Finite Group Databasebeta

← C3×#25623 #10644230

Subgroups — C3×#25623 #10644230

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2C3×#3152C211yesmaximal, characteristic
3C3×#3139C211yesmaximal, characteristic, complemented
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2005C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic
8C3×#1882C214yesmaximal, complemented
9C3×C4×#101C211yesmaximal, characteristic, complemented
10#25623C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137C2214yes—
12C3×#559—21no—
13C3×#557—21no—
14C3×#556—21no—
15C3×#556—22no—
16C3×#554—21no—
17C3×#552—22no—
18C3×#539—22no—
19C3×#536C2212yes—
20C3×#533C2211yescharacteristic
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2212yes—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C3×#529—22no—
25C3×#520C2212yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C3×#497—22no—
28C3×#494—21no—
29C3×#492—21no—
30C3×#491—22no—
31C3×#487—21no—
32C3×#487—21no—
33C2×C3×#104C2214yes—
34C3×#419C2214yes—
35C2×C3×#101C2212yes—
36C3×C4×M16C2211yescharacteristic
37C3×C4×M16C2211yescharacteristic
38C3×#398—22no—
39C3×#391C2212yes—
40C3×#389—22no—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3507C2×C311yescharacteristic
45#3152C2×C311yescharacteristic
46#3139C2×C311yescharacteristic, complemented
47#2546C2×C311yescharacteristic, complemented
48#2005C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic
51#1882C2×C314yescomplemented
52C4×#101C2×C311yescharacteristic, complemented
53C2×C3×#41C2312yes—
54C22×C3×Q8C2312yes—
55C22×C3×Q8C2311yescharacteristic
56C3×#137—28no—
57C3×#137—44no—
58C2×C3×Q16—24no—
59C2×C3×SD16—24no—
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—44no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×C4⋊Q8D411yescharacteristic
71C3×#127D411yescharacteristic
72C3×#126—24no—
73C3×#126—24no—
74C3×#125—22no—
75C3×#125—24no—
76C3×#125D412yes—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124D412yes—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×#121—21no—
88C3×#121—21no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—24no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×#117—24no—
103C3×#117—22no—
104C3×#117C2312yes—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—24no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4D412yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C42D412yes—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115C3×#104—44no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#101—24no—
119C3×#97—21no—
120C3×#97—21no—
121C3×C4×C8—21no—
122C3×C4×C8—21no—
123C3×#95—21no—
124C3×#95—24no—
125C2×#137C22×C314yes—
126#559—21no—
127#557—21no—
128#556—21no—
129#556—22no—
130#554—21no—
131#552—22no—
132#539—22no—
133#536C22×C312yes—
134#533C22×C311yescharacteristic
135#532C22×C311yescharacteristic
136C2×C4⋊Q8C22×C311yescharacteristic
137C2×C4⋊Q8C22×C312yes—
138#529—22no—
139#520C22×C312yes—
140C2×C4×Q8C22×C311yescharacteristic
141#497—22no—
142#494—21no—
143#492—21no—
144#491—22no—
145#487—21no—
146#487—21no—
147C2×#104C22×C314yes—
148#419C22×C314yes—
149C2×#101C22×C312yes—
150C4×M16C22×C311yescharacteristic
151C4×M16C22×C311yescharacteristic
152#398—22no—
153#391C22×C312yes—
154#389—22no—
155C4×C4⋊C4C22×C311yescharacteristic
156#343C22×C312yes—
157#339C22×C314yes—
158C3×#41—24no—
159C3×#41—24no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—24no—
169C2×C3×Q8—24no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C3×Q16—48no—
185C3×SD16—48no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16#4114yes—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—48no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—41no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C42—24no—
220C3×C42—22no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42—24no—
224C3×C42C2×D411yescharacteristic
225C3×C42C2×D411yescharacteristic
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C2×#41C23×C312yes—
229C22×Q8C23×C312yes—
230C22×Q8C23×C311yescharacteristic
231#137—28no—
232#137—44no—
233C2×Q16—24no—
234C2×SD16—24no—
235C2×M16C23×C312yes—
236C2×M16C23×C312yes—
237C4⋊Q8—22no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—44no—
243C4⋊Q8C3×D411yescharacteristic
244C4⋊Q8C3×D411yescharacteristic
245C4⋊Q8C3×D411yescharacteristic
246#127C3×D411yescharacteristic
247#126—24no—
248#126—24no—
249#125—22no—
250#125—24no—
251#125C3×D412yes—
252#124—22no—
253#124—22no—
254#124C3×D412yes—
255#123—24no—
256#123—24no—
257C22⋊Q8—24no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—21no—
263#121—21no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—21no—
276C4×D4—21no—
277#117—24no—
278#117—22no—
279#117C23×C312yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C3×D412yes—
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C42C3×D412yes—
288C2×C42C23×C312yes—
289C2×C42C23×C311yescharacteristic
290#104—44no—
291#103—24no—
292#103—24no—
293#101—24no—
294#97—21no—
295#97—21no—
296C4×C8—21no—
297C4×C8—21no—
298#95—21no—
299#95—24no—
300C23×C3—22no—
301C23×C3C22×D411yescharacteristic
302C3×Q8—24no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—42no—
306C3×Q8—44no—
307C3×Q8—44no—
308C3×Q8—81no—
309C3×D4—24no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—81no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
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—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—84no—
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4112yes—
339C2×C3×C4H(5,𝔽2)12yes—
340C3×C8—24no—
341C3×C8—42no—
342#41—24no—
343#41—24no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8C2×C3×D412yes—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4C2×C3×D412yes—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4C2×C3×D412yes—
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C24×C311yescharacteristic
368Q16—48no—
369SD16—48no—
370M16—22no—
371M16—22no—
372M16—22no—
373M16C3×#4114yes—
374C2×C8—22no—
375C2×C8—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—48no—
391C4⋊C4—41no—
392C4⋊C4—41no—
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—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42C2×C3×D411yescharacteristic
409C42C2×C3×D411yescharacteristic
410C42C2×C3×D411yescharacteristic
411C42C2×C3×D411yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3C2×#12011yescharacteristic
418C22×C3#53211yescharacteristic
419C22×C3#53511yescharacteristic
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—44no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—82no—
430C3×C4#45612yes—
431C3×C4#54612yes—
432C3×C4#55012yes—
433C23—22no—
434C23C22×C3×D411yescharacteristic
435Q8—24no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—44no—
440Q8—44no—
441Q8—81no—
442D4—24no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—81no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—84no—
465C2×C4C3×#12011yescharacteristic
466C2×C4C3×#12011yescharacteristic
467C2×C4C3×#12011yescharacteristic
468C2×C4C3×#12011yescharacteristic
469C2×C4C22×C3×D411yescharacteristic
470C2×C4C22×C3×D411yescharacteristic
471C2×C4C2×C3×#4112yes—
472C2×C4C3×H(5,𝔽2)12yes—
473C8—24no—
474C8—42no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—82no—
478C2×C3#243111yescharacteristic
479C2×C3C2×#45611yescharacteristic
480C2×C3#302211yescharacteristic
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22C2×C3×#12011yescharacteristic
487C22C3×#53211yescharacteristic
488C22C3×#53511yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—82no—
499C4C3×#45612yes—
500C4C3×#54612yes—
501C4C3×#55012yes—
502C3#2562311yescharacteristic, direct product factor, Sylow
503C2—21no—
504C2—21no—
505C2—82no—
506C2C3×#243111yescharacteristic
507C2C2×C3×#45611yescharacteristic
508C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#532C22
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
3. C3×#3139C2semidirect
4. C3×#2546C2semidirect
5. C3×#2005C2semidirect
6. C3×#1986C2semidirect
8. C3×#1882C2semidirect
9. C3×C4×#101C2semidirect
10. #25623C3direct (N × H)
46. #3139C2×C3semidirect
47. #2546C2×C3semidirect
48. #2005C2×C3semidirect
49. #1986C2×C3semidirect
51. #1882C2×C3semidirect
52. C4×#101C2×C3semidirect
502. C3#25623direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#31522
C3×#31392
C3×#25462
C3×#20052
C3×#19862
C3×#19802
C3×#18822
C3×C4×#1012
#256233