Finite Group Databasebeta

← C3×#33023 #10651630

Subgroups — C3×#33023 #10651630

Order 768.

1314 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic
2C3×#2484C212yesmaximal, complemented
3C3×#2406C211yesmaximal, characteristic
4C3×#1890C214yesmaximal, complemented
5C3×#1867C214yesmaximal, complemented
6C3×#1820C212yesmaximal, complemented
7C3×#1782C211yesmaximal, characteristic
8#33023C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#123C2212yes—
11C3×#520—22no—
12C3×#520—22no—
13C3×#519—21no—
14C3×#518—21no—
15C3×#518C2214yes—
16C3×C4×#41—24no—
17C2×C3×#117—22no—
18C2×C3×#117C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×C3×#99C2214yes—
21C3×#400—22no—
22C3×#398—22no—
23C3×#395—21no—
24C3×#394—21no—
25C3×#387—24no—
26C3×#386C2212yes—
27C3×#385—22no—
28C3×#385C2214yes—
29C3×#381C2214yes—
30C3×#381C2211yescharacteristic
31C3×#381C2212yes—
32C3×#342C2218yes—
33#3428C2×C311yescharacteristic
34#2484C2×C312yescomplemented
35#2406C2×C311yescharacteristic
36#1890C2×C314yescomplemented
37#1867C2×C314yescomplemented
38#1820C2×C312yescomplemented
39#1782C2×C311yescharacteristic
40C2×C3×#41—28no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C22×C3×Q8D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C22×C3×D4C2312yes—
46C23×C3×C4D412yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C3×#123—28no—
51C3×C4×Q8—24no—
52C3×C4×Q8—24no—
53C3×C4×D4—24no—
54C3×C4×D4—24no—
55C3×C4×D4—28no—
56C3×C4×D4—28no—
57C3×#117—24no—
58C3×#117—24no—
59C3×#117—22no—
60C3×#117—24no—
61C3×#117—22no—
62C3×#117—22no—
63C3×#117—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4C2314yes—
78C2×C3×C22⋊C4C2312yes—
79C2×C3×C22⋊C4C2314yes—
80C2×C3×C42—24no—
81C2×C3×C42D414yes—
82C3×#99—28no—
83C3×#95—21no—
84C3×#95—22no—
85C3×#95—24no—
86C3×#95—24no—
87C3×#95—21no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—24no—
91C22×#41C22×C311yescharacteristic
92C2×#123C22×C312yes—
93#520—22no—
94#520—22no—
95#519—21no—
96#518—21no—
97#518C22×C314yes—
98C4×#41—24no—
99C2×#117—22no—
100C2×#117C22×C312yes—
101C22×C4⋊C4C22×C311yescharacteristic
102C2×#99C22×C314yes—
103#400—22no—
104#398—22no—
105#395—21no—
106#394—21no—
107#387—24no—
108#386C22×C312yes—
109#385—22no—
110#385C22×C314yes—
111#381C22×C311yescharacteristic
112#381C22×C312yes—
113#381C22×C314yes—
114#342C22×C318yes—
115C24×C3C2×D412yes—
116C24×C3C2411yescharacteristic
117C3×#41—216no—
118C3×#41—216no—
119C2×C3×Q8—22no—
120C2×C3×Q8—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—22no—
125C2×C3×D4#4118yes—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—24no—
140C22×C3×C4—42no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D414yes—
147C22×C3×C4C2×D414yes—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—48no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—44no—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—24no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—44no—
166C3×C42—28no—
167C3×C42—24no—
168C3×C42—24no—
169C2×#41—28no—
170C2×#41—22no—
171C2×#41—22no—
172C22×Q8C3×D411yescharacteristic
173C22×D4C3×D411yescharacteristic
174C22×D4C23×C312yes—
175C23×C4C3×D412yes—
176C23×C4C23×C311yescharacteristic
177C23×C4C23×C311yescharacteristic
178C23×C4C23×C311yescharacteristic
179#123—28no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—28no—
185C4×D4—28no—
186#117—24no—
187#117—24no—
188#117—22no—
189#117—24no—
190#117—22no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4C3×D412yes—
202C2×C4⋊C4C3×D412yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C23×C312yes—
207C2×C22⋊C4C23×C314yes—
208C2×C22⋊C4C23×C314yes—
209C2×C42—24no—
210C2×C42C3×D414yes—
211#99—28no—
212#95—21no—
213#95—22no—
214#95—24no—
215#95—24no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—24no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3#12314yes—
227C23×C3C22×D411yescharacteristic
228C23×C3C22×D412yes—
229C23×C3C2×#4114yes—
230C3×Q8—28no—
231C3×D4—28no—
232C3×D4—216no—
233C2×C3×C4—28no—
234C2×C3×C4—28no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—44no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—44no—
252C2×C3×C4—44no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—84no—
260C2×C3×C4—84no—
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12012yes—
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12314yes—
265C24C2×C3×D412yes—
266C24C24×C311yescharacteristic
267#41—216no—
268#41—216no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4C3×#4118yes—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4C2×C3×D412yes—
292C22×C4C2×C3×D412yes—
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D412yes—
296C22×C4C2×C3×D414yes—
297C22×C4C2×C3×D414yes—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—48no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—44no—
309C22⋊C4—28no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—24no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C42—28no—
317C42—24no—
318C42—24no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—21no—
323C22×C3—24no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3C2×#12011yescharacteristic
330C22×C3C2×#12312yes—
331C22×C3#52514yes—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—24no—
336C3×C4—42no—
337C3×C4—44no—
338C3×C4—44no—
339C3×C4—82no—
340C3×C4—84no—
341C3×C4—81no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23C3×#12314yes—
349C23C22×C3×D411yescharacteristic
350C23C22×C3×D412yes—
351C23C2×C3×#4114yes—
352Q8—28no—
353D4—28no—
354D4—216no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4C3×#12011yescharacteristic
384C2×C4C3×#12012yes—
385C2×C4C3×#12011yescharacteristic
386C2×C4C3×#12314yes—
387C2×C3—22no—
388C2×C3—24no—
389C2×C3—44no—
390C2×C3#284911yescharacteristic
391C2×C3#302511yescharacteristic
392C2×C3#302811yescharacteristic
393C22—22no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22C2×C3×#12011yescharacteristic
404C22C2×C3×#12312yes—
405C22C3×#52514yes—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—24no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—82no—
414C4—84no—
415C4—81no—
416C3#3302311yescharacteristic, direct product factor, Sylow
417C2—22no—
418C2—24no—
419C2—44no—
420C2C3×#284911yescharacteristic
421C2C3×#302511yescharacteristic
422C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2484C2semidirect
4. C3×#1890C2semidirect
5. C3×#1867C2semidirect
6. C3×#1820C2semidirect
8. #33023C3direct (N × H)
34. #2484C2×C3semidirect
36. #1890C2×C3semidirect
37. #1867C2×C3semidirect
38. #1820C2×C3semidirect
416. C3#33023direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34282
C3×#24842
C3×#24062
C3×#18902
C3×#18672
C3×#18202
C3×#17822
#330233