Finite Group Databasebeta

← C3×#78697 #11857660

Subgroups — C3×#78697 #11857660

Order 1152.

1605 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1004C211yesmaximal, characteristic
2C3×#5745C212yesmaximal, complemented
3C2×C32×#100C211yesmaximal, characteristic, complemented
4C2×C3×#833C211yesmaximal, characteristic, complemented
5C3×#5132C212yesmaximal, complemented
6C3×#4745C214yesmaximal, complemented
7C3×D3×#100C214yesmaximal, complemented
8C3×#2888—31nomaximal
9#78697C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C22×C3×#67C414yes—
12C3×#1004—24no—
13C3×#1004—24no—
14C2×C3×#237C412yes—
15C2×C3×#237C2212yes—
16C2×C3×D3×D4C412yes—
17C2×C3×D3×D4C2212yes—
18C2×C3×#235C2211yescharacteristic
19C2×C3×#235C2211yescharacteristic
20C2×C32×M16C2212yescomplemented
21C2×C3×#208C2212yescomplemented
22C3×#907—24no—
23C3×D3×M16C2218yescomplemented
24C3×#901—24no—
25C32×#100C2214yescomplemented
26C3×#833C2214yescomplemented
27C3×#824C2214yescomplemented
28C3×#823C2214yescomplemented
29C2×C3×H(5,𝔽2)—31no—
30C2×#1004C2×C311yescharacteristic
31C3×#568—32no—
32#5745C2×C312yescomplemented
33C3×#413—32no—
34C3×#413—34no—
35C2×C3×#100—21no—
36C2×C3×#100—31no—
37C2×C3×#100—34no—
38C2×C3×#100D311yescharacteristic, complemented
39C2×C3×#100C2×C311yescharacteristic, complemented
40C2×#833C2×C311yescharacteristic, complemented
41#5132C2×C312yescomplemented
42#4745C2×C314yescomplemented
43D3×#100C2×C314yescomplemented
44C24×C32C2×C412yes—
45C23×C3×D3C2×C414yes—
46C2×C32×D4—22no—
47C2×C32×D4—22no—
48C2×C32×D4C2×C412yes—
49C2×C32×D4C2312yes—
50C22×C32×C4C2311yescharacteristic
51C2×C3×#67—28no—
52C2×C3×#67—28no—
53C2×C3×#67C2×C412yes—
54C2×C3×#67C2×C414yes—
55C2×C3×#67C2×C414yes—
56C2×C3×#67C2×C412yes—
57C22×C3×Dic3C2×C414yes—
58C3×#237—28no—
59C3×#237—28no—
60C3×D3×D4—28no—
61C3×D3×D4—28no—
62C3×#235—22no—
63C3×#235—22no—
64C3×#235D414yes—
65C3×#235D414yes—
66C2×C3×D12C2×C411yescharacteristic
67C2×C3×D12C2311yescharacteristic
68C2×C3×C4×D3C2×C412yes—
69C2×C3×C4×D3C2312yes—
70C2×C3×Dic6C2×C411yescharacteristic
71C2×C3×Dic6C2311yescharacteristic
72C32×M16—22no—
73C32×M16C2314yes—
74C2×C32×C8—22no—
75C3×#208—22no—
76C3×#208C2314yes—
77C2×C3×C3⋊C8—22no—
78C3×ΓL(1,𝔽25)—28no—
79C3×D3×C8—28no—
80#2888—31noSylow
81C3×H(5,𝔽2)—64no—
82C3×H(5,𝔽2)—64no—
83C2×C3×#41—32no—
84C2×C3×#41—32no—
85C2×C3×#41—31no—
86C2×C3×#41—31no—
87C22×C3×D4—21no—
88C22×C3×D4—34no—
89C22×C3×D4—32no—
90C22×C3×D4—32no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C22×C311yescharacteristic
93C22×#67C3×C414yes—
94#1004—24no—
95#1004—24no—
96C2×#237C3×C412yes—
97C2×#237C22×C312yes—
98C2×D3×D4C3×C412yes—
99C2×D3×D4C22×C312yes—
100C2×#235C22×C311yescharacteristic
101C2×#235C22×C311yescharacteristic
102C3×#131—64no—
103C3×#131—64no—
104C2×C3×M16—22no—
105C2×C3×M16—32no—
106C2×C3×M16—38no—
107C2×C3×M16C2×D312yescomplemented
108C2×C3×M16C22×C312yescomplemented
109C2×#208C22×C312yescomplemented
110#907—24no—
111D3×M16C22×C318yescomplemented
112#901—24no—
113C3×#101—34no—
114C3×#100—24no—
115C3×#100—34no—
116C3×#100—34no—
117C3×#100C2×D314yescomplemented
118C3×#100C22×C314yescomplemented
119#833C22×C314yescomplemented
120#824C22×C314yescomplemented
121#823C22×C314yescomplemented
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—82no—
125C23×C32C22×C414yes—
126C23×C32C22×C411yescharacteristic
127C22×C3×D3—28no—
128C22×C3×D3—84no—
129C22×C3×D3C22×C412yes—
130C22×C3×D3C22×C412yes—
131C32×D4—24no—
132C32×D4—24no—
133C2×C32×C4C22×C411yescharacteristic
134C2×C32×C4C2×D412yes—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2411yescharacteristic
137C3×#67—22no—
138C3×#67—22no—
139C3×#67—28no—
140C3×#67—28no—
141C3×#67C22⋊C414yes—
142C3×#67C22⋊C414yes—
143C2×C3×Dic3—28no—
144C2×C3×Dic3C22×C412yes—
145C2×C3×Dic3C22×C412yes—
146C3×D12—22no—
147C3×D12C22⋊C414yes—
148C3×C4×D3—24no—
149C3×C4×D3C2×D418yes—
150C3×Dic6—22no—
151C3×Dic6C22⋊C414yes—
152C32×C8—24no—
153C3×C3⋊C8—24no—
154C2×H(5,𝔽2)—31no—
155#568—32no—
156#413—32no—
157#413—34no—
158C2×#100—31no—
159C2×#100—34no—
160C2×#100C3×D311yescharacteristic, complemented
161C24×C3—22no—
162C24×C3—34no—
163C24×C3C4×D312yes—
164C24×C3C2×C3×C412yes—
165C23×D3C2×C3×C414yes—
166C3×#41—34no—
167C3×#41—34no—
168C3×#41—68no—
169C3×#41—68no—
170C3×#41—62no—
171C3×#41—62no—
172C2×C3×Q8—31no—
173C2×C3×Q8—31no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—32no—
181C2×C3×D4—34no—
182C2×C3×D4—34no—
183C2×C3×D4—32no—
184C2×C3×D4—31no—
185C2×C3×D4—31no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—68no—
189C2×C3×D4—68no—
190C2×C3×D4—68no—
191C2×C3×D4—68no—
192C2×C3×D4C4×D312yes—
193C2×C3×D4C2×C3×C412yes—
194C2×C3×D4C22×D312yes—
195C2×C3×D4C23×C312yes—
196C22×C3×C4—21no—
197C22×C3×C4—32no—
198C22×C3×C4—34no—
199C22×C3×C4—32no—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C311yescharacteristic
202C2×#67—28no—
203C2×#67—28no—
204C2×#67C2×C3×C412yes—
205C2×#67C2×C3×C414yes—
206C2×#67C2×C3×C414yes—
207C2×#67C2×C3×C412yes—
208C22×Dic3C2×C3×C414yes—
209#237—28no—
210#237—28no—
211D3×D4—28no—
212D3×D4—28no—
213#235—22no—
214#235—22no—
215#235C3×D414yes—
216#235C3×D414yes—
217C2×D12C2×C3×C411yescharacteristic
218C2×D12C23×C311yescharacteristic
219C2×C4×D3C2×C3×C412yes—
220C2×C4×D3C23×C312yes—
221C2×Dic6C2×C3×C411yescharacteristic
222C2×Dic6C23×C311yescharacteristic
223C3×M16—22no—
224C3×M16—22no—
225C3×M16—24no—
226C3×M16—34no—
227C3×M16—42no—
228C3×M16—68no—
229C3×M16—62no—
230C3×M16C22×D314yes—
231C3×M16C23×C314yes—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—42no—
235C2×C3×C8—68no—
236C2×C3×C8—62no—
237#208—22no—
238#208C23×C314yes—
239C2×C3⋊C8—22no—
240ΓL(1,𝔽25)—28no—
241D3×C8—28no—
242C22×C32—24no—
243C22×C32—41no—
244C22×C32—44no—
245C22×C32—82no—
246C22×C32—82no—
247C22×C32C2×C22⋊C411yescharacteristic
248C22×C32C2×C22⋊C411yescharacteristic
249C22×C32C23×C411yescharacteristic
250C2×C3×D3—22no—
251C2×C3×D3—42no—
252C2×C3×D3—42no—
253C2×C3×D3—88no—
254C2×C3×D3C2×C22⋊C414yes—
255C32×C4C2×C22⋊C412yes—
256C32×C4C22×D412yes—
257C3×Dic3—22no—
258C3×Dic3C2×C22⋊C414yes—
259H(5,𝔽2)—64no—
260H(5,𝔽2)—64no—
261C2×#41—32no—
262C2×#41—32no—
263C2×#41—31no—
264C2×#41—31no—
265C22×D4—34no—
266C22×D4—32no—
267C22×D4—32no—
268C22×D4C2×C3×D311yescharacteristic
269#131—64no—
270#131—64no—
271C2×M16—32no—
272C2×M16—38no—
273C2×M16C2×C3×D312yescomplemented
274#101—34no—
275#100—34no—
276#100—34no—
277#100C2×C3×D314yescomplemented
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—21no—
283C23×C3—24no—
284C23×C3—32no—
285C23×C3—32no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—68no—
289C23×C3—82no—
290C23×C3—82no—
291C23×C3—84no—
292C23×C3—244no—
293C23×C3C2×C4×D314yes—
294C23×C3C2×C4×D311yescharacteristic
295C23×C3C22×C3×C414yes—
296C23×C3C22×C3×C411yescharacteristic
297C22×D3—28no—
298C22×D3—84no—
299C22×D3C22×C3×C412yes—
300C22×D3C22×C3×C412yes—
301C3×Q8—34no—
302C3×Q8—62no—
303C3×D4—24no—
304C3×D4—24no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—34no—
308C3×D4—34no—
309C3×D4—34no—
310C3×D4—44no—
311C3×D4—44no—
312C3×D4—68no—
313C3×D4—68no—
314C3×D4—62no—
315C3×D4—62no—
316C3×D4—62no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—32no—
322C2×C3×C4—38no—
323C2×C3×C4—32no—
324C2×C3×C4—68no—
325C2×C3×C4—64no—
326C2×C3×C4C2×C4×D311yescharacteristic
327C2×C3×C4D3×D412yes—
328C2×C3×C4D3×D412yes—
329C2×C3×C4C22×C3×C411yescharacteristic
330C2×C3×C4C2×C3×D412yes—
331C2×C3×C4C2×C3×D412yes—
332C2×C3×C4C23×D311yescharacteristic
333C2×C3×C4C24×C311yescharacteristic
334#67—22no—
335#67—22no—
336#67—28no—
337#67—28no—
338#67C3×C22⋊C414yes—
339#67C3×C22⋊C414yes—
340C2×Dic3—28no—
341C2×Dic3C22×C3×C412yes—
342C2×Dic3C22×C3×C412yes—
343D12—22no—
344D12C3×C22⋊C414yes—
345C4×D3—24no—
346C4×D3C2×C3×D418yes—
347Dic6—22no—
348Dic6C3×C22⋊C414yes—
349C3×C8—24no—
350C3×C8—24no—
351C3×C8—44no—
352C3×C8—64no—
353C3⋊C8—24no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—44no—
358C2×C32C22×C22⋊C411yescharacteristic
359C3×D3—24no—
360C3×D3—42no—
361C24—34no—
362C24C3×C4×D312yes—
363#41—34no—
364#41—34no—
365#41—68no—
366#41—68no—
367#41—62no—
368#41—62no—
369C2×Q8—31no—
370C2×Q8—31no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—32no—
374C2×D4—34no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—31no—
378C2×D4—31no—
379C2×D4—68no—
380C2×D4—68no—
381C2×D4—68no—
382C2×D4—68no—
383C2×D4C3×C4×D312yes—
384C2×D4C22×C3×D312yes—
385C22×C4—32no—
386C22×C4—34no—
387C22×C4—32no—
388C22×C4C22×C3×D311yescharacteristic
389M16—22no—
390M16—34no—
391M16—68no—
392M16—62no—
393M16C22×C3×D314yes—
394C2×C8—22no—
395C2×C8—68no—
396C2×C8—62no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—34no—
403C22×C3—44no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—42no—
409C22×C3—62no—
410C22×C3—84no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—82no—
415C22×C3—84no—
416C22×C3—84no—
417C22×C3—122no—
418C22×C3—122no—
419C22×C3—248no—
420C22×C3D3×C22⋊C411yescharacteristic
421C22×C3D3×C22⋊C411yescharacteristic
422C22×C3C2×C3×C22⋊C411yescharacteristic
423C22×C3C2×C3×C22⋊C411yescharacteristic
424C22×C3C22×C4×D311yescharacteristic
425C22×C3C23×C3×C411yescharacteristic
426C2×D3—22no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—88no—
430C2×D3C2×C3×C22⋊C414yes—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—34no—
434C3×C4—62no—
435C3×C4D3×C22⋊C412yes—
436C3×C4C2×C3×C22⋊C412yes—
437C3×C4C2×D3×D412yes—
438C3×C4C22×C3×D412yes—
439Dic3—22no—
440Dic3C2×C3×C22⋊C414yes—
441C32#288811yescharacteristic, complemented, Sylow
442C23—22no—
443C23—22no—
444C23—32no—
445C23—32no—
446C23—68no—
447C23—82no—
448C23—244no—
449C23C2×C3×C4×D314yes—
450C23C2×C3×C4×D311yescharacteristic
451Q8—34no—
452Q8—62no—
453D4—24no—
454D4—24no—
455D4—34no—
456D4—34no—
457D4—34no—
458D4—68no—
459D4—68no—
460D4—62no—
461D4—62no—
462D4—62no—
463C2×C4—32no—
464C2×C4—38no—
465C2×C4—32no—
466C2×C4—68no—
467C2×C4—64no—
468C2×C4C2×C3×C4×D311yescharacteristic
469C2×C4C3×D3×D412yes—
470C2×C4C3×D3×D412yes—
471C2×C4C23×C3×D311yescharacteristic
472C8—24no—
473C8—64no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—44no—
485C2×C3—44no—
486C2×C3—64no—
487C2×C3—84no—
488C2×C3—122no—
489C2×C3C2×D3×C22⋊C411yescharacteristic
490C2×C3C22×C3×C22⋊C411yescharacteristic
491D3—24no—
492D3—42no—
493C22—24no—
494C22—34no—
495C22—41no—
496C22—44no—
497C22—62no—
498C22—82no—
499C22—82no—
500C22—122no—
501C22—122no—
502C22—248no—
503C22C3×D3×C22⋊C411yescharacteristic
504C22C3×D3×C22⋊C411yescharacteristic
505C22C22×C3×C4×D311yescharacteristic
506C4—34no—
507C4—62no—
508C4C3×D3×C22⋊C412yes—
509C4C2×C3×D3×D412yes—
510C3—21no—
511C3#7869711yescharacteristic, direct product factor
512C3C3×#288811yescharacteristic, complemented
513C2—21no—
514C2—21no—
515C2—21no—
516C2—44no—
517C2—64no—
518C2—122no—
519C2C2×C3×D3×C22⋊C411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×C22⋊C4
commutatorC22×C3C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#100C2
socleC2×C32C22×C22⋊C4
prime radicalC2×C3×#1004C2
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#100D3
hypocenterC3C3×#2888
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
2. C3×#5745C2semidirect
3. C2×C32×#100C2semidirect
4. C2×C3×#833C2semidirect
5. C3×#5132C2semidirect
6. C3×#4745C2semidirect
7. C3×D3×#100C2semidirect
9. #78697C3direct (N × H)
20. C2×C32×M16C22semidirect
21. C2×C3×#208C22semidirect
23. C3×D3×M16C22semidirect
25. C32×#100C22semidirect
26. C3×#833C22semidirect
27. C3×#824C22semidirect
28. C3×#823C22semidirect
32. #5745C2×C3semidirect
38. C2×C3×#100D3semidirect
39. C2×C3×#100C2×C3semidirect
40. C2×#833C2×C3semidirect
41. #5132C2×C3semidirect
42. #4745C2×C3semidirect
43. D3×#100C2×C3semidirect
107. C2×C3×M16C2×D3semidirect
108. C2×C3×M16C22×C3semidirect
109. C2×#208C22×C3semidirect
111. D3×M16C22×C3semidirect
117. C3×#100C2×D3semidirect
118. C3×#100C22×C3semidirect
119. #833C22×C3semidirect
120. #824C22×C3semidirect
121. #823C22×C3semidirect
160. C2×#100C3×D3semidirect
273. C2×M16C2×C3×D3semidirect
277. #100C2×C3×D3semidirect
441. C32#2888semidirect
511. C3#78697direct (N × H)
512. C3C3×#2888semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10042
C3×#57452
C2×C32×#1002
C2×C3×#8332
C3×#51322
C3×#47452
C3×D3×#1002
C3×#28883
#786973