Finite Group Databasebeta

← C3×#15069 #10633676

Subgroups — C3×#15069 #10633676

Order 768.

2770 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
1C22×C3×#123C212yesmaximal
2C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#386C211yesmaximal, characteristic
4C2×C3×#381C212yesmaximal, complemented
5C3×#1444C214yesmaximal, complemented
6C3×#1439C214yesmaximal, complemented
7C3×C22⋊#95C211yesmaximal, characteristic, complemented
8#15069C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C24×C3×C4C2212yes—
11C24×C3×C4C2211yescharacteristic
12C2×C3×#123—28no—
13C2×C3×#123—28no—
14C2×C3×#123—28no—
15C22×C3×C4⋊C4—28no—
16C22×C3×C4⋊C4—24no—
17C22×C3×C4⋊C4C2214yes—
18C22×C3×C4⋊C4C2214yes—
19C22×C3×C22⋊C4C2214yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2212yes—
23C3×#386—24no—
24C3×#381—28no—
25C2×C3×#95C2218yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2214yes—
28C22×#123C2×C312yes—
29C23×C4⋊C4C2×C311yescharacteristic, complemented
30C2×#386C2×C311yescharacteristic
31C2×#381C2×C312yescomplemented
32#1444C2×C314yescomplemented
33#1439C2×C314yescomplemented
34C22⋊#95C2×C311yescharacteristic, complemented
35C25×C3C2311yescharacteristic
36C25×C3C2311yescharacteristic
37C22×C3×D4—22no—
38C22×C3×D4—24no—
39C22×C3×D4—24no—
40C22×C3×D4—22no—
41C22×C3×D4D414yes—
42C23×C3×C4—24no—
43C23×C3×C4—28no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—24no—
50C23×C3×C4D414yes—
51C23×C3×C4D418yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2314yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2314yes—
56C23×C3×C4C2312yes—
57C3×#123—264no—
58C2×C3×C4⋊C4—232no—
59C2×C3×C4⋊C4—216no—
60C2×C3×C4⋊C4—28no—
61C2×C3×C4⋊C4—28no—
62C2×C3×C4⋊C4—28no—
63C2×C3×C4⋊C4—28no—
64C2×C3×C22⋊C4—28no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—28no—
68C2×C3×C22⋊C4—28no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—28no—
73C2×C3×C22⋊C4D418yes—
74C3×#95—216no—
75C3×#95—24no—
76C3×#95—28no—
77C23×D4C22×C311yescharacteristic
78C24×C4C22×C312yes—
79C24×C4C22×C311yescharacteristic
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—28no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4C22×C314yes—
86C22×C4⋊C4C22×C314yes—
87C22×C22⋊C4C22×C314yes—
88C22×C22⋊C4C22×C312yes—
89C22×C22⋊C4C22×C311yescharacteristic
90C22×C22⋊C4C22×C312yes—
91#386—24no—
92#381—28no—
93C2×#95C22×C314yes—
94C2×#95C22×C318yes—
95C2×#95C22×C312yes—
96C24×C3—24no—
97C24×C3—24no—
98C24×C3—22no—
99C24×C3—22no—
100C24×C3—21no—
101C24×C3—24no—
102C24×C3—21no—
103C24×C3—24no—
104C24×C3C2×D412yes—
105C24×C3C2×D412yes—
106C24×C3C2×D414yes—
107C24×C3C2×D414yes—
108C24×C3C2×D412yes—
109C24×C3#4112yes—
110C24×C3C2411yescharacteristic
111C2×C3×D4—216no—
112C2×C3×D4—28no—
113C2×C3×D4—44no—
114C2×C3×D4—48no—
115C2×C3×D4—44no—
116C22×C3×C4—232no—
117C22×C3×C4—28no—
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4—216no—
121C22×C3×C4—216no—
122C22×C3×C4—216no—
123C22×C3×C4—216no—
124C22×C3×C4—24no—
125C22×C3×C4—21no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—28no—
135C22×C3×C4—24no—
136C22×C3×C4—21no—
137C22×C3×C4—28no—
138C22×C3×C4—24no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—48no—
143C22×C3×C4—48no—
144C22×C3×C4—48no—
145C22×C3×C4—48no—
146C22×C3×C4C2×D418yes—
147C22×C3×C4C2×D418yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D414yes—
150C22×C3×C4#4118yes—
151C3×C4⋊C4—232no—
152C3×C4⋊C4—232no—
153C3×C22⋊C4—232no—
154C3×C22⋊C4—216no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—48no—
157C25C23×C311yescharacteristic
158C25C23×C311yescharacteristic
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C3×D414yes—
164C23×C4—24no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4C3×D418yes—
173C23×C4C3×D414yes—
174C23×C4C23×C312yes—
175C23×C4C23×C314yes—
176C23×C4C23×C314yes—
177C23×C4C23×C311yescharacteristic
178C23×C4C23×C312yes—
179#123—264no—
180C2×C4⋊C4—232no—
181C2×C4⋊C4—216no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4C3×D418yes—
196#95—216no—
197#95—24no—
198#95—28no—
199C23×C3—24no—
200C23×C3—28no—
201C23×C3—28no—
202C23×C3—24no—
203C23×C3—28no—
204C23×C3—216no—
205C23×C3—28no—
206C23×C3—24no—
207C23×C3—22no—
208C23×C3—28no—
209C23×C3—41no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—44no—
213C23×C3—41no—
214C23×C3—44no—
215C23×C3—42no—
216C23×C3—48no—
217C23×C3—44no—
218C23×C3#12018yes—
219C23×C3#12014yes—
220C23×C3#12118yes—
221C23×C3#12114yes—
222C23×C3#12714yes—
223C23×C3C22×D412yes—
224C23×C3C22×D411yescharacteristic
225C23×C3C22×D411yescharacteristic
226C23×C3C22×D412yes—
227C23×C3C2×#4111yescharacteristic
228C23×C3C2×#4114yes—
229C23×C32−1+414yes—
230C3×D4—416no—
231C2×C3×C4—232no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—216no—
235C2×C3×C4—28no—
236C2×C3×C4—28no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—28no—
240C2×C3×C4—22no—
241C2×C3×C4—216no—
242C2×C3×C4—44no—
243C2×C3×C4—41no—
244C2×C3×C4—44no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—41no—
250C2×C3×C4—44no—
251C2×C3×C4—48no—
252C2×C3×C4—432no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—48no—
257C2×C3×C4#12318yescomplemented
258C24—24no—
259C24—24no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—24no—
264C24—21no—
265C24—24no—
266C24C2×C3×D412yes—
267C24C2×C3×D412yes—
268C24C2×C3×D414yes—
269C24C2×C3×D414yes—
270C24C2×C3×D412yes—
271C24C3×#4112yes—
272C24C24×C311yescharacteristic
273C2×D4—216no—
274C2×D4—28no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—44no—
278C22×C4—232no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—216no—
283C22×C4—216no—
284C22×C4—216no—
285C22×C4—216no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4C2×C3×D418yes—
309C22×C4C2×C3×D418yes—
310C22×C4C2×C3×D412yes—
311C22×C4C2×C3×D414yes—
312C22×C4C3×#4118yes—
313C4⋊C4—232no—
314C4⋊C4—232no—
315C22⋊C4—232no—
316C22⋊C4—216no—
317C22⋊C4—44no—
318C22⋊C4—48no—
319C22×C3—24no—
320C22×C3—24no—
321C22×C3—216no—
322C22×C3—216no—
323C22×C3—28no—
324C22×C3—28no—
325C22×C3—42no—
326C22×C3—48no—
327C22×C3—44no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—84no—
331C22×C3—82no—
332C22×C3#39218yes—
333C22×C3C2×#12011yescharacteristic
334C22×C3C2×#12012yes—
335C22×C3C2×#12112yes—
336C22×C3C2×#12111yescharacteristic
337C22×C3C2×#12312yes—
338C22×C3#52514yes—
339C22×C3C2×#12711yescharacteristic
340C22×C3#53614yes—
341C22×C3#53612yes—
342C22×C3#54714yes—
343C22×C3#54718yes—
344C22×C3#55214yes—
345C3×C4—28no—
346C3×C4—48no—
347C3×C4—42no—
348C3×C4—48no—
349C3×C4—82no—
350C23—24no—
351C23—28no—
352C23—28no—
353C23—24no—
354C23—28no—
355C23—216no—
356C23—28no—
357C23—24no—
358C23—22no—
359C23—28no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—41no—
365C23—44no—
366C23—42no—
367C23—48no—
368C23—44no—
369C23C3×#12018yes—
370C23C3×#12014yes—
371C23C3×#12118yes—
372C23C3×#12114yes—
373C23C3×#12714yes—
374C23C22×C3×D412yes—
375C23C22×C3×D411yescharacteristic
376C23C22×C3×D411yescharacteristic
377C23C22×C3×D412yes—
378C23C2×C3×#4111yescharacteristic
379C23C2×C3×#4114yes—
380C23C3×2−1+414yes—
381D4—416no—
382C2×C4—232no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—216no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—22no—
392C2×C4—216no—
393C2×C4—432no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—41no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4C3×#12318yescomplemented
409C2×C3—28no—
410C2×C3—82no—
411C2×C3C2×#39211yescharacteristic
412C2×C3#240614yes—
413C2×C3#241314yes—
414C2×C3#248612yes—
415C2×C3#250112yes—
416C2×C3#261411yescharacteristic
417C2×C3#284911yescharacteristic
418C22—216no—
419C22—216no—
420C22—28no—
421C22—28no—
422C22—24no—
423C22—24no—
424C22—42no—
425C22—48no—
426C22—44no—
427C22—84no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22C3×#39218yes—
432C22C2×C3×#12012yes—
433C22C2×C3×#12011yescharacteristic
434C22C2×C3×#12112yes—
435C22C2×C3×#12111yescharacteristic
436C22C2×C3×#12312yes—
437C22C3×#52514yes—
438C22C2×C3×#12711yescharacteristic
439C22C3×#53614yes—
440C22C3×#53612yes—
441C22C3×#54714yes—
442C22C3×#54718yes—
443C22C3×#55214yes—
444C4—28no—
445C4—48no—
446C4—42no—
447C4—48no—
448C4—82no—
449C3#1506911yescharacteristic, direct product factor, Sylow
450C2—28no—
451C2—82no—
452C2C2×C3×#39211yescharacteristic
453C2C3×#240614yes—
454C2C3×#241314yes—
455C2C3×#248612yes—
456C2C3×#250112yes—
457C2C3×#261411yescharacteristic
458C2C3×#284911yescharacteristic
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
2. C23×C3×C4⋊C4C2semidirect
4. C2×C3×#381C2semidirect
5. C3×#1444C2semidirect
6. C3×#1439C2semidirect
7. C3×C22⋊#95C2semidirect
8. #15069C3direct (N × H)
29. C23×C4⋊C4C2×C3semidirect
31. C2×#381C2×C3semidirect
32. #1444C2×C3semidirect
33. #1439C2×C3semidirect
34. C22⋊#95C2×C3semidirect
257. C2×C3×C4#123semidirect
408. C2×C4C3×#123semidirect
449. C3#15069direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C23×C3×C4⋊C42
C2×C3×#3862
C2×C3×#3812
C3×#14442
C3×#14392
C3×C22⋊#952
#150693