Finite Group Databasebeta

← #13155602

Subgroups — #13155602

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
1C4×#6295C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic, complemented
3C2×C3×#3712C211yesmaximal, characteristic, complemented
4#11722051C211yesmaximal, characteristic, complemented
5#11722023C211yesmaximal, characteristic, complemented
6C3×#63557C211yesmaximal, characteristic, complemented
7C3×#63553C211yesmaximal, characteristic, complemented
8#10590735—31nomaximal
9#10590730—31nomaximal
10#10590729—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6295C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815C411yescharacteristic
20#88814C411yescharacteristic
21#88814C411yescharacteristic
22#88787—22no—
23C3×#3712—22no—
24C2×#3743—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26C2×C3×ΓL(1,𝔽25)—31no—
27C4×#626—31no—
28C4×#626—31no—
29C4×#625—31no—
30C2×#3712D311yescharacteristic, complemented
31C2×#3711—31no—
32C2×#3711—31no—
33C2×#3711—61no—
34#63647—31no—
35#63641—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×(C2×D3)⋊C8—31no—
38C3×C8×Dic3—31no—
39C3×C8×Dic3—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—61no—
43#63557D311yescharacteristic, complemented
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—31no—
46(C2×D3)⋊(C3⋊C8)—61no—
47#63553D311yescharacteristic, complemented
48C22×C3×C3⋊D3C2×C411yescharacteristic
49C22×C3×C3⋊D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊Dic3C2×C411yescharacteristic
53C2×C3×C3⋊Dic3C2×C411yescharacteristic
54C2×C3×C3⋊Dic3C2×C411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59#6295—22no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C32×C3⋊C8—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8D412yescomplemented
64#5128—91no—
65#4729—91no—
66#4688—91no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—61no—
75C2×#626—31no—
76C2×#626—31no—
77C2×#625—31no—
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C2×C32×C8—31no—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—121no—
89#3743—62no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—61no—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8C2×D311yescharacteristic
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C3×ΓL(1,𝔽25)—62no—
103C3×ΓL(1,𝔽25)—62no—
104#3724—31no—
105Dic3⋊Dic3—31no—
106Dic3⋊Dic3—31no—
107#3722—31no—
108#3722—31no—
109(C2×D3)⋊Dic3—31no—
110(C2×D3)⋊Dic3—31no—
111(C2×D3)⋊Dic3—31no—
112(C2×D3)⋊Dic3—31no—
113#3712—22no—
114#3711—64no—
115#3711—62no—
116#3711—62no—
117C22×C33C22×C411yescharacteristic
118C2×C3×C3⋊D3—21no—
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—21no—
123C2×C3×C3⋊D3—21no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3M1612yescomplemented
130C2×C3×M16—91no—
131C4×#67—91no—
132C4×#67—91no—
133C2×#208—91no—
134C2×ΓL(1,𝔽25)—31no—
135C2×ΓL(1,𝔽25)—31no—
136C2×ΓL(1,𝔽25)—91no—
137C2×ΓL(1,𝔽25)—91no—
138C2×ΓL(1,𝔽25)—181no—
139C4×D12—91no—
140C3×C22⋊C8—91no—
141C3×C4×C8—91no—
142#830—91no—
143(C2×D3)⋊C8—31no—
144(C2×D3)⋊C8—31no—
145(C2×D3)⋊C8—91no—
146(C2×D3)⋊C8—91no—
147(C2×D3)⋊C8—181no—
148Dic3⋊C8—91no—
149Dic3⋊C8—91no—
150Dic3⋊C8—181no—
151C8×Dic3—31no—
152C8×Dic3—31no—
153C4⋊(C3⋊C8)—91no—
154C22×C3⋊D3C4×D311yescharacteristic
155C22×C3⋊D3C4×D311yescharacteristic
156C22×C3×D3—31no—
157C22×C3×D3—31no—
158C22×C3×D3—31no—
159C22×C3×D3—31no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3C4×D311yescharacteristic
170C2×C3⋊Dic3C4×D311yescharacteristic
171C2×C3⋊Dic3C4×D311yescharacteristic
172C2×C3⋊Dic3C4×D311yescharacteristic
173C4×C3⋊D3—21no—
174C4×C3⋊D3—21no—
175C4×C3⋊D3—21no—
176C4×C3⋊D3—21no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C3×C4×D3—62no—
190C3×C4×D3—62no—
191C3×C4×D3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—121no—
200C3×C4×D3—121no—
201#626—62no—
202#626—62no—
203#625—62no—
204C32×C8—32no—
205C32×C8—61no—
206C32×C8—61no—
207C3⋊(C3⋊C8)—61no—
208C3⋊(C3⋊C8)—62no—
209C3⋊(C3⋊C8)—61no—
210C3⋊(C3⋊C8)—61no—
211C3⋊(C3⋊C8)—121no—
212C3⋊(C3⋊C8)—121no—
213C3⋊(C3⋊C8)—122no—
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—22no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8—41no—
221C3×C3⋊C8D1212yescomplemented
222C3×C3⋊C8#6712yescomplemented
223#436—271noSylow
224C2×C33C4×D411yescharacteristic
225C2×C33C2×M1611yescharacteristic
226C2×C33#13111yescharacteristic
227C3×C3⋊D3—41no—
228C3×C3⋊D3—41no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C2×#67—91no—
232C2×#67—91no—
233C2×D12—91no—
234C2×C4×D3—31no—
235C2×C4×D3—31no—
236C2×C4×D3—31no—
237C2×C4×D3—31no—
238C2×C4×D3—61no—
239C2×C4×D3—61no—
240C3×M16—182no—
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—61no—
248C2×C3×C8—61no—
249C2×C3×C8—61no—
250C3×C42—91no—
251#217—91no—
252#217—91no—
253(C2×D3)⋊C4—91no—
254(C2×D3)⋊C4—91no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—91no—
257C4⋊Dic3—91no—
258Dic3⋊C4—91no—
259Dic3⋊C4—91no—
260C4×Dic3—31no—
261C4×Dic3—31no—
262C4×Dic3—61no—
263#208—182no—
264C2×C3⋊C8—91no—
265C2×C3⋊C8—91no—
266C2×C3⋊C8—91no—
267C2×C3⋊C8—181no—
268C2×C3⋊C8—181no—
269C2×C3⋊C8—181no—
270C2×C3⋊C8—361no—
271C2×C3⋊C8D3211yescharacteristic
272C2×C3⋊C8D3211yescharacteristic
273C2×C3⋊C8D3211yescharacteristic
274ΓL(1,𝔽25)—62no—
275ΓL(1,𝔽25)—62no—
276ΓL(1,𝔽25)—182no—
277ΓL(1,𝔽25)—182no—
278ΓL(1,𝔽25)—184no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32C2×C4×D311yescharacteristic
284C22×C32C2×C4×D311yescharacteristic
285C22×C32C2×C4×D311yescharacteristic
286C2×C3⋊D3—21no—
287C2×C3⋊D3—21no—
288C2×C3⋊D3—21no—
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—61no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
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—41no—
317C32×C4—41no—
318C32×C4C2×D1211yescharacteristic
319C32×C4#23511yescharacteristic
320C32×C4#23511yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4#23711yescharacteristic
323C32×C4C2×#6711yescharacteristic
324C3⋊Dic3—21no—
325C3⋊Dic3—21no—
326C3⋊Dic3—21no—
327C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
328C3×Dic3—32no—
329C3×Dic3—32no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C2×M16—91no—
341C2×M16—271no—
342C4×D4—271no—
343C4⋊C8—271no—
344C22⋊C8—91no—
345C22⋊C8—271no—
346C4×C8—91no—
347C33#43611yescharacteristic, complemented, Sylow
348C23×C3—91no—
349C23×C3—91no—
350C22×D3—31no—
351C22×D3—31no—
352C22×D3—31no—
353C22×D3—31no—
354C22×D3—61no—
355C22×D3—61no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371#67—182no—
372#67—182no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385D12—182no—
386C4×D3—62no—
387C4×D3—62no—
388C4×D3—61no—
389C4×D3—61no—
390C4×D3—61no—
391C4×D3—61no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—61no—
395C4×D3—61no—
396C4×D3—121no—
397C4×D3—121no—
398C3×C8—32no—
399C3×C8—32no—
400C3×C8—62no—
401C3×C8—61no—
402C3×C8—61no—
403C3×C8—61no—
404C3×C8—61no—
405C3×C8—121no—
406C3×C8—121no—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—182no—
410C3⋊C8—181no—
411C3⋊C8—181no—
412C3⋊C8—181no—
413C3⋊C8—361no—
414C3⋊C8—361no—
415C3⋊C8—362no—
416C3⋊C8#62612yescomplemented
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×C32C4×D1211yescharacteristic
430C2×C32#67⋊C411yescharacteristic
431C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
432C2×C32#90111yescharacteristic
433C2×C32#90111yescharacteristic
434C2×C32D3×M1611yescharacteristic
435C2×C32D3×M1611yescharacteristic
436C2×C32#90711yescharacteristic
437C2×C32C4×#6711yescharacteristic
438C3⋊D3—41no—
439C3⋊D3—41no—
440C3×D3—122no—
441C3×D3—122no—
442C3×D3—121no—
443C3×D3—121no—
444C3×D3—121no—
445C3×D3—121no—
446C2×D4—271no—
447C22×C4—91no—
448C22×C4—91no—
449M16—182no—
450M16—542no—
451C2×C8—31no—
452C2×C8—31no—
453C2×C8—31no—
454C2×C8—271no—
455C4⋊C4—271no—
456C22⋊C4—271no—
457C22⋊C4—271no—
458C42—91no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3—181no—
468C22×C3—181no—
469C22×C3C4×D3211yescharacteristic
470C22×C3C4×D3211yescharacteristic
471C22×C3C4×D3211yescharacteristic
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—61no—
475C2×D3—61no—
476C2×D3—61no—
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—121no—
487C2×D3—121no—
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—41no—
497C3×C4—41no—
498C3×C4—92no—
499C3×C4—181no—
500C3×C4—181no—
501C3×C4—181no—
502C3×C4#379511yescharacteristic
503C3×C4#379811yescharacteristic
504C3×C4D3×D1211yescharacteristic
505C3×C4#380411yescharacteristic
506C3×C4C2×#62611yescharacteristic
507C3×C4D3×#6711yescharacteristic
508Dic3—32no—
509Dic3—32no—
510Dic3—62no—
511Dic3—62no—
512Dic3—61no—
513Dic3—61no—
514Dic3—61no—
515Dic3—61no—
516Dic3—61no—
517Dic3—61no—
518Dic3—121no—
519Dic3—121no—
520C32—21no—
521C32—21no—
522C32—21no—
523C32—41no—
524C32#468811yescharacteristic, complemented
525C32#472911yescharacteristic, complemented
526C32#512811yescharacteristic, complemented
527C23—91no—
528C23—91no—
529D4—542no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—91no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4D3311yescharacteristic
539C8—32no—
540C8—61no—
541C8—61no—
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—361no—
556C2×C3—361no—
557C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
558C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
559C2×C3#6379111yescharacteristic
560C2×C3#6380311yescharacteristic
561C2×C3#6380411yescharacteristic
562C2×C3#6380411yescharacteristic
563C2×C3#6387911yescharacteristic
564C2×C3#6388011yescharacteristic
565C2×C3C4×#62611yescharacteristic
566D3—122no—
567D3—122no—
568D3—121no—
569D3—121no—
570D3—121no—
571D3—121no—
572C22—181no—
573C22—181no—
574C22—181no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22#8894911yescharacteristic
579C4—92no—
580C4—181no—
581C4—181no—
582C4—181no—
583C4D3×#62611yescharacteristic
584C4#8896011yescharacteristic
585C3—21no—
586C3—21no—
587C3—21no—
588C3—41no—
589C3#1059072911yescharacteristic, complemented
590C3#1059073011yescharacteristic, complemented
591C3#1059073511yescharacteristic, complemented
592C2—361no—
593C2—361no—
594C2#1172282011yescharacteristic
595C2#1172282611yescharacteristic
596C2#1172293411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×#6295C4
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88787C2semidirect
3. C2×C3×#3712C2semidirect
4. #11722051C2semidirect
5. #11722023C2semidirect
6. C3×#63557C2semidirect
7. C3×#63553C2semidirect
30. C2×#3712D3semidirect
43. #63557D3semidirect
47. #63553D3semidirect
63. C32×C3⋊C8D4semidirect
129. C3×C3⋊Dic3M16semidirect
221. C3×C3⋊C8D12semidirect
222. C3×C3⋊C8#67semidirect
327. C3⋊Dic3ΓL(1,𝔽25)semidirect
347. C33#436semidirect
416. C3⋊C8#626semidirect
524. C32#4688semidirect
525. C32#4729semidirect
526. C32#5128semidirect
589. C3#10590729semidirect
590. C3#10590730semidirect
591. C3#10590735semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
C2×#887872
C2×C3×#37122
#117220512
#117220232
C3×#635572
C3×#635532
#105907353
#105907303
#105907293