Finite Group Databasebeta

← C3×#32978 #10651585

Subgroups — C3×#32978 #10651585

Order 768.

1812 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#480C212yesmaximal, complemented
3C3×#2847C211yesmaximal, characteristic
4C3×#1874C214yesmaximal, complemented
5C3×#1852C214yesmaximal, complemented
6C3×#1823C212yesmaximal, complemented
7C3×#1779C211yesmaximal, characteristic, complemented
8#32978C311yesmaximal, characteristic, direct product factor, Sylow
9C24×C3×C4—21no—
10C24×C3×C4C2211yescharacteristic
11C2×C3×#121C2212yes—
12C2×C3×#121C2211yescharacteristic
13C2×C3×#120—21no—
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4—22no—
16C22×C3×C4⋊C4—28no—
17C22×C3×C4⋊C4C2214yes—
18C22×C3×C4⋊C4C2212yes—
19C3×#480—28no—
20C2×C3×#107C2214yescomplemented
21C2×C3×#102C2214yescomplemented
22C2×C3×C22⋊C8C2212yes—
23C2×C3×C22⋊C8C2214yes—
24C3×#386—22no—
25C3×#386—21no—
26C3×#380C2211yescharacteristic
27C3×#380C2212yes—
28C3×#340C2218yescomplemented
29C23×C4⋊C4C2×C311yescharacteristic, complemented
30C2×#480C2×C312yescomplemented
31#2847C2×C311yescharacteristic
32#1874C2×C314yescomplemented
33#1852C2×C314yescomplemented
34#1823C2×C312yescomplemented
35#1779C2×C311yescharacteristic, complemented
36C25×C3D411yescharacteristic
37C22×C3×D4—21no—
38C22×C3×D4—22no—
39C22×C3×D4C2311yescharacteristic
40C23×C3×C4—24no—
41C23×C3×C4—28no—
42C23×C3×C4—28no—
43C23×C3×C4—24no—
44C23×C3×C4—22no—
45C23×C3×C4—44no—
46C23×C3×C4—44no—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C22×C3×C8C2314yes—
52C3×#121—24no—
53C3×#121—28no—
54C3×#120—44no—
55C2×C3×C4⋊C4—28no—
56C2×C3×C4⋊C4—216no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—216no—
61C2×C3×C4⋊C4D418yescomplemented
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2314yes—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C3×#107—28no—
70C3×#102—28no—
71C3×C22⋊C8—24no—
72C3×C22⋊C8—28no—
73C3×#95—21no—
74C3×#95—22no—
75C24×C4—21no—
76C24×C4C22×C311yescharacteristic
77C2×#121C22×C311yescharacteristic
78C2×#121C22×C312yes—
79C2×#120—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4C22×C314yes—
84C22×C4⋊C4C22×C312yes—
85#480—28no—
86C2×#107C22×C314yescomplemented
87C2×#102C22×C314yescomplemented
88C2×C22⋊C8C22×C312yes—
89C2×C22⋊C8C22×C314yes—
90#386—22no—
91#386—21no—
92#380C22×C311yescharacteristic
93#380C22×C312yes—
94#340C22×C318yescomplemented
95C24×C3—22no—
96C24×C3—24no—
97C24×C3—22no—
98C24×C3—24no—
99C24×C3—21no—
100C24×C3D814yes—
101C24×C3C2×D411yescharacteristic
102C24×C3C2×D412yes—
103C2×C3×D4—22no—
104C2×C3×D4—24no—
105C2×C3×D4—42no—
106C2×C3×D4—44no—
107C2×C3×D4—42no—
108C2×C3×D4—44no—
109C2×C3×D4—42no—
110C2×C3×D4—44no—
111C22×C3×C4—232no—
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—216no—
121C22×C3×C4—28no—
122C22×C3×C4—24no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—48no—
126C22×C3×C4—432no—
127C22×C3×C4—48no—
128C22×C3×C4—44no—
129C22×C3×C4—44no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D414yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4#4118yes—
134C22×C3×C4C2411yescharacteristic
135C2×C3×C8—24no—
136C2×C3×C8—28no—
137C3×C4⋊C4—216no—
138C3×C4⋊C4—28no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4C2×D418yes—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—44no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—44no—
147C25C3×D411yescharacteristic
148C22×D4—21no—
149C22×D4—22no—
150C22×D4C23×C311yescharacteristic
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4C3×D411yescharacteristic
159C23×C4C3×D412yes—
160C23×C4C23×C311yescharacteristic
161C23×C4C23×C312yes—
162C22×C8C23×C314yes—
163#121—24no—
164#121—28no—
165#120—44no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—216no—
172C2×C4⋊C4C3×D418yescomplemented
173C2×C4⋊C4C23×C312yes—
174C2×C4⋊C4C23×C311yescharacteristic
175C2×C4⋊C4C23×C314yes—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180#107—28no—
181#102—28no—
182C22⋊C8—24no—
183C22⋊C8—28no—
184#95—21no—
185#95—22no—
186C23×C3—28no—
187C23×C3—216no—
188C23×C3—28no—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—28no—
196C23×C3—216no—
197C23×C3—41no—
198C23×C3—42no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—42no—
202C23×C3#12012yes—
203C23×C3#12314yes—
204C23×C3C2×D812yes—
205C23×C3C2×D814yes—
206C23×C3AGL(1,ℤ/8)12yes—
207C23×C3#13714yes—
208C23×C3C22×D411yescharacteristic
209C3×D4—44no—
210C3×D4—84no—
211C3×D4—88no—
212C2×C3×C4—216no—
213C2×C3×C4—28no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—28no—
218C2×C3×C4—216no—
219C2×C3×C4—28no—
220C2×C3×C4—432no—
221C2×C3×C4—44no—
222C2×C3×C4—44no—
223C2×C3×C4—416no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4#12012yes—
231C2×C3×C4#12314yes—
232C2×C3×C4C22×D412yes—
233C2×C3×C4C2×#4114yes—
234C3×C8—44no—
235C24—22no—
236C24—24no—
237C24—22no—
238C24—24no—
239C24—21no—
240C24C3×D814yes—
241C24C2×C3×D411yescharacteristic
242C24C2×C3×D412yes—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—44no—
251C22×C4—232no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—216no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—48no—
266C22×C4—432no—
267C22×C4—48no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4C2×C3×D411yescharacteristic
271C22×C4C2×C3×D414yes—
272C22×C4C2×C3×D412yes—
273C22×C4C3×#4118yes—
274C22×C4C24×C311yescharacteristic
275C2×C8—24no—
276C2×C8—28no—
277C4⋊C4—216no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4C2×C3×D418yes—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22×C3—232no—
288C22×C3—24no—
289C22×C3—24no—
290C22×C3—24no—
291C22×C3—24no—
292C22×C3—24no—
293C22×C3—22no—
294C22×C3—24no—
295C22×C3—24no—
296C22×C3—22no—
297C22×C3—24no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3#44714yes—
304C22×C3#48018yescomplemented
305C22×C3C2×#12011yescharacteristic
306C22×C3C2×#12312yes—
307C22×C3C22×D811yescharacteristic
308C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C3C2×#13712yes—
310C3×C4—22no—
311C3×C4—24no—
312C3×C4—48no—
313C3×C4—82no—
314C3×C4—81no—
315C3×C4#52514yes—
316C23—28no—
317C23—28no—
318C23—216no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—28no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—216no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23C3×#12012yes—
333C23C3×#12314yes—
334C23C2×C3×D812yes—
335C23C2×C3×D814yes—
336C23C3×AGL(1,ℤ/8)12yes—
337C23C3×#13714yes—
338C23C22×C3×D411yescharacteristic
339D4—44no—
340D4—84no—
341D4—88no—
342C2×C4—216no—
343C2×C4—28no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—216no—
349C2×C4—28no—
350C2×C4—432no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—416no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4C3×#12012yes—
361C2×C4C3×#12314yes—
362C2×C4C22×C3×D412yes—
363C2×C4C2×C3×#4114yes—
364C8—44no—
365C2×C3—24no—
366C2×C3—28no—
367C2×C3—161no—
368C2×C3#284911yescharacteristic
369C2×C3C2×#44711yescharacteristic
370C2×C3#300411yescharacteristic
371C2×C3C2×#48012yes—
372C2×C3#309212yes—
373C22—232no—
374C22—24no—
375C22—22no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—22no—
383C22—24no—
384C22—81no—
385C22—81no—
386C22—82no—
387C22—82no—
388C22—81no—
389C22C3×#44714yes—
390C22C3×#48018yescomplemented
391C22C2×C3×#12011yescharacteristic
392C22C2×C3×#12312yes—
393C22C22×C3×D811yescharacteristic
394C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
395C22C2×C3×#13712yes—
396C4—22no—
397C4—24no—
398C4—48no—
399C4—81no—
400C4—82no—
401C4C3×#52514yes—
402C3#3297811yescharacteristic, direct product factor, Sylow
403C2—24no—
404C2—28no—
405C2—161no—
406C2C3×#284911yescharacteristic
407C2C2×C3×#44711yescharacteristic
408C2C3×#300411yescharacteristic
409C2C2×C3×#48012yes—
410C2C3×#309212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×C4⋊C4C2semidirect
2. C2×C3×#480C2semidirect
4. C3×#1874C2semidirect
5. C3×#1852C2semidirect
6. C3×#1823C2semidirect
7. C3×#1779C2semidirect
8. #32978C3direct (N × H)
20. C2×C3×#107C22semidirect
21. C2×C3×#102C22semidirect
28. C3×#340C22semidirect
29. C23×C4⋊C4C2×C3semidirect
30. C2×#480C2×C3semidirect
32. #1874C2×C3semidirect
33. #1852C2×C3semidirect
34. #1823C2×C3semidirect
35. #1779C2×C3semidirect
61. C2×C3×C4⋊C4D4semidirect
86. C2×#107C22×C3semidirect
87. C2×#102C22×C3semidirect
94. #340C22×C3semidirect
172. C2×C4⋊C4C3×D4semidirect
304. C22×C3#480semidirect
390. C22C3×#480semidirect
402. C3#32978direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×C3×#4802
C3×#28472
C3×#18742
C3×#18522
C3×#18232
C3×#17792
#329783