Finite Group Databasebeta

← #13155566

Subgroups — #13155566

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88786C211yesmaximal, characteristic, complemented
2C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C3×C4×#625C211yesmaximal, characteristic
4#11722078C211yesmaximal, characteristic, complemented
5#11722020C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591158—31nomaximal
9#10591151—31nomaximal
10#10590725—31nomaximal
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#625C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88786—22no—
20D3×C3⋊(C3⋊C8)—22no—
21C3×#3724C411yescharacteristic
22C3×(C2×D3)⋊Dic3C411yescharacteristic
23C3×(C2×D3)⋊Dic3C411yescharacteristic
24C2×#3748—31no—
25C22×C3⋊(C3⋊C8)—31no—
26C3×C4×#67—31no—
27C3×C4×#67—31no—
28C4×#625D311yescharacteristic
29C2×#3711—31no—
30C2×#3711—31no—
31C2×#3711—61no—
32C2×D3×C3⋊C8—31no—
33C2×D3×C3⋊C8—31no—
34C2×D3×C3⋊C8—61no—
35#63642—31no—
36C3×#830—31no—
37C3×#830—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×(C2×D3)⋊C8—31no—
40#63572—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
44(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
45#63554—31no—
46#63554—31no—
47#63554—61no—
48C22×C32×D3C2×C411yescharacteristic
49C22×C32×D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C3×#625—22no—
60C3⋊(C3⋊(C3⋊C8))D412yescomplemented
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5011—91no—
65#4730—91no—
66#4728—91no—
67C22×C32×C4—31no—
68C22×C32×C4—31no—
69C2×C3×#67—31no—
70C2×C3×#67—31no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3C2×D311yescharacteristic
74C2×C3×C4×D3C2×D311yescharacteristic
75C2×C3×C4×D3C2×D311yescharacteristic
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×#625Dic311yescharacteristic
78C2×C32×C8—31no—
79C4×C3⋊Dic3C2×D311yescharacteristic
80#3748—62no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—62no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—121no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×#217—31no—
92C3×#217—31no—
93C3×(C2×D3)⋊C4—31no—
94C3×(C2×D3)⋊C4—31no—
95C3×Dic3⋊C4—31no—
96C3×Dic3⋊C4—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109#3724Dic311yescharacteristic
110(C2×D3)⋊Dic3Dic311yescharacteristic
111(C2×D3)⋊Dic3Dic311yescharacteristic
112#3711—62no—
113#3711—62no—
114#3711—64no—
115D3×C3⋊C8—62no—
116D3×C3⋊C8—62no—
117D3×C3⋊C8—64no—
118C22×C33C22×C411yescharacteristic
119C2×C32×D3—21no—
120C2×C32×D3—21no—
121C2×C32×D3—21no—
122C2×C32×D3—21no—
123C2×C32×D3—21no—
124C2×C32×D3M1612yescomplemented
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3—21no—
129C32×Dic3—21no—
130C32×Dic3—21no—
131C3×C4×D4—91no—
132C4×#67—31no—
133C4×#67—31no—
134C2×#208—91no—
135C2×#208—91no—
136C2×#208—181no—
137C22×C3⋊C8—91no—
138C22×C3⋊C8—91no—
139C22×C3⋊C8—181no—
140C2×ΓL(1,𝔽25)—91no—
141C2×D3×C8—91no—
142C3×C22⋊C8—91no—
143C3×C22⋊C8—91no—
144#830—31no—
145#830—31no—
146(C2×D3)⋊C8—31no—
147(C2×D3)⋊C8—31no—
148#815—91no—
149#815—91no—
150#815—181no—
151Dic3⋊C8—91no—
152Dic3⋊C8—91no—
153C4⋊(C3⋊C8)—91no—
154#803—91no—
155C23×C32—31no—
156C23×C32—31no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3C4×D311yescharacteristic
160C22×C3×D3C4×D311yescharacteristic
161C22×C3×D3C2×Dic311yescharacteristic
162C22×C3×D3C2×Dic311yescharacteristic
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3C2×Dic311yescharacteristic
177C2×C3⋊Dic3C2×Dic311yescharacteristic
178C3×#67—62no—
179C3×#67—62no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3C4×D311yescharacteristic
189C2×C3×Dic3C4×D311yescharacteristic
190C2×C3×Dic3C2×Dic311yescharacteristic
191C2×C3×Dic3C2×Dic311yescharacteristic
192C3×C4×D3—22no—
193C3×C4×D3—22no—
194C3×C4×D3—22no—
195C3×C4×D3—22no—
196C3×C4×D3—21no—
197C3×C4×D3—21no—
198C3×C4×D3—21no—
199C3×C4×D3—21no—
200C3×C4×D3—21no—
201C3×C4×D3—21no—
202C3×C4×D3—21no—
203C3×C4×D3—21no—
204#625—22no—
205C32×C8—61no—
206C3⋊(C3⋊C8)—21no—
207C3⋊(C3⋊C8)—21no—
208C3⋊(C3⋊C8)—32no—
209C3⋊(C3⋊C8)—32no—
210C3⋊(C3⋊C8)—32no—
211C3⋊(C3⋊C8)—62no—
212C3⋊(C3⋊C8)—62no—
213C3⋊(C3⋊C8)—62no—
214C3⋊(C3⋊C8)—122no—
215C3×C3⋊C8—22no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—62no—
219C3×C3⋊C8—62no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224#435—271noSylow
225C2×C33C4×D411yescharacteristic
226C2×C33C2×M1611yescharacteristic
227C2×C33#13111yescharacteristic
228C32×D3—22no—
229C32×D3—41no—
230C2×C3×D4—91no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—61no—
236C22×C3×C4—61no—
237C2×#67—31no—
238C2×#67—31no—
239C2×C4×D3D3211yescharacteristic
240C2×C4×D3D3211yescharacteristic
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—61no—
244C2×C3×C8—91no—
245C2×C3×C8—91no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C42—91no—
250#217—31no—
251#217—31no—
252(C2×D3)⋊C4—31no—
253(C2×D3)⋊C4—31no—
254Dic3⋊C4—31no—
255Dic3⋊C4—31no—
256C4×Dic3—31no—
257C4×Dic3—31no—
258C4×Dic3—61no—
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—182no—
275C2×C3⋊C8—182no—
276C2×C3⋊C8—184no—
277C2×C3⋊C8—361no—
278C2×C3⋊C8D3211yescharacteristic
279ΓL(1,𝔽25)—182no—
280D3×C8—182no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—32no—
285C22×C32—41no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32—61no—
289C22×C32—61no—
290C22×C32—61no—
291C22×C32C2×C4×D311yescharacteristic
292C22×C32C2×C4×D311yescharacteristic
293C22×C32C22×Dic311yescharacteristic
294C2×C3×D3—22no—
295C2×C3×D3—22no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3—22no—
301C2×C3×D3—21no—
302C2×C3×D3—21no—
303C2×C3×D3—21no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3—22no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3ΓL(1,𝔽25)12yescomplemented
311C2×C3×D3#20812yescomplemented
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—61no—
321C32×C4—61no—
322C32×C4D3×D411yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4#23711yescharacteristic
326C32×C4#23711yescharacteristic
327C32×C4#23711yescharacteristic
328C3⋊Dic3—21no—
329C3⋊Dic3—21no—
330C3×Dic3—22no—
331C3×Dic3—22no—
332C3×Dic3—21no—
333C3×Dic3—21no—
334C3×Dic3—21no—
335C3×Dic3—21no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—121no—
341C3×Dic3—121no—
342C2×M16—271no—
343C22×C8—271no—
344C4×D4—91no—
345C4⋊C8—271no—
346C22⋊C8—91no—
347C22⋊C8—91no—
348#97—271no—
349C33#43511yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—31no—
353C23×C3—31no—
354C23×C3—61no—
355C23×C3—61no—
356C22×D3D3×Dic311yescharacteristic
357C22×D3D3×Dic311yescharacteristic
358C3×D4—182no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
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—62no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386#67—62no—
387#67—62no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3D3×Dic311yescharacteristic
395C2×Dic3D3×Dic311yescharacteristic
396C4×D3—21no—
397C4×D3—21no—
398C4×D3—21no—
399C4×D3—21no—
400C3×C8—62no—
401C3×C8—61no—
402C3×C8—61no—
403C3×C8—181no—
404C3×C8—181no—
405C3⋊C8—21no—
406C3⋊C8—62no—
407C3⋊C8—62no—
408C3⋊C8—61no—
409C3⋊C8—61no—
410C3⋊C8—61no—
411C3⋊C8—61no—
412C3⋊C8—92no—
413C3⋊C8—92no—
414C3⋊C8—92no—
415C3⋊C8—182no—
416C3⋊C8—182no—
417C3⋊C8—182no—
418C3⋊C8—362no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—62no—
432C2×C32—121no—
433C2×C32#67⋊C411yescharacteristic
434C2×C32#67⋊C411yescharacteristic
435C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
436C2×C32#90111yescharacteristic
437C2×C32D3×M1611yescharacteristic
438C2×C32#90711yescharacteristic
439C2×C32C2×#20811yescharacteristic
440C2×C32D4×Dic311yescharacteristic
441C2×C32#94811yescharacteristic
442C3×D3—24no—
443C3×D3—22no—
444C3×D3—22no—
445C3×D3—42no—
446C3×D3—41no—
447C3×D3—41no—
448C2×D4—91no—
449C22×C4—31no—
450C22×C4—31no—
451M16—542no—
452C2×C8—31no—
453C2×C8—91no—
454C2×C8—91no—
455C2×C8—271no—
456C2×C8—542no—
457C4⋊C4—91no—
458C22⋊C4—91no—
459C22⋊C4—91no—
460C42—91no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—32no—
465C22×C3—32no—
466C22×C3—41no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—62no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3—61no—
479C22×C3—61no—
480C22×C3—61no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3C4×D3211yescharacteristic
484C22×C3C2×D3×Dic311yescharacteristic
485C22×C3C2×D3×Dic311yescharacteristic
486C2×D3—21no—
487C2×D3—21no—
488C2×D3—21no—
489C2×D3—21no—
490C2×D3—21no—
491C2×D3#371112yescomplemented
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
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—181no—
507C3×C4—181no—
508C3×C4#380211yescharacteristic
509C3×C4#380511yescharacteristic
510C3×C4#380511yescharacteristic
511C3×C4#380511yescharacteristic
512C3×C4#381111yescharacteristic
513C3×C4#381111yescharacteristic
514Dic3—21no—
515Dic3—21no—
516Dic3—61no—
517Dic3—61no—
518Dic3—61no—
519Dic3—61no—
520Dic3—121no—
521Dic3—121no—
522C32—21no—
523C32—21no—
524C32—21no—
525C32—41no—
526C32#472811yescharacteristic, complemented
527C32#473011yescharacteristic, complemented
528C32#501111yescharacteristic, complemented
529C23—31no—
530C23—31no—
531D4—182no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—91no—
539C2×C4—91no—
540C2×C4D3311yescharacteristic
541C8—61no—
542C8—181no—
543C8—181no—
544C8—272no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—64no—
558C2×C3—62no—
559C2×C3—62no—
560C2×C3—122no—
561C2×C3—121no—
562C2×C3—121no—
563C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
564C2×C3#6380411yescharacteristic
565C2×C3D3×#20811yescharacteristic
566C2×C3#6381411yescharacteristic
567C2×C3#6381511yescharacteristic
568C2×C3C2×#371111yescharacteristic
569C2×C3#6390211yescharacteristic
570C2×C3#67⋊Dic311yescharacteristic
571C2×C3#67⋊Dic311yescharacteristic
572D3—22no—
573D3—41no—
574C22—32no—
575C22—61no—
576C22—61no—
577C22—61no—
578C22—61no—
579C22—61no—
580C22D32×Dic311yescharacteristic
581C4—61no—
582C4—61no—
583C4—181no—
584C4—181no—
585C4#8895411yescharacteristic
586C4#8896311yescharacteristic
587C3—21no—
588C3—21no—
589C3—21no—
590C3—41no—
591C3#1059072511yescharacteristic, complemented
592C3#1059115111yescharacteristic, complemented
593C3#1059115811yescharacteristic, complemented
594C2—62no—
595C2—121no—
596C2D3×#371111yescharacteristic
597C2#1172282211yescharacteristic
598C2#1172292511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×C3×#625C4
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88786C2semidirect
2. C2×D3×C3⋊(C3⋊C8)C2semidirect
4. #11722078C2semidirect
5. #11722020C2semidirect
6. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
43. (C2×D3)⋊(C3⋊C8)D3semidirect
44. (C2×D3)⋊(C3⋊C8)D3semidirect
60. C3⋊(C3⋊(C3⋊C8))D4semidirect
124. C2×C32×D3M16semidirect
310. C2×C3×D3ΓL(1,𝔽25)semidirect
311. C2×C3×D3#208semidirect
349. C33#435semidirect
491. C2×D3#3711semidirect
526. C32#4728semidirect
527. C32#4730semidirect
528. C32#5011semidirect
591. C3#10590725semidirect
592. C3#10591151semidirect
593. C3#10591158semidirect

Maximal subgroups

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

subgroupindex
C2×#887862
C2×D3×C3⋊(C3⋊C8)2
C3×C4×#6252
#117220782
#117220202
C3×(C2×D3)⋊(C3⋊C8)2
#105911583
#105911513
#105907253