Finite Group Databasebeta

← C3×#25694 #10644301

Subgroups — C3×#25694 #10644301

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3141C211yesmaximal, characteristic, complemented
4C3×#2542C211yesmaximal, characteristic, complemented
5C3×#2037C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic, complemented
8C3×#1859C214yesmaximal, complemented
9C3×C4×#100C211yesmaximal, characteristic, complemented
10#25694C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—22no—
12C3×#556—21no—
13C3×#555—21no—
14C3×#553—22no—
15C3×#550—21no—
16C3×#547—21no—
17C3×#538—22no—
18C3×#535C2212yes—
19C3×#533C2211yescharacteristic
20C3×#532C2211yescharacteristic
21C2×C3×C4⋊Q8C2211yescharacteristic
22C3×#525—22no—
23C2×C3×C22⋊Q8—22no—
24C3×#520C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C3×#501—22no—
27C3×#500—21no—
28C3×#498—21no—
29C3×#489—22no—
30C3×#487—21no—
31C3×#486—21no—
32C3×#428C2214yes—
33C2×C3×#104C2214yes—
34C3×#417C2214yes—
35C2×C3×#100C2212yes—
36C3×C4×M16C2211yescharacteristic
37C3×C4×M16C2211yescharacteristic
38C3×#399—22no—
39C3×#390C2212yes—
40C3×#384C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3506C2×C311yescharacteristic
45#3165C2×C311yescharacteristic, complemented
46#3141C2×C311yescharacteristic, complemented
47#2542C2×C311yescharacteristic, complemented
48#2037C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic, complemented
51#1859C2×C314yescomplemented
52C4×#100C2×C311yescharacteristic, complemented
53C2×C3×#41—22no—
54C2×C3×#41C2312yes—
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C2×C3×M16C2312yes—
59C2×C3×M16C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×C4⋊Q8D411yescharacteristic
66C3×C4⋊Q8D411yescharacteristic
67C3×#127—22no—
68C3×#127D411yescharacteristic
69C3×#126—24no—
70C3×#125D412yes—
71C3×#124—24no—
72C3×#124—22no—
73C3×#124D412yes—
74C3×#123—24no—
75C3×#123—24no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—21no—
78C3×C22⋊Q8—21no—
79C3×C22⋊Q8—48no—
80C3×#121—22no—
81C3×#121—24no—
82C3×#121—21no—
83C3×#121—21no—
84C3×#121—22no—
85C3×#121—22no—
86C3×#120—24no—
87C3×C4×Q8—22no—
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×D4—24no—
91C3×C4×D4—22no—
92C3×C4×D4—24no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×#117—22no—
100C3×#117—22no—
101C3×#117C2312yes—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—24no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—24no—
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42C2312yes—
111C2×C3×C42C2311yescharacteristic
112C3×#107—24no—
113C3×#106—24no—
114C3×#104—28no—
115C3×#103—24no—
116C3×#102—24no—
117C3×#100—24no—
118C3×#97—21no—
119C3×#97—21no—
120C3×C4×C8—21no—
121C3×C4×C8—21no—
122C3×#95—21no—
123#559—22no—
124#556—21no—
125#555—21no—
126#553—22no—
127#550—21no—
128#547—21no—
129#538—22no—
130#535C22×C312yes—
131#533C22×C311yescharacteristic
132#532C22×C311yescharacteristic
133C2×C4⋊Q8C22×C311yescharacteristic
134#525—22no—
135C2×C22⋊Q8—22no—
136#520C22×C312yes—
137C2×C4×D4C22×C311yescharacteristic
138#501—22no—
139#500—21no—
140#498—21no—
141#489—22no—
142#487—21no—
143#486—21no—
144#428C22×C314yes—
145C2×#104C22×C314yes—
146#417C22×C314yes—
147C2×#100C22×C312yes—
148C4×M16C22×C311yescharacteristic
149C4×M16C22×C311yescharacteristic
150#399—22no—
151#390C22×C312yes—
152#384C22×C312yes—
153C4×C4⋊C4C22×C311yescharacteristic
154#343C22×C312yes—
155#339C22×C314yes—
156C24×C3C2×D412yes—
157C3×#41—24no—
158C3×#41—24no—
159C3×#41—84no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—44no—
163C2×C3×Q8—41no—
164C2×C3×Q8—41no—
165C2×C3×Q8C2×D412yes—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4C2×D412yes—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—44no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16#4114yescomplemented
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—48no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—48no—
219C3×C42—24no—
220C3×C42—22no—
221C3×C42—24no—
222C3×C42C2×D411yescharacteristic
223C3×C42C2×D411yescharacteristic
224C3×C42C2×D411yescharacteristic
225C3×C42C2×D411yescharacteristic
226C2×#41—22no—
227C2×#41C23×C312yes—
228C22×Q8—21no—
229C22×D4C23×C311yescharacteristic
230C23×C4C3×D412yes—
231C2×M16C23×C312yes—
232C2×M16C23×C312yes—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8C3×D411yescharacteristic
238C4⋊Q8C3×D411yescharacteristic
239C4⋊Q8C3×D411yescharacteristic
240#127—22no—
241#127C3×D411yescharacteristic
242#126—24no—
243#125C3×D412yes—
244#124—24no—
245#124—22no—
246#124C3×D412yes—
247#123—24no—
248#123—24no—
249C22⋊Q8—22no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—48no—
253#121—22no—
254#121—24no—
255#121—21no—
256#121—21no—
257#121—22no—
258#121—22no—
259#120—24no—
260C4×Q8—22no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—24no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272#117—22no—
273#117—22no—
274#117C23×C312yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C23×C312yes—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C22⋊C4—24no—
282C2×C22⋊C4C3×D412yes—
283C2×C42C23×C312yes—
284C2×C42C23×C311yescharacteristic
285#107—24no—
286#106—24no—
287#104—28no—
288#103—24no—
289#102—24no—
290#100—24no—
291#97—21no—
292#97—21no—
293C4×C8—21no—
294C4×C8—21no—
295#95—21no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C3×Q8—24no—
304C3×Q8—42no—
305C3×Q8—81no—
306C3×Q8—82no—
307C3×Q8—82no—
308C3×D4—24no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—81no—
315C3×D4—84no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×C3×C4—42no—
329C2×C3×C4—44no—
330C2×C3×C4—42no—
331C2×C3×C4—82no—
332C2×C3×C4—84no—
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4#12011yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4112yes—
340C2×C3×C42−1+412yes—
341C3×C8—24no—
342C3×C8—42no—
343C24C2×C3×D412yes—
344#41—24no—
345#41—24no—
346#41—84no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—44no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8C2×C3×D412yes—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4C2×C3×D412yes—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C24×C311yescharacteristic
374M16—22no—
375M16—22no—
376M16—22no—
377M16C3×#4114yescomplemented
378C2×C8—22no—
379C2×C8—22no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—48no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4C2×C3×D412yes—
396C4⋊C4C2×C3×D412yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—48no—
406C42—24no—
407C42—22no—
408C42—24no—
409C42C2×C3×D411yescharacteristic
410C42C2×C3×D411yescharacteristic
411C42C2×C3×D411yescharacteristic
412C42C2×C3×D411yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—82no—
423C22×C3C2×#12011yescharacteristic
424C22×C3#53211yescharacteristic
425C22×C3#53611yescharacteristic
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—44no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—82no—
435C3×C4#45312yes—
436C3×C4#54712yes—
437C3×C4#54812yes—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23C22×C3×D411yescharacteristic
445Q8—24no—
446Q8—42no—
447Q8—81no—
448Q8—82no—
449Q8—82no—
450D4—24no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—81no—
457D4—84no—
458D4—84no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—84no—
475C2×C4C3×#12011yescharacteristic
476C2×C4C3×#12011yescharacteristic
477C2×C4C3×#12011yescharacteristic
478C2×C4C3×#12011yescharacteristic
479C2×C4C22×C3×D411yescharacteristic
480C2×C4C22×C3×D411yescharacteristic
481C2×C4C2×C3×#4112yes—
482C2×C4C3×2−1+412yes—
483C8—24no—
484C8—42no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—82no—
488C2×C3—82no—
489C2×C3#243211yescharacteristic
490C2×C3C2×#45311yescharacteristic
491C2×C3#302211yescharacteristic
492C22—21no—
493C22—21no—
494C22—44no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22C2×C3×#12011yescharacteristic
503C22C3×#53211yescharacteristic
504C22C3×#53611yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—44no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—82no—
514C4C3×#45312yes—
515C4C3×#54712yes—
516C4C3×#54812yes—
517C3#2569411yescharacteristic, direct product factor, Sylow
518C2—21no—
519C2—21no—
520C2—82no—
521C2—82no—
522C2C3×#243211yescharacteristic
523C2C2×C3×#45311yescharacteristic
524C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3506C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3165C2semidirect
3. C3×#3141C2semidirect
4. C3×#2542C2semidirect
5. C3×#2037C2semidirect
6. C3×#1986C2semidirect
7. C3×#1980C2semidirect
8. C3×#1859C2semidirect
9. C3×C4×#100C2semidirect
10. #25694C3direct (N × H)
45. #3165C2×C3semidirect
46. #3141C2×C3semidirect
47. #2542C2×C3semidirect
48. #2037C2×C3semidirect
49. #1986C2×C3semidirect
50. #1980C2×C3semidirect
51. #1859C2×C3semidirect
52. C4×#100C2×C3semidirect
190. C3×M16#41semidirect
377. M16C3×#41semidirect
517. C3#25694direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#31652
C3×#31412
C3×#25422
C3×#20372
C3×#19862
C3×#19802
C3×#18592
C3×C4×#1002
#256943