Finite Group Databasebeta

← #13161482

Subgroups — #13161482

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722080C212yesmaximal, complemented
5#11722075C212yesmaximal
6#10590313—31nomaximal
7#10590272—31nomaximal
8#10590269—31nomaximal
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C4×C3⋊D3—21no—
11C2×C3×#634—21no—
12C33×C42C2211yescharacteristic
13C3×#3752—21no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C4×C4⋊(C3⋊D3)D311yescharacteristic
19#64078D311yescharacteristic, complemented
20C3×C4×D12—31no—
21C3×C4×D12—31no—
22C3×C4×D12—61no—
23C3×#869—31no—
24C3×#869—31no—
25C3×#869—61no—
26C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
27C3×C4⋊(C3⋊C8)—31no—
28C3×C4⋊(C3⋊C8)—31no—
29C3×C4⋊(C3⋊C8)—61no—
30#63576—32no—
31#63575—32no—
32#63575—32no—
33#63564—32no—
34#63563—32no—
35#63562—32no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)Q812yescomplemented
44C3×C4×C3⋊D3—42no—
45C3×#634—42no—
46C3×C3⋊(C3⋊C8)—22no—
47#5087—91no—
48#5007—91no—
49#4846—91no—
50C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C4×C3⋊D3—21no—
52C2×#634—21no—
53C2×C3×D12—31no—
54C2×C3×D12—31no—
55C2×C3×D12—61no—
56C2×C3×C4×D3—62no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—62no—
60C2×C3×Dic6—61no—
61C2×C3×Dic6—61no—
62C32×C42—21no—
63C32×C42—21no—
64C32×C42—21no—
65C32×C42—41no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68C32×C42C2×D311yescharacteristic
69#3752—21no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)C2×D312yes—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D312yes—
74C3×(C2×D3)⋊C4—62no—
75C3×(C2×D3)⋊C4—61no—
76C3×(C2×D3)⋊C4—61no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—32no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—62no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—62no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—62no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4—21no—
94C33×C4D812yes—
95C33×C4C2×Q811yescharacteristic
96C33×C4#4111yescharacteristic
97C3×C3⋊Dic3—42no—
98C3×C3⋊Dic3—41no—
99C3×C4⋊Q8—91no—
100C3×C4×D4—91no—
101C4×D12—31no—
102C4×D12—31no—
103C4×D12—61no—
104#869—31no—
105#869—31no—
106#869—61no—
107C3×C4⋊C8—91no—
108#832—92no—
109#821—92no—
110#818—92no—
111#809—92no—
112#807—92no—
113#807—92no—
114C4⋊(C3⋊C8)—31no—
115C4⋊(C3⋊C8)—31no—
116C4⋊(C3⋊C8)—61no—
117C22×C3⋊D3—21no—
118C22×C3×D3—62no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4D1211yescharacteristic
134C2×C32×C4D1211yescharacteristic
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—21no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)Dic612yescomplemented
146C4×C3⋊D3—42no—
147#634—42no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—61no—
154C3×D12—32no—
155C3×D12—32no—
156C3×D12—62no—
157C3×D12—62no—
158C3×D12—61no—
159C3×D12—61no—
160C3×C4×D3—124no—
161C3×C4×D3—122no—
162C3×C4×D3—122no—
163C3×Dic6—124no—
164C3×Dic6—122no—
165C3×Dic6—122no—
166C3⋊(C3⋊C8)—22no—
167C3×C3⋊C8—62no—
168C3×C3⋊C8—62no—
169C3×C3⋊C8—122no—
170#474—271noSylow
171C2×C33C22⋊Q811yescharacteristic
172C2×C33C2×D811yescharacteristic
173C2×C33#13711yescharacteristic
174C3×C3⋊D3—42no—
175C2×C3×Q8—181no—
176C2×C3×D4—91no—
177C22×C3×C4—181no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—61no—
181C2×C4×D3—62no—
182C2×C4×D3—61no—
183C2×C4×D3—61no—
184C2×Dic6—62no—
185C2×Dic6—61no—
186C2×Dic6—61no—
187C2×C3×C8—92no—
188C3×C4⋊C4—91no—
189C3×C4⋊C4—92no—
190C3×C4⋊C4—181no—
191C3×C22⋊C4—181no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—41no—
196C3×C42D3211yescharacteristic
197C3×C42D3211yescharacteristic
198C3×C42D3211yescharacteristic
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—61no—
201(C2×D3)⋊C4—61no—
202C4⋊Dic3—31no—
203C4⋊Dic3—32no—
204C4⋊Dic3—31no—
205C4⋊Dic3—32no—
206C4⋊Dic3—62no—
207C4⋊Dic3—61no—
208C4⋊Dic3—62no—
209C4⋊Dic3—61no—
210C4⋊Dic3—61no—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—32no—
213C2×C3⋊C8—62no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32C2×D1211yescharacteristic
219C22×C32C2×#6711yescharacteristic
220C22×C32C2×#6711yescharacteristic
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3×D3—64no—
225C2×C3×D3—62no—
226C2×C3×D3—62no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C32×C4—21no—
234C32×C4—22no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—42no—
247C32×C4—41no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4—41no—
251C32×C4—42no—
252C32×C4D2412yes—
253C32×C4#21312yes—
254C32×C4#21312yes—
255C32×C4C2×Dic611yescharacteristic
256C32×C4#23511yescharacteristic
257C32×C4#23711yescharacteristic
258C32×C4D3×Q811yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4#23911yescharacteristic
261C3⋊Dic3—42no—
262C3⋊Dic3—41no—
263C3×Dic3—124no—
264C3×Dic3—122no—
265C3×Dic3—122no—
266C3×Dic3—121no—
267C3×Dic3—122no—
268C3×Dic3—121no—
269C4⋊Q8—91no—
270C4×D4—91no—
271#107—272no—
272C4⋊C8—91no—
273#102—272no—
274C33#47411yescharacteristic, complemented, Sylow
275C23×C3—181no—
276C22×D3—62no—
277C22×D3—61no—
278C22×D3—61no—
279C3×Q8—362no—
280C3×D4—92no—
281C3×D4—181no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—181no—
295C2×C3×C4—182no—
296C2×C3×C4—362no—
297C2×C3×C4#62511yescharacteristic
298C2×C3×C4#62511yescharacteristic
299C2×C3×C4#62611yescharacteristic
300C2×C3×C4#62611yescharacteristic
301C2×C3×C4#62611yescharacteristic
302C2×C3×C4#62611yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×C3×C4C2×D3211yescharacteristic
306C2×Dic3—64no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3—61no—
310C2×Dic3—62no—
311C2×Dic3—61no—
312D12—32no—
313D12—32no—
314D12—62no—
315D12—62no—
316D12—61no—
317D12—61no—
318C4×D3—124no—
319C4×D3—122no—
320C4×D3—122no—
321Dic6—124no—
322Dic6—122no—
323Dic6—122no—
324C3×C8—182no—
325C3⋊C8—62no—
326C3⋊C8—62no—
327C3⋊C8—122no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32#89711yescharacteristic
341C2×C32C2×D2411yescharacteristic
342C2×C32#90911yescharacteristic
343C2×C32#92411yescharacteristic
344C2×C32C2×#21311yescharacteristic
345C2×C32C2×#21311yescharacteristic
346C2×C32#94211yescharacteristic
347C2×C32#94611yescharacteristic
348C2×C32#95111yescharacteristic
349C3⋊D3—42no—
350C3×D3—124no—
351C3×D3—122no—
352C3×D3—122no—
353C2×Q8—181no—
354C2×D4—91no—
355C22×C4—181no—
356C2×C8—92no—
357C4⋊C4—92no—
358C4⋊C4—91no—
359C4⋊C4—181no—
360C22⋊C4—181no—
361C42#111311yescharacteristic
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—182no—
367C22×C3—361no—
368C22×C3—361no—
369C22×C3C2×#62511yescharacteristic
370C22×C3C2×#62611yescharacteristic
371C22×C3C2×#62611yescharacteristic
372C2×D3—64no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—122no—
376C2×D3—122no—
377C2×D3—121no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—121no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—42no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—42no—
400C3×C4—361no—
401C3×C4—362no—
402C3×C4#371312yes—
403C3×C4#371412yes—
404C3×C4#371412yes—
405C3×C4D3×Dic611yescharacteristic
406C3×C4D3×Dic611yescharacteristic
407C3×C4#379511yescharacteristic
408C3×C4#379611yescharacteristic
409C3×C4#379711yescharacteristic
410C3×C4#379911yescharacteristic
411Dic3—124no—
412Dic3—122no—
413Dic3—122no—
414Dic3—121no—
415Dic3—122no—
416Dic3—121no—
417C32—21no—
418C32—21no—
419C32—21no—
420C32—41no—
421C32#484611yescharacteristic, complemented
422C32#500711yescharacteristic, complemented
423C32#508711yescharacteristic, complemented
424C23—181no—
425Q8—362no—
426D4—92no—
427D4—181no—
428C2×C4—181no—
429C2×C4—182no—
430C2×C4—362no—
431C2×C4#629511yescharacteristic
432C2×C4#629511yescharacteristic
433C2×C4C2×#111311yescharacteristic
434C8—182no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—362no—
448C2×C3C2×#371311yescharacteristic
449C2×C3C2×#371411yescharacteristic
450C2×C3C2×#371411yescharacteristic
451C2×C3#6382711yescharacteristic
452C2×C3#6383111yescharacteristic
453C2×C3#6383311yescharacteristic
454C2×C3#6385611yescharacteristic
455C2×C3#6385711yescharacteristic
456C2×C3#6387111yescharacteristic
457D3—124no—
458D3—122no—
459D3—122no—
460C22—182no—
461C22—361no—
462C22—361no—
463C22C2×#629511yescharacteristic
464C4—21no—
465C4—361no—
466C4—362no—
467C4#8881012yes—
468C4#8904011yescharacteristic
469C4#8904111yescharacteristic
470C3—21no—
471C3—21no—
472C3—21no—
473C3—41no—
474C3#1059026911yescharacteristic, complemented
475C3#1059027211yescharacteristic, complemented
476C3#1059031311yescharacteristic, complemented
477C2—362no—
478C2C2×#8881011yescharacteristic
479C2#1172366311yescharacteristic
480C2#1172367411yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42#1113
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64078C2semidirect
3. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722080C2semidirect
19. #64078D3semidirect
26. C4⋊(C3⋊(C3⋊C8))D3semidirect
43. C3×C4⋊(C3⋊D3)Q8semidirect
145. C4⋊(C3⋊D3)Dic6semidirect
274. C33#474semidirect
421. C32#4846semidirect
422. C32#5007semidirect
423. C32#5087semidirect
474. C3#10590269semidirect
475. C3#10590272semidirect
476. C3#10590313semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C3×#640782
C3×C4⋊(C3⋊(C3⋊C8))2
#117220802
#117220752
#105903133
#105902723
#105902693