Finite Group Databasebeta

← #11792022

Subgroups — #11792022

Order 1152.

841 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#897C211yesmaximal, characteristic
2C2×#63856C211yesmaximal, characteristic
3C22×#3718C211yesmaximal, characteristic, complemented
4C2×C3×#810C211yesmaximal, characteristic, complemented
5C2×#63566C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
7#10589395C211yesmaximal, characteristic, complemented
8#71700—31nomaximal
9#70273—31nomaximal
10C22×C3×C4×D3C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×C4⋊Dic3—21no—
18C22×C3×C3⋊C8C2211yescharacteristic
19C3×#897—22no—
20C3×#897—22no—
21C2×#3724—21no—
22C2×(C2×D3)⋊Dic3—21no—
23#63856—22no—
24#63856—22no—
25C2×#3718—22no—
26C2×#3718—22no—
27C2×#3718—22no—
28C3×#810—22no—
29#63566—22no—
30(C2×D3)⋊(C3⋊C8)—22no—
31C2×C3×C22⋊Q8—31no—
32C2×#946—31no—
33C22×#216—31no—
34C22×#216—31no—
35C2×#924—31no—
36C2×#897D311yescharacteristic
37C2×C3×#103—31no—
38C2×#835—31no—
39C2×#830—31no—
40C2×(C2×D3)⋊C8—31no—
41C2×#816—31no—
42C2×#810D311yescharacteristic, complemented
43#4529—31no—
44#4529—31no—
45#4529—61no—
46C23×C3×D3D411yescharacteristic
47C22×C32×C4—21no—
48C22×C32×C4C2311yescharacteristic
49C22×C3⋊Dic3—21no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—21no—
55C2×C3×C4×D3—21no—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6D412yes—
70C2×C3×Dic6D412yes—
71C32×C4⋊C4—21no—
72C32×C4⋊C4D412yes—
73C4⋊(C3⋊Dic3)—21no—
74C4⋊(C3⋊Dic3)D412yes—
75C2×C3⋊(C3⋊C8)—21no—
76C2×C3⋊(C3⋊C8)—21no—
77C2×C3⋊(C3⋊C8)D412yes—
78C3×(C2×D3)⋊C4—24no—
79C3×C4⋊Dic3—24no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83#3724—24no—
84(C2×D3)⋊Dic3—24no—
85#3718—28no—
86#2001—91noSylow
87C22×C3×Q8—31no—
88C22×C3×Q8—31no—
89C23×C3×C4—31no—
90C22×C4×D3C2×D311yescharacteristic
91C22×Dic6C2×D311yescharacteristic
92C22×Dic6C2×D311yescharacteristic
93C22×C3×C8—31no—
94C3×C22⋊Q8—62no—
95C3×C22⋊Q8—62no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—61no—
98C2×C3×C4⋊C4C2×D311yescharacteristic
99C2×C3×C4⋊C4C2×D311yescharacteristic
100C2×C3×C22⋊C4—61no—
101C2×#217—61no—
102#946—62no—
103#946—62no—
104C2×#216—62no—
105C2×#216—62no—
106C2×#216—62no—
107C2×#216—62no—
108C2×#216—62no—
109C2×#216—62no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—61no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—31no—
115C2×C4⋊Dic3—61no—
116#924—62no—
117#924—62no—
118C2×Dic3⋊C4—61no—
119C2×Dic3⋊C4—61no—
120C22×C3⋊C8—31no—
121C22×C3⋊C8—31no—
122C22×C3⋊C8—61no—
123C22×C3⋊C8C2×D311yescharacteristic
124#897—22no—
125#897—22no—
126C3×#103—62no—
127#835—62no—
128#830—62no—
129(C2×D3)⋊C8—62no—
130#816—62no—
131#810—22no—
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—21no—
134C22×C3×D3—21no—
135C22×C3×D3—21no—
136C22×C3×D3—21no—
137C22×C3×D3—21no—
138C22×C3×D3—21no—
139C22×C3×D3Q1612yescomplemented
140C2×C32×C4—22no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4#4111yescharacteristic
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—41no—
151C2×C3⋊Dic3—41no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—41no—
158C2×C3×Dic3—41no—
159C2×C3×Dic3—41no—
160C2×C3×Dic3—41no—
161C3×C4×D3—41no—
162C3×C4×D3—41no—
163C3×C4×D3—41no—
164C3×C4×D3—41no—
165C3×Dic6—22no—
166C3×Dic6—22no—
167C3×Dic6—22no—
168C3×Dic6—22no—
169C3×Dic6—41no—
170C3×Dic6—41no—
171C3×Dic6—41no—
172C3×Dic6—41no—
173C3⋊(C3⋊C8)—22no—
174C3×C3⋊C8—41no—
175C22×Q16—91no—
176C2×C22⋊Q8—31no—
177C2×C22⋊Q8—91no—
178C2×#103—31no—
179C2×#103—91no—
180C2×C22⋊C8—91no—
181#340—91no—
182C24×C3—31no—
183C23×D3#6711yescharacteristic
184C2×C3×Q8—32no—
185C2×C3×Q8—32no—
186C2×C3×Q8—61no—
187C2×C3×Q8—61no—
188C2×C3×Q8—61no—
189C2×C3×Q8—61no—
190C2×C3×Q8—61no—
191C2×C3×Q8—61no—
192C2×C3×Q8—61no—
193C2×C3×Q8—61no—
194C2×C3×Q8—61no—
195C2×C3×Q8—61no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—31no—
200C22×C3×C4—41no—
201C22×C3×C4—61no—
202C22×C3×C4—61no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4—61no—
207C22×C3×C4—61no—
208C22×C3×C4—61no—
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C22×D311yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—61no—
214C22×Dic3—61no—
215C22×Dic3—121no—
216C22×Dic3#6711yescharacteristic
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6D1212yes—
234C2×Dic6#6712yes—
235C2×C3×C8—61no—
236C2×C3×C8—61no—
237C2×C3×C8—61no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—64no—
243C3×C4⋊C4#6712yes—
244C3×C4⋊C4#6712yes—
245C3×C22⋊C4—64no—
246#217—64no—
247#216—68no—
248#216—68no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—64no—
251C4⋊Dic3—24no—
252C4⋊Dic3—32no—
253C4⋊Dic3—32no—
254C4⋊Dic3—62no—
255C4⋊Dic3—61no—
256C4⋊Dic3—61no—
257C4⋊Dic3—121no—
258Dic3⋊C4—64no—
259Dic3⋊C4—64no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—32no—
264C2×C3⋊C8—32no—
265C2×C3⋊C8—62no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—61no—
269C2×C3⋊C8—61no—
270C2×C3⋊C8—121no—
271C2×C3⋊C8—121no—
272C22×C32#12011yescharacteristic
273C22×C32#12011yescharacteristic
274C22×C32#12111yescharacteristic
275C22×C32C2×Q1611yescharacteristic
276C22×C32#13511yescharacteristic
277C22×C32#13711yescharacteristic
278C22×C32#13711yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C32×C4—42no—
287C32×C4#12011yescharacteristic
288C32×C4#12111yescharacteristic
289C32×C4#12111yescharacteristic
290C32×C4#12711yescharacteristic
291C3⋊Dic3—42no—
292C3×Dic3—42no—
293C3×Dic3—42no—
294C3×Dic3—41no—
295C22×Q8—31no—
296C22×Q8—31no—
297C23×C4—31no—
298C2×Q16—182no—
299C2×Q16—182no—
300C2×Q16—182no—
301C22×C8—31no—
302C22×C8—91no—
303C22⋊Q8—62no—
304C22⋊Q8—62no—
305C22⋊Q8—182no—
306C22⋊Q8—182no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—91no—
309C2×C4⋊C4—181no—
310C2×C4⋊C4D3211yescharacteristic
311C2×C22⋊C4—61no—
312C2×C22⋊C4—181no—
313#103—62no—
314#103—182no—
315C22⋊C8—182no—
316C23×C3—21no—
317C23×C3—32no—
318C23×C3—61no—
319C23×C3—61no—
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—61no—
323C23×C3—61no—
324C23×C3D3×D411yescharacteristic
325C23×C3C2×#6711yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3#21612yescomplemented
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—62no—
336C3×Q8—62no—
337C3×Q8—121no—
338C3×Q8—121no—
339C3×Q8—121no—
340C3×Q8—121no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—81no—
360C2×C3×C4—81no—
361C2×C3×C4—121no—
362C2×C3×C4—121no—
363C2×C3×C4—121no—
364C2×C3×C4—121no—
365C2×C3×C4—121no—
366C2×C3×C4—121no—
367C2×C3×C4—121no—
368C2×C3×C4—121no—
369C2×C3×C4C2×D1211yescharacteristic
370C2×C3×C4#23511yescharacteristic
371C2×C3×C4D3×D411yescharacteristic
372C2×C3×C4D3×D411yescharacteristic
373C2×C3×C4D3×D411yescharacteristic
374C2×C3×C4D3×D411yescharacteristic
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4#23711yescharacteristic
378C2×C3×C4C2×#6711yescharacteristic
379C2×C3×C4C2×#6711yescharacteristic
380C2×C3×C4C2×#6711yescharacteristic
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—122no—
393C2×Dic3—121no—
394C2×Dic3—121no—
395C2×Dic3—121no—
396C2×Dic3—121no—
397C2×Dic3—241no—
398C2×Dic3—241no—
399C4×D3—41no—
400C4×D3—41no—
401C4×D3—41no—
402C4×D3—41no—
403Dic6—22no—
404Dic6—22no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411C3×C8—121no—
412C3⋊C8—41no—
413C3⋊C8—62no—
414C3⋊C8—62no—
415C3⋊C8—122no—
416C2×C32#39211yescharacteristic
417C2×C32#45111yescharacteristic
418C2×C32#45111yescharacteristic
419C2×C32#45211yescharacteristic
420C2×C32#45211yescharacteristic
421C2×C32#46711yescharacteristic
422C2×C32#47011yescharacteristic
423C3×D3—42no—
424C24—31no—
425C2×Q8—32no—
426C2×Q8—32no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—181no—
448C22×C4C2×D3211yescharacteristic
449Q16—188no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—92no—
454C2×C8—181no—
455C2×C8—181no—
456C4⋊C4—21no—
457C4⋊C4—64no—
458C4⋊C4—92no—
459C4⋊C4—184no—
460C4⋊C4—181no—
461C4⋊C4#62512yes—
462C22⋊C4—64no—
463C22⋊C4—184no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—62no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3#88211yescharacteristic
479C22×C3#88411yescharacteristic
480C22×C3#91511yescharacteristic
481C22×C3#91611yescharacteristic
482C22×C3D3×Q1611yescharacteristic
483C22×C3#91811yescharacteristic
484C22×C3#93711yescharacteristic
485C22×C3#93711yescharacteristic
486C22×C3#93911yescharacteristic
487C22×C3#94211yescharacteristic
488C22×C3C2×#21611yescharacteristic
489C22×C3#95011yescharacteristic
490C22×C3#95111yescharacteristic
491C22×C3#95311yescharacteristic
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—41no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—82no—
506C3×C4—121no—
507C3×C4—122no—
508C3×C4—122no—
509C3×C4#88211yescharacteristic
510C3×C4#88411yescharacteristic
511C3×C4#93011yescharacteristic
512C3×C4#93711yescharacteristic
513C3×C4#93811yescharacteristic
514C3×C4#93911yescharacteristic
515C3×C4#94011yescharacteristic
516C3×C4#94011yescharacteristic
517Dic3—42no—
518Dic3—42no—
519Dic3—41no—
520Dic3—122no—
521Dic3—122no—
522Dic3—242no—
523C32#200111yescharacteristic, complemented, Sylow
524C23—32no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—61no—
531C23D3×#6711yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—62no—
535Q8—62no—
536Q8—121no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—61no—
546C2×C4—62no—
547C2×C4—61no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—182no—
557C2×C4—361no—
558C2×C4—361no—
559C2×C4#379511yescharacteristic
560C2×C4D3×D1211yescharacteristic
561C2×C4#380211yescharacteristic
562C2×C4#380211yescharacteristic
563C2×C4C2×#62511yescharacteristic
564C2×C4D3×#6711yescharacteristic
565C8—121no—
566C8—182no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—122no—
575C2×C3#480811yescharacteristic
576C2×C3#480911yescharacteristic
577C2×C3#481311yescharacteristic
578C2×C3#481611yescharacteristic
579C2×C3#496011yescharacteristic
580C2×C3#504711yescharacteristic
581C2×C3#505011yescharacteristic
582C2×C3#505111yescharacteristic
583C2×C3#505211yescharacteristic
584C2×C3#517311yescharacteristic
585C2×C3#518611yescharacteristic
586C2×C3#522311yescharacteristic
587C2×C3#522811yescharacteristic
588C2×C3#523911yescharacteristic
589D3—42no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22—121no—
597C22#6388511yescharacteristic
598C22#6392211yescharacteristic
599C22#6392911yescharacteristic
600C22#6393511yescharacteristic
601C22D3×#21611yescharacteristic
602C22#6394811yescharacteristic
603C22#6397611yescharacteristic
604C4—42no—
605C4—121no—
606C4—122no—
607C4—122no—
608C4—362no—
609C4#6388511yescharacteristic
610C4#6390811yescharacteristic
611C4#6391411yescharacteristic
612C4#6392211yescharacteristic
613C3—21no—
614C3#7027311yescharacteristic, complemented
615C3#7170011yescharacteristic, complemented
616C2—122no—
617C2#1059051811yescharacteristic
618C2#1059052011yescharacteristic
619C2#1059060411yescharacteristic
620C2#1059061611yescharacteristic
621C2#1059138911yescharacteristic
622C2#1059146111yescharacteristic
623C2#1059159411yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
hypercenterC2×C4⋊C4D32
hypocenterC32#2001
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×#3718C2semidirect
4. C2×C3×#810C2semidirect
5. C2×#63566C2semidirect
7. #10589395C2semidirect
42. C2×#810D3semidirect
139. C22×C3×D3Q16semidirect
332. C22×D3#216semidirect
523. C32#2001semidirect
614. C3#70273semidirect
615. C3#71700semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8972
C2×#638562
C22×#37182
C2×C3×#8102
C2×#635662
C2×(C2×D3)⋊(C3⋊C8)2
#105893952
#717003
#702733