Finite Group Databasebeta

← C3×#25854 #10644461

Subgroups — C3×#25854 #10644461

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic
2C3×#2583C212yesmaximal, complemented
3C3×#2579C211yesmaximal, characteristic
4C3×#2062C212yesmaximal, complemented
5C3×#2062C212yesmaximal, complemented
6C3×#1996C212yesmaximal, complemented
7C3×#1867C212yesmaximal
8C3×#1867C212yesmaximal
9C3×C4×#99C211yesmaximal, characteristic
10#25854C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—22no—
13C3×D4×Q8—22no—
14C3×#544—21no—
15C3×#543—21no—
16C3×#543C2212yes—
17C3×#533—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C3×#518C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C2×C3×#99C2212yes—
24C3×#398—21no—
25C3×#398—22no—
26C3×#396—21no—
27C3×#395—22no—
28C3×#389—24no—
29C3×#387—21no—
30C3×#386C2211yescharacteristic
31C3×#386C2211yescharacteristic
32C3×#385—21no—
33C3×#385C2211yescharacteristic
34C3×#385C2211yescharacteristic
35C3×#385C2214yes—
36C3×#385C2214yes—
37C3×#380C2211yescharacteristic
38C3×C4×C22⋊C4C2212yes—
39C3×#342C2214yes—
40C3×#342C2214yes—
41C3×#342C2212yes—
42#3501C2×C311yescharacteristic
43#2583C2×C312yescomplemented
44#2579C2×C311yescharacteristic
45#2062C2×C312yescomplemented
46#2062C2×C312yescomplemented
47#1996C2×C312yescomplemented
48#1867C2×C312yes—
49#1867C2×C312yes—
50C4×#99C2×C311yescharacteristic
51C22×C3×Q8—21no—
52C22×C3×Q8D412yes—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—24no—
61C3×#125—22no—
62C3×#125—24no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—24no—
66C3×C22⋊Q8—24no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—24no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×#117—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—24no—
93C2×C3×C4⋊C4D414yes—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D412yes—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4C2311yescharacteristic
102C2×C3×C22⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C22⋊C4C2314yes—
105C2×C3×C42—22no—
106C2×C3×C42D411yescharacteristic
107C3×#99Q818yescomplemented
108C3×#95—22no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—24no—
113C3×#95—24no—
114C3×#95—21no—
115C3×#95—21no—
116#554—24no—
117#554—22no—
118D4×Q8—22no—
119#544—21no—
120#543—21no—
121#543C22×C312yes—
122#533—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C22×C312yes—
125#518C22×C312yes—
126C2×C4×D4C22×C311yescharacteristic
127C22×C4⋊C4C22×C311yescharacteristic
128C2×#99C22×C312yes—
129#398—21no—
130#398—22no—
131#396—21no—
132#395—22no—
133#389—24no—
134#387—21no—
135#386C22×C311yescharacteristic
136#386C22×C311yescharacteristic
137#385—21no—
138#385C22×C311yescharacteristic
139#385C22×C311yescharacteristic
140#385C22×C314yes—
141#385C22×C314yes—
142#380C22×C311yescharacteristic
143C4×C22⋊C4C22×C312yes—
144#342C22×C312yes—
145#342C22×C314yes—
146#342C22×C314yes—
147C24×C3C2×D411yescharacteristic
148C24×C3C2411yescharacteristic
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—82no—
154C2×C3×Q8—82no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4C2×Q814yes—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—42no—
170C22×C3×C4—81no—
171C22×C3×C4—81no—
172C22×C3×C4—81no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C3×C4⋊C4—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4—48no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—81no—
199C3×C4⋊C4—84no—
200C3×C4⋊C4—81no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—81no—
211C3×C22⋊C4C2×Q818yescomplemented
212C3×C42—24no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—82no—
216C22×Q8—21no—
217C22×Q8C3×D412yes—
218C22×D4C23×C311yescharacteristic
219C23×C4C3×D411yescharacteristic
220C23×C4C23×C311yescharacteristic
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C311yescharacteristic
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—24no—
226#125—22no—
227#125—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—24no—
245C4×D4—24no—
246#117—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4C3×D411yescharacteristic
259C2×C4⋊C4C3×D414yes—
260C2×C4⋊C4C3×D412yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4C3×D411yescharacteristic
265C2×C22⋊C4C23×C311yescharacteristic
266C2×C22⋊C4C23×C311yescharacteristic
267C2×C22⋊C4C23×C311yescharacteristic
268C2×C22⋊C4C23×C314yes—
269C2×C22⋊C4C23×C314yes—
270C2×C42—22no—
271C2×C42C3×D411yescharacteristic
272#99C3×Q818yescomplemented
273#95—22no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—24no—
278#95—24no—
279#95—21no—
280#95—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3C4⋊Q812yes—
288C23×C3C22×D411yescharacteristic
289C23×C3C22×D412yes—
290C23×C3C22×Q812yes—
291C23×C32−1+412yes—
292C3×Q8—44no—
293C3×Q8—82no—
294C3×Q8—82no—
295C3×Q8—82no—
296C3×D4—28no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—21no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—21no—
305C2×C3×C4—28no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—44no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—44no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—44no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—81no—
331C2×C3×C4—82no—
332C2×C3×C4—82no—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12012yes—
336C2×C3×C4C4⋊Q812yes—
337C24C2×C3×D411yescharacteristic
338C24C24×C311yescharacteristic
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—82no—
344C2×Q8—82no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4C2×C3×Q814yes—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D414yes—
366C22×C4C2×C3×D414yes—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C2×C3×D412yes—
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D411yescharacteristic
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—81no—
389C4⋊C4—84no—
390C4⋊C4—81no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—22no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—42no—
400C22⋊C4—81no—
401C22⋊C4C2×C3×Q818yescomplemented
402C42—24no—
403C42—21no—
404C42—21no—
405C42—82no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—22no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—81no—
417C22×C3C2×#12011yescharacteristic
418C22×C3C2×C4⋊Q811yescharacteristic
419C22×C3#53611yescharacteristic
420C22×C3D4×Q814yes—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—48no—
426C3×C4—42no—
427C3×C4—44no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—44no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4—81no—
434C23—21no—
435C23—21no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C3×C4⋊Q812yes—
441C23C22×C3×D411yescharacteristic
442C23C22×C3×D412yes—
443C23C22×C3×Q812yes—
444C23C3×2−1+412yes—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449D4—28no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—28no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4C3×#12011yescharacteristic
487C2×C4C3×#12011yescharacteristic
488C2×C4C3×#12012yes—
489C2×C4C3×C4⋊Q812yes—
490C2×C3—24no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—42no—
494C2×C3#243411yescharacteristic
495C2×C3#302511yescharacteristic
496C2×C3#302611yescharacteristic
497C22—22no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—81no—
508C22C2×C3×#12011yescharacteristic
509C22C2×C3×C4⋊Q811yescharacteristic
510C22C3×#53611yescharacteristic
511C22C3×D4×Q814yes—
512C4—22no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—48no—
517C4—42no—
518C4—44no—
519C4—41no—
520C4—41no—
521C4—44no—
522C4—81no—
523C4—81no—
524C4—81no—
525C3#2585411yescharacteristic, direct product factor, Sylow
526C2—24no—
527C2—21no—
528C2—21no—
529C2—42no—
530C2C3×#243411yescharacteristic
531C2C3×#302511yescharacteristic
532C2C3×#302611yescharacteristic
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×#2583C2semidirect
4. C3×#2062C2semidirect
5. C3×#2062C2semidirect
6. C3×#1996C2semidirect
10. #25854C3direct (N × H)
43. #2583C2×C3semidirect
45. #2062C2×C3semidirect
46. #2062C2×C3semidirect
47. #1996C2×C3semidirect
107. C3×#99Q8semidirect
211. C3×C22⋊C4C2×Q8semidirect
272. #99C3×Q8semidirect
401. C22⋊C4C2×C3×Q8semidirect
525. C3#25854direct (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×#25832
C3×#25792
C3×#20622
C3×#19962
C3×#18672
C3×C4×#992
#258543