Finite Group Databasebeta

← C3×#10980 #10629587

Subgroups — C3×#10980 #10629587

Order 768.

1946 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C211yesmaximal, characteristic
2C22×C3×#99C214yesmaximal, complemented
3C3×C22⋊#95C212yesmaximal, complemented
4#10980C311yesmaximal, characteristic, direct product factor, Sylow
5C23×C3×D4C411yescharacteristic, complemented
6C23×C3×D4C2212yes—
7C24×C3×C4C2211yescharacteristic
8C2×C3×#121—28no—
9C2×C3×#121—24no—
10C22×C3×C4⋊C4C411yescharacteristic, complemented
11C22×C3×C22⋊C4C412yescomplemented
12C22×C3×C22⋊C4C2214yes—
13C2×C3×#99—28no—
14C2×C3×#99—216no—
15C2×C3×#95—24no—
16C22×#121C2×C311yescharacteristic
17C22×#99C2×C314yescomplemented
18C22⋊#95C2×C312yescomplemented
19C25×C3C2×C412yes—
20C25×C3C2311yescharacteristic
21C22×C3×D4—24no—
22C22×C3×D4—28no—
23C22×C3×D4—24no—
24C22×C3×D4—24no—
25C22×C3×D4—28no—
26C22×C3×D4—22no—
27C22×C3×D4—44no—
28C22×C3×D4D418yes—
29C23×C3×C4—24no—
30C23×C3×C4—24no—
31C23×C3×C4—44no—
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C412yes—
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4D414yes—
36C3×#121—232no—
37C2×C3×C4⋊C4—24no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—24no—
41C2×C3×C22⋊C4—24no—
42C2×C3×C22⋊C4—28no—
43C2×C3×C22⋊C4—28no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C22⋊C4D418yescomplemented
46C3×#99—232no—
47C3×#95—48no—
48C23×D4C3×C411yescharacteristic, complemented
49C23×D4C22×C312yes—
50C24×C4C22×C311yescharacteristic
51C2×#121—28no—
52C2×#121—24no—
53C22×C4⋊C4C3×C411yescharacteristic, complemented
54C22×C22⋊C4C3×C412yescomplemented
55C22×C22⋊C4C22×C314yes—
56C2×#99—28no—
57C2×#99—216no—
58C2×#95—24no—
59C24×C3—22no—
60C24×C3—22no—
61C24×C3—28no—
62C24×C3—42no—
63C24×C3—42no—
64C24×C3—44no—
65C24×C3—44no—
66C24×C3C22⋊C414yes—
67C24×C3C22×C411yescharacteristic
68C24×C3C2×D412yes—
69C24×C3C2×D414yes—
70C24×C3C2×D414yes—
71C24×C3#4114yes—
72C2×C3×D4—28no—
73C2×C3×D4—28no—
74C2×C3×D4—216no—
75C2×C3×D4—216no—
76C2×C3×D4—28no—
77C2×C3×D4—416no—
78C2×C3×D4—44no—
79C2×C3×D4—44no—
80C2×C3×D4—48no—
81C2×C3×D4—48no—
82C2×C3×D4—44no—
83C2×C3×D4—44no—
84C2×C3×D4—44no—
85C2×C3×D4—44no—
86C22×C3×C4—22no—
87C22×C3×C4—24no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—28no—
92C22×C3×C4—28no—
93C22×C3×C4—28no—
94C22×C3×C4—28no—
95C22×C3×C4—42no—
96C22×C3×C4—44no—
97C22×C3×C4—416no—
98C22×C3×C4—48no—
99C22×C3×C4—48no—
100C22×C3×C4—44no—
101C22×C3×C4—44no—
102C22×C3×C4C22⋊C414yes—
103C22×C3×C4C22⋊C412yes—
104C22×C3×C4C22⋊C412yes—
105C3×C4⋊C4—28no—
106C3×C22⋊C4—216no—
107C3×C22⋊C4—216no—
108C3×C22⋊C4—48no—
109C25C2×C3×C412yes—
110C25C23×C311yescharacteristic
111C22×D4—24no—
112C22×D4—24no—
113C22×D4—24no—
114C22×D4—28no—
115C22×D4—28no—
116C22×D4—22no—
117C22×D4—44no—
118C22×D4C3×D418yes—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—44no—
122C23×C4C2×C3×C412yes—
123C23×C4C2×C3×C411yescharacteristic
124C23×C4C2×C3×C411yescharacteristic
125C23×C4C3×D414yes—
126#121—232no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4C3×D418yescomplemented
136#99—232no—
137#95—48no—
138C23×C3—24no—
139C23×C3—22no—
140C23×C3—24no—
141C23×C3—22no—
142C23×C3—24no—
143C23×C3—28no—
144C23×C3—44no—
145C23×C3—48no—
146C23×C3—44no—
147C23×C3—48no—
148C23×C3—44no—
149C23×C3—44no—
150C23×C3—44no—
151C23×C3—44no—
152C23×C3—44no—
153C23×C3—44no—
154C23×C3—44no—
155C23×C3—44no—
156C23×C3—816no—
157C23×C3C2×C22⋊C412yes—
158C23×C3C2×C22⋊C411yescharacteristic
159C23×C3C4×D414yes—
160C23×C3#12014yes—
161C23×C3#12014yes—
162C23×C3#12114yes—
163C23×C3#12118yes—
164C23×C3#12314yes—
165C23×C3#12412yes—
166C23×C3#12712yes—
167C3×D4—416no—
168C3×D4—416no—
169C3×D4—416no—
170C2×C3×C4—28no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—24no—
175C2×C3×C4—28no—
176C2×C3×C4—24no—
177C2×C3×C4—416no—
178C2×C3×C4—44no—
179C2×C3×C4—44no—
180C2×C3×C4—44no—
181C2×C3×C4—416no—
182C2×C3×C4—48no—
183C2×C3×C4—88no—
184C2×C3×C4—84no—
185C2×C3×C4—84no—
186C2×C3×C4#9914yescomplemented
187C24—22no—
188C24—22no—
189C24—28no—
190C24—42no—
191C24—42no—
192C24—44no—
193C24—44no—
194C24C3×C22⋊C414yes—
195C24C22×C3×C411yescharacteristic
196C24C2×C3×D412yes—
197C24C2×C3×D414yes—
198C24C2×C3×D414yes—
199C24C3×#4114yes—
200C2×D4—28no—
201C2×D4—28no—
202C2×D4—216no—
203C2×D4—216no—
204C2×D4—28no—
205C2×D4—416no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—416no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4C3×C22⋊C412yes—
231C22×C4C3×C22⋊C414yes—
232C22×C4C3×C22⋊C412yes—
233C4⋊C4—28no—
234C22⋊C4—216no—
235C22⋊C4—216no—
236C22⋊C4—48no—
237C22×C3—28no—
238C22×C3—24no—
239C22×C3—28no—
240C22×C3—24no—
241C22×C3—22no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—22no—
245C22×C3—44no—
246C22×C3—48no—
247C22×C3—44no—
248C22×C3—44no—
249C22×C3—44no—
250C22×C3—44no—
251C22×C3—88no—
252C22×C3—88no—
253C22×C3—88no—
254C22×C3#37911yescharacteristic
255C22×C3#38614yes—
256C22×C3#39012yes—
257C22×C3#39214yes—
258C22×C3#39414yes—
259C22×C3C2×#9912yes—
260C22×C3#41012yes—
261C3×C4—24no—
262C3×C4—44no—
263C3×C4—42no—
264C3×C4—88no—
265C23—24no—
266C23—22no—
267C23—24no—
268C23—22no—
269C23—24no—
270C23—28no—
271C23—44no—
272C23—44no—
273C23—48no—
274C23—48no—
275C23—44no—
276C23—44no—
277C23—44no—
278C23—44no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—816no—
284C23C2×C3×C22⋊C412yes—
285C23C2×C3×C22⋊C411yescharacteristic
286C23C3×C4×D414yes—
287C23C3×#12014yes—
288C23C3×#12014yes—
289C23C3×#12114yes—
290C23C3×#12118yes—
291C23C3×#12314yes—
292C23C3×#12412yes—
293C23C3×#12712yes—
294D4—416no—
295D4—416no—
296D4—416no—
297C2×C4—28no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—28no—
303C2×C4—24no—
304C2×C4—416no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—416no—
309C2×C4—48no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—88no—
313C2×C4C3×#9914yescomplemented
314C2×C3—24no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—84no—
318C2×C3#143811yescharacteristic
319C2×C3#178111yescharacteristic
320C2×C3#178211yescharacteristic
321C2×C3#196414yes—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—22no—
326C22—28no—
327C22—24no—
328C22—28no—
329C22—24no—
330C22—44no—
331C22—48no—
332C22—44no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—88no—
337C22—88no—
338C22—88no—
339C22C3×#37911yescharacteristic
340C22C3×#38614yes—
341C22C3×#39012yes—
342C22C3×#39214yes—
343C22C3×#39414yes—
344C22C2×C3×#9912yes—
345C22C3×#41012yes—
346C4—24no—
347C4—42no—
348C4—44no—
349C4—88no—
350C3#1098011yescharacteristic, direct product factor, Sylow
351C2—24no—
352C2—24no—
353C2—24no—
354C2—84no—
355C2C3×#143811yescharacteristic
356C2C3×#178111yescharacteristic
357C2C3×#178211yescharacteristic
358C2C3×#196414yes—
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC24C22×C3×C4
Frattini, cosocleC25C23×C3
prime radicalC22×C3×#121C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#99C2semidirect
3. C3×C22⋊#95C2semidirect
4. #10980C3direct (N × H)
5. C23×C3×D4C4semidirect
10. C22×C3×C4⋊C4C4semidirect
11. C22×C3×C22⋊C4C4semidirect
17. C22×#99C2×C3semidirect
18. C22⋊#95C2×C3semidirect
45. C2×C3×C22⋊C4D4semidirect
48. C23×D4C3×C4semidirect
53. C22×C4⋊C4C3×C4semidirect
54. C22×C22⋊C4C3×C4semidirect
135. C2×C22⋊C4C3×D4semidirect
186. C2×C3×C4#99semidirect
313. C2×C4C3×#99semidirect
350. C3#10980direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C22×C3×#992
C3×C22⋊#952
#109803