Finite Group Databasebeta

← #13166643

Subgroups — #13166643

Order 1728.

1152 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×#157C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×#67C211yesmaximal, characteristic
3C2×#88994C211yesmaximal, characteristic, complemented
4C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
6#11722722C211yesmaximal, characteristic, complemented
7#11722722C211yesmaximal, characteristic, complemented
8#10593826—31nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×D3×C3⋊D3C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3⋊#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×D3×#157—22no—
16C2×D3×#157—22no—
17C2×D3×#157—22no—
18C2×#6312C411yescharacteristic, complemented
19C2×C32×#67C411yescharacteristic, complemented
20C3⋊D3×#67—22no—
21C3⋊D3×#67—22no—
22C2×#6292C411yescharacteristic, complemented
23C2×#6290C411yescharacteristic, complemented
24C2×Dic3×C3⋊D3C2211yescharacteristic
25#88994—22no—
26C3×C22⋊#157—22no—
27(C2×D3)⋊#157—22no—
28C23×#157—31no—
29C2×D4×C3⋊D3—31no—
30C2×D3×#67—62no—
31C2×C22⋊#157—31no—
32C2×C22⋊#157—31no—
33C2×C22⋊#157D311yescharacteristic, complemented
34#63781—31no—
35#63781—31no—
36C23×C33C2×C411yescharacteristic, complemented
37C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
45C22×C3×C3⋊D3C2311yescharacteristic
46C22×C32×D3C2×C411yescharacteristic
47C2×D3×C3⋊D3—21no—
48C2×D3×C3⋊D3—21no—
49C2×D3×C3⋊D3—21no—
50C2×D3×C3⋊D3—21no—
51C2×D3×C3⋊D3D412yes—
52C2×D3×C3⋊D3D412yes—
53C2×C3⋊#157—21no—
54C2×C3⋊#157—21no—
55C2×C3⋊#157—21no—
56C2×C3⋊#157—21no—
57C2×C3⋊#157—21no—
58C2×C3⋊#157D412yescomplemented
59C2×C3×#157—21no—
60C2×C3×#157—21no—
61C2×C3×#157—21no—
62C2×C3×#157—21no—
63C2×C3×#157—21no—
64C2×C3×#157D412yescomplemented
65D3×#157—28no—
66#6312—22no—
67C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic, complemented
68C32×#67—22no—
69C2×C32×Dic3C2×C411yescharacteristic, complemented
70#6292—22no—
71#6290—22no—
72Dic3×C3⋊D3—21no—
73Dic3×C3⋊D3—21no—
74#4956—91nomaximal
75C23×C3⋊D3—31no—
76C23×C3⋊D3C2×D311yescharacteristic
77C23×C3×D3—62no—
78C22×D32—62no—
79C22×#157—31no—
80C22×#157—31no—
81C22×#157—62no—
82C22×#157—62no—
83C22×#157—62no—
84C22×#157C2×D311yescharacteristic
85C22×#157C2×D311yescharacteristic
86C2×C32×D4—31no—
87C2×#638—31no—
88C2×#638—31no—
89C2×#638—62no—
90C2×#638—122no—
91D4×C3⋊D3—62no—
92D4×C3⋊D3—62no—
93C2×C4⋊(C3⋊D3)—31no—
94C2×C4×C3⋊D3—31no—
95C2×C3×#67—22no—
96D3×#67—68no—
97D3×#67—68no—
98C2×#626—62no—
99C2×#625—62no—
100C2×D3×Dic3—62no—
101C22⋊#157—22no—
102C22⋊#157—62no—
103C22⋊#157—62no—
104C22×C33—21no—
105C22×C33—21no—
106C22×C33—21no—
107C22×C33C22×C411yescharacteristic
108C2×C3⋊(C3⋊D3)—21no—
109C2×C3⋊(C3⋊D3)—21no—
110C2×C3⋊(C3⋊D3)C22⋊C412yescomplemented
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3—21no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C2×C3×C3⋊D3—41no—
118C2×C3×C3⋊D3C2×D411yescharacteristic
119C2×C3×C3⋊D3C2×D411yescharacteristic
120C2×C3×C3⋊D3C2×D411yescharacteristic
121C2×C3×C3⋊D3C2×D411yescharacteristic
122C2×C3×C3⋊D3#4111yescharacteristic
123C2×C3×C3⋊D3#4111yescharacteristic
124C2×C32×D3—21no—
125C2×C32×D3—21no—
126C2×C32×D3C22⋊C412yescomplemented
127D3×C3⋊D3—22no—
128D3×C3⋊D3—22no—
129D3×C3⋊D3—22no—
130D3×C3⋊D3—22no—
131C3⋊#157—22no—
132C3⋊#157—41no—
133C3×#157—22no—
134C3×#157—41no—
135C3⋊(C3⋊Dic3)—21no—
136C32×Dic3—21no—
137C22×#67—91no—
138C2×D3×D4—182no—
139C22×C4×D3—91no—
140C2×C3×C22⋊C4—91no—
141C2×#217—91no—
142C2×(C2×D3)⋊C4—91no—
143C3⋊#95—91no—
144C3⋊#95—91no—
145C23×C32—22no—
146C23×C32—31no—
147C23×C32—42no—
148C23×C32C4×D311yescharacteristic, complemented
149C22×C3⋊D3—21no—
150C22×C3⋊D3—21no—
151C22×C3⋊D3—21no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—21no—
155C22×C3⋊D3—31no—
156C22×C3⋊D3—32no—
157C22×C3⋊D3—32no—
158C22×C3⋊D3—62no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—61no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—122no—
164C22×C3⋊D3C4×D311yescharacteristic, complemented
165C22×C3⋊D3C22×D311yescharacteristic
166C22×C3×D3—22no—
167C22×C3×D3—62no—
168C22×C3×D3—62no—
169C22×C3×D3—64no—
170C22×C3×D3—64no—
171C22×C3×D3—122no—
172C22×C3×D3—122no—
173C22×C3×D3—122no—
174C22×C3×D3—122no—
175C2×D32—64no—
176C2×D32—64no—
177C2×D32—122no—
178C2×D32—122no—
179C2×D32—122no—
180C2×D32—122no—
181C2×#157—21no—
182C2×#157—21no—
183C2×#157—21no—
184C2×#157—21no—
185C2×#157—21no—
186C2×#157—32no—
187C2×#157—68no—
188C2×#157—61no—
189C2×#157—61no—
190C2×#157—61no—
191C2×#157—61no—
192C2×#157—61no—
193C2×#157#6712yescomplemented
194C32×D4—62no—
195C2×C32×C4—31no—
196#638—62no—
197#638—62no—
198#638—68no—
199#638—128no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—62no—
202C2×C3⋊Dic3—122no—
203C4⋊(C3⋊D3)—62no—
204C4×C3⋊D3—61no—
205C4×C3⋊D3—61no—
206C3×#67—28no—
207C2×C3×Dic3—22no—
208#626—68no—
209#625—68no—
210D3×Dic3—64no—
211D3×Dic3—64no—
212#386—271noSylow
213C2×C33—41no—
214C2×C33C2×C22⋊C411yescharacteristic
215C2×C33C4×D411yescharacteristic
216C2×C33C4×D411yescharacteristic
217C3⋊(C3⋊D3)—22no—
218C3×C3⋊D3—41no—
219C3×C3⋊D3#12011yescharacteristic
220C3×C3⋊D3#12111yescharacteristic
221C3×C3⋊D3#12111yescharacteristic
222C3×C3⋊D3#12311yescharacteristic
223C32×D3—22no—
224C24×C3—91no—
225C23×D3—62no—
226C23×D3—91no—
227C23×D3—182no—
228C2×C3×D4—62no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C2×#67—91no—
232C2×#67—91no—
233C2×#67—91no—
234C2×#67—182no—
235C2×#67—182no—
236C2×#67—182no—
237C2×#67—182no—
238C2×#67—362no—
239C2×#67#15711yescharacteristic, complemented
240C22×Dic3—91no—
241C22×Dic3—91no—
242C22×Dic3—91no—
243D3×D4—188no—
244D3×D4—188no—
245C2×D12—182no—
246C2×C4×D3—182no—
247C2×C4×D3—182no—
248C2×C4×D3—182no—
249C2×C4×D3—182no—
250C3×C22⋊C4—182no—
251#217—182no—
252(C2×D3)⋊C4—182no—
253C22×C32—21no—
254C22×C32—24no—
255C22×C32—22no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—32no—
259C22×C32—44no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—42no—
263C22×C32—61no—
264C22×C32—61no—
265C22×C32—82no—
266C22×C32—82no—
267C22×C32C2×C4×D311yescharacteristic
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3⋊D3—21no—
271C2×C3⋊D3—32no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C2×C3⋊D3—41no—
275C2×C3⋊D3—41no—
276C2×C3⋊D3—64no—
277C2×C3⋊D3—61no—
278C2×C3⋊D3—61no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—124no—
284C2×C3⋊D3—122no—
285C2×C3⋊D3—122no—
286C2×C3⋊D3—242no—
287C2×C3⋊D3—242no—
288C2×C3⋊D3#23511yescharacteristic
289C2×C3⋊D3D3×D411yescharacteristic
290C2×C3⋊D3D3×D411yescharacteristic
291C2×C3⋊D3D3×D411yescharacteristic
292C2×C3⋊D3#23711yescharacteristic
293C2×C3⋊D3C2×#6711yescharacteristic
294C2×C3×D3—24no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—64no—
304C2×C3×D3—122no—
305C2×C3×D3—122no—
306C2×C3×D3—124no—
307C2×C3×D3—124no—
308C2×C3×D3—124no—
309C2×C3×D3—124no—
310D32—124no—
311D32—124no—
312D32—124no—
313D32—124no—
314#157—22no—
315#157—41no—
316#157—62no—
317#157—121no—
318C32×C4—61no—
319C3⋊Dic3—61no—
320C3⋊Dic3—64no—
321C3⋊Dic3—124no—
322C3×Dic3—24no—
323C22×D4—271no—
324C23×C4—271no—
325C2×C22⋊C4—91no—
326C2×C22⋊C4—271no—
327C2×C22⋊C4—271no—
328#95—271no—
329#95—271no—
330C33#38611yescharacteristic, complemented, Sylow
331C23×C3—22no—
332C23×C3—42no—
333C23×C3—62no—
334C23×C3—91no—
335C23×C3—91no—
336C23×C3—181no—
337C23×C3—181no—
338C23×C3—181no—
339C23×C3—181no—
340C23×C3—181no—
341C23×C3—181no—
342C23×C3C2×#15711yescharacteristic, complemented
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—64no—
346C22×D3—64no—
347C22×D3—91no—
348C22×D3—92no—
349C22×D3—92no—
350C22×D3—122no—
351C22×D3—122no—
352C22×D3—122no—
353C22×D3—122no—
354C22×D3—184no—
355C22×D3—184no—
356C22×D3—182no—
357C22×D3—181no—
358C22×D3—181no—
359C22×D3—181no—
360C22×D3—181no—
361C22×D3—362no—
362C22×D3—362no—
363C22×D3—362no—
364C22×D3—362no—
365C22×D3—362no—
366C22×D3C2×#15711yescharacteristic
367C3×D4—68no—
368C2×C3×C4—62no—
369C2×C3×C4—92no—
370C2×C3×C4—181no—
371C2×C3×C4—181no—
372C2×C3×C4—181no—
373C2×C3×C4—181no—
374C2×C3×C4—181no—
375#67—22no—
376#67—182no—
377#67—188no—
378#67—182no—
379#67—188no—
380#67—182no—
381#67—368no—
382C2×Dic3—92no—
383C2×Dic3—91no—
384C2×Dic3—182no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—362no—
393C2×Dic3C2×#15711yescharacteristic, complemented
394D12—188no—
395C4×D3—184no—
396C4×D3—184no—
397C4×D3—188no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—42no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—41no—
405C2×C32—44no—
406C2×C32—62no—
407C2×C32—84no—
408C2×C32D3×C22⋊C411yescharacteristic
409C2×C32#67⋊C411yescharacteristic
410C2×C32C4×#6711yescharacteristic
411C3⋊D3—41no—
412C3⋊D3—62no—
413C3⋊D3—124no—
414C3⋊D3—244no—
415C3⋊D3#88311yescharacteristic
416C3⋊D3#88411yescharacteristic
417C3⋊D3#93711yescharacteristic
418C3⋊D3#93811yescharacteristic
419C3×D3—44no—
420C3×D3—62no—
421C3×D3—62no—
422C3×D3—62no—
423C3×D3—62no—
424C3×D3—124no—
425C24—91no—
426C24—271no—
427C2×D4—31no—
428C2×D4—271no—
429C2×D4—271no—
430C2×D4—271no—
431C2×D4—542no—
432C2×D4—542no—
433C22×C4—91no—
434C22×C4—91no—
435C22×C4—271no—
436C22×C4—271no—
437C22×C4—271no—
438C22×C4—542no—
439C22×C4—542no—
440C22×C4—542no—
441C22⋊C4—182no—
442C22⋊C4—542no—
443C22⋊C4—542no—
444C22×C3—24no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—44no—
452C22×C3—42no—
453C22×C3—64no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—91no—
457C22×C3—91no—
458C22×C3—91no—
459C22×C3—91no—
460C22×C3—91no—
461C22×C3—91no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3—361no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3—361no—
471C22×C3C22×#15711yescharacteristic
472C2×D3—21no—
473C2×D3—21no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—64no—
481C2×D3—92no—
482C2×D3—122no—
483C2×D3—122no—
484C2×D3—124no—
485C2×D3—124no—
486C2×D3—124no—
487C2×D3—124no—
488C2×D3—184no—
489C2×D3—182no—
490C2×D3—182no—
491C2×D3—182no—
492C2×D3—182no—
493C2×D3—181no—
494C2×D3—181no—
495C2×D3—364no—
496C2×D3—364no—
497C2×D3—364no—
498C2×D3—364no—
499C2×D3—364no—
500C2×D3—362no—
501C2×D3—362no—
502C2×D3—722no—
503C2×D3—722no—
504C2×D3C22⋊#15712yescomplemented
505C3×C4—64no—
506C3×C4—182no—
507C3×C4—361no—
508Dic3—21no—
509Dic3—184no—
510Dic3—182no—
511Dic3—181no—
512Dic3—364no—
513Dic3—361no—
514C32—22no—
515C32—42no—
516C32#495611yescharacteristic, complemented
517C23—31no—
518C23—91no—
519C23—91no—
520C23—181no—
521C23—181no—
522C23—181no—
523C23—181no—
524C23—181no—
525C23—181no—
526C23—272no—
527C23—272no—
528C23—271no—
529C23—541no—
530C23—541no—
531C23—541no—
532C23—541no—
533C23D3×#15711yescharacteristic, complemented
534D4—62no—
535D4—542no—
536D4—542no—
537D4—542no—
538C2×C4—31no—
539C2×C4—92no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4—181no—
543C2×C4—181no—
544C2×C4—181no—
545C2×C4—272no—
546C2×C4—271no—
547C2×C4—541no—
548C2×C4—541no—
549C2×C4—548no—
550C2×C4—541no—
551C2×C4—541no—
552C2×C4—541no—
553C2×C4—541no—
554C2×C4—541no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—44no—
559C2×C3—41no—
560C2×C3—42no—
561C2×C3—42no—
562C2×C3—42no—
563C2×C3—84no—
564C2×C3—91no—
565C2×C3—91no—
566C2×C3—91no—
567C2×C3—91no—
568C2×C3—124no—
569C2×C3—361no—
570C2×C3D4×#15711yescharacteristic
571C2×C3D4×#15711yescharacteristic
572C2×C3C2×C22⋊#15711yescharacteristic
573D3—22no—
574D3—62no—
575D3—62no—
576D3—62no—
577D3—62no—
578D3—124no—
579D3—182no—
580D3—364no—
581D3—724no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—32no—
586C22—61no—
587C22—61no—
588C22—91no—
589C22—91no—
590C22—91no—
591C22—91no—
592C22—91no—
593C22—91no—
594C22—181no—
595C22—181no—
596C22—181no—
597C22—272no—
598C22—361no—
599C22—361no—
600C22—361no—
601C22—361no—
602C22—542no—
603C22—542no—
604C22—542no—
605C22—542no—
606C22—541no—
607C22—541no—
608C22C2×D3×#15711yescharacteristic
609C4—61no—
610C4—182no—
611C4—361no—
612C4—542no—
613C4—541no—
614C4—1081no—
615C3—22no—
616C3—42no—
617C3#1059382611yescharacteristic, complemented
618C2—41no—
619C2—62no—
620C2—91no—
621C2—91no—
622C2—91no—
623C2—91no—
624C2—361no—
625C2—542no—
626C2D3×C22⋊#15711yescharacteristic
627C2#67×#15711yescharacteristic
628C2#67⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC2×C3⋊D3×#67C2
cosocleC22×C3×C3⋊D3C23
hypercenterC23D3×#157
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×D3×#157C2semidirect
3. C2×#88994C2semidirect
4. C2×C3×C22⋊#157C2semidirect
5. C2×(C2×D3)⋊#157C2semidirect
6. #11722722C2semidirect
7. #11722722C2semidirect
18. C2×#6312C4semidirect
19. C2×C32×#67C4semidirect
22. C2×#6292C4semidirect
23. C2×#6290C4semidirect
33. C2×C22⋊#157D3semidirect
36. C23×C33C2×C4semidirect
44. C22×C3×C3⋊D3C2×C4semidirect
58. C2×C3⋊#157D4semidirect
64. C2×C3×#157D4semidirect
67. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
69. C2×C32×Dic3C2×C4semidirect
110. C2×C3⋊(C3⋊D3)C22⋊C4semidirect
126. C2×C32×D3C22⋊C4semidirect
148. C23×C32C4×D3semidirect
164. C22×C3⋊D3C4×D3semidirect
193. C2×#157#67semidirect
239. C2×#67#157semidirect
330. C33#386semidirect
342. C23×C3C2×#157semidirect
393. C2×Dic3C2×#157semidirect
504. C2×D3C22⋊#157semidirect
516. C32#4956semidirect
533. C23D3×#157semidirect
617. C3#10593826semidirect

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×C3⋊D3×#672
C2×#889942
C2×C3×C22⋊#1572
C2×(C2×D3)⋊#1572
#117227222
#105938263
#49569