Finite Group Databasebeta

← C3×#68195 #11779304

Subgroups — C3×#68195 #11779304

Order 1152.

669 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×D12C211yesmaximal, characteristic
2C3×#5008C211yesmaximal, characteristic, complemented
3C3×#4842C211yesmaximal, characteristic, complemented
4C3×#4693C211yesmaximal, characteristic, complemented
5C32×D4⋊C8C211yesmaximal, characteristic, complemented
6C3×#4485C211yesmaximal, characteristic, complemented
7C3×#4459C211yesmaximal, characteristic, complemented
8C3×#1629—31nomaximal
9#68195C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D4—21no—
11C22×C3×D12—21no—
12C32×C4×D4C2211yescharacteristic
13C3×#930—21no—
14C2×C3×#204—22no—
15C3×#895—21no—
16C3×#882—22no—
17C3×#874C2211yescharacteristic
18C3×C4×D12C2211yescharacteristic
19C3×#869C2211yescharacteristic, complemented
20C32×C4⋊C8C2211yescharacteristic, complemented
21C32×C4×C8C2211yescharacteristic, complemented
22C3×#832—21no—
23C3×#821—21no—
24C3×#817—21no—
25C3×#808—21no—
26C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
27C3×D42—31no—
28D4×D12C2×C311yescharacteristic
29C3×#492—31no—
30C3×#476—31no—
31C3×#476—31no—
32#5008C2×C311yescharacteristic, complemented
33#4842C2×C311yescharacteristic, complemented
34#4693C2×C311yescharacteristic, complemented
35C3×#331—31no—
36C3×D4⋊C8—21no—
37C3×D4⋊C8—31no—
38C3×D4⋊C8D311yescharacteristic, complemented
39C3×D4⋊C8C2×C311yescharacteristic, complemented
40#4485C2×C311yescharacteristic, complemented
41#4459C2×C311yescharacteristic, complemented
42C23×C3×D3—22no—
43C2×C32×D4D411yescharacteristic
44C22×C32×C4—21no—
45C2×C3×#67—22no—
46C3×D3×D4—42no—
47C3×D3×D4—42no—
48C2×C3×D12—22no—
49C2×C3×D12—21no—
50C2×C3×D12—21no—
51C2×C3×D12—42no—
52C2×C3×D12—42no—
53C2×C3×D12D411yescharacteristic
54C2×C3×D12D412yes—
55C2×C3×C4×D3—21no—
56C2×C3×Dic6—21no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C32×C4⋊C4D411yescharacteristic
60C32×C22⋊C4—21no—
61C32×C42C2311yescharacteristic
62C3×(C2×D3)⋊C4—22no—
63C3×(C2×D3)⋊C4—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3D411yescharacteristic
67C2×C3×C3⋊C8—21no—
68C3×#204—44no—
69#1629—31noSylow
70C22×C3×D4—61no—
71C22×C3×D4—61no—
72C2×D3×D4—21no—
73C22×D12—21no—
74C2×C3×SD16—62no—
75C3×C4⋊Q8—31no—
76C3×#127—31no—
77C3×#121—61no—
78C3×#121—61no—
79C3×#120—62no—
80C3×C4×D4—21no—
81C3×C4×D4—31no—
82C3×C4×D4C2×D311yescharacteristic
83C3×C4×D4C22×C311yescharacteristic
84#930—21no—
85C2×#204—22no—
86#895—21no—
87#882—22no—
88#874C22×C311yescharacteristic
89C4×D12C22×C311yescharacteristic
90#869C22×C311yescharacteristic, complemented
91C3×#106—61no—
92C3×#106—61no—
93C3×C4⋊C8—21no—
94C3×C4⋊C8—31no—
95C3×C4⋊C8C2×D311yescharacteristic, complemented
96C3×C4⋊C8C22×C311yescharacteristic, complemented
97C3×#102—61no—
98C3×#102—61no—
99C3×C4×C8—21no—
100C3×C4×C8C2×D311yescharacteristic, complemented
101C3×C4×C8C22×C311yescharacteristic, complemented
102#832—21no—
103#821—21no—
104#817—21no—
105#808—21no—
106C4⋊(C3⋊C8)C22×C311yescharacteristic, complemented
107C23×C32—21no—
108C22×C3×D3—21no—
109C22×C3×D3—22no—
110C22×C3×D3—44no—
111C22×C3×D3—44no—
112C22×C3×D3—82no—
113C32×D4—21no—
114C32×D4SD1612yescomplemented
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C3×#67—44no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—41no—
124C3×D12—21no—
125C3×D12—42no—
126C3×D12—44no—
127C3×D12—41no—
128C3×D12—41no—
129C3×D12—41no—
130C3×D12—41no—
131C3×D12SD1612yescomplemented
132C3×C4×D3—41no—
133C3×C4×D3—41no—
134C3×Dic6—42no—
135C32×C8—22no—
136C32×C8—41no—
137C3×C3⋊C8—41no—
138D42—31no—
139#492—31no—
140#476—31no—
141#476—31no—
142#331—31no—
143D4⋊C8—31no—
144D4⋊C8C3×D311yescharacteristic, complemented
145C24×C3—62no—
146C23×D3—22no—
147C2×C3×Q8—61no—
148C2×C3×D4—21no—
149C2×C3×D4—31no—
150C2×C3×D4—32no—
151C2×C3×D4—62no—
152C2×C3×D4—62no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4—122no—
156C2×C3×D4—122no—
157C2×C3×D4—122no—
158C2×C3×D4—122no—
159C2×C3×D4D1211yescharacteristic
160C2×C3×D4C3×D411yescharacteristic
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—41no—
164C22×C3×C4—61no—
165C2×#67—22no—
166D3×D4—42no—
167D3×D4—42no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—42no—
172C2×D12—42no—
173C2×D12C3×D411yescharacteristic
174C2×D12C3×D412yes—
175C2×C4×D3—21no—
176C2×Dic6—21no—
177C3×SD16—124no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—41no—
183C2×C3×C8—41no—
184C2×C3×C8—61no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—61no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4D1211yescharacteristic
190C3×C4⋊C4C3×D411yescharacteristic
191C3×C22⋊C4—21no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—41no—
194C3×C22⋊C4—62no—
195C3×C22⋊C4—61no—
196C3×C42—21no—
197C3×C42C22×D311yescharacteristic
198C3×C42C23×C311yescharacteristic
199(C2×D3)⋊C4—22no—
200(C2×D3)⋊C4—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203C4⋊Dic3C3×D411yescharacteristic
204C2×C3⋊C8—21no—
205#204—44no—
206C22×C32—22no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32#12011yescharacteristic
210C2×C3×D3—22no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—42no—
214C2×C3×D3—42no—
215C2×C3×D3—42no—
216C2×C3×D3—88no—
217C32×C4—21no—
218C32×C4—41no—
219C32×C4C2×SD1611yescharacteristic
220C32×C4C2×SD1611yescharacteristic
221C32×C4AGL(1,ℤ/8)11yescharacteristic
222C32×C4AGL(1,ℤ/8)11yescharacteristic
223C3×Dic3—41no—
224C3×Dic3—81no—
225C22×D4—61no—
226C22×D4—61no—
227C2×SD16—62no—
228C4⋊Q8—31no—
229#127—31no—
230#121—61no—
231#121—61no—
232#120—62no—
233C4×D4—31no—
234C4×D4C2×C3×D311yescharacteristic
235#106—61no—
236#106—61no—
237C4⋊C8—31no—
238C4⋊C8C2×C3×D311yescharacteristic, complemented
239#102—61no—
240#102—61no—
241C4×C8C2×C3×D311yescharacteristic, complemented
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—41no—
245C23×C3—61no—
246C23×C3—62no—
247C23×C3—124no—
248C23×C3—124no—
249C23×C3—242no—
250C22×D3—21no—
251C22×D3—22no—
252C22×D3—44no—
253C22×D3—44no—
254C22×D3—82no—
255C3×Q8—122no—
256C3×D4—21no—
257C3×D4—21no—
258C3×D4—22no—
259C3×D4—32no—
260C3×D4—41no—
261C3×D4—61no—
262C3×D4—122no—
263C3×D4—124no—
264C3×D4—124no—
265C3×D4—121no—
266C3×D4—121no—
267C3×D4—121no—
268C3×D4—121no—
269C3×D4#20412yescomplemented
270C3×D4C3×SD1612yescomplemented
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—61no—
282C2×C3×C4—81no—
283C2×C3×C4—81no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4—121no—
287C2×C3×C4C2×D1211yescharacteristic
288C2×C3×C4D3×D411yescharacteristic
289C2×C3×C4D3×D411yescharacteristic
290C2×C3×C4C2×C3×D411yescharacteristic
291C2×C3×C4C2×C3×D411yescharacteristic
292C2×C3×C4C2×C3×D411yescharacteristic
293#67—44no—
294C2×Dic3—21no—
295C2×Dic3—41no—
296D12—21no—
297D12—42no—
298D12—44no—
299D12—41no—
300D12—41no—
301D12—41no—
302D12—41no—
303D12C3×SD1612yescomplemented
304C4×D3—41no—
305C4×D3—41no—
306Dic6—42no—
307C3×C8—22no—
308C3×C8—22no—
309C3×C8—42no—
310C3×C8—41no—
311C3×C8—41no—
312C3×C8—81no—
313C3×C8—121no—
314C3⋊C8—41no—
315C2×C32—42no—
316C2×C32#45011yescharacteristic
317C2×C32#45011yescharacteristic
318C2×C32#45311yescharacteristic
319C3×D3—42no—
320C3×D3—82no—
321C24—62no—
322C2×Q8—61no—
323C2×D4—31no—
324C2×D4—32no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—122no—
330C2×D4—122no—
331C2×D4—122no—
332C2×D4—122no—
333C2×D4C3×D1211yescharacteristic
334C22×C4—21no—
335C22×C4—61no—
336SD16—124no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—61no—
340C4⋊C4—31no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4C3×D1211yescharacteristic
344C22⋊C4—21no—
345C22⋊C4—62no—
346C22⋊C4—61no—
347C42C22×C3×D311yescharacteristic
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—21no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—62no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—122no—
362C22×C3—122no—
363C22×C3—122no—
364C22×C3—248no—
365C22×C3#88211yescharacteristic
366C22×C3C3×#12011yescharacteristic
367C2×D3—22no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—42no—
373C2×D3—88no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—81no—
384C3×C4—121no—
385C3×C4—241no—
386C3×C4C2×#20411yescharacteristic
387C3×C4#90811yescharacteristic
388C3×C4#91111yescharacteristic
389C3×C4D3×SD1611yescharacteristic
390C3×C4C2×C3×SD1611yescharacteristic
391C3×C4C2×C3×SD1611yescharacteristic
392C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
393C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
394Dic3—41no—
395Dic3—81no—
396C32#162911yescharacteristic, complemented, Sylow
397C23—21no—
398C23—61no—
399C23—62no—
400C23—124no—
401C23—124no—
402C23—242no—
403Q8—122no—
404D4—21no—
405D4—32no—
406D4—61no—
407D4—122no—
408D4—124no—
409D4—124no—
410D4—121no—
411D4—121no—
412D4—121no—
413D4—121no—
414D4C3×#20412yescomplemented
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—61no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4C2×C3×D1211yescharacteristic
423C2×C4C3×D3×D411yescharacteristic
424C2×C4C3×D3×D411yescharacteristic
425C8—22no—
426C8—41no—
427C8—121no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—42no—
432C2×C3—42no—
433C2×C3—82no—
434C2×C3—122no—
435C2×C3—242no—
436C2×C3#473111yescharacteristic
437C2×C3#477611yescharacteristic
438C2×C3#482211yescharacteristic
439C2×C3C3×#45011yescharacteristic
440C2×C3C3×#45011yescharacteristic
441C2×C3C3×#45311yescharacteristic
442D3—42no—
443D3—82no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—62no—
448C22—121no—
449C22—121no—
450C22—122no—
451C22—122no—
452C22—122no—
453C22—248no—
454C22C3×#88211yescharacteristic
455C4—21no—
456C4—41no—
457C4—121no—
458C4—241no—
459C4C2×C3×#20411yescharacteristic
460C4C3×#90811yescharacteristic
461C4C3×#91111yescharacteristic
462C4C3×D3×SD1611yescharacteristic
463C3—21no—
464C3#6819511yescharacteristic, direct product factor
465C3C3×#162911yescharacteristic, complemented
466C2—42no—
467C2—122no—
468C2—242no—
469C2C3×#473111yescharacteristic
470C2C3×#477611yescharacteristic
471C2C3×#482211yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×D4⋊C8C2
socleC22×C32#120
prime radicalC3×D4×D12C2
hypercenterC3×D4⋊C8D3
hypocenterC3C3×#1629
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#5008C2semidirect
3. C3×#4842C2semidirect
4. C3×#4693C2semidirect
5. C32×D4⋊C8C2semidirect
6. C3×#4485C2semidirect
7. C3×#4459C2semidirect
9. #68195C3direct (N × H)
19. C3×#869C22semidirect
20. C32×C4⋊C8C22semidirect
21. C32×C4×C8C22semidirect
26. C3×C4⋊(C3⋊C8)C22semidirect
32. #5008C2×C3semidirect
33. #4842C2×C3semidirect
34. #4693C2×C3semidirect
38. C3×D4⋊C8D3semidirect
39. C3×D4⋊C8C2×C3semidirect
40. #4485C2×C3semidirect
41. #4459C2×C3semidirect
90. #869C22×C3semidirect
95. C3×C4⋊C8C2×D3semidirect
96. C3×C4⋊C8C22×C3semidirect
100. C3×C4×C8C2×D3semidirect
101. C3×C4×C8C22×C3semidirect
106. C4⋊(C3⋊C8)C22×C3semidirect
114. C32×D4SD16semidirect
131. C3×D12SD16semidirect
144. D4⋊C8C3×D3semidirect
238. C4⋊C8C2×C3×D3semidirect
241. C4×C8C2×C3×D3semidirect
269. C3×D4#204semidirect
270. C3×D4C3×SD16semidirect
303. D12C3×SD16semidirect
396. C32#1629semidirect
414. D4C3×#204semidirect
464. C3#68195direct (N × H)
465. C3C3×#1629semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D122
C3×#50082
C3×#48422
C3×#46932
C32×D4⋊C82
C3×#44852
C3×#44592
C3×#16293
#681953