Finite Group Databasebeta

← C3×#29636 #10648243

Subgroups — C3×#29636 #10648243

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3449C211yesmaximal, characteristic, complemented
2C3×#2482C211yesmaximal, characteristic, complemented
3C3×#2404C212yesmaximal, complemented
4C3×#2044C214yesmaximal, complemented
5C3×#2021C214yesmaximal, complemented
6C3×#1820C211yesmaximal, characteristic, complemented
7C3×#1782C212yesmaximal, complemented
8#29636C311yesmaximal, 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×#120C2214yes—
18C2×C3×#117C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C2×C3×#99C2218yes—
21C3×#399—21no—
22C3×#397—21no—
23C3×#393—22no—
24C3×#392—22no—
25C3×#388—22no—
26C3×#386—24no—
27C3×#386C2214yes—
28C3×#385C2212yes—
29C3×#381C2212yes—
30C3×#381C2211yescharacteristic
31C3×#379C2212yes—
32C3×#342C2214yes—
33#3449C2×C311yescharacteristic, complemented
34#2482C2×C311yescharacteristic, complemented
35#2404C2×C312yescomplemented
36#2044C2×C314yescomplemented
37#2021C2×C314yescomplemented
38#1820C2×C311yescharacteristic, complemented
39#1782C2×C312yescomplemented
40C25×C3C2311yescharacteristic
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—48no—
46C22×C3×Q8D412yescomplemented
47C22×C3×D4—24no—
48C22×C3×D4D412yescomplemented
49C22×C3×D4C2314yes—
50C23×C3×C4D414yescomplemented
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C3×C4⋊Q8—22no—
54C3×#127—22no—
55C3×#124—24no—
56C3×#123—28no—
57C3×C22⋊Q8—24no—
58C3×C22⋊Q8—24no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#121—24no—
63C3×#120—28no—
64C3×#120—48no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C3×#117—22no—
68C3×#117—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D411yescharacteristic, complemented
71C2×C3×C4⋊C4D411yescharacteristic, complemented
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4C2312yes—
81C2×C3×C22⋊C4C2314yes—
82C2×C3×C22⋊C4C2312yes—
83C2×C3×C42D412yescomplemented
84C3×#99—48no—
85C3×#95—22no—
86C3×#95—21no—
87C3×#95—22no—
88C3×#95—21no—
89C3×#95—22no—
90C3×#95—24no—
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×#120C22×C314yes—
100C2×#117C22×C311yescharacteristic
101C22×C22⋊C4C22×C311yescharacteristic
102C2×#99C22×C318yes—
103#399—21no—
104#397—21no—
105#393—22no—
106#392—22no—
107#388—22no—
108#386—24no—
109#386C22×C314yes—
110#385C22×C312yes—
111#381C22×C312yes—
112#381C22×C311yescharacteristic
113#379C22×C312yes—
114#342C22×C314yes—
115C24×C3—21no—
116C24×C3—22no—
117C24×C3—24no—
118C24×C3—42no—
119C24×C3—81no—
120C24×C3C2×D414yes—
121C24×C3C2411yescharacteristic
122C3×#41—48no—
123C3×#41—48no—
124C3×#41—48no—
125C3×#41—48no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—44no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4—28no—
132C2×C3×D4—28no—
133C2×C3×D4—24no—
134C2×C3×D4—24no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—44no—
138C2×C3×D4—48no—
139C2×C3×D4—48no—
140C2×C3×D4—84no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4—81no—
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—82no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—28no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—28no—
168C3×C22⋊C4—48no—
169C3×C22⋊C4—44no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—82no—
174C3×C22⋊C4—84no—
175C3×C42—22no—
176C3×C42—22no—
177C25C23×C311yescharacteristic
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—22no—
182C2×#41—48no—
183C22×Q8C3×D412yescomplemented
184C22×D4—24no—
185C22×D4C3×D412yescomplemented
186C22×D4C23×C314yes—
187C23×C4C3×D414yescomplemented
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C311yescharacteristic
190C4⋊Q8—22no—
191#127—22no—
192#124—24no—
193#123—28no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196#121—24no—
197#121—24no—
198#121—24no—
199#121—24no—
200#120—28no—
201#120—48no—
202C4×D4—24no—
203C4×D4—24no—
204#117—22no—
205#117—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C3×D411yescharacteristic, complemented
208C2×C4⋊C4C3×D411yescharacteristic, complemented
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4C23×C314yes—
218C2×C22⋊C4C23×C312yes—
219C2×C22⋊C4C23×C312yes—
220C2×C42C3×D412yescomplemented
221#99—48no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—24no—
228C23×C3—22no—
229C23×C3—24no—
230C23×C3—24no—
231C23×C3—24no—
232C23×C3—24no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—44no—
236C23×C3—44no—
237C23×C3—42no—
238C23×C3—44no—
239C23×C3—41no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—44no—
243C23×C3—42no—
244C23×C3—88no—
245C23×C3C22×D412yes—
246C23×C3C22×D411yescharacteristic
247C23×C3H(5,𝔽2)14yes—
248C3×Q8—44no—
249C3×Q8—44no—
250C3×D4—44no—
251C3×D4—44no—
252C3×D4—48no—
253C3×D4—48no—
254C3×D4—84no—
255C3×D4—84no—
256C3×D4—816no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—48no—
267C2×C3×C4—48no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—41no—
277C2×C3×C4—44no—
278C2×C3×C4—42no—
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12012yes—
282C24—21no—
283C24—22no—
284C24—24no—
285C24—42no—
286C24—81no—
287C24C2×C3×D414yes—
288C24C24×C311yescharacteristic
289#41—48no—
290#41—48no—
291#41—48no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—44no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—84no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—81no—
318C22×C4C2×C3×D414yes—
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C2×C3×D411yescharacteristic
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D412yes—
324C22×C4C2×C3×D412yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—82no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—28no—
335C22⋊C4—48no—
336C22⋊C4—44no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—82no—
341C22⋊C4—84no—
342C42—22no—
343C42—22no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—44no—
349C22×C3—48no—
350C22×C3—48no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—44no—
356C22×C3—84no—
357C22×C3—84no—
358C22×C3—82no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3C2×#12011yescharacteristic
362C22×C3#53414yes—
363C22×C3#53512yes—
364C3×C4—21no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—44no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—82no—
371C3×C4—84no—
372C3×C4—81no—
373C23—22no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—41no—
379C23—41no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—44no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—44no—
388C23—42no—
389C23—88no—
390C23C22×C3×D412yes—
391C23C22×C3×D411yescharacteristic
392C23C3×H(5,𝔽2)14yes—
393Q8—44no—
394Q8—44no—
395D4—44no—
396D4—44no—
397D4—48no—
398D4—48no—
399D4—84no—
400D4—84no—
401D4—816no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4C3×#12011yescharacteristic
425C2×C4C3×#12011yescharacteristic
426C2×C4C3×#12012yes—
427C2×C3—24no—
428C2×C3—22no—
429C2×C3—82no—
430C2×C3—84no—
431C2×C3#261311yescharacteristic
432C2×C3#302511yescharacteristic
433C2×C3#302711yescharacteristic
434C22—21no—
435C22—22no—
436C22—21no—
437C22—22no—
438C22—44no—
439C22—48no—
440C22—48no—
441C22—44no—
442C22—44no—
443C22—42no—
444C22—44no—
445C22—44no—
446C22—84no—
447C22—84no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22C2×C3×#12011yescharacteristic
452C22C3×#53414yes—
453C22C3×#53512yes—
454C4—21no—
455C4—22no—
456C4—21no—
457C4—42no—
458C4—42no—
459C4—44no—
460C4—84no—
461C4—82no—
462C4—81no—
463C3#2963611yescharacteristic, direct product factor, Sylow
464C2—24no—
465C2—22no—
466C2—82no—
467C2—84no—
468C2C3×#261311yescharacteristic
469C2C3×#302511yescharacteristic
470C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
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
1. C3×#3449C2semidirect
2. C3×#2482C2semidirect
3. C3×#2404C2semidirect
4. C3×#2044C2semidirect
5. C3×#2021C2semidirect
6. C3×#1820C2semidirect
7. C3×#1782C2semidirect
8. #29636C3direct (N × H)
33. #3449C2×C3semidirect
34. #2482C2×C3semidirect
35. #2404C2×C3semidirect
36. #2044C2×C3semidirect
37. #2021C2×C3semidirect
38. #1820C2×C3semidirect
39. #1782C2×C3semidirect
46. C22×C3×Q8D4semidirect
48. C22×C3×D4D4semidirect
50. C23×C3×C4D4semidirect
70. C2×C3×C4⋊C4D4semidirect
71. C2×C3×C4⋊C4D4semidirect
83. C2×C3×C42D4semidirect
183. C22×Q8C3×D4semidirect
185. C22×D4C3×D4semidirect
187. C23×C4C3×D4semidirect
207. C2×C4⋊C4C3×D4semidirect
208. C2×C4⋊C4C3×D4semidirect
220. C2×C42C3×D4semidirect
463. C3#29636direct (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×#24822
C3×#24042
C3×#20442
C3×#20212
C3×#18202
C3×#17822
#296363