Finite Group Databasebeta

← C3×#70139 #11784966

Subgroups — C3×#70139 #11784966

Order 1152.

1251 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1009C211yesmaximal, characteristic
2C32×#413C211yesmaximal, characteristic, complemented
3C3×#5240C211yesmaximal, characteristic, complemented
4C3×#5139C212yesmaximal, complemented
5C3×#5131C212yesmaximal, complemented
6C3×#1795—31nomaximal
7#70139C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C3×#1009—24no—
10C3×#1009—24no—
11C2×C3×#239—21no—
12C2×C3×D3×D4—21no—
13C2×C3×D3×D4—22no—
14C2×C3×#235C412yescomplemented
15C2×C3×#235C2211yescharacteristic
16C22×C3×D12C412yescomplemented
17C22×C3×D12C2211yescharacteristic
18C2×C32×M16C2212yescomplemented
19C3×#926C2212yescomplemented
20C32×#101—21no—
21C32×#100—21no—
22C3×#834—22no—
23C3×#825—24no—
24C3×#821—24no—
25C2×C3×H(5,𝔽2)—31no—
26C2×#1009C2×C311yescharacteristic
27C3×#421—32no—
28C3×#418—32no—
29C3×#413—21no—
30C3×#413D311yescharacteristic, complemented
31C3×#413C2×C311yescharacteristic, complemented
32C3×#410—31no—
33#5240C2×C311yescharacteristic, complemented
34#5139C2×C312yescomplemented
35#5131C2×C312yescomplemented
36C23×C3×D3—22no—
37C23×C3×D3—21no—
38C32×#41—22no—
39C32×#41—22no—
40C2×C32×Q8D411yescharacteristic
41C2×C32×D4D412yes—
42C2×C32×D4D411yescharacteristic
43C22×C32×C4C2×C412yes—
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—22no—
46C2×C3×#67—21no—
47C3×#239—28no—
48C3×D3×D4—216no—
49C3×D3×D4—28no—
50C3×#235—22no—
51C3×#235—24no—
52C3×#235—24no—
53C3×#235D414yes—
54C2×C3×D12—22no—
55C2×C3×D12—21no—
56C2×C3×D12—22no—
57C2×C3×D12—24no—
58C2×C3×D12—24no—
59C2×C3×D12C2×C411yescharacteristic
60C2×C3×D12C2×C412yes—
61C2×C3×D12D414yes—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3—21no—
64C2×C3×Dic6C2×C411yescharacteristic
65C32×M16—22no—
66C32×M16—22no—
67C2×C32×C8—22no—
68C3×#217—22no—
69C3×C4⋊Dic3—22no—
70C3×C4×Dic3—22no—
71#1795—31noSylow
72C3×H(5,𝔽2)—64no—
73C3×H(5,𝔽2)—64no—
74C2×C3×#41—21no—
75C2×C3×#41—32no—
76C2×C3×#41—31no—
77C2×C3×#41—61no—
78C2×C3×#41C2×D311yescharacteristic
79C2×C3×#41C22×C311yescharacteristic
80C22×C3×D4—32no—
81C22×C3×D4—31no—
82C22×C3×D4—62no—
83C22×C3×D4—61no—
84#1009—24no—
85#1009—24no—
86C2×#239—21no—
87C2×D3×D4—22no—
88C2×D3×D4—21no—
89C2×#235C3×C412yescomplemented
90C2×#235C22×C311yescharacteristic
91C22×D12C3×C412yescomplemented
92C22×D12C22×C311yescharacteristic
93C2×C3×M16—22no—
94C2×C3×M16C2×D312yescomplemented
95C2×C3×M16C22×C312yescomplemented
96C3×#117—32no—
97#926C22×C312yescomplemented
98C3×#104—64no—
99C3×#102—64no—
100C3×#101—22no—
101C3×#101—21no—
102C3×#101—21no—
103C3×#100—22no—
104C3×#100—21no—
105C3×#100—21no—
106C3×#99—62no—
107#834—22no—
108#825—24no—
109#821—24no—
110C23×C32C22⋊C412yes—
111C23×C32C2×D411yescharacteristic
112C22×C3×D3—21no—
113C22×C3×D3—22no—
114C22×C3×D3—28no—
115C22×C3×D3—24no—
116C22×C3×D3—82no—
117C22×C3×D3—84no—
118C32×Q8—22no—
119C32×D4—24no—
120C32×D4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4C22⋊C412yes—
124C2×C32×C4C22×C411yescharacteristic
125C2×C32×C4C2×D412yes—
126C2×C32×C4C2×D412yes—
127C2×C32×C4C2×D411yescharacteristic
128C3×#67—28no—
129C3×#67—24no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—42no—
132C3×D12—28no—
133C3×D12—21no—
134C3×D12—22no—
135C3×D12—24no—
136C3×D12C22⋊C412yes—
137C3×D12C22⋊C414yes—
138C3×C4×D3—28no—
139C3×C4×D3—24no—
140C3×Dic6—21no—
141C3×Dic6C22⋊C412yes—
142C32×C8—42no—
143C2×H(5,𝔽2)—31no—
144#421—32no—
145#418—32no—
146#413C3×D311yescharacteristic, complemented
147#410—31no—
148C24×C3—62no—
149C24×C3—61no—
150C23×D3—22no—
151C23×D3—21no—
152C3×#41—22no—
153C3×#41—22no—
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—34no—
157C3×#41—42no—
158C3×#41—42no—
159C3×#41—64no—
160C3×#41—64no—
161C3×#41—68no—
162C3×#41—62no—
163C2×C3×Q8—21no—
164C2×C3×Q8—31no—
165C2×C3×Q8#6711yescharacteristic
166C2×C3×Q8C3×D411yescharacteristic
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—32no—
170C2×C3×D4—31no—
171C2×C3×D4—34no—
172C2×C3×D4—64no—
173C2×C3×D4—64no—
174C2×C3×D4—616no—
175C2×C3×D4—68no—
176C2×C3×D4—62no—
177C2×C3×D4—61no—
178C2×C3×D4—62no—
179C2×C3×D4—62no—
180C2×C3×D4—61no—
181C2×C3×D4D1212yes—
182C2×C3×D4#6711yescharacteristic
183C2×C3×D4C3×D412yes—
184C2×C3×D4C3×D411yescharacteristic
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—62no—
188C22×C3×C4—61no—
189C22×C3×C4C4×D312yes—
190C22×C3×C4C2×C3×C412yes—
191C22×C3×C4C22×D311yescharacteristic
192C22×C3×C4C23×C311yescharacteristic
193C2×#67—22no—
194C2×#67—21no—
195#239—28no—
196D3×D4—216no—
197D3×D4—28no—
198#235—22no—
199#235—24no—
200#235—24no—
201#235C3×D414yes—
202C2×D12—22no—
203C2×D12—21no—
204C2×D12—22no—
205C2×D12—24no—
206C2×D12—24no—
207C2×D12C2×C3×C411yescharacteristic
208C2×D12C2×C3×C412yes—
209C2×D12C3×D414yes—
210C2×C4×D3—22no—
211C2×C4×D3—21no—
212C2×Dic6C2×C3×C411yescharacteristic
213C3×M16—24no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C3×M16—22no—
218C3×M16—42no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—42no—
222C3×C4⋊C4—62no—
223C3×C22⋊C4—62no—
224C3×C42—62no—
225#217—22no—
226C4⋊Dic3—22no—
227C4×Dic3—22no—
228C22×C32—22no—
229C22×C32—41no—
230C22×C32—42no—
231C22×C32C2×C22⋊C411yescharacteristic
232C22×C32#12012yes—
233C2×C3×D3—24no—
234C2×C3×D3—22no—
235C2×C3×D3—44no—
236C2×C3×D3—42no—
237C2×C3×D3—84no—
238C2×C3×D3—88no—
239C32×C4—22no—
240C32×C4C2×C22⋊C411yescharacteristic
241C32×C4C2×C22⋊C411yescharacteristic
242C32×C4#12012yes—
243C3×Dic3—22no—
244C3×Dic3—82no—
245H(5,𝔽2)—64no—
246H(5,𝔽2)—64no—
247C2×#41—32no—
248C2×#41—31no—
249C2×#41—61no—
250C2×#41C2×C3×D311yescharacteristic
251C22×D4—32no—
252C22×D4—31no—
253C22×D4—62no—
254C22×D4—61no—
255C2×M16C2×C3×D312yescomplemented
256#117—32no—
257#104—64no—
258#102—64no—
259#101—21no—
260#100—21no—
261#99—62no—
262C23×C3—22no—
263C23×C3—21no—
264C23×C3—68no—
265C23×C3—62no—
266C23×C3—61no—
267C23×C3—64no—
268C23×C3—242no—
269C23×C3—244no—
270C23×C3(C2×D3)⋊C412yes—
271C23×C3C3×C22⋊C412yes—
272C23×C3C2×D1211yescharacteristic
273C23×C3C2×C3×D411yescharacteristic
274C22×D3—22no—
275C22×D3—21no—
276C22×D3—28no—
277C22×D3—24no—
278C22×D3—82no—
279C22×D3—84no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×Q8—32no—
283C3×Q8—42no—
284C3×Q8—61no—
285C3×D4—24no—
286C3×D4—24no—
287C3×D4—22no—
288C3×D4—22no—
289C3×D4—34no—
290C3×D4—32no—
291C3×D4—42no—
292C3×D4—44no—
293C3×D4—68no—
294C3×D4—68no—
295C3×D4—64no—
296C3×D4—61no—
297C3×D4—62no—
298C3×D4—64no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—68no—
311C2×C3×C4—61no—
312C2×C3×C4—64no—
313C2×C3×C4—122no—
314C2×C3×C4(C2×D3)⋊C412yes—
315C2×C3×C4C3×C22⋊C412yes—
316C2×C3×C4C2×C4×D311yescharacteristic
317C2×C3×C4D3×D412yes—
318C2×C3×C4D3×D412yes—
319C2×C3×C4C2×#6711yescharacteristic
320C2×C3×C4C22×C3×C411yescharacteristic
321C2×C3×C4C2×C3×D412yes—
322C2×C3×C4C2×C3×D412yes—
323C2×C3×C4C2×C3×D411yescharacteristic
324#67—28no—
325#67—24no—
326C2×Dic3—21no—
327C2×Dic3—42no—
328D12—28no—
329D12—21no—
330D12—22no—
331D12—24no—
332D12C3×C22⋊C412yes—
333D12C3×C22⋊C414yes—
334C4×D3—28no—
335C4×D3—24no—
336Dic6—21no—
337Dic6C3×C22⋊C412yes—
338C3×C8—44no—
339C3×C8—42no—
340C3×C8—42no—
341C2×C32—22no—
342C2×C32—21no—
343C2×C32—42no—
344C2×C32#37911yescharacteristic
345C3×D3—44no—
346C3×D3—42no—
347C24—62no—
348C24—61no—
349#41—22no—
350#41—22no—
351#41—34no—
352#41—64no—
353#41—64no—
354#41—68no—
355#41—62no—
356C2×Q8—31no—
357C2×Q8C3×#6711yescharacteristic
358C2×D4—32no—
359C2×D4—31no—
360C2×D4—34no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—616no—
364C2×D4—68no—
365C2×D4—62no—
366C2×D4—61no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—61no—
370C2×D4C3×D1212yes—
371C2×D4C3×#6711yescharacteristic
372C22×C4—62no—
373C22×C4—61no—
374C22×C4C3×C4×D312yes—
375C22×C4C22×C3×D311yescharacteristic
376M16—22no—
377M16—22no—
378C2×C8—22no—
379C4⋊C4—62no—
380C22⋊C4—62no—
381C42—62no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—21no—
386C22×C3—42no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—64no—
392C22×C3—62no—
393C22×C3—81no—
394C22×C3—82no—
395C22×C3—124no—
396C22×C3—122no—
397C22×C3—244no—
398C22×C3—248no—
399C22×C3#88212yes—
400C22×C3C2×(C2×D3)⋊C411yescharacteristic
401C22×C3C2×C3×C22⋊C411yescharacteristic
402C22×C3C3×#12012yes—
403C2×D3—24no—
404C2×D3—22no—
405C2×D3—44no—
406C2×D3—42no—
407C2×D3—84no—
408C2×D3—88no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—62no—
416C3×C4—242no—
417C3×C4D3×C22⋊C411yescharacteristic
418C3×C4D3×C22⋊C411yescharacteristic
419C3×C4#93712yes—
420C3×C4C2×C3×C22⋊C411yescharacteristic
421C3×C4C2×C3×C22⋊C411yescharacteristic
422C3×C4C3×#12012yes—
423Dic3—22no—
424Dic3—82no—
425C32#179511yescharacteristic, complemented, Sylow
426C23—68no—
427C23—62no—
428C23—61no—
429C23—64no—
430C23—242no—
431C23—244no—
432C23C3×(C2×D3)⋊C412yes—
433C23C2×C3×D1211yescharacteristic
434Q8—22no—
435Q8—32no—
436Q8—61no—
437D4—24no—
438D4—22no—
439D4—34no—
440D4—32no—
441D4—68no—
442D4—68no—
443D4—64no—
444D4—61no—
445D4—62no—
446D4—64no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—68no—
450C2×C4—61no—
451C2×C4—64no—
452C2×C4—122no—
453C2×C4C3×(C2×D3)⋊C412yes—
454C2×C4C2×C3×C4×D311yescharacteristic
455C2×C4C3×D3×D412yes—
456C2×C4C3×D3×D412yes—
457C2×C4C2×C3×#6711yescharacteristic
458C8—42no—
459C2×C3—22no—
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—41no—
465C2×C3—42no—
466C2×C3—42no—
467C2×C3—42no—
468C2×C3—82no—
469C2×C3—124no—
470C2×C3—122no—
471C2×C3#495511yescharacteristic
472C2×C3C3×#37911yescharacteristic
473D3—44no—
474D3—42no—
475C22—22no—
476C22—41no—
477C22—42no—
478C22—64no—
479C22—62no—
480C22—124no—
481C22—122no—
482C22—244no—
483C22—248no—
484C22C3×#88212yes—
485C22C2×C3×(C2×D3)⋊C411yescharacteristic
486C4—22no—
487C4—62no—
488C4—242no—
489C4C3×D3×C22⋊C411yescharacteristic
490C4C3×D3×C22⋊C411yescharacteristic
491C4C3×#93712yes—
492C3—21no—
493C3#7013911yescharacteristic, direct product factor
494C3C3×#179511yescharacteristic, complemented
495C2—22no—
496C2—21no—
497C2—42no—
498C2—124no—
499C2—122no—
500C2C3×#495511yescharacteristic
subgroupG/H
centerC2×C3#4955
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#413C2
socleC2×C32#379
prime radicalC2×C3×#1009C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#413D3
hypocenterC3C3×#1795
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#413

Complements

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

NH ≅ G/Ntype
2. C32×#413C2semidirect
3. C3×#5240C2semidirect
4. C3×#5139C2semidirect
5. C3×#5131C2semidirect
7. #70139C3direct (N × H)
14. C2×C3×#235C4semidirect
16. C22×C3×D12C4semidirect
18. C2×C32×M16C22semidirect
19. C3×#926C22semidirect
30. C3×#413D3semidirect
31. C3×#413C2×C3semidirect
33. #5240C2×C3semidirect
34. #5139C2×C3semidirect
35. #5131C2×C3semidirect
89. C2×#235C3×C4semidirect
91. C22×D12C3×C4semidirect
94. C2×C3×M16C2×D3semidirect
95. C2×C3×M16C22×C3semidirect
97. #926C22×C3semidirect
146. #413C3×D3semidirect
255. C2×M16C2×C3×D3semidirect
425. C32#1795semidirect
493. C3#70139direct (N × H)
494. C3C3×#1795semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10092
C32×#4132
C3×#52402
C3×#51392
C3×#51312
C3×#17953
#701393