Finite Group Databasebeta

← #13155533

Subgroups — #13155533

Order 1728.

538 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63855C211yesmaximal, characteristic
2#11722071C211yesmaximal, characteristic, complemented
3Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591056—31nomaximal
9#10590698—31nomaximal
10#10590253—31nomaximal
11C2×C32×C4×D3C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C3×Dic3⋊Dic3C411yescharacteristic
19C3×Dic3⋊Dic3C411yescharacteristic
20C3×(C2×D3)⋊Dic3C411yescharacteristic
21C3×(C2×D3)⋊Dic3C411yescharacteristic
22C3×#926—31no—
23C3×#872—31no—
24#63855D311yescharacteristic
25#63657—31no—
26C8×C3⋊Dic3—31no—
27C4⋊(C3⋊(C3⋊C8))—31no—
28C3×#830—31no—
29C3×(C2×D3)⋊C8—31no—
30C3×Dic3⋊C8—31no—
31C3×C4⋊(C3⋊C8)—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34Dic3⋊(C3⋊C8)—61no—
35Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
36(C2×D3)⋊(C3⋊C8)—31no—
37(C2×D3)⋊(C3⋊C8)—31no—
38(C2×D3)⋊(C3⋊C8)—61no—
39(C2×D3)⋊(C3⋊C8)D311yescharacteristic
40#63555—31no—
41#63554—31no—
42#63554—31no—
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C22×C32×D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C2×C3×C3⋊Dic3C2×C411yescharacteristic
50C2×C32×Dic3C2×C411yescharacteristic
51C2×C32×Dic3C2×C411yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5000—91no—
60#4840—91no—
61#4720—91no—
62C22×C32×C4—31no—
63C2×C3×C4×D3—21no—
64C2×C3×C4×D3C2×D311yescharacteristic
65C2×C3×C4×D3C2×D311yescharacteristic
66C2×C32×C8—31no—
67C32×C42—31no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—61no—
74C2×C3⋊(C3⋊C8)—61no—
75C2×C3⋊(C3⋊C8)—121no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×#217—31no—
79C3×#217—31no—
80C3×(C2×D3)⋊C4—31no—
81C3×(C2×D3)⋊C4—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×Dic3⋊C4—31no—
85C3×Dic3⋊C4—31no—
86C3×C4×Dic3—21no—
87C3×C4×Dic3—31no—
88C3×C4×Dic3—31no—
89C3×C4×Dic3—61no—
90C3×C4×Dic3C2×D311yescharacteristic
91C3×C4×Dic3C2×D311yescharacteristic
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—61no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101Dic3⋊Dic3Dic311yescharacteristic
102Dic3⋊Dic3Dic311yescharacteristic
103(C2×D3)⋊Dic3Dic311yescharacteristic
104(C2×D3)⋊Dic3Dic311yescharacteristic
105C22×C33C22×C411yescharacteristic
106C2×C32×D3—21no—
107C2×C32×D3—21no—
108C2×C32×D3—21no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—21no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—21no—
116C3×#117—91no—
117#926—31no—
118#872—31no—
119C3×C4⋊C8—91no—
120C3×C22⋊C8—91no—
121#830—31no—
122#830—91no—
123#830—91no—
124#830—181no—
125(C2×D3)⋊C8—31no—
126(C2×D3)⋊C8—91no—
127#815—91no—
128#815—91no—
129Dic3⋊C8—31no—
130Dic3⋊C8—91no—
131C8×Dic3—91no—
132C8×Dic3—91no—
133C8×Dic3—181no—
134C4⋊(C3⋊C8)—31no—
135C4⋊(C3⋊C8)—91no—
136C4⋊(C3⋊C8)—91no—
137C4⋊(C3⋊C8)—181no—
138#803—91no—
139C4×C3⋊C8—91no—
140C23×C32—31no—
141C22×C3×D3—21no—
142C22×C3×D3C4×D311yescharacteristic
143C22×C3×D3C2×Dic311yescharacteristic
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3C2×Dic311yescharacteristic
157C2×C3⋊Dic3C2×Dic311yescharacteristic
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3C4×D311yescharacteristic
168C2×C3×Dic3C4×D311yescharacteristic
169C2×C3×Dic3C4×D311yescharacteristic
170C2×C3×Dic3C2×Dic311yescharacteristic
171C2×C3×Dic3C2×Dic311yescharacteristic
172C2×C3×Dic3C2×Dic311yescharacteristic
173C3×C4×D3—21no—
174C3×C4×D3—21no—
175C3×C4×D3—21no—
176C3×C4×D3—21no—
177C3×C4×D3—41no—
178C3×C4×D3—41no—
179C32×C8—61no—
180C3⋊(C3⋊C8)—21no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—62no—
184C3⋊(C3⋊C8)—61no—
185C3⋊(C3⋊C8)—61no—
186C3⋊(C3⋊C8)—121no—
187C3⋊(C3⋊C8)—121no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—61no—
193C3×C3⋊C8—61no—
194C3×C3⋊C8—61no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—121no—
197C3×C3⋊C8—121no—
198#433—271noSylow
199C2×C33#11711yescharacteristic
200C2×C33#13111yescharacteristic
201C2×C33#13111yescharacteristic
202C32×D3—41no—
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—61no—
206C2×C4×D3D3211yescharacteristic
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—61no—
210C2×C3×C8—91no—
211C2×C3×C8—91no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—91no—
214C3×C22⋊C4—91no—
215C3×C22⋊C4—91no—
216C3×C42—31no—
217C3×C42—31no—
218C3×C42—61no—
219C3×C42—91no—
220#217—31no—
221#217—31no—
222(C2×D3)⋊C4—31no—
223(C2×D3)⋊C4—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3—31no—
226Dic3⋊C4—31no—
227Dic3⋊C4—31no—
228C4×Dic3—31no—
229C4×Dic3—31no—
230C4×Dic3—61no—
231C4×Dic3D3211yescharacteristic
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—61no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—91no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—181no—
243C2×C3⋊C8—181no—
244C2×C3⋊C8—361no—
245C2×C3⋊C8D3211yescharacteristic
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—61no—
251C22×C32—61no—
252C22×C32—61no—
253C22×C32C2×C4×D311yescharacteristic
254C22×C32C2×C4×D311yescharacteristic
255C22×C32C22×Dic311yescharacteristic
256C2×C3×D3—22no—
257C2×C3×D3—21no—
258C2×C3×D3—21no—
259C2×C3×D3—21no—
260C2×C3×D3—21no—
261C2×C3×D3—21no—
262C2×C3×D3—21no—
263C2×C3×D3—41no—
264C2×C3×D3—41no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—61no—
274C32×C4—61no—
275C32×C4—61no—
276C32×C4#23511yescharacteristic
277C32×C4#23511yescharacteristic
278C32×C4#23711yescharacteristic
279C32×C4#23711yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4#23911yescharacteristic
282C3⋊Dic3—21no—
283C3⋊Dic3—21no—
284C3×Dic3—22no—
285C3×Dic3—21no—
286C3×Dic3—21no—
287C3×Dic3—21no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299#117—91no—
300C4⋊C8—91no—
301C4⋊C8—271no—
302C22⋊C8—91no—
303C22⋊C8—271no—
304#97—271no—
305C4×C8—271no—
306C33#43311yescharacteristic, complemented, Sylow
307C23×C3—31no—
308C23×C3—31no—
309C23×C3—61no—
310C22×D3D3×Dic311yescharacteristic
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—41no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—121no—
331C2×C3×C4—121no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—31no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3D3×Dic311yescharacteristic
342C2×Dic3D3×Dic311yescharacteristic
343C2×Dic3D3×Dic311yescharacteristic
344C4×D3—21no—
345C4×D3—21no—
346C3×C8—62no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—181no—
350C3×C8—181no—
351C3⋊C8—21no—
352C3⋊C8—61no—
353C3⋊C8—61no—
354C3⋊C8—61no—
355C3⋊C8—61no—
356C3⋊C8—121no—
357C3⋊C8—121no—
358C3⋊C8—182no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C3⋊C8—361no—
363C3⋊C8—361no—
364C3⋊C8—362no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—121no—
378C2×C32#87711yescharacteristic
379C2×C32#89211yescharacteristic
380C2×C32#90111yescharacteristic
381C2×C32#90111yescharacteristic
382C2×C32#90711yescharacteristic
383C2×C32#90711yescharacteristic
384C2×C32#92611yescharacteristic
385C2×C32#94811yescharacteristic
386C2×C32#94811yescharacteristic
387C3×D3—42no—
388C3×D3—41no—
389C3×D3—41no—
390C22×C4—31no—
391C2×C8—31no—
392C2×C8—91no—
393C2×C8—91no—
394C2×C8—271no—
395C4⋊C4—91no—
396C4⋊C4—91no—
397C22⋊C4—91no—
398C22⋊C4—91no—
399C42—31no—
400C42—91no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—62no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3—61no—
409C22×C3—61no—
410C22×C3—61no—
411C22×C3—61no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3C4×D3211yescharacteristic
415C22×C3C2×D3×Dic311yescharacteristic
416C22×C3C2×D3×Dic311yescharacteristic
417C2×D3—21no—
418C2×D3—21no—
419C2×D3—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—62no—
429C3×C4—61no—
430C3×C4—61no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—181no—
438C3×C4—181no—
439C3×C4#379511yescharacteristic
440C3×C4#379611yescharacteristic
441C3×C4#379911yescharacteristic
442C3×C4#380411yescharacteristic
443C3×C4#380411yescharacteristic
444C3×C4#380511yescharacteristic
445Dic3—21no—
446Dic3—21no—
447Dic3—21no—
448Dic3—62no—
449Dic3—62no—
450Dic3—61no—
451Dic3—61no—
452Dic3—61no—
453Dic3—61no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#472011yescharacteristic, complemented
459C32#484011yescharacteristic, complemented
460C32#500011yescharacteristic, complemented
461C23—31no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—91no—
468C2×C4—91no—
469C2×C4D3311yescharacteristic
470C8—61no—
471C8—181no—
472C8—181no—
473C8—541no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—122no—
487C2×C3—121no—
488C2×C3—121no—
489C2×C3#6380311yescharacteristic
490C2×C3#6380511yescharacteristic
491C2×C3#6381411yescharacteristic
492C2×C3#6381411yescharacteristic
493C2×C3#6381511yescharacteristic
494C2×C3#6381511yescharacteristic
495C2×C3#6384111yescharacteristic
496C2×C3#6387811yescharacteristic
497C2×C3#6395511yescharacteristic
498D3—41no—
499C22—61no—
500C22—61no—
501C22—61no—
502C22D32×Dic311yescharacteristic
503C4—61no—
504C4—61no—
505C4—61no—
506C4—181no—
507C4—181no—
508C4#8896111yescharacteristic
509C4#8896211yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059025311yescharacteristic, complemented
515C3#1059069811yescharacteristic, complemented
516C3#1059105611yescharacteristic, complemented
517C2—121no—
518C2#1172282111yescharacteristic
519C2#1172282311yescharacteristic
520C2#1172291411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C32×D3C2×C4
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722071C2semidirect
3. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
35. Dic3⋊(C3⋊C8)D3semidirect
306. C33#433semidirect
458. C32#4720semidirect
459. C32#4840semidirect
460. C32#5000semidirect
514. C3#10590253semidirect
515. C3#10590698semidirect
516. C3#10591056semidirect

Maximal subgroups

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

subgroupindex
C3×#638552
#117220712
Dic3⋊(C3⋊(C3⋊C8))2
(C2×D3)⋊(C3⋊(C3⋊C8))2
C3⋊C8×C3⋊Dic32
C3×Dic3⋊(C3⋊C8)2
C3×(C2×D3)⋊(C3⋊C8)2
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