Finite Group Databasebeta

← C3×#29637 #10648244

Subgroups — C3×#29637 #10648244

Order 768.

1358 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic
2C3×#2484C211yesmaximal, characteristic, complemented
3C3×#2406C212yesmaximal, complemented
4C3×#2044C214yesmaximal, complemented
5C3×#2022C214yesmaximal, complemented
6C3×#1820C211yesmaximal, characteristic, complemented
7C3×#1782C212yesmaximal, complemented
8#29637C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2212yes—
10C3×#537—21no—
11C3×#537—21no—
12C3×#536—22no—
13C3×#535—22no—
14C3×#534C2214yes—
15C3×#532—22no—
16C3×#525—24no—
17C2×C3×#123C2214yes—
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×C3×#99C2218yes—
21C3×#400—21no—
22C3×#398—21no—
23C3×#395—22no—
24C3×#394—22no—
25C3×#387—22no—
26C3×#386C2214yes—
27C3×#385—24no—
28C3×#385C2212yes—
29C3×#381C2212yes—
30C3×#381C2212yes—
31C3×#381C2211yescharacteristic
32C3×#342C2214yes—
33#3449C2×C311yescharacteristic
34#2484C2×C311yescharacteristic, complemented
35#2406C2×C312yescomplemented
36#2044C2×C314yescomplemented
37#2022C2×C314yescomplemented
38#1820C2×C311yescharacteristic, complemented
39#1782C2×C312yescomplemented
40C2×C3×#41—22no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—48no—
45C22×C3×Q8D412yes—
46C22×C3×D4D412yes—
47C22×C3×D4C2314yes—
48C23×C3×C4D414yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C3×C4⋊Q8—22no—
53C3×#127—22no—
54C3×#124—24no—
55C3×#123—28no—
56C3×#123—48no—
57C3×C22⋊Q8—24no—
58C3×C22⋊Q8—24no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#121—24no—
63C3×#120—28no—
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×#117—22no—
67C3×#117—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—24no—
78C2×C3×C22⋊C4C2312yes—
79C2×C3×C22⋊C4C2314yes—
80C2×C3×C22⋊C4C2312yes—
81C2×C3×C42D412yes—
82C3×#99—48no—
83C3×#95—22no—
84C3×#95—21no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—21no—
89C3×#95—24no—
90C3×#95—22no—
91C22×#41C22×C312yes—
92#537—21no—
93#537—21no—
94#536—22no—
95#535—22no—
96#534C22×C314yes—
97#532—22no—
98#525—24no—
99C2×#123C22×C314yes—
100C2×#117C22×C311yescharacteristic
101C22×C4⋊C4C22×C311yescharacteristic
102C2×#99C22×C318yes—
103#400—21no—
104#398—21no—
105#395—22no—
106#394—22no—
107#387—22no—
108#386C22×C314yes—
109#385—24no—
110#385C22×C312yes—
111#381C22×C312yes—
112#381C22×C312yes—
113#381C22×C311yescharacteristic
114#342C22×C314yes—
115C24×C3C2×D414yes—
116C24×C3C2411yescharacteristic
117C3×#41—48no—
118C3×#41—48no—
119C3×#41—48no—
120C3×#41—48no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—44no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—28no—
127C2×C3×D4—28no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—44no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—42no—
144C22×C3×C4—44no—
145C22×C3×C4—44no—
146C22×C3×C4C2×D414yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—44no—
158C3×C4⋊C4—48no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—42no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—24no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—28no—
167C3×C22⋊C4—48no—
168C3×C22⋊C4—48no—
169C3×C42—22no—
170C3×C42—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—48no—
176C22×Q8C3×D412yes—
177C22×D4C3×D412yes—
178C22×D4C23×C314yes—
179C23×C4C3×D414yes—
180C23×C4C23×C311yescharacteristic
181C23×C4C23×C311yescharacteristic
182C23×C4C23×C311yescharacteristic
183C4⋊Q8—22no—
184#127—22no—
185#124—24no—
186#123—28no—
187#123—48no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190#121—24no—
191#121—24no—
192#121—24no—
193#121—24no—
194#120—28no—
195C4×D4—24no—
196C4×D4—24no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C3×D411yescharacteristic
206C2×C4⋊C4C3×D411yescharacteristic
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4C23×C314yes—
210C2×C22⋊C4C23×C312yes—
211C2×C22⋊C4C23×C312yes—
212C2×C42C3×D412yes—
213#99—48no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—24no—
221#95—22no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—24no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—44no—
228C23×C3—44no—
229C23×C3C22×D412yes—
230C23×C3C22×D411yescharacteristic
231C23×C3H(5,𝔽2)14yes—
232C3×Q8—44no—
233C3×Q8—44no—
234C3×D4—44no—
235C3×D4—44no—
236C3×D4—48no—
237C3×D4—48no—
238C3×D4—84no—
239C3×D4—84no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—48no—
252C2×C3×C4—48no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—44no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—41no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12012yes—
270C24C2×C3×D414yes—
271C24C24×C311yescharacteristic
272#41—48no—
273#41—48no—
274#41—48no—
275#41—48no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—44no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—28no—
282C2×D4—28no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4C2×C3×D414yes—
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D412yes—
305C22×C4C2×C3×D412yes—
306C22×C4C2×C3×D412yes—
307C22×C4C2×C3×D412yes—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—28no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C42—22no—
325C42—22no—
326C22×C3—21no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—22no—
330C22×C3—44no—
331C22×C3—48no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—44no—
335C22×C3—84no—
336C22×C3C2×#12011yescharacteristic
337C22×C3#53414yes—
338C22×C3#53512yes—
339C3×C4—21no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—42no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—84no—
347C3×C4—82no—
348C3×C4—81no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—41no—
353C23—41no—
354C23—44no—
355C23—44no—
356C23C22×C3×D412yes—
357C23C22×C3×D411yescharacteristic
358C23C3×H(5,𝔽2)14yes—
359Q8—44no—
360Q8—44no—
361D4—44no—
362D4—44no—
363D4—48no—
364D4—48no—
365D4—84no—
366D4—84no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4C3×#12011yescharacteristic
395C2×C4C3×#12011yescharacteristic
396C2×C4C3×#12012yes—
397C2×C3—24no—
398C2×C3—22no—
399C2×C3—84no—
400C2×C3#261311yescharacteristic
401C2×C3#302511yescharacteristic
402C2×C3#302811yescharacteristic
403C22—21no—
404C22—22no—
405C22—21no—
406C22—22no—
407C22—44no—
408C22—48no—
409C22—44no—
410C22—44no—
411C22—44no—
412C22—84no—
413C22C2×C3×#12011yescharacteristic
414C22C3×#53414yes—
415C22C3×#53512yes—
416C4—21no—
417C4—22no—
418C4—21no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—44no—
423C4—84no—
424C4—82no—
425C4—81no—
426C3#2963711yescharacteristic, direct product factor, Sylow
427C2—24no—
428C2—22no—
429C2—84no—
430C2C3×#261311yescharacteristic
431C2C3×#302511yescharacteristic
432C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#3449C2
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
3. C3×#2406C2semidirect
4. C3×#2044C2semidirect
5. C3×#2022C2semidirect
6. C3×#1820C2semidirect
7. C3×#1782C2semidirect
8. #29637C3direct (N × H)
34. #2484C2×C3semidirect
35. #2406C2×C3semidirect
36. #2044C2×C3semidirect
37. #2022C2×C3semidirect
38. #1820C2×C3semidirect
39. #1782C2×C3semidirect
426. C3#29637direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34492
C3×#24842
C3×#24062
C3×#20442
C3×#20222
C3×#18202
C3×#17822
#296373