yes—
260C24C2×C3×D412yes—
261C24C2×C3×Q812yes—
262C24C2×C3×Q812yes—
263C24C3×#4112yes—
264C24C3×#4114yes—
265C24C24×C311yescharacteristic
266C2×D4—216no—
267C2×D4—28no—
268C2×D4—48no—
269C2×D4—44no—
270C2×D4—44no—
271C22×C4—216no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—48no—
298C22×C4—416no—
299C22×C4—416no—
300C22×C4—48no—
301C22×C4—416no—
302C22×C4—48no—
303C22×C4C2×C3×D414yes—
304C22×C4C2×C3×D418yes—
305C22×C4C2×C3×Q812yes—
306C22×C4C3×#4118yes—
307C4⋊C4—416no—
308C22⋊C4—216no—
309C22⋊C4—416no—
310C22⋊C4—48no—
311C22⋊C4—44no—
312C22×C3—28no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—216no—
316C22×C3—216no—
317C22×C3—28no—
318C22×C3—44no—
319C22×C3—42no—
320C22×C3—48no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—84no—
324C22×C3—82no—
325C22×C3#39918yes—
326C22×C3C2×C22⋊Q811yescharacteristic
327C22×C3C2×C22⋊Q812yes—
328C22×C3C2×#12311yescharacteristic
329C22×C3C2×#12312yes—
330C22×C3#52512yes—
331C22×C3#52914yes—
332C22×C3C2×#12711yescharacteristic
333C22×C3#53412yes—
334C22×C3#53814yes—
335C22×C3#54614yes—
336C22×C3#54614yes—
337C22×C3#55314yes—
338C22×C3#55414yes—
339C3×C4—44no—
340C3×C4—42no—
341C3×C4—48no—
342C3×C4—48no—
343C3×C4—82no—
344C23—24no—
345C23—28no—
346C23—24no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—28no—
351C23—28no—
352C23—24no—
353C23—22no—
354C23—28no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—41no—
360C23—42no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23C3×C22⋊Q818yes—
366C23C3×C22⋊Q814yes—
367C23C3×#12318yes—
368C23C3×#12314yes—
369C23C3×#12714yes—
370C23C22×C3×D411yescharacteristic
371C23C22×C3×D412yes—
372C23C22×C3×Q811yescharacteristic
373C23C2×C3×#4112yes—
374C23C2×C3×#4111yescharacteristic
375C23C2×C3×#4114yes—
376C23C3×H(5,𝔽2)14yes—
377D4—416no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—416no—
390C2×C4—432no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C3—28no—
409C2×C3—82no—
410C2×C3C2×#39911yescharacteristic
411C2×C3#244414yes—
412C2×C3#245014yes—
413C2×C3#253912yes—
414C2×C3#257912yes—
415C2×C3#262211yescharacteristic
416C2×C3#285311yescharacteristic
417C22—24no—
418C22—216no—
419C22—216no—
420C22—28no—
421C22—28no—
422C22—24no—
423C22—42no—
424C22—48no—
425C22—44no—
426C22—81no—
427C22—84no—
428C22—82no—
429C22—81no—
430C22C3×#39918yes—
431C22C2×C3×C22⋊Q812yes—
432C22C2×C3×C22⋊Q811yescharacteristic
433C22C2×C3×#12311yescharacteristic
434C22C2×C3×#12312yes—
435C22C3×#52512yes—
436C22C3×#52914yes—
437C22C2×C3×#12711yescharacteristic
438C22C3×#53412yes—
439C22C3×#53814yes—
440C22C3×#54614yes—
441C22C3×#54614yes—
442C22C3×#55314yes—
443C22C3×#55414yes—
444C4—44no—
445C4—42no—
446C4—48no—
447C4—48no—
448C4—82no—
449C3#1494611yescharacteristic, direct product factor, Sylow
450C2—28no—
451C2—82no—
452C2C2×C3×#39911yescharacteristic
453C2C3×#244414yes—
454C2C3×#245014yes—
455C2C3×#253912yes—
456C2C3×#257912yes—
457C2C3×#262211yescharacteristic
458C2C3×#285311yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#385C2semidirect
4. C22×C3×#95C2semidirect
5. C3×#1445C2semidirect
6. C3×#1439C2semidirect
7. C3×C22⋊#95C2semidirect
8. #14946C3direct (N × H)
28. C2×#385C2×C3semidirect
29. C22×#95C2×C3semidirect
30. #1445C2×C3semidirect
31. #1439C2×C3semidirect
32. C22⋊#95C2×C3semidirect
449. C3#14946direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3862
C2×C3×#3852
C22×C3×#952
C3×#14452
C3×#14392
C3×C22⋊#952
#149463