Finite Group Databasebeta

← C3×#23916 #10642523

Subgroups — C3×#23916 #10642523

Order 768.

1694 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3487C211yesmaximal, characteristic
2C2×C3×#536C211yesmaximal, characteristic, complemented
3C2×C3×#456C212yesmaximal, complemented
4C3×#3004C212yesmaximal, complemented
5C3×#2004C214yesmaximal, complemented
6C3×#1897C212yesmaximal, complemented
7C3×#1886C212yesmaximal, complemented
8C3×#1785C211yesmaximal, characteristic, complemented
9#23916C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C2×C3×#137C2214yes—
14C3×#550—24no—
15C3×#548—24no—
16C3×#536—28no—
17C3×#535—22no—
18C3×#535C2212yes—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×#124C2212yes—
21C3×#525—22no—
22C3×#525C2212yes—
23C2×C3×#123—22no—
24C2×C3×C22⋊Q8—22no—
25C3×#519C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#456—28no—
28C3×#451—24no—
29C3×#448—24no—
30C2×C3×#104C2214yes—
31C3×#419C2214yes—
32C2×C3×#101C2212yes—
33C2×C3×#100C2212yes—
34C3×#407C2212yes—
35C3×#339C2214yes—
36#3487C2×C311yescharacteristic
37C2×#536C2×C311yescharacteristic, complemented
38C2×#456C2×C312yescomplemented
39#3004C2×C312yescomplemented
40#2004C2×C314yescomplemented
41#1897C2×C312yescomplemented
42#1886C2×C312yescomplemented
43#1785C2×C311yescharacteristic, complemented
44C3×2−1+4—24no—
45C3×2−1+4—42no—
46C2×C3×#41—24no—
47C2×C3×#41—22no—
48C2×C3×#41—24no—
49C2×C3×#41—22no—
50C2×C3×#41—42no—
51C2×C3×#41C2312yes—
52C22×C3×Q8—24no—
53C22×C3×Q8—22no—
54C22×C3×Q8—21no—
55C22×C3×Q8—24no—
56C22×C3×Q8—42no—
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D412yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4D412yes—
63C23×C3×C4C2311yescharacteristic
64C3×#137—28no—
65C3×#137—44no—
66C2×C3×Q16—24no—
67C2×C3×SD16—24no—
68C2×C3×M16C2314yes—
69C3×C4⋊Q8—24no—
70C3×C4⋊Q8D418yescomplemented
71C3×#127—24no—
72C3×#127—22no—
73C3×#124—24no—
74C3×#124—24no—
75C3×#124—22no—
76C3×#124—24no—
77C3×#123—216no—
78C3×#123—22no—
79C3×C22⋊Q8—216no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8D414yes—
82C3×#121—24no—
83C3×#121—28no—
84C3×#121—24no—
85C3×#121D414yes—
86C3×#120—24no—
87C3×#120—22no—
88C3×C4×Q8—24no—
89C3×C4×D4—28no—
90C3×C4×D4—24no—
91C3×#117—22no—
92C3×#117—24no—
93C3×#117C2312yes—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—24no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C42C2312yes—
99C3×#104—28no—
100C3×#103—28no—
101C3×#101—24no—
102C3×#100—42no—
103C3×C22⋊C8—24no—
104C2×2−1+4C22×C311yescharacteristic
105C22×#41C22×C311yescharacteristic
106C23×Q8C22×C311yescharacteristic
107C2×#137C22×C314yes—
108#550—24no—
109#548—24no—
110#536—28no—
111#535—22no—
112#535C22×C312yes—
113C2×C4⋊Q8C22×C312yes—
114C2×#124C22×C312yes—
115#525—22no—
116#525C22×C312yes—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119#519C22×C311yescharacteristic
120C2×#117C22×C311yescharacteristic
121#456—28no—
122#451—24no—
123#448—24no—
124C2×#104C22×C314yes—
125#419C22×C314yes—
126C2×#101C22×C312yes—
127C2×#100C22×C312yes—
128#407C22×C312yes—
129#339C22×C314yes—
130C24×C3C2×D412yes—
131C24×C3C2×D411yescharacteristic
132C3×#41—24no—
133C3×#41—24no—
134C3×#41—416no—
135C3×#41—44no—
136C3×#41—48no—
137C3×#41—44no—
138C2×C3×Q8—24no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—24no—
144C2×C3×Q8—24no—
145C2×C3×Q8—416no—
146C2×C3×Q8—42no—
147C2×C3×Q8—48no—
148C2×C3×Q8—42no—
149C2×C3×Q8C2×D412yes—
150C2×C3×Q8C2×D414yes—
151C2×C3×Q8C2×D414yes—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—28no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—42no—
162C2×C3×D4C2×D412yes—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2411yescharacteristic
179C3×Q16—48no—
180C3×SD16—48no—
181C3×M16—24no—
182C3×M16—24no—
183C2×C3×C8—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—216no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4C2×D414yes—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—216no—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—24no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D414yes—
1992−1+4—24no—
2002−1+4—42no—
201C2×#41—24no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—24no—
205C2×#41—42no—
206C2×#41C23×C312yes—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8C23×C312yes—
213C22×Q8C23×C311yescharacteristic
214C22×D4C3×D412yes—
215C22×D4C23×C311yescharacteristic
216C23×C4—21no—
217C23×C4C3×D412yes—
218C23×C4C23×C311yescharacteristic
219#137—28no—
220#137—44no—
221C2×Q16—24no—
222C2×SD16—24no—
223C2×M16C23×C314yes—
224C4⋊Q8—24no—
225C4⋊Q8C3×D418yescomplemented
226#127—24no—
227#127—22no—
228#124—24no—
229#124—24no—
230#124—22no—
231#124—24no—
232#123—216no—
233#123—22no—
234C22⋊Q8—216no—
235C22⋊Q8—24no—
236C22⋊Q8C3×D414yes—
237#121—24no—
238#121—28no—
239#121—24no—
240#121C3×D414yes—
241#120—24no—
242#120—22no—
243C4×Q8—24no—
244C4×D4—28no—
245C4×D4—24no—
246#117—22no—
247#117—24no—
248#117C23×C312yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C42C23×C312yes—
254#104—28no—
255#103—28no—
256#101—24no—
257#100—42no—
258C22⋊C8—24no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3#12012yes—
268C23×C3C22×D411yescharacteristic
269C3×Q8—28no—
270C3×Q8—28no—
271C3×Q8—24no—
272C3×Q8—48no—
273C3×Q8—44no—
274C3×Q8—44no—
275C3×D4—24no—
276C3×D4—48no—
277C3×D4—44no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—44no—
281C3×D4—84no—
282C3×D4—88no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—28no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—48no—
293C2×C3×C4—48no—
294C2×C3×C4—44no—
295C2×C3×C4—48no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4#12014yes—
300C2×C3×C4#12012yes—
301C2×C3×C4C22×D412yes—
302C2×C3×C4C22×D412yes—
303C2×C3×C4H(5,𝔽2)12yes—
304C3×C8—44no—
305C24C2×C3×D412yes—
306C24C2×C3×D411yescharacteristic
307#41—24no—
308#41—24no—
309#41—416no—
310#41—44no—
311#41—48no—
312#41—44no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—416no—
321C2×Q8—42no—
322C2×Q8—48no—
323C2×Q8—42no—
324C2×Q8C2×C3×D412yes—
325C2×Q8C2×C3×D414yes—
326C2×Q8C2×C3×D414yes—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—28no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4C2×C3×D412yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C2×C3×D412yes—
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D414yes—
353C22×C4C24×C311yescharacteristic
354Q16—48no—
355SD16—48no—
356M16—24no—
357M16—24no—
358C2×C8—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—216no—
362C4⋊C4—24no—
363C4⋊C4C2×C3×D414yes—
364C22⋊C4—24no—
365C22⋊C4—216no—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C42—24no—
371C42—22no—
372C42—22no—
373C42C2×C3×D414yes—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—24no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—82no—
385C22×C3C2×#12011yescharacteristic
386C22×C3C2×#12011yescharacteristic
387C22×C3#53411yescharacteristic
388C3×C4—22no—
389C3×C4—24no—
390C3×C4—44no—
391C3×C4—42no—
392C3×C4—44no—
393C3×C4—44no—
394C3×C4D4214yes—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—44no—
403C23C3×#12012yes—
404C23C22×C3×D411yescharacteristic
405Q8—28no—
406Q8—28no—
407Q8—24no—
408Q8—48no—
409Q8—44no—
410Q8—44no—
411D4—24no—
412D4—48no—
413D4—44no—
414D4—42no—
415D4—42no—
416D4—44no—
417D4—84no—
418D4—88no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4C3×#12012yes—
436C2×C4C3×#12014yes—
437C2×C4C22×C3×D412yes—
438C2×C4C22×C3×D412yes—
439C2×C4C3×H(5,𝔽2)12yes—
440C8—44no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—42no—
444C2×C3—82no—
445C2×C3—82no—
446C2×C3#240311yescharacteristic
447C2×C3#302212yes—
448C22—22no—
449C22—22no—
450C22—24no—
451C22—42no—
452C22—42no—
453C22—44no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22—82no—
459C22C2×C3×#12011yescharacteristic
460C22C2×C3×#12011yescharacteristic
461C22C3×#53411yescharacteristic
462C4—22no—
463C4—24no—
464C4—44no—
465C4—42no—
466C4—44no—
467C4—44no—
468C4C3×D4214yes—
469C3#2391611yescharacteristic, direct product factor, Sylow
470C2—21no—
471C2—21no—
472C2—42no—
473C2—82no—
474C2—82no—
475C2C3×#240311yescharacteristic
476C2C3×#302212yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3487C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#536C2semidirect
3. C2×C3×#456C2semidirect
4. C3×#3004C2semidirect
5. C3×#2004C2semidirect
6. C3×#1897C2semidirect
7. C3×#1886C2semidirect
8. C3×#1785C2semidirect
9. #23916C3direct (N × H)
37. C2×#536C2×C3semidirect
38. C2×#456C2×C3semidirect
39. #3004C2×C3semidirect
40. #2004C2×C3semidirect
41. #1897C2×C3semidirect
42. #1886C2×C3semidirect
43. #1785C2×C3semidirect
70. C3×C4⋊Q8D4semidirect
225. C4⋊Q8C3×D4semidirect
469. C3#23916direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34872
C2×C3×#5362
C2×C3×#4562
C3×#30042
C3×#20042
C3×#18972
C3×#18862
C3×#17852
#239163