Finite Group Databasebeta

← #13155592

Subgroups — #13155592

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88809C211yesmaximal, characteristic, complemented
2C2×#88786C211yesmaximal, characteristic, complemented
3C3×C4×#626C211yesmaximal, characteristic
4Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5#11722023C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591067—31nomaximal
9#10590730—31nomaximal
10#10590337—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#626C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88809—22no—
20#88786—22no—
21C3×Dic3⋊Dic3C411yescharacteristic
22C3×#3722C411yescharacteristic
23C3×(C2×D3)⋊Dic3C411yescharacteristic
24C2×#3748—31no—
25C3×C4×#67—31no—
26C3×C4×D12—31no—
27C4×#626D311yescharacteristic
28C2×#3712—31no—
29C2×#3711—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32C2×#3711—31no—
33C2×#3711—61no—
34#63647—31no—
35C4⋊(C3⋊(C3⋊C8))—31no—
36C3×#830—31no—
37C3×(C2×D3)⋊C8—31no—
38C3×C8×Dic3—31no—
39C3×C4×C3⋊C8—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—61no—
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—61no—
46(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
47Dic3×C3⋊C8D311yescharacteristic, complemented
48C22×C3×C3⋊D3C2×C411yescharacteristic
49C22×C32×D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3C2×C411yescharacteristic
55C2×C32×Dic3C2×C411yescharacteristic
56C2×C32×Dic3C2×C411yescharacteristic
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C3×#626—22no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)D412yescomplemented
63C32×C3⋊C8—21no—
64#5012—91no—
65#4729—91no—
66#4688—91no—
67C22×C32×C4—31no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×#67—31no—
70C2×C3×D12—31no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×C3×C4×D3C2×D311yescharacteristic
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×#626Dic311yescharacteristic
78C2×C32×C8—31no—
79C32×C42—31no—
80#3748—62no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—121no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#217—31no—
91C3×(C2×D3)⋊C4—31no—
92C3×(C2×D3)⋊C4—31no—
93C3×(C2×D3)⋊C4—31no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—31no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108Dic3⋊Dic3Dic311yescharacteristic
109#3722Dic311yescharacteristic
110(C2×D3)⋊Dic3Dic311yescharacteristic
111#3712—62no—
112#3711—62no—
113#3711—62no—
114#3711—62no—
115#3711—64no—
116#3711—62no—
117C22×C33C22×C411yescharacteristic
118C2×C3×C3⋊D3—21no—
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C32×D3—21no—
122C2×C32×D3—21no—
123C2×C32×D3—21no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—21no—
127C32×Dic3—21no—
128C32×Dic3—21no—
129C32×Dic3M1612yescomplemented
130C3×C4×D4—91no—
131C4×#67—31no—
132C2×#208—91no—
133C2×#208—91no—
134C2×#208—91no—
135C2×#208—181no—
136C2×ΓL(1,𝔽25)—91no—
137C2×ΓL(1,𝔽25)—91no—
138C2×ΓL(1,𝔽25)—91no—
139C4×D12—31no—
140C3×C22⋊C8—91no—
141C3×C4×C8—91no—
142#830—31no—
143#830—91no—
144(C2×D3)⋊C8—31no—
145(C2×D3)⋊C8—91no—
146(C2×D3)⋊C8—91no—
147(C2×D3)⋊C8—181no—
148Dic3⋊C8—91no—
149C8×Dic3—31no—
150C4⋊(C3⋊C8)—91no—
151C4⋊(C3⋊C8)—91no—
152C4⋊(C3⋊C8)—181no—
153C4×C3⋊C8—31no—
154C23×C32—31no—
155C22×C3⋊D3C2×Dic311yescharacteristic
156C22×C3×D3—21no—
157C22×C3×D3—31no—
158C22×C3×D3—31no—
159C22×C3×D3—61no—
160C22×C3×D3C4×D311yescharacteristic
161C22×C3×D3C2×Dic311yescharacteristic
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—31no—
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3C2×Dic311yescharacteristic
175C4×C3⋊D3—21no—
176C4×C3⋊D3—21no—
177C3×#67—62no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3C4×D311yescharacteristic
185C2×C3×Dic3C4×D311yescharacteristic
186C2×C3×Dic3C4×D311yescharacteristic
187C2×C3×Dic3C2×Dic311yescharacteristic
188C2×C3×Dic3C2×Dic311yescharacteristic
189C2×C3×Dic3C2×Dic311yescharacteristic
190C3×D12—62no—
191C3×C4×D3—22no—
192C3×C4×D3—22no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—21no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—61no—
200C3×C4×D3—61no—
201C3×C4×D3—121no—
202C3×C4×D3—121no—
203#626—22no—
204C32×C8—61no—
205C3⋊(C3⋊C8)—21no—
206C3⋊(C3⋊C8)—61no—
207C3⋊(C3⋊C8)—62no—
208C3⋊(C3⋊C8)—61no—
209C3⋊(C3⋊C8)—61no—
210C3⋊(C3⋊C8)—121no—
211C3⋊(C3⋊C8)—121no—
212C3⋊(C3⋊C8)—122no—
213C3⋊(C3⋊C8)D1212yescomplemented
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—32no—
217C3×C3⋊C8—32no—
218C3×C3⋊C8—41no—
219C3×C3⋊C8—62no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#436—271noSylow
224C2×C33C4×D411yescharacteristic
225C2×C33C2×M1611yescharacteristic
226C2×C33#13111yescharacteristic
227C3×C3⋊D3—41no—
228C32×D3—41no—
229C2×C3×D4—91no—
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—61no—
233C22×C3×C4—91no—
234C2×#67—31no—
235C2×D12—31no—
236C2×C4×D3—31no—
237C2×C4×D3—31no—
238C2×C4×D3—61no—
239C2×C4×D3D3211yescharacteristic
240C2×C3×C8—31no—
241C2×C3×C8—31no—
242C2×C3×C8—61no—
243C2×C3×C8—91no—
244C2×C3×C8—91no—
245C3×C4⋊C4—91no—
246C3×C22⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C42—31no—
249C3×C42—31no—
250C3×C42—61no—
251#217—31no—
252(C2×D3)⋊C4—31no—
253(C2×D3)⋊C4—31no—
254(C2×D3)⋊C4—31no—
255C4⋊Dic3—31no—
256Dic3⋊C4—31no—
257C4×Dic3D3211yescharacteristic
258#208—182no—
259#208—182no—
260#208—182no—
261#208—184no—
262C2×C3⋊C8—31no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—91no—
269C2×C3⋊C8—91no—
270C2×C3⋊C8—91no—
271C2×C3⋊C8—181no—
272C2×C3⋊C8—181no—
273C2×C3⋊C8—181no—
274C2×C3⋊C8—361no—
275C2×C3⋊C8D3211yescharacteristic
276ΓL(1,𝔽25)—182no—
277ΓL(1,𝔽25)—182no—
278ΓL(1,𝔽25)—182no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32—61no—
284C22×C32—61no—
285C22×C32—61no—
286C22×C32C2×C4×D311yescharacteristic
287C22×C32C2×C4×D311yescharacteristic
288C22×C32C22×Dic311yescharacteristic
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3×D3—22no—
293C2×C3×D3—21no—
294C2×C3×D3—21no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—62no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—32no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4—61no—
320C32×C4—61no—
321C32×C4C2×D1211yescharacteristic
322C32×C4#23511yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4#23711yescharacteristic
326C32×C4#23711yescharacteristic
327C3⋊Dic3—21no—
328C3×Dic3—22no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—22no—
334C3×Dic3—41no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—121no—
338C3×Dic3ΓL(1,𝔽25)12yescomplemented
339C3×Dic3#20812yescomplemented
340C2×M16—271no—
341C2×M16—271no—
342C4×D4—91no—
343C4⋊C8—271no—
344C22⋊C8—91no—
345C22⋊C8—271no—
346C4×C8—91no—
347C33#43611yescharacteristic, complemented, Sylow
348C23×C3—31no—
349C23×C3—31no—
350C23×C3—61no—
351C23×C3—91no—
352C22×D3—31no—
353C22×D3—31no—
354C22×D3—61no—
355C22×D3D3×Dic311yescharacteristic
356C3×D4—182no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—41no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—91no—
377C2×C3×C4—181no—
378C2×C3×C4—181no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382#67—62no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—61no—
386C2×Dic3D3×Dic311yescharacteristic
387C2×Dic3D3×Dic311yescharacteristic
388C2×Dic3D3×Dic311yescharacteristic
389D12—62no—
390C4×D3—21no—
391C4×D3—21no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—61no—
395C4×D3—61no—
396C4×D3—121no—
397C4×D3—121no—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—92no—
401C3×C8—121no—
402C3×C8—181no—
403C3⋊C8—21no—
404C3⋊C8—32no—
405C3⋊C8—32no—
406C3⋊C8—62no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C3⋊C8—121no—
410C3⋊C8—182no—
411C3⋊C8—181no—
412C3⋊C8—181no—
413C3⋊C8—181no—
414C3⋊C8—361no—
415C3⋊C8—361no—
416C3⋊C8—362no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—121no—
430C2×C32C4×D1211yescharacteristic
431C2×C32#67⋊C411yescharacteristic
432C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
433C2×C32#90111yescharacteristic
434C2×C32#90111yescharacteristic
435C2×C32D3×M1611yescharacteristic
436C2×C32C2×#20811yescharacteristic
437C2×C32D4×Dic311yescharacteristic
438C2×C32#94811yescharacteristic
439C3⋊D3—41no—
440C3×D3—42no—
441C3×D3—41no—
442C3×D3—41no—
443C3×D3—122no—
444C3×D3—121no—
445C3×D3—121no—
446C2×D4—91no—
447C22×C4—31no—
448C22×C4—91no—
449M16—542no—
450M16—542no—
451C2×C8—31no—
452C2×C8—91no—
453C2×C8—91no—
454C2×C8—271no—
455C4⋊C4—91no—
456C22⋊C4—91no—
457C22⋊C4—91no—
458C42—31no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—62no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3—181no—
475C22×C3C4×D3211yescharacteristic
476C22×C3C2×D3×Dic311yescharacteristic
477C22×C3C2×D3×Dic311yescharacteristic
478C2×D3—21no—
479C2×D3—21no—
480C2×D3—21no—
481C2×D3—62no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—61no—
487C2×D3—61no—
488C2×D3—121no—
489C2×D3—121no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—32no—
497C3×C4—32no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—62no—
501C3×C4—62no—
502C3×C4—61no—
503C3×C4—61no—
504C3×C4—61no—
505C3×C4—61no—
506C3×C4—121no—
507C3×C4—181no—
508C3×C4#379511yescharacteristic
509C3×C4D3×D1211yescharacteristic
510C3×C4D3×D1211yescharacteristic
511C3×C4#380411yescharacteristic
512C3×C4#380511yescharacteristic
513C3×C4#381111yescharacteristic
514Dic3—21no—
515Dic3—21no—
516Dic3—61no—
517Dic3—61no—
518Dic3—121no—
519Dic3#371112yescomplemented
520C32—21no—
521C32—21no—
522C32—21no—
523C32—41no—
524C32#468811yescharacteristic, complemented
525C32#472911yescharacteristic, complemented
526C32#501211yescharacteristic, complemented
527C23—31no—
528C23—91no—
529D4—182no—
530C2×C4—31no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—91no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4D3311yescharacteristic
539C8—61no—
540C8—92no—
541C8—181no—
542C8—541no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—41no—
555C2×C3—122no—
556C2×C3—121no—
557C2×C3—121no—
558C2×C3—361no—
559C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
560C2×C3#6380311yescharacteristic
561C2×C3D3×#20811yescharacteristic
562C2×C3#6381411yescharacteristic
563C2×C3#6381411yescharacteristic
564C2×C3C2×#371111yescharacteristic
565C2×C3#6388011yescharacteristic
566C2×C3Dic3×D1211yescharacteristic
567C2×C3#67⋊Dic311yescharacteristic
568D3—41no—
569D3—122no—
570D3—121no—
571D3—121no—
572C22—61no—
573C22—61no—
574C22—61no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22D32×Dic311yescharacteristic
579C4—32no—
580C4—61no—
581C4—61no—
582C4—181no—
583C4#8895511yescharacteristic
584C4#8896111yescharacteristic
585C3—21no—
586C3—21no—
587C3—21no—
588C3—41no—
589C3#1059033711yescharacteristic, complemented
590C3#1059073011yescharacteristic, complemented
591C3#1059106711yescharacteristic, complemented
592C2—121no—
593C2—361no—
594C2D3×#371111yescharacteristic
595C2#1172282111yescharacteristic
596C2#1172292811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×C3×#626C4
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88809C2semidirect
2. C2×#88786C2semidirect
4. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
5. #11722023C2semidirect
6. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
46. (C2×D3)⋊(C3⋊C8)D3semidirect
47. Dic3×C3⋊C8D3semidirect
62. C3×C3⋊(C3⋊C8)D4semidirect
129. C32×Dic3M16semidirect
213. C3⋊(C3⋊C8)D12semidirect
338. C3×Dic3ΓL(1,𝔽25)semidirect
339. C3×Dic3#208semidirect
347. C33#436semidirect
519. Dic3#3711semidirect
524. C32#4688semidirect
525. C32#4729semidirect
526. C32#5012semidirect
589. C3#10590337semidirect
590. C3#10590730semidirect
591. C3#10591067semidirect

Maximal subgroups

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

subgroupindex
C2×#888092
C2×#887862
C3×C4×#6262
Dic3⋊(C3⋊(C3⋊C8))2
#117220232
C3×(C2×D3)⋊(C3⋊C8)2
C3×Dic3×C3⋊C82
#105910673
#105907303
#105903373