Finite Group Databasebeta

← C2×C3×#5222 #11808135

Subgroups — C2×C3×#5222 #11808135

Order 1152.

3185 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×D4C211yesmaximal, characteristic
2C22×C3×#217C212yesmaximal, complemented
3C22×C3×#217C212yesmaximal, complemented
4C22×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C3×#5222C218yesmaximal, direct product factor
6C2×C3×C3⋊#95C211yesmaximal, characteristic, complemented
7C2×C3×#394—31nomaximal
8C2×#5222C311yesmaximal, characteristic, direct product factor
9C25×C32C2212yes—
10C22×C32×D4—24no—
11C22×C32×D4—28no—
12C22×C32×D4C2214yes—
13C23×C32×C4C2211yescharacteristic
14C23×C3×Dic3C2211yescharacteristic
15C23×C3×Dic3C2212yes—
16C23×C3×Dic3C2211yescharacteristic
17C2×C3×#217—28no—
18C2×C3×#217—28no—
19C2×C3×#217C2218yescomplemented
20C2×C3×#217C2218yescomplemented
21C2×C3×Dic3⋊C4—24no—
22C2×C3×Dic3⋊C4C2214yescomplemented
23C3×C3⋊#95C2214yescomplemented
24C23×C3×D4—21no—
25C23×C3×D4D311yescharacteristic
26C23×C3×D4C2×C311yescharacteristic
27C22×C3×C4⋊C4—31no—
28C22×C3×C22⋊C4—32no—
29C22×C3×C22⋊C4—32no—
30C22×#217C2×C312yescomplemented
31C22×#217C2×C312yescomplemented
32C22×Dic3⋊C4C2×C311yescharacteristic, complemented
33C3×#394—38no—
34C2×C3×#95—31no—
35#5222C2×C318yesdirect product factor
36C2×C3⋊#95C2×C311yescharacteristic, complemented
37C24×C32—24no—
38C24×C32—24no—
39C24×C32—24no—
40C24×C32—24no—
41C24×C32—24no—
42C24×C32—24no—
43C24×C32D418yes—
44C24×C32C2314yes—
45C24×C32C2311yescharacteristic
46C2×C32×D4—232no—
47C2×C32×D4—216no—
48C2×C32×D4—28no—
49C22×C32×C4—24no—
50C22×C32×C4D414yes—
51C22×C32×C4C2312yes—
52C22×C3×Dic3—22no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—24no—
56C22×C3×Dic3—24no—
57C22×C3×Dic3—24no—
58C22×C3×Dic3—24no—
59C22×C3×Dic3D414yescomplemented
60C22×C3×Dic3C2312yes—
61C22×C3×Dic3C2314yes—
62C22×C3×Dic3C2312yes—
63C3×#217—216no—
64C3×#217—216no—
65C3×Dic3⋊C4—28no—
66C2×#394—31noSylow
67C25×C3—22no—
68C25×C3C2×D312yes—
69C25×C3C22×C312yes—
70C22×C3×D4—216no—
71C22×C3×D4—28no—
72C22×C3×D4—28no—
73C22×C3×D4—28no—
74C22×C3×D4—24no—
75C22×C3×D4—24no—
76C22×C3×D4—24no—
77C22×C3×D4C2×D314yes—
78C22×C3×D4C22×C314yes—
79C23×C3×C4—21no—
80C23×C3×C4—31no—
81C23×C3×C4—32no—
82C23×C3×C4—31no—
83C23×C3×C4C2×D311yescharacteristic
84C23×C3×C4C22×C311yescharacteristic
85C23×Dic3C22×C312yes—
86C23×Dic3C22×C311yescharacteristic
87C23×Dic3C22×C311yescharacteristic
88C2×C3×C4⋊C4—34no—
89C2×C3×C4⋊C4—64no—
90C2×C3×C22⋊C4—38no—
91C2×C3×C22⋊C4—38no—
92C2×C3×C22⋊C4—68no—
93C2×C3×C22⋊C4—68no—
94C2×#217—28no—
95C2×#217—28no—
96C2×#217C22×C318yescomplemented
97C2×#217C22×C318yescomplemented
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4C22×C314yescomplemented
100C3×#95—34no—
101C3⋊#95C22×C314yescomplemented
102C23×C32—216no—
103C23×C32—24no—
104C23×C32—24no—
105C23×C32—216no—
106C23×C32—216no—
107C23×C32—24no—
108C23×C32—24no—
109C23×C32—24no—
110C23×C32—24no—
111C23×C32—48no—
112C23×C32—48no—
113C23×C32—48no—
114C23×C32—48no—
115C23×C32C2×D412yes—
116C23×C32C2×D418yes—
117C23×C32C2×D414yes—
118C23×C32C2×D412yes—
119C23×C32#4112yes—
120C23×C32#4114yes—
121C23×C32C2411yescharacteristic
122C32×D4—232no—
123C2×C32×C4—28no—
124C2×C32×C4—24no—
125C2×C32×C4C2×D414yes—
126C2×C3×Dic3—28no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—44no—
135C2×C3×Dic3—48no—
136C2×C3×Dic3C2×D414yescomplemented
137C23×D4C3×D311yescharacteristic
138C22×C4⋊C4—31no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141#394—38no—
142C2×#95—31no—
143C24×C3—28no—
144C24×C3—28no—
145C24×C3—24no—
146C24×C3—24no—
147C24×C3—24no—
148C24×C3—24no—
149C24×C3—24no—
150C24×C3—24no—
151C24×C3—24no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—24no—
157C24×C3—28no—
158C24×C3—21no—
159C24×C3—24no—
160C24×C3—44no—
161C24×C3—44no—
162C24×C3—44no—
163C24×C3—44no—
164C24×C3#6718yes—
165C24×C3C3×D418yes—
166C24×C3C22×D314yes—
167C24×C3C22×D311yescharacteristic
168C24×C3C23×C314yes—
169C24×C3C23×C311yescharacteristic
170C2×C3×D4—264no—
171C2×C3×D4—232no—
172C2×C3×D4—232no—
173C2×C3×D4—232no—
174C2×C3×D4—216no—
175C2×C3×D4—216no—
176C2×C3×D4—216no—
177C2×C3×D4—28no—
178C2×C3×D4—28no—
179C22×C3×C4—28no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—34no—
185C22×C3×C4—32no—
186C22×C3×C4—34no—
187C22×C3×C4—32no—
188C22×C3×C4—62no—
189C22×C3×C4—62no—
190C22×C3×C4—64no—
191C22×C3×C4—64no—
192C22×C3×C4—64no—
193C22×C3×C4—64no—
194C22×C3×C4—62no—
195C22×C3×C4#6714yes—
196C22×C3×C4C3×D414yes—
197C22×C3×C4C22×D312yes—
198C22×C3×C4C23×C312yes—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3C3×D414yescomplemented
207C22×Dic3C23×C314yes—
208C22×Dic3C23×C312yes—
209C22×Dic3C23×C312yes—
210C3×C4⋊C4—68no—
211C3×C22⋊C4—616no—
212C3×C22⋊C4—616no—
213#217—216no—
214#217—216no—
215Dic3⋊C4—28no—
216C22×C32—28no—
217C22×C32—28no—
218C22×C32—28no—
219C22×C32—44no—
220C22×C32—44no—
221C22×C32—44no—
222C22×C32—44no—
223C22×C32—432no—
224C22×C32#12014yes—
225C22×C32#12114yes—
226C22×C32#12118yes—
227C22×C32#12318yes—
228C22×C32#12414yes—
229C22×C32C22×D411yescharacteristic
230C22×C32C22×D412yes—
231C22×C32C22×D411yescharacteristic
232C22×C32C2×#4111yescharacteristic
233C22×C32C2×#4112yes—
234C32×C4—24no—
235C3×Dic3—24no—
236C3×Dic3—42no—
237C3×Dic3—44no—
238C25C2×C3×D312yes—
239C22×D4—28no—
240C22×D4—24no—
241C22×D4C2×C3×D314yes—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4C2×C3×D311yescharacteristic
246C2×C4⋊C4—34no—
247C2×C4⋊C4—64no—
248C2×C22⋊C4—38no—
249C2×C22⋊C4—38no—
250C2×C22⋊C4—68no—
251C2×C22⋊C4—68no—
252#95—34no—
253C23×C3—232no—
254C23×C3—28no—
255C23×C3—28no—
256C23×C3—216no—
257C23×C3—216no—
258C23×C3—216no—
259C23×C3—216no—
260C23×C3—216no—
261C23×C3—216no—
262C23×C3—24no—
263C23×C3—24no—
264C23×C3—24no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3—24no—
270C23×C3—24no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—22no—
280C23×C3—28no—
281C23×C3—416no—
282C23×C3—416no—
283C23×C3—44no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3—44no—
287C23×C3—416no—
288C23×C3—416no—
289C23×C3—416no—
290C23×C3—416no—
291C23×C3—48no—
292C23×C3—48no—
293C23×C3—48no—
294C23×C3—48no—
295C23×C3—48no—
296C23×C3—48no—
297C23×C3—48no—
298C23×C3—48no—
299C23×C3D3×D412yes—
300C23×C3#23712yes—
301C23×C3#23714yes—
302C23×C3C2×#6718yes—
303C23×C3C2×#6714yes—
304C23×C3C2×#6712yes—
305C23×C3C2×C3×D412yes—
306C23×C3C2×C3×D418yes—
307C23×C3C2×C3×D414yes—
308C23×C3C2×C3×D412yes—
309C23×C3C3×#4112yes—
310C23×C3C3×#4114yes—
311C23×C3C23×D311yescharacteristic
312C23×C3C24×C311yescharacteristic
313C3×D4—264no—
314C3×D4—232no—
315C3×D4—232no—
316C2×C3×C4—216no—
317C2×C3×C4—28no—
318C2×C3×C4—28no—
319C2×C3×C4—28no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—34no—
324C2×C3×C4—68no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—64no—
330C2×C3×C4—64no—
331C2×C3×C4—62no—
332C2×C3×C4—128no—
333C2×C3×C4—124no—
334C2×C3×C4C2×#6714yes—
335C2×C3×C4C2×C3×D414yes—
336C2×Dic3—28no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—44no—
345C2×Dic3—48no—
346C2×Dic3C2×C3×D414yescomplemented
347C2×C32—48no—
348C2×C32#39418yescomplemented
349C2×C32C2×#12011yescharacteristic
350C2×C32C2×#12111yescharacteristic
351C2×C32C2×#12112yes—
352C2×C32C2×#12312yes—
353C2×C32C2×#12411yescharacteristic
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24C3×#6718yes—
361C24C22×C3×D314yes—
362C24C22×C3×D311yescharacteristic
363C2×D4—232no—
364C2×D4—216no—
365C2×D4—28no—
366C22×C4—24no—
367C22×C4—32no—
368C22×C4—34no—
369C22×C4—34no—
370C22×C4—32no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4—64no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4C3×#6714yes—
379C22×C4C22×C3×D312yes—
380C4⋊C4—68no—
381C22⋊C4—616no—
382C22⋊C4—616no—
383C22×C3—216no—
384C22×C3—28no—
385C22×C3—28no—
386C22×C3—28no—
387C22×C3—28no—
388C22×C3—28no—
389C22×C3—28no—
390C22×C3—28no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—24no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—48no—
401C22×C3—48no—
402C22×C3—48no—
403C22×C3—48no—
404C22×C3—48no—
405C22×C3—48no—
406C22×C3—464no—
407C22×C3—432no—
408C22×C3—432no—
409C22×C3—44no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3#93518yes—
418C22×C3#93614yes—
419C22×C3#93814yes—
420C22×C3#93918yes—
421C22×C3#95314yes—
422C22×C3C3×#12014yes—
423C22×C3C3×#12114yes—
424C22×C3C3×#12118yes—
425C22×C3C3×#12318yes—
426C22×C3C3×#12414yes—
427C22×C3C2×D3×D411yescharacteristic
428C22×C3C2×#23711yescharacteristic
429C22×C3C2×#23712yes—
430C22×C3C22×#6712yes—
431C22×C3C22×#6711yescharacteristic
432C22×C3C22×C3×D411yescharacteristic
433C22×C3C22×C3×D412yes—
434C22×C3C22×C3×D411yescharacteristic
435C22×C3C2×C3×#4111yescharacteristic
436C22×C3C2×C3×#4112yes—
437C3×C4—28no—
438C3×C4—24no—
439C3×C4—24no—
440C3×C4—64no—
441C3×C4—124no—
442C3×C4—122no—
443Dic3—24no—
444Dic3—42no—
445Dic3—44no—
446C32C2×#39411yescharacteristic, complemented, Sylow
447C23—216no—
448C23—216no—
449C23—216no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—48no—
457C23—48no—
458C23—48no—
459C23—48no—
460C23C3×D3×D412yes—
461C23C3×#23712yes—
462C23C3×#23714yes—
463C23C2×C3×#6718yes—
464C23C2×C3×#6714yes—
465C23C2×C3×#6712yes—
466C23C23×C3×D311yescharacteristic
467D4—232no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—34no—
471C2×C4—68no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—64no—
476C2×C4—64no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—128no—
480C2×C4—124no—
481C2×C4C2×C3×#6714yes—
482C2×C3—22no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—22no—
486C2×C3—28no—
487C2×C3—21no—
488C2×C3—416no—
489C2×C3—48no—
490C2×C3—48no—
491C2×C3#522218yesdirect product factor
492C2×C3C3×#39418yescomplemented
493C2×C3C2×#93512yes—
494C2×C3C2×#93611yescharacteristic
495C2×C3C2×#93811yescharacteristic
496C2×C3C2×#93912yes—
497C2×C3C2×#95311yescharacteristic
498C2×C3C2×C3×#12011yescharacteristic
499C2×C3C2×C3×#12111yescharacteristic
500C2×C3C2×C3×#12112yes—
501C2×C3C2×C3×#12312yes—
502C2×C3C2×C3×#12411yescharacteristic
503C22—28no—
504C22—28no—
505C22—28no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—432no—
511C22C3×#93518yes—
512C22C3×#93614yes—
513C22C3×#93814yes—
514C22C3×#93918yes—
515C22C3×#95314yes—
516C22C2×C3×D3×D411yescharacteristic
517C22C2×C3×#23711yescharacteristic
518C22C2×C3×#23712yes—
519C22C22×C3×#6712yes—
520C22C22×C3×#6711yescharacteristic
521C4—24no—
522C4—64no—
523C4—124no—
524C4—122no—
525C3—21no—
526C3C2×#522211yescharacteristic, direct product factor
527C3C2×C3×#39411yescharacteristic, complemented
528C2—48no—
529C2C3×#522218yesdirect product factor
530C2C2×C3×#93512yes—
531C2C2×C3×#93611yescharacteristic
532C2C2×C3×#93811yescharacteristic
533C2C2×C3×#93912yes—
534C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
Fitting, prime radicalC23×C32×D4C2
socleC24×C32C23
hypercenterC23×C3×D4D3
hypocenterC3C2×C3×#394
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C23×C3×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#217C2semidirect
3. C22×C3×#217C2semidirect
4. C22×C3×Dic3⋊C4C2semidirect
5. C3×#5222C2direct (N × H)
6. C2×C3×C3⋊#95C2semidirect
8. C2×#5222C3direct (N × H)
19. C2×C3×#217C22semidirect
20. C2×C3×#217C22semidirect
22. C2×C3×Dic3⋊C4C22semidirect
23. C3×C3⋊#95C22semidirect
30. C22×#217C2×C3semidirect
31. C22×#217C2×C3semidirect
32. C22×Dic3⋊C4C2×C3semidirect
35. #5222C2×C3direct (N × H)
36. C2×C3⋊#95C2×C3semidirect
59. C22×C3×Dic3D4semidirect
96. C2×#217C22×C3semidirect
97. C2×#217C22×C3semidirect
99. C2×Dic3⋊C4C22×C3semidirect
101. C3⋊#95C22×C3semidirect
136. C2×C3×Dic3C2×D4semidirect
206. C22×Dic3C3×D4semidirect
346. C2×Dic3C2×C3×D4semidirect
348. C2×C32#394semidirect
446. C32C2×#394semidirect
491. C2×C3#5222direct (N × H)
492. C2×C3C3×#394semidirect
526. C3C2×#5222direct (N × H)
527. C3C2×C3×#394semidirect
529. C2C3×#5222direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×D42
C22×C3×#2172
C22×C3×Dic3⋊C42
C3×#52222
C2×C3×C3⋊#952
C2×C3×#3943
C2×#52223