Finite Group Databasebeta

← C3×#25852 #10644459

Subgroups — C3×#25852 #10644459

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic
2C3×#2434C212yesmaximal, complemented
3C3×#2409C211yesmaximal, characteristic
4C3×#2062C214yesmaximal, complemented
5C3×#1996C212yesmaximal, complemented
6C3×#1867C214yesmaximal
7C3×C4×#99C211yesmaximal, characteristic
8#25852C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2211yescharacteristic
10C3×#554—22no—
11C3×#554—24no—
12C3×D4×Q8—22no—
13C3×#544—21no—
14C3×#543—21no—
15C3×#543C2212yes—
16C3×#533—21no—
17C2×C3×C4⋊Q8—22no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C3×#518C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#99C2212yes—
24C3×#395—21no—
25C3×#395—24no—
26C3×#393—21no—
27C3×#391—22no—
28C3×#391—21no—
29C3×#386C2212yes—
30C3×#385C2212yes—
31C3×#385C2218yes—
32C3×#380C2211yescharacteristic
33C3×C4×C22⋊C4C2212yes—
34C3×#342C2218yes—
35C3×#342C2212yes—
36#3501C2×C311yescharacteristic
37#2434C2×C312yescomplemented
38#2409C2×C311yescharacteristic
39#2062C2×C314yescomplemented
40#1996C2×C312yescomplemented
41#1867C2×C314yes—
42C4×#99C2×C311yescharacteristic
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—22no—
46C22×C3×Q8—21no—
47C22×C3×Q8—22no—
48C22×C3×Q8—44no—
49C22×C3×Q8D412yes—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8—24no—
57C3×C4⋊Q8—48no—
58C3×#125—22no—
59C3×#125—24no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—24no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—44no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—24no—
78C3×C4×D4—24no—
79C3×#117—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—28no—
85C2×C3×C4⋊C4D412yes—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D414yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C22⋊C4C2312yes—
92C2×C3×C22⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2314yes—
95C2×C3×C42—22no—
96C2×C3×C42D411yescharacteristic
97C3×#99Q818yescomplemented
98C3×#95—22no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—28no—
102C3×#95—22no—
103C23×Q8C22×C311yescharacteristic
104#554—22no—
105#554—24no—
106D4×Q8—22no—
107#544—21no—
108#543—21no—
109#543C22×C312yes—
110#533—21no—
111C2×C4⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C22×C312yes—
115#518C22×C312yes—
116C2×C4×D4C22×C311yescharacteristic
117C2×#99C22×C312yes—
118#395—21no—
119#395—24no—
120#393—21no—
121#391—22no—
122#391—21no—
123#386C22×C312yes—
124#385C22×C312yes—
125#385C22×C318yes—
126#380C22×C311yescharacteristic
127C4×C22⋊C4C22×C312yes—
128#342C22×C312yes—
129#342C22×C318yes—
130C24×C3C2×D411yescharacteristic
131C24×C3C2411yescharacteristic
132C2×C3×Q8—24no—
133C2×C3×Q8—28no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—44no—
139C2×C3×Q8—42no—
140C2×C3×Q8—44no—
141C2×C3×Q8—48no—
142C2×C3×Q8—44no—
143C2×C3×Q8—41no—
144C2×C3×Q8—42no—
145C2×C3×Q8—41no—
146C2×C3×Q8—82no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4C2×Q814yes—
150C22×C3×C4—24no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—44no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—44no—
183C3×C4⋊C4—416no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—44no—
192C3×C22⋊C4C2×Q818yescomplemented
193C3×C42—24no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—44no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—22no—
200C22×Q8—21no—
201C22×Q8—22no—
202C22×Q8—44no—
203C22×Q8C3×D412yes—
204C22×D4C23×C311yescharacteristic
205C23×C4C3×D411yescharacteristic
206C23×C4C23×C312yes—
207C23×C4C23×C311yescharacteristic
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—24no—
211C4⋊Q8—48no—
212#125—22no—
213#125—24no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—44no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—24no—
232C4×D4—24no—
233#117—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4C3×D411yescharacteristic
240C2×C4⋊C4C3×D412yes—
241C2×C4⋊C4C3×D414yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C3×D411yescharacteristic
245C2×C22⋊C4C23×C312yes—
246C2×C22⋊C4C23×C311yescharacteristic
247C2×C22⋊C4C23×C314yes—
248C2×C22⋊C4C23×C314yes—
249C2×C42—22no—
250C2×C42C3×D411yescharacteristic
251#99C3×Q818yescomplemented
252#95—22no—
253#95—21no—
254#95—22no—
255#95—28no—
256#95—22no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C4⋊Q812yescomplemented
264C23×C3C22×D411yescharacteristic
265C23×C3C22×D412yes—
266C23×C3C22×Q812yes—
267C23×C32−1+412yes—
268C3×Q8—44no—
269C3×Q8—48no—
270C3×Q8—44no—
271C3×Q8—44no—
272C3×Q8—82no—
273C3×Q8—82no—
274C3×Q8—82no—
275C3×D4—28no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—28no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—28no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—44no—
290C2×C3×C4—44no—
291C2×C3×C4—44no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—81no—
303C2×C3×C4—82no—
304C2×C3×C4—82no—
305C2×C3×C4#12011yescharacteristic
306C2×C3×C4#12012yes—
307C2×C3×C4#12011yescharacteristic
308C2×C3×C4C4⋊Q812yescomplemented
309C24C2×C3×D411yescharacteristic
310C24C24×C311yescharacteristic
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—82no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4C2×C3×Q814yes—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D414yes—
345C22×C4C2×C3×D414yes—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D412yes—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—416no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—22no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C22⋊C4—44no—
371C22⋊C4C2×C3×Q818yescomplemented
372C42—24no—
373C42—21no—
374C42—21no—
375C42—44no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—81no—
387C22×C3C2×#12011yescharacteristic
388C22×C3C2×C4⋊Q811yescharacteristic
389C22×C3#53611yescharacteristic
390C22×C3D4×Q814yes—
391C3×C4—22no—
392C3×C4—21no—
393C3×C4—22no—
394C3×C4—48no—
395C3×C4—44no—
396C3×C4—41no—
397C3×C4—44no—
398C3×C4—44no—
399C3×C4—41no—
400C3×C4—82no—
401C3×C4B(4,𝔽2)12yes—
402C23—21no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23C3×C4⋊Q812yescomplemented
409C23C22×C3×D411yescharacteristic
410C23C22×C3×D412yes—
411C23C22×C3×Q812yes—
412C23C3×2−1+412yes—
413Q8—44no—
414Q8—48no—
415Q8—44no—
416Q8—44no—
417Q8—82no—
418Q8—82no—
419Q8—82no—
420D4—28no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—28no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4C3×#12011yescharacteristic
451C2×C4C3×#12012yes—
452C2×C4C3×#12011yescharacteristic
453C2×C4C3×C4⋊Q812yescomplemented
454C2×C3—24no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—42no—
458C2×C3#243411yescharacteristic
459C2×C3C2×B(4,𝔽2)11yescharacteristic
460C2×C3#302811yescharacteristic
461C22—22no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—22no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—41no—
470C22—41no—
471C22—81no—
472C22C2×C3×#12011yescharacteristic
473C22C2×C3×C4⋊Q811yescharacteristic
474C22C3×#53611yescharacteristic
475C22C3×D4×Q814yes—
476C4—22no—
477C4—21no—
478C4—22no—
479C4—48no—
480C4—44no—
481C4—41no—
482C4—44no—
483C4—44no—
484C4—41no—
485C4—82no—
486C4C3×B(4,𝔽2)12yes—
487C3#2585211yescharacteristic, direct product factor, Sylow
488C2—24no—
489C2—21no—
490C2—21no—
491C2—42no—
492C2C3×#243411yescharacteristic
493C2C2×C3×B(4,𝔽2)11yescharacteristic
494C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×D4C23
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×#2434C2semidirect
4. C3×#2062C2semidirect
5. C3×#1996C2semidirect
8. #25852C3direct (N × H)
37. #2434C2×C3semidirect
39. #2062C2×C3semidirect
40. #1996C2×C3semidirect
97. C3×#99Q8semidirect
192. C3×C22⋊C4C2×Q8semidirect
251. #99C3×Q8semidirect
263. C23×C3C4⋊Q8semidirect
308. C2×C3×C4C4⋊Q8semidirect
371. C22⋊C4C2×C3×Q8semidirect
408. C23C3×C4⋊Q8semidirect
453. C2×C4C3×C4⋊Q8semidirect
487. C3#25852direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35012
C3×#24342
C3×#24092
C3×#20622
C3×#19962
C3×#18672
C3×C4×#992
#258523