Finite Group Databasebeta

← C3×#33019 #10651626

Subgroups — C3×#33019 #10651626

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic
3C2×C3×#451C211yesmaximal, characteristic
4C3×#2625C211yesmaximal, characteristic
5C3×#1878C212yesmaximal, complemented
6C3×#1878C212yesmaximal, complemented
7C3×#1865C212yesmaximal, complemented
8C3×#1863C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic
10C3×#1818C211yesmaximal, characteristic
11C3×#1787C211yesmaximal, characteristic
12#33019C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C3×#519—24no—
17C3×#519—24no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×#117—22no—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C3×#485—24no—
26C3×#451—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#400—21no—
37C3×#398—21no—
38C3×#397—22no—
39C3×#391C2212yes—
40C3×#389C2212yes—
41C3×#380C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#519C2×C311yescharacteristic
47C2×#485C2×C311yescharacteristic
48C2×#451C2×C311yescharacteristic
49#2625C2×C311yescharacteristic
50#1878C2×C312yescomplemented
51#1878C2×C312yescomplemented
52#1865C2×C312yescomplemented
53#1863C2×C312yescomplemented
54#1825C2×C311yescharacteristic
55#1818C2×C311yescharacteristic
56#1787C2×C311yescharacteristic
57C22×C3×Q8—24no—
58C22×C3×Q8—24no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2312yes—
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C4×Q8—28no—
75C3×C4×Q8—28no—
76C3×C4×Q8—24no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8—24no—
80C3×#117—216no—
81C3×#117—22no—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117D414yes—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D412yes—
99C2×C3×C42—22no—
100C2×C3×C42—22no—
101C2×C3×C42—21no—
102C2×C3×C42C2312yes—
103C3×#107—22no—
104C3×#106—22no—
105C3×#103—24no—
106C3×#103—24no—
107C3×#103—24no—
108C3×#103—22no—
109C3×#103—22no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113C3×#95—22no—
114C3×#95—22no—
115C3×#95—22no—
116C23×Q8C22×C311yescharacteristic
117C22×Q16C22×C312yes—
118C2×C22⋊Q8C22×C311yescharacteristic
119#519—24no—
120#519—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C22×C312yes—
124C2×C4×Q8C22×C312yes—
125C2×#117—22no—
126C2×#117C22×C311yescharacteristic
127C2×#117C22×C311yescharacteristic
128#485—24no—
129#451—24no—
130C2×#107C22×C311yescharacteristic
131C2×#106C22×C311yescharacteristic
132C2×#103C22×C311yescharacteristic
133C2×#103C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#400—21no—
140#398—21no—
141#397—22no—
142#391C22×C312yes—
143#389C22×C312yes—
144#380C22×C311yescharacteristic
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C312yes—
148#336C22×C312yes—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—28no—
151C2×C3×Q8—28no—
152C2×C3×Q8—28no—
153C2×C3×Q8—216no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8C2×D414yes—
163C2×C3×Q8#4114yes—
164C2×C3×Q8#4114yes—
165C22×C3×C4—28no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C3×Q16—48no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—82no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4C2×D412yes—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—42no—
206C3×C22⋊C4—81no—
207C3×C42—28no—
208C3×C42—28no—
209C3×C42—24no—
210C3×C42—22no—
211C3×C42—22no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8C3×D414yes—
217C22×Q8C23×C311yescharacteristic
218C22×Q8C23×C312yes—
219C23×C4—21no—
220C23×C4C23×C311yescharacteristic
221C2×Q16—24no—
222C2×Q16—24no—
223C2×Q16—24no—
224C22×C8C23×C312yes—
225C22×C8C23×C311yescharacteristic
226C22×C8C23×C311yescharacteristic
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C4×Q8—28no—
230C4×Q8—28no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235#117—216no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117C3×D414yes—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C3×D411yescharacteristic
245C2×C4⋊C4C3×D411yescharacteristic
246C2×C4⋊C4C23×C311yescharacteristic
247C2×C4⋊C4C23×C312yes—
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C3×D412yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42C23×C312yes—
258#107—22no—
259#106—22no—
260#103—24no—
261#103—24no—
262#103—24no—
263#103—22no—
264#103—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—21no—
275C23×C3#12012yes—
276C23×C3C22×D411yescharacteristic
277C3×Q8—216no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×Q8—24no—
281C3×Q8—42no—
282C3×Q8—41no—
283C3×Q8—41no—
284C3×Q8#12318yes—
285C2×C3×C4—216no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4#12012yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C3×C8—42no—
317C3×C8—41no—
318C3×C8—41no—
319C24C2×C3×D411yescharacteristic
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—28no—
323C2×Q8—216no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8C2×C3×D414yes—
333C2×Q8C3×#4114yes—
334C2×Q8C3×#4114yes—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D412yes—
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D412yes—
350C22×C4C24×C311yescharacteristic
351Q16—48no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—82no—
371C4⋊C4C2×C3×D412yes—
372C4⋊C4C2×C3×D412yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—81no—
377C42—28no—
378C42—28no—
379C42—24no—
380C42—22no—
381C42—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3C2×#12011yescharacteristic
388C22×C3#52512yes—
389C22×C3#57711yescharacteristic
390C22×C3#57812yes—
391C22×C3#57811yescharacteristic
392C3×C4—28no—
393C3×C4—22no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—42no—
397C3×C4—42no—
398C3×C4—81no—
399C3×C4—82no—
400C3×C4—81no—
401C3×C4C2×#12312yes—
402C3×C4#52511yescharacteristic
403C3×C4#52511yescharacteristic
404C23—21no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23C3×#12012yes—
409C23C22×C3×D411yescharacteristic
410Q8—216no—
411Q8—24no—
412Q8—24no—
413Q8—24no—
414Q8—42no—
415Q8—41no—
416Q8—41no—
417Q8C3×#12318yes—
418C2×C4—216no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4C3×#12012yes—
445C2×C4C22×C3×D411yescharacteristic
446C2×C4C22×C3×D411yescharacteristic
447C2×C4C2×C3×#4112yes—
448C2×C4C2×C3×#4112yes—
449C8—42no—
450C8—41no—
451C8—41no—
452C2×C3—42no—
453C2×C3#284911yescharacteristic
454C2×C3#301111yescharacteristic
455C2×C3#301311yescharacteristic
456C2×C3#312112yes—
457C2×C3#312712yes—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22C2×C3×#12011yescharacteristic
464C22C3×#52512yes—
465C22C3×#57711yescharacteristic
466C22C3×#57812yes—
467C22C3×#57811yescharacteristic
468C4—28no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—82no—
477C4C2×C3×#12312yes—
478C4C3×#52511yescharacteristic
479C4C3×#52511yescharacteristic
480C3#3301911yescharacteristic, direct product factor, Sylow
481C2—42no—
482C2C3×#284911yescharacteristic
483C2C3×#301111yescharacteristic
484C2C3×#301311yescharacteristic
485C2C3×#312112yes—
486C2C3×#312712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#1878C2semidirect
6. C3×#1878C2semidirect
7. C3×#1865C2semidirect
8. C3×#1863C2semidirect
12. #33019C3direct (N × H)
50. #1878C2×C3semidirect
51. #1878C2×C3semidirect
52. #1865C2×C3semidirect
53. #1863C2×C3semidirect
480. C3#33019direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×C3×#4852
C2×C3×#4512
C3×#26252
C3×#18782
C3×#18652
C3×#18632
C3×#18252
C3×#18182
C3×#17872
#330193