Finite Group Databasebeta

← C3×#33045 #10651652

Subgroups — C3×#33045 #10651652

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic, complemented
2C2×C3×#405C211yesmaximal, characteristic, complemented
3C3×#2628C211yesmaximal, characteristic, complemented
4C3×#2619C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6C3×#1961C211yesmaximal, characteristic, complemented
7C3×#1936C212yesmaximal, complemented
8C3×#1934C212yesmaximal, complemented
9C3×#1895C212yesmaximal, complemented
10C3×#1893C212yesmaximal, complemented
11C3×#1835C211yesmaximal, characteristic, complemented
12#33045C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121C2212yes—
18C2×C3×#117C2211yescharacteristic
19C3×#486—24no—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C3×#405—24no—
25C2×C3×#97C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#400—21no—
30C3×#399—21no—
31C3×#395—21no—
32C3×#394—21no—
33C3×#394—21no—
34C3×#393—21no—
35C3×#392—21no—
36C3×#392—21no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#382C2211yescharacteristic
40C3×#382C2211yescharacteristic
41C3×#380C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#486C2×C311yescharacteristic, complemented
47C2×#405C2×C311yescharacteristic, complemented
48#2628C2×C311yescharacteristic, complemented
49#2619C2×C311yescharacteristic
50#1993C2×C311yescharacteristic, complemented
51#1961C2×C311yescharacteristic, complemented
52#1936C2×C312yescomplemented
53#1934C2×C312yescomplemented
54#1895C2×C312yescomplemented
55#1893C2×C312yescomplemented
56#1835C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C2×C3×M16—22no—
64C2×C3×M16—22no—
65C2×C3×M16—22no—
66C22×C3×C8—22no—
67C22×C3×C8—22no—
68C22×C3×C8—22no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×#127—22no—
76C3×#127—22no—
77C3×#121—24no—
78C3×#121—24no—
79C3×#117—22no—
80C3×#117—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C3×#102—24no—
97C3×#102—24no—
98C3×#102—24no—
99C3×#102—24no—
100C3×#97—22no—
101C3×#97—22no—
102C3×C4×C8—22no—
103C3×C4×C8D414yescomplemented
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—22no—
110C22×M16C22×C311yescharacteristic
111C23×C8C22×C311yescharacteristic
112C2×C4⋊Q8C22×C311yescharacteristic
113C2×#127C22×C311yescharacteristic
114C2×#121C22×C312yes—
115C2×#117C22×C311yescharacteristic
116#486—24no—
117C2×#102C22×C312yes—
118C2×#102C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121#405—24no—
122C2×#97C22×C311yescharacteristic
123C2×#97C22×C311yescharacteristic
124C2×C4×C8C22×C311yescharacteristic
125C2×C4×C8C22×C311yescharacteristic
126#400—21no—
127#399—21no—
128#395—21no—
129#394—21no—
130#394—21no—
131#393—21no—
132#392—21no—
133#392—21no—
134#390C22×C312yes—
135#389C22×C312yes—
136#382C22×C311yescharacteristic
137#382C22×C311yescharacteristic
138#380C22×C312yes—
139#340C22×C312yes—
140#340C22×C312yes—
141#340C22×C312yes—
142#340C22×C312yes—
143C24×C3—22no—
144C24×C3C2×D411yescharacteristic
145C2×C3×Q8—42no—
146C2×C3×Q8—82no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—44no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—82no—
157C2×C3×D4—82no—
158C2×C3×D4—82no—
159C2×C3×D4—82no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4#4112yes—
173C22×C3×C4#4112yes—
174C22×C3×C4C2411yescharacteristic
175C3×M16—28no—
176C2×C3×C8—28no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8C2×D412yes—
186C2×C3×C8C2×D412yes—
187C2×C3×C8#4112yes—
188C2×C3×C8#4112yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—82no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—82no—
203C3×C22⋊C4—82no—
204C3×C22⋊C4—82no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42C2×D412yes—
208C3×C42#4112yes—
209C22×Q8—21no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C23×C312yes—
214C23×C4C23×C311yescharacteristic
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8C23×C311yescharacteristic
222C22×C8C23×C311yescharacteristic
223C22×C8C23×C311yescharacteristic
224C22×C8C23×C311yescharacteristic
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227#127—22no—
228#127—22no—
229#121—24no—
230#121—24no—
231#117—22no—
232#117—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C3×D411yescharacteristic
236C2×C4⋊C4C3×D411yescharacteristic
237C2×C4⋊C4C23×C312yes—
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C312yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C3×D411yescharacteristic
245C2×C22⋊C4C3×D411yescharacteristic
246C2×C42C23×C311yescharacteristic
247C2×C42C23×C311yescharacteristic
248#102—24no—
249#102—24no—
250#102—24no—
251#102—24no—
252#97—22no—
253#97—22no—
254C4×C8—22no—
255C4×C8C3×D414yescomplemented
256#95—22no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—22no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3#12412yes—
274C23×C3C22×D411yescharacteristic
275C3×Q8—84no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—84no—
281C3×D4—84no—
282C3×D4—88no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12112yes—
305C2×C3×C4#12112yes—
306C2×C3×C4#12412yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C3×C8—22no—
314C3×C8—22no—
315C3×C8—22no—
316C3×C8—22no—
317C24—22no—
318C24C2×C3×D411yescharacteristic
319C2×Q8—42no—
320C2×Q8—82no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C3×#4112yes—
347C22×C4C3×#4112yes—
348C22×C4C24×C311yescharacteristic
349M16—28no—
350C2×C8—28no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8C2×C3×D412yes—
360C2×C8C2×C3×D412yes—
361C2×C8C3×#4112yes—
362C2×C8C3×#4112yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—44no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—21no—
380C42—21no—
381C42C2×C3×D412yes—
382C42C3×#4112yes—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3C2×#12111yescharacteristic
396C22×C3C2×#12411yescharacteristic
397C22×C3#53211yescharacteristic
398C22×C3#57612yes—
399C22×C3#57712yes—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—82no—
406C3×C4—82no—
407C3×C4—82no—
408C3×C4#52511yescharacteristic
409C3×C4#52511yescharacteristic
410C3×C4#52911yescharacteristic
411C3×C4#52911yescharacteristic
412C23—22no—
413C23—22no—
414C23—21no—
415C23—21no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C3×#12412yes—
424C23C22×C3×D411yescharacteristic
425Q8—84no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—84no—
431D4—84no—
432D4—88no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4C3×#12112yes—
455C2×C4C3×#12112yes—
456C2×C4C3×#12412yes—
457C2×C4C22×C3×D411yescharacteristic
458C2×C4C2×C3×#4111yescharacteristic
459C2×C4C2×C3×#4111yescharacteristic
460C2×C4C2×C3×#4111yescharacteristic
461C2×C4C2×C3×#4111yescharacteristic
462C2×C4C2×C3×#4111yescharacteristic
463C8—22no—
464C8—22no—
465C8—22no—
466C8—22no—
467C2×C3—42no—
468C2×C3—162no—
469C2×C3#285011yescharacteristic
470C2×C3#305711yescharacteristic
471C2×C3#305811yescharacteristic
472C2×C3#313612yes—
473C2×C3#317912yes—
474C22—24no—
475C22—22no—
476C22—42no—
477C22—41no—
478C22—41no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22C2×C3×#12111yescharacteristic
487C22C2×C3×#12411yescharacteristic
488C22C3×#53211yescharacteristic
489C22C3×#57612yes—
490C22C3×#57712yes—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—41no—
495C4—41no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4C3×#52511yescharacteristic
500C4C3×#52511yescharacteristic
501C4C3×#52911yescharacteristic
502C4C3×#52911yescharacteristic
503C3#3304511yescharacteristic, direct product factor, Sylow
504C2—42no—
505C2—162no—
506C2C3×#285011yescharacteristic
507C2C3×#305711yescharacteristic
508C2C3×#305811yescharacteristic
509C2C3×#313612yes—
510C2C3×#317912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2619C2
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
1. C2×C3×#486C2semidirect
2. C2×C3×#405C2semidirect
3. C3×#2628C2semidirect
5. C3×#1993C2semidirect
6. C3×#1961C2semidirect
7. C3×#1936C2semidirect
8. C3×#1934C2semidirect
9. C3×#1895C2semidirect
10. C3×#1893C2semidirect
11. C3×#1835C2semidirect
12. #33045C3direct (N × H)
46. C2×#486C2×C3semidirect
47. C2×#405C2×C3semidirect
48. #2628C2×C3semidirect
50. #1993C2×C3semidirect
51. #1961C2×C3semidirect
52. #1936C2×C3semidirect
53. #1934C2×C3semidirect
54. #1895C2×C3semidirect
55. #1893C2×C3semidirect
56. #1835C2×C3semidirect
103. C3×C4×C8D4semidirect
255. C4×C8C3×D4semidirect
503. C3#33045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×#4052
C3×#26282
C3×#26192
C3×#19932
C3×#19612
C3×#19362
C3×#19342
C3×#18952
C3×#18932
C3×#18352
#330453