Finite Group Databasebeta

← #13155513

Subgroups — #13155513

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#624C211yesmaximal, characteristic
2#11722072C211yesmaximal, characteristic
3Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722023C211yesmaximal, characteristic
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590643—31nomaximal
9#10590261—31nomaximal
10#10590248—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#624C411yescharacteristic
13C2×C3×#624C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic32C411yescharacteristic
21C3×Dic32C411yescharacteristic
22C3×C42×D3—31no—
23C3×C42×D3—31no—
24C4×#624D311yescharacteristic
25#63647—31no—
26C4⋊(C3⋊(C3⋊C8))—31no—
27C4×C3⋊(C3⋊C8)—31no—
28C3×Dic3⋊C8—31no—
29C3×C8×Dic3—31no—
30C3×C4⋊(C3⋊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#63557—31no—
37(C2×D3)⋊(C3⋊C8)—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40(C2×D3)⋊(C3⋊C8)—31no—
41(C2×D3)⋊(C3⋊C8)—61no—
42Dic3×C3⋊C8—31no—
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—61no—
45Dic3×C3⋊C8D311yescharacteristic, complemented
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3C2×C411yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C3×#624—22no—
56C3×#624C814yescomplemented
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#4925—91no—
62#4833—91no—
63#4685—91no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—31no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—61no—
72C2×#624Dic311yescharacteristic
73C2×#624Dic311yescharacteristic
74C2×C32×C8—31no—
75C32×C42—31no—
76C32×C42—31no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—121no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×C4×Dic3—21no—
87C3×C4×Dic3—21no—
88C3×C4×Dic3—31no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105Dic32Dic311yescharacteristic
106Dic32Dic311yescharacteristic
107C22×C33C22×C411yescharacteristic
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3C2×C812yes—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3C2×C812yes—
116C32×Dic3C2×C812yes—
117C32×Dic3M1612yes—
118C2×C3×C42—91no—
119C42×D3—31no—
120C42×D3—31no—
121C3×C4⋊C8—91no—
122C3×C4×C8—91no—
123#830—91no—
124#830—91no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—181no—
130Dic3⋊C8—31no—
131Dic3⋊C8—91no—
132C8×Dic3—31no—
133C8×Dic3—91no—
134C4⋊(C3⋊C8)—31no—
135C4⋊(C3⋊C8)—91no—
136C4⋊(C3⋊C8)—91no—
137C4⋊(C3⋊C8)—181no—
138C4×C3⋊C8—31no—
139C4×C3⋊C8—91no—
140C4×C3⋊C8—91no—
141C4×C3⋊C8—181no—
142C22×C3⋊D3C2×Dic311yescharacteristic
143C22×C3×D3—31no—
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C32×C4—31no—
152C2×C32×C4—31no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3C2×Dic311yescharacteristic
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3C4×D311yescharacteristic
168C2×C3×Dic3C4×D311yescharacteristic
169C2×C3×Dic3C4×D311yescharacteristic
170C2×C3×Dic3C4×D311yescharacteristic
171C2×C3×Dic3C2×Dic311yescharacteristic
172C2×C3×Dic3C2×Dic311yescharacteristic
173C2×C3×Dic3C2×Dic311yescharacteristic
174C2×C3×Dic3C2×Dic311yescharacteristic
175C3×C4×D3—34no—
176C3×C4×D3—34no—
177C3×C4×D3—62no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—61no—
181C3×C4×D3—61no—
182C3×C4×D3—62no—
183C3×C4×D3—121no—
184C3×C4×D3—121no—
185#624—22no—
186#624C3⋊C814yescomplemented
187C32×C8—61no—
188C3⋊(C3⋊C8)—21no—
189C3⋊(C3⋊C8)—21no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—62no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—61no—
194C3⋊(C3⋊C8)—121no—
195C3⋊(C3⋊C8)—121no—
196C3⋊(C3⋊C8)—122no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—41no—
200C3×C3⋊C8—62no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#431—271noSylow
207C2×C33#11711yescharacteristic
208C2×C33C22×C811yescharacteristic
209C2×C33C2×M1611yescharacteristic
210C3×C3⋊D3—22no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C2×C4×D3—31no—
215C2×C4×D3—31no—
216C2×C4×D3—31no—
217C2×C4×D3—31no—
218C2×C4×D3—31no—
219C2×C4×D3—31no—
220C2×C4×D3—61no—
221C2×C3×C8—31no—
222C2×C3×C8—31no—
223C2×C3×C8—61no—
224C2×C3×C8—91no—
225C2×C3×C8—91no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—31no—
229C3×C42—31no—
230C3×C42—61no—
231C3×C42—61no—
232C3×C42—91no—
233C3×C42—91no—
234C4×Dic3—31no—
235C4×Dic3—31no—
236C4×Dic3—31no—
237C4×Dic3—31no—
238C4×Dic3D3211yescharacteristic
239C4×Dic3D3211yescharacteristic
240C2×C3⋊C8—31no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—91no—
248C2×C3⋊C8—91no—
249C2×C3⋊C8—181no—
250C2×C3⋊C8—181no—
251C2×C3⋊C8—181no—
252C2×C3⋊C8—361no—
253C2×C3⋊C8D3211yescharacteristic
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32C2×C4×D311yescharacteristic
259C22×C32C2×C4×D311yescharacteristic
260C22×C32C22×Dic311yescharacteristic
261C2×C3⋊D3—21no—
262C2×C3⋊D3—21no—
263C2×C3⋊D3C2×C3⋊C812yes—
264C2×C3×D3—32no—
265C2×C3×D3—32no—
266C2×C3×D3—62no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—32no—
280C32×C4—32no—
281C32×C4—32no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—61no—
285C32×C4#23511yescharacteristic
286C32×C4#23511yescharacteristic
287C32×C4#23511yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23911yescharacteristic
291C3⋊Dic3—21no—
292C3×Dic3—21no—
293C3×Dic3—21no—
294C3×Dic3—22no—
295C3×Dic3—22no—
296C3×Dic3—22no—
297C3×Dic3—41no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C3×Dic3—121no—
301C3×Dic3D3×C812yes—
302C3×Dic3D3×C812yes—
303C3×Dic3ΓL(1,𝔽25)12yes—
304C3×Dic3C2×C3⋊C812yes—
305C3×Dic3C2×C3⋊C812yes—
306C3×Dic3#20812yes—
307C2×C42—91no—
308C4⋊C8—91no—
309C4⋊C8—271no—
310C22⋊C8—271no—
311C22⋊C8—271no—
312C4×C8—91no—
313C4×C8—271no—
314C33#43111yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C22×D3—31no—
317C22×D3—31no—
318C22×D3—61no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—41no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—94no—
336C2×C3×C4—91no—
337C2×C3×C4—182no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—61no—
346C2×Dic3D3×Dic311yescharacteristic
347C2×Dic3D3×Dic311yescharacteristic
348C2×Dic3D3×Dic311yescharacteristic
349C2×Dic3D3×Dic311yescharacteristic
350C4×D3—34no—
351C4×D3—34no—
352C4×D3—62no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—62no—
358C4×D3—121no—
359C4×D3—121no—
360C3×C8—61no—
361C3×C8—61no—
362C3×C8—121no—
363C3×C8—181no—
364C3×C8—181no—
365C3⋊C8—21no—
366C3⋊C8—62no—
367C3⋊C8—61no—
368C3⋊C8—61no—
369C3⋊C8—61no—
370C3⋊C8—61no—
371C3⋊C8—121no—
372C3⋊C8—182no—
373C3⋊C8—181no—
374C3⋊C8—181no—
375C3⋊C8—181no—
376C3⋊C8—361no—
377C3⋊C8—361no—
378C3⋊C8—362no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32#87211yescharacteristic
392C2×C32#89211yescharacteristic
393C2×C32C2×D3×C811yescharacteristic
394C2×C32C2×D3×C811yescharacteristic
395C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
396C2×C32D3×M1611yescharacteristic
397C2×C32C22×C3⋊C811yescharacteristic
398C2×C32C2×#20811yescharacteristic
399C2×C32#92611yescharacteristic
400C3⋊D3—22no—
401C3×D3—62no—
402C3×D3—62no—
403C3×D3—122no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—91no—
407C2×C8—31no—
408C2×C8—91no—
409C2×C8—91no—
410C2×C8—271no—
411C42—31no—
412C42—31no—
413C42—91no—
414C42—91no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3—92no—
420C22×C3—181no—
421C22×C3—181no—
422C22×C3C4×D3211yescharacteristic
423C22×C3C2×D3×Dic311yescharacteristic
424C22×C3C2×D3×Dic311yescharacteristic
425C2×D3—32no—
426C2×D3—32no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—121no—
433C2×D3—121no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—32no—
443C3×C4—32no—
444C3×C4—32no—
445C3×C4—32no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—121no—
454C3×C4—181no—
455C3×C4#379511yescharacteristic
456C3×C4#379811yescharacteristic
457C3×C4#379811yescharacteristic
458C3×C4#379911yescharacteristic
459C3×C4#379911yescharacteristic
460C3×C4#380411yescharacteristic
461Dic3—21no—
462Dic3—61no—
463Dic3—61no—
464Dic3—121no—
465Dic3D3×C3⋊C812yes—
466Dic3D3×C3⋊C812yes—
467Dic3#371112yes—
468C32—21no—
469C32—21no—
470C32—21no—
471C32—41no—
472C32#468511yescharacteristic, complemented
473C32#483311yescharacteristic, complemented
474C32#492511yescharacteristic, complemented
475C23—91no—
476C2×C4—31no—
477C2×C4—31no—
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—94no—
481C2×C4—91no—
482C2×C4—182no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4D3311yescharacteristic
486C8—61no—
487C8—181no—
488C8—181no—
489C8—541no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—182no—
503C2×C3D32×C811yescharacteristic
504C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
505C2×C3C2×D3×C3⋊C811yescharacteristic
506C2×C3C2×D3×C3⋊C811yescharacteristic
507C2×C3D3×#20811yescharacteristic
508C2×C3C2×#371111yescharacteristic
509C2×C3#6383611yescharacteristic
510C2×C3#6384111yescharacteristic
511C2×C3#6385511yescharacteristic
512D3—62no—
513D3—62no—
514D3—122no—
515C22—92no—
516C22—181no—
517C22—181no—
518C22D32×Dic311yescharacteristic
519C4—32no—
520C4—32no—
521C4—32no—
522C4—61no—
523C4—181no—
524C4#8896011yescharacteristic
525C4#8896511yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059024811yescharacteristic, complemented
531C3#1059026111yescharacteristic, complemented
532C3#1059064311yescharacteristic, complemented
533C2—182no—
534C2D32×C3⋊C811yescharacteristic
535C2D3×#371111yescharacteristic
536C2#1172291511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C22×C8
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
35. Dic3⋊(C3⋊C8)D3semidirect
45. Dic3×C3⋊C8D3semidirect
56. C3×#624C8semidirect
186. #624C3⋊C8semidirect
314. C33#431semidirect
472. C32#4685semidirect
473. C32#4833semidirect
474. C32#4925semidirect
530. C3#10590248semidirect
531. C3#10590261semidirect
532. C3#10590643semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6242
#117220722
Dic3⋊(C3⋊(C3⋊C8))2
#117220232
Dic3×C3⋊(C3⋊C8)2
C3×Dic3⋊(C3⋊C8)2
C3×Dic3×C3⋊C82
#105906433
#105902613
#105902483