Finite Group Databasebeta

← C3×#27614 #10646221

Subgroups — C3×#27614 #10646221

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C3×#2285C211yesmaximal, characteristic
3C3×#2100C212yesmaximal, complemented
4C3×#2083C212yesmaximal, complemented
5C3×#1921C214yesmaximal, complemented
6C3×#1852C212yesmaximal
7C3×#1824C212yesmaximal
8C3×#1817C211yesmaximal, characteristic
9#27614C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×#117—21no—
11C22×C3×C4⋊C4—21no—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×C3×#107C2214yes—
16C2×C3×C22⋊C8C2212yes—
17C3×#389—21no—
18C3×#389—21no—
19C3×#389C2212yes—
20C3×#389C2212yes—
21C3×#384C2211yescharacteristic
22C3×#384C2211yescharacteristic
23C3×#382—21no—
24C3×#380—21no—
25C3×#380C2212yes—
26C3×C4×C4⋊C4—21no—
27C3×C4×C4⋊C4C2212yes—
28C2×C3×#95—21no—
29C3×#340C2214yes—
30C3×#340C2214yes—
31C3×#340C2214yes—
32C3×#336C2212yes—
33C3×#336C2212yes—
34C2×#389C2×C311yescharacteristic
35#2285C2×C311yescharacteristic
36#2100C2×C312yescomplemented
37#2083C2×C312yescomplemented
38#1921C2×C314yescomplemented
39#1852C2×C312yes—
40#1824C2×C312yes—
41#1817C2×C311yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C23×C3×C4D411yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C22×C3×C8C2314yes—
48C3×#117—44no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—44no—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C22⋊C4—22no—
65C2×C3×C42—24no—
66C2×C3×C42—22no—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C3×#107—28no—
72C3×C22⋊C8—24no—
73C3×#95—22no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C22×C311yescharacteristic
80C22×C4⋊C4C22×C311yescharacteristic
81C22×C42C22×C311yescharacteristic
82C2×#107C22×C314yes—
83C2×C22⋊C8C22×C312yes—
84#389—21no—
85#389—21no—
86#389C22×C312yes—
87#389C22×C312yes—
88#384C22×C311yescharacteristic
89#384C22×C311yescharacteristic
90#382—21no—
91#380—21no—
92#380C22×C312yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C22×C312yes—
95C2×#95—21no—
96#340C22×C314yes—
97#340C22×C314yes—
98#340C22×C314yes—
99#336C22×C312yes—
100#336C22×C312yes—
101C24×C3C2×D411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—42no—
119C22×C3×C4Q1614yes—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4114yes—
126C22×C3×C4C2411yescharacteristic
127C2×C3×C8—24no—
128C2×C3×C8—24no—
129C2×C3×C8—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—42no—
136C3×C4⋊C4—44no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4—42no—
141C3×C4⋊C4—44no—
142C3×C4⋊C4#4114yes—
143C3×C22⋊C4—44no—
144C3×C42—22no—
145C3×C42—22no—
146C3×C42—22no—
147C3×C42—22no—
148C3×C42—44no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C3×D411yescharacteristic
152C23×C4C3×D411yescharacteristic
153C23×C4C23×C311yescharacteristic
154C22×C8C23×C314yes—
155#117—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C23×C312yes—
168C2×C4⋊C4C23×C312yes—
169C2×C4⋊C4C23×C312yes—
170C2×C4⋊C4C23×C312yes—
171C2×C22⋊C4—22no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42C3×D411yescharacteristic
175C2×C42C3×D411yescharacteristic
176C2×C42C23×C311yescharacteristic
177C2×C42C23×C311yescharacteristic
178#107—28no—
179C22⋊C8—24no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3#12312yes—
189C23×C3C2×Q1612yes—
190C23×C3AGL(1,ℤ/8)12yes—
191C23×C3C22×D411yescharacteristic
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—28no—
201C2×C3×C4—28no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—44no—
207C2×C3×C4—48no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—84no—
220C2×C3×C4#12312yes—
221C2×C3×C4C2×Q1612yes—
222C2×C3×C4C2×Q1612yes—
223C2×C3×C4AGL(1,ℤ/8)12yes—
224C2×C3×C4C2×#4112yes—
225C2×C3×C4C2×#4112yes—
226C2×C3×C42−1+412yes—
227C3×C8—44no—
228C24C2×C3×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4C3×Q1614yes—
247C22×C4C2×C3×D411yescharacteristic
248C22×C4C2×C3×D411yescharacteristic
249C22×C4C2×C3×D411yescharacteristic
250C22×C4C2×C3×D411yescharacteristic
251C22×C4C2×C3×D411yescharacteristic
252C22×C4C3×#4114yes—
253C22×C4C24×C311yescharacteristic
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4C3×#4114yes—
270C22⋊C4—44no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—44no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3#48414yes—
283C22×C3C2×#12311yescharacteristic
284C22×C3#52511yescharacteristic
285C22×C3#53611yescharacteristic
286C22×C3C22×Q1611yescharacteristic
287C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C3#57511yescharacteristic
290C3×C4—22no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—41no—
295C3×C4—44no—
296C3×C4—82no—
297C3×C4—82no—
298C3×C4—82no—
299C3×C4#55214yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23C3×#12312yes—
305C23C2×C3×Q1612yes—
306C23C3×AGL(1,ℤ/8)12yes—
307C23C22×C3×D411yescharacteristic
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—48no—
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×#12312yes—
337C2×C4C2×C3×Q1612yes—
338C2×C4C2×C3×Q1612yes—
339C2×C4C3×AGL(1,ℤ/8)12yes—
340C2×C4C2×C3×#4112yes—
341C2×C4C2×C3×#4112yes—
342C2×C4C3×2−1+412yes—
343C8—44no—
344C2×C3—24no—
345C2×C3#248411yescharacteristic
346C2×C3C2×#48411yescharacteristic
347C2×C3#309411yescharacteristic
348C2×C3#310411yescharacteristic
349C2×C3#310911yescharacteristic
350C2×C3#323011yescharacteristic
351C2×C3#323411yescharacteristic
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22C3×#48414yes—
359C22C2×C3×#12311yescharacteristic
360C22C3×#52511yescharacteristic
361C22C3×#53611yescharacteristic
362C22C22×C3×Q1611yescharacteristic
363C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
364C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
365C22C3×#57511yescharacteristic
366C4—22no—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—41no—
371C4—44no—
372C4—82no—
373C4—82no—
374C4—82no—
375C4C3×#55214yes—
376C3#2761411yescharacteristic, direct product factor, Sylow
377C2—24no—
378C2C3×#248411yescharacteristic
379C2C2×C3×#48411yescharacteristic
380C2C3×#309411yescharacteristic
381C2C3×#310411yescharacteristic
382C2C3×#310911yescharacteristic
383C2C3×#323011yescharacteristic
384C2C3×#323411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2100C2semidirect
4. C3×#2083C2semidirect
5. C3×#1921C2semidirect
9. #27614C3direct (N × H)
36. #2100C2×C3semidirect
37. #2083C2×C3semidirect
38. #1921C2×C3semidirect
376. C3#27614direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C3×#22852
C3×#21002
C3×#20832
C3×#19212
C3×#18522
C3×#18242
C3×#18172
#276143