Finite Group Databasebeta

← C3×#29341 #10647948

Subgroups — C3×#29341 #10647948

Order 768.

730 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2621C211yesmaximal, characteristic
2C3×#2316C211yesmaximal, characteristic
3C3×#2101C212yesmaximal, complemented
4C3×#2043C212yesmaximal, complemented
5C3×#1935C212yesmaximal, complemented
6C3×#1933C212yesmaximal, complemented
7C3×#1873C212yesmaximal
8C3×#1872C212yesmaximal
9C3×#1750C211yesmaximal, characteristic
10#29341C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×C3×#103C2212yes—
17C2×C3×#103C2212yes—
18C2×C3×#103C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×C22⋊C8C2212yes—
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#398—22no—
24C3×#397—22no—
25C3×#391C2212yes—
26C3×#389—22no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×#382C2211yescharacteristic
30C3×#380—22no—
31C3×#340C2212yes—
32C3×#340C2212yes—
33C3×#340C2212yes—
34C3×#340C2212yes—
35C3×#336C2212yes—
36#2621C2×C311yescharacteristic
37#2316C2×C311yescharacteristic
38#2101C2×C312yescomplemented
39#2043C2×C312yescomplemented
40#1935C2×C312yescomplemented
41#1933C2×C312yescomplemented
42#1873C2×C312yes—
43#1872C2×C312yes—
44#1750C2×C311yescharacteristic
45C22×C3×Q8C2312yes—
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C22×C3×C8C2312yes—
49C22×C3×C8C2312yes—
50C3×C4⋊Q8—41no—
51C3×C4⋊Q8—41no—
52C3×C22⋊Q8—24no—
53C3×C22⋊Q8—24no—
54C3×#117—41no—
55C3×#117—41no—
56C3×#117—41no—
57C3×#117—41no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42—21no—
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C3×#103—24no—
75C3×#103—24no—
76C3×#103—42no—
77C3×#103—42no—
78C3×C22⋊C8—24no—
79C3×#97—41no—
80C3×C4×C8—41no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—24no—
85C2×C4⋊Q8C22×C311yescharacteristic
86C2×C22⋊Q8C22×C312yes—
87C2×#117C22×C311yescharacteristic
88C2×#117C22×C311yescharacteristic
89C22×C4⋊C4C22×C311yescharacteristic
90C2×#103C22×C312yes—
91C2×#103C22×C312yes—
92C2×#103C22×C312yes—
93C2×#103C22×C312yes—
94C2×C22⋊C8C22×C312yes—
95C2×#97C22×C311yescharacteristic
96C2×C4×C8C22×C311yescharacteristic
97#398—22no—
98#397—22no—
99#391C22×C312yes—
100#389—22no—
101#389C22×C312yes—
102#389C22×C312yes—
103#382C22×C311yescharacteristic
104#380—22no—
105#340C22×C312yes—
106#340C22×C312yes—
107#340C22×C312yes—
108#340C22×C312yes—
109#336C22×C312yes—
110C24×C3C2×D411yescharacteristic
111C2×C3×Q8—22no—
112C2×C3×Q8—22no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—82no—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4114yes—
130C22×C3×C4C2411yescharacteristic
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—42no—
144C3×C4⋊C4—44no—
145C3×C4⋊C4—44no—
146C3×C4⋊C4—82no—
147C3×C4⋊C4—82no—
148C3×C4⋊C4#4114yes—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—42no—
151C3×C22⋊C4—44no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42—21no—
156C3×C42—42no—
157C22×Q8C23×C312yes—
158C23×C4—21no—
159C23×C4C23×C311yescharacteristic
160C22×C8C23×C312yes—
161C22×C8C23×C312yes—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C3×D412yes—
177C2×C4⋊C4C23×C312yes—
178C2×C4⋊C4C23×C312yes—
179C2×C4⋊C4C23×C312yes—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C3×D412yes—
183C2×C42—21no—
184C2×C42C23×C311yescharacteristic
185C2×C42C23×C311yescharacteristic
186#103—24no—
187#103—24no—
188#103—42no—
189#103—42no—
190C22⋊C8—24no—
191#97—41no—
192C4×C8—41no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—24no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3#12412yes—
202C23×C3C22×D411yescharacteristic
203C3×Q8—42no—
204C3×Q8—42no—
205C3×Q8—42no—
206C3×Q8—42no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—24no—
211C2×C3×C4—24no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—84no—
230C2×C3×C4#12412yes—
231C2×C3×C4C2×#4112yes—
232C2×C3×C4C2×#4112yes—
233C2×C3×C4H(5,𝔽2)11yescharacteristic
234C2×C3×C4H(5,𝔽2)11yescharacteristic
235C3×C8—42no—
236C3×C8—42no—
237C24C2×C3×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—82no—
253C22×C4C2×C3×D412yes—
254C22×C4C2×C3×D412yes—
255C22×C4C2×C3×D411yescharacteristic
256C22×C4C3×#4114yes—
257C22×C4C24×C311yescharacteristic
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4C3×#4114yes—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—42no—
284C22×C3—22no—
285C22×C3—24no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3#52511yescharacteristic
290C22×C3C2×#12411yescharacteristic
291C22×C3#53411yescharacteristic
292C22×C3#57712yes—
293C22×C3#57812yes—
294C3×C4—22no—
295C3×C4—44no—
296C3×C4—42no—
297C3×C4—42no—
298C3×C4—82no—
299C3×C4—82no—
300C3×C4—82no—
301C3×C4#55312yes—
302C3×C4#55312yes—
303C23—22no—
304C23—22no—
305C23—21no—
306C23—21no—
307C23C3×#12412yes—
308C23C22×C3×D411yescharacteristic
309Q8—42no—
310Q8—42no—
311Q8—42no—
312Q8—42no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—84no—
336C2×C4C3×#12412yes—
337C2×C4C2×C3×#4112yes—
338C2×C4C2×C3×#4112yes—
339C2×C4C3×H(5,𝔽2)11yescharacteristic
340C2×C4C3×H(5,𝔽2)11yescharacteristic
341C8—42no—
342C8—42no—
343C2×C3—42no—
344C2×C3#259111yescharacteristic
345C2×C3#312311yescharacteristic
346C2×C3#312711yescharacteristic
347C2×C3#313712yes—
348C2×C3#320112yes—
349C22—24no—
350C22—22no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22C3×#52511yescharacteristic
355C22C2×C3×#12411yescharacteristic
356C22C3×#53411yescharacteristic
357C22C3×#57712yes—
358C22C3×#57812yes—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—44no—
363C4—82no—
364C4—82no—
365C4—82no—
366C4C3×#55312yes—
367C4C3×#55312yes—
368C3#2934111yescharacteristic, direct product factor, Sylow
369C2—42no—
370C2C3×#259111yescharacteristic
371C2C3×#312311yescharacteristic
372C2C3×#312711yescharacteristic
373C2C3×#313712yes—
374C2C3×#320112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2101C2semidirect
4. C3×#2043C2semidirect
5. C3×#1935C2semidirect
6. C3×#1933C2semidirect
10. #29341C3direct (N × H)
38. #2101C2×C3semidirect
39. #2043C2×C3semidirect
40. #1935C2×C3semidirect
41. #1933C2×C3semidirect
368. C3#29341direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26212
C3×#23162
C3×#21012
C3×#20432
C3×#19352
C3×#19332
C3×#18732
C3×#18722
C3×#17502
#293413