Finite Group Databasebeta

← #13166606

Subgroups — #13166606

Order 1728.

1096 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1096 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
1C2×C3⋊D3×#67C211yesmaximal, characteristic
2C2×#88994C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4C2×Dic3⋊#157C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
7C2×Dic3×#157C211yesmaximal, characteristic, complemented
8#10593827—31nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×D3×C3⋊D3C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3⋊#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×#6312C411yescharacteristic, complemented
16C2×C32×#67C411yescharacteristic, complemented
17C3⋊D3×#67—22no—
18C3⋊D3×#67—22no—
19C2×#6292C411yescharacteristic, complemented
20C2×#6290C411yescharacteristic, complemented
21C2×Dic3×C3⋊D3C2211yescharacteristic
22#88994—22no—
23C3×C22⋊#157—22no—
24Dic3⋊#157—22no—
25(C2×D3)⋊#157—22no—
26(C2×D3)⋊#157—22no—
27Dic3×#157—22no—
28C2×D4×C3⋊D3—31no—
29C2×D3×#67—62no—
30C2×C22⋊#157—31no—
31C2×C22⋊#157—31no—
32C2×C22⋊#157—31no—
33C2×C22⋊#157D311yescharacteristic, complemented
34C2×C4⋊#157—31no—
35C2×C4×#157—31no—
36C23×C33C2×C411yescharacteristic
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
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⋊D3—21no—
52C2×D3×C3⋊D3—21no—
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
65#6312—22no—
66C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
67C32×#67—22no—
68C2×C32×Dic3C2×C411yescharacteristic
69#6292—22no—
70#6290—22no—
71Dic3×C3⋊D3D412yes—
72Dic3×C3⋊D3D412yes—
73#4957—91nomaximal
74C23×C3⋊D3—31no—
75C23×C3⋊D3C2×D311yescharacteristic
76C23×C3×D3—62no—
77C22×D32—62no—
78C22×#157—31no—
79C22×#157—31no—
80C22×#157C2×D311yescharacteristic
81C22×#157C2×D311yescharacteristic
82C2×C32×D4—31no—
83C2×#638—31no—
84C2×#638—31no—
85C2×#638—62no—
86C2×#638—122no—
87D4×C3⋊D3—62no—
88D4×C3⋊D3—62no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4×C3⋊D3—31no—
91C2×C3×#67—22no—
92D3×#67—68no—
93D3×#67—68no—
94C2×#626—62no—
95C2×#625—62no—
96C2×D3×Dic3—62no—
97C22⋊#157—22no—
98C22⋊#157—62no—
99C22⋊#157—62no—
100C22⋊#157—62no—
101C4⋊#157—62no—
102C4×#157—62no—
103C22×C33—21no—
104C22×C33—21no—
105C22×C33—21no—
106C22×C33C22×C411yescharacteristic
107C2×C3⋊(C3⋊D3)—21no—
108C2×C3⋊(C3⋊D3)—21no—
109C2×C3⋊(C3⋊D3)—21no—
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C2×C3×C3⋊D3C2×D411yescharacteristic
118C2×C3×C3⋊D3C2×D411yescharacteristic
119C2×C3×C3⋊D3C2×D411yescharacteristic
120C2×C3×C3⋊D3C2×D411yescharacteristic
121C2×C3×C3⋊D3#4111yescharacteristic
122C2×C3×C3⋊D3#4111yescharacteristic
123C2×C32×D3—21no—
124C2×C32×D3—21no—
125C2×C32×D3—21no—
126D3×C3⋊D3—41no—
127D3×C3⋊D3—41no—
128D3×C3⋊D3—41no—
129D3×C3⋊D3—41no—
130C3⋊#157—22no—
131C3⋊#157—41no—
132C3×#157—22no—
133C3×#157—41no—
134C3⋊(C3⋊Dic3)C22⋊C412yescomplemented
135C32×Dic3C22⋊C412yescomplemented
136C22×#67—91no—
137C2×D3×D4—182no—
138C2×C3×C22⋊C4—91no—
139C2×#217—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C2×Dic3⋊C4—91no—
143C2×C4×Dic3—91no—
144C23×C32—22no—
145C23×C32—31no—
146C23×C32—42no—
147C23×C32C4×D311yescharacteristic
148C22×C3⋊D3—21no—
149C22×C3⋊D3—21no—
150C22×C3⋊D3—21no—
151C22×C3⋊D3—21no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—31no—
155C22×C3⋊D3—62no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—61no—
158C22×C3⋊D3—61no—
159C22×C3⋊D3—61no—
160C22×C3⋊D3—61no—
161C22×C3⋊D3—61no—
162C22×C3⋊D3—122no—
163C22×C3⋊D3C4×D311yescharacteristic
164C22×C3⋊D3C22×D311yescharacteristic
165C22×C3×D3—22no—
166C22×C3×D3—62no—
167C22×C3×D3—62no—
168C22×C3×D3—64no—
169C22×C3×D3—64no—
170C22×C3×D3—122no—
171C22×C3×D3—122no—
172C22×C3×D3—122no—
173C22×C3×D3—122no—
174C2×D32—64no—
175C2×D32—64no—
176C2×D32—122no—
177C2×D32—122no—
178C2×D32—122no—
179C2×D32—122no—
180C2×#157—21no—
181C2×#157—21no—
182C2×#157—21no—
183C2×#157—21no—
184C2×#157—21no—
185C2×#157—32no—
186C2×#157—61no—
187C2×#157—61no—
188C2×#157—61no—
189C2×#157—61no—
190C2×#157—61no—
191C2×#157#6712yescomplemented
192C32×D4—62no—
193C2×C32×C4—31no—
194#638—62no—
195#638—62no—
196#638—68no—
197#638—128no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—62no—
200C2×C3⋊Dic3—122no—
201C4⋊(C3⋊D3)—62no—
202C4×C3⋊D3—32no—
203C4×C3⋊D3—32no—
204C3×#67—28no—
205C2×C3×Dic3—22no—
206#626—68no—
207#625—68no—
208D3×Dic3—64no—
209D3×Dic3—64no—
210#390—271noSylow
211C2×C33—41no—
212C2×C33C2×C22⋊C411yescharacteristic
213C2×C33C4×D411yescharacteristic
214C2×C33C4×D411yescharacteristic
215C3⋊(C3⋊D3)—41no—
216C3×C3⋊D3—41no—
217C3×C3⋊D3#12111yescharacteristic
218C3×C3⋊D3#12111yescharacteristic
219C3×C3⋊D3#12411yescharacteristic
220C3×C3⋊D3#12711yescharacteristic
221C32×D3—41no—
222C24×C3—91no—
223C23×D3—62no—
224C23×D3—91no—
225C23×D3—182no—
226C2×C3×D4—62no—
227C22×C3×C4—91no—
228C22×C3×C4—91no—
229C2×#67—91no—
230C2×#67—91no—
231C2×#67—91no—
232C2×#67—182no—
233C2×#67—182no—
234C2×#67—182no—
235C2×#67—182no—
236C2×#67—362no—
237C2×#67#15711yescharacteristic, complemented
238C22×Dic3—91no—
239C22×Dic3—91no—
240C22×Dic3—91no—
241D3×D4—188no—
242D3×D4—188no—
243C2×D12—182no—
244C2×C4×D3—182no—
245C3×C22⋊C4—182no—
246#217—182no—
247(C2×D3)⋊C4—182no—
248(C2×D3)⋊C4—182no—
249Dic3⋊C4—182no—
250C4×Dic3—182no—
251C22×C32—21no—
252C22×C32—24no—
253C22×C32—22no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—44no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32—82no—
264C22×C32—82no—
265C22×C32C2×C4×D311yescharacteristic
266C2×C3⋊D3—21no—
267C2×C3⋊D3—21no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—61no—
274C2×C3⋊D3—64no—
275C2×C3⋊D3—61no—
276C2×C3⋊D3—61no—
277C2×C3⋊D3—124no—
278C2×C3⋊D3—122no—
279C2×C3⋊D3—122no—
280C2×C3⋊D3—121no—
281C2×C3⋊D3—121no—
282C2×C3⋊D3—121no—
283C2×C3⋊D3—121no—
284C2×C3⋊D3—242no—
285C2×C3⋊D3—242no—
286C2×C3⋊D3#23511yescharacteristic
287C2×C3⋊D3D3×D411yescharacteristic
288C2×C3⋊D3D3×D411yescharacteristic
289C2×C3⋊D3D3×D411yescharacteristic
290C2×C3⋊D3#23711yescharacteristic
291C2×C3⋊D3C2×#6711yescharacteristic
292C2×C3×D3—24no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—64no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
304C2×C3×D3—124no—
305C2×C3×D3—124no—
306C2×C3×D3—124no—
307C2×C3×D3—124no—
308D32—124no—
309D32—124no—
310D32—124no—
311D32—124no—
312#157—22no—
313#157—41no—
314#157—62no—
315#157—121no—
316C32×C4—32no—
317C3⋊Dic3—32no—
318C3⋊Dic3—64no—
319C3⋊Dic3—124no—
320C3×Dic3—24no—
321C22×D4—271no—
322C2×C4⋊C4—271no—
323C2×C22⋊C4—91no—
324C2×C22⋊C4—271no—
325C2×C22⋊C4—271no—
326C2×C22⋊C4—271no—
327C2×C42—271no—
328C33#39011yescharacteristic, complemented, Sylow
329C23×C3—22no—
330C23×C3—42no—
331C23×C3—62no—
332C23×C3—91no—
333C23×C3—91no—
334C23×C3—181no—
335C23×C3—181no—
336C23×C3—181no—
337C23×C3—181no—
338C23×C3—181no—
339C23×C3—181no—
340C23×C3C2×#15711yescharacteristic
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—64no—
344C22×D3—64no—
345C22×D3—91no—
346C22×D3—122no—
347C22×D3—122no—
348C22×D3—122no—
349C22×D3—122no—
350C22×D3—182no—
351C22×D3—181no—
352C22×D3—181no—
353C22×D3—181no—
354C22×D3—181no—
355C22×D3—181no—
356C22×D3—181no—
357C22×D3—184no—
358C22×D3—184no—
359C22×D3—362no—
360C22×D3—362no—
361C22×D3—362no—
362C22×D3—362no—
363C22×D3—362no—
364C22×D3C2×#15711yescharacteristic
365C3×D4—68no—
366C2×C3×C4—62no—
367C2×C3×C4—92no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4—181no—
372C2×C3×C4—181no—
373#67—22no—
374#67—182no—
375#67—182no—
376#67—188no—
377#67—188no—
378#67—182no—
379#67—368no—
380C2×Dic3—91no—
381C2×Dic3—92no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—182no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—362no—
391C2×Dic3C2×#15711yescharacteristic
392D12—188no—
393C4×D3—184no—
394C4×D3—184no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—22no—
398C2×C32—42no—
399C2×C32—42no—
400C2×C32—42no—
401C2×C32—41no—
402C2×C32—44no—
403C2×C32—84no—
404C2×C32—121no—
405C2×C32D3×C22⋊C411yescharacteristic
406C2×C32#67⋊C411yescharacteristic
407C2×C32C4×#6711yescharacteristic
408C3⋊D3—41no—
409C3⋊D3—121no—
410C3⋊D3—124no—
411C3⋊D3—244no—
412C3⋊D3#88411yescharacteristic
413C3⋊D3#88511yescharacteristic
414C3⋊D3#93911yescharacteristic
415C3⋊D3#94011yescharacteristic
416C3×D3—44no—
417C3×D3—62no—
418C3×D3—62no—
419C3×D3—62no—
420C3×D3—62no—
421C3×D3—124no—
422C24—91no—
423C24—271no—
424C2×D4—31no—
425C2×D4—271no—
426C2×D4—271no—
427C2×D4—271no—
428C2×D4—542no—
429C2×D4—542no—
430C22×C4—91no—
431C22×C4—91no—
432C22×C4—271no—
433C22×C4—271no—
434C22×C4—271no—
435C4⋊C4—542no—
436C22⋊C4—182no—
437C22⋊C4—542no—
438C22⋊C4—542no—
439C22⋊C4—542no—
440C42—542no—
441C22×C3—24no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—44no—
449C22×C3—42no—
450C22×C3—64no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—91no—
454C22×C3—91no—
455C22×C3—91no—
456C22×C3—91no—
457C22×C3—91no—
458C22×C3—91no—
459C22×C3—122no—
460C22×C3—122no—
461C22×C3—181no—
462C22×C3—181no—
463C22×C3—181no—
464C22×C3—361no—
465C22×C3—361no—
466C22×C3—361no—
467C22×C3—361no—
468C22×C3C22×#15711yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3—21no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—64no—
479C2×D3—122no—
480C2×D3—122no—
481C2×D3—124no—
482C2×D3—124no—
483C2×D3—124no—
484C2×D3—124no—
485C2×D3—184no—
486C2×D3—181no—
487C2×D3—181no—
488C2×D3—181no—
489C2×D3—362no—
490C2×D3—362no—
491C2×D3—364no—
492C2×D3—361no—
493C2×D3—361no—
494C2×D3—361no—
495C2×D3—361no—
496C2×D3—364no—
497C2×D3—364no—
498C2×D3—364no—
499C2×D3—364no—
500C2×D3—722no—
501C2×D3—722no—
502C3×C4—64no—
503C3×C4—182no—
504C3×C4—361no—
505Dic3—92no—
506Dic3—184no—
507Dic3—182no—
508Dic3—364no—
509Dic3—361no—
510Dic3C22⋊#15712yescomplemented
511C32—22no—
512C32—42no—
513C32#495711yescharacteristic, complemented
514C23—31no—
515C23—91no—
516C23—91no—
517C23—181no—
518C23—181no—
519C23—181no—
520C23—181no—
521C23—181no—
522C23—181no—
523C23—271no—
524C23—541no—
525C23—541no—
526C23—541no—
527C23—541no—
528C23—541no—
529C23—541no—
530C23D3×#15711yescharacteristic
531D4—62no—
532D4—542no—
533D4—542no—
534D4—542no—
535C2×C4—31no—
536C2×C4—92no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4—272no—
543C2×C4—272no—
544C2×C4—271no—
545C2×C4—272no—
546C2×C4—541no—
547C2×C4—541no—
548C2×C4—541no—
549C2×C4—541no—
550C2×C4—541no—
551C2×C3—22no—
552C2×C3—22no—
553C2×C3—22no—
554C2×C3—44no—
555C2×C3—41no—
556C2×C3—42no—
557C2×C3—42no—
558C2×C3—42no—
559C2×C3—84no—
560C2×C3—91no—
561C2×C3—91no—
562C2×C3—91no—
563C2×C3—91no—
564C2×C3—124no—
565C2×C3—361no—
566C2×C3D4×#15711yescharacteristic
567C2×C3D4×#15711yescharacteristic
568C2×C3C2×C22⋊#15711yescharacteristic
569D3—41no—
570D3—62no—
571D3—62no—
572D3—62no—
573D3—62no—
574D3—124no—
575D3—364no—
576D3—361no—
577D3—724no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—61no—
582C22—61no—
583C22—61no—
584C22—91no—
585C22—91no—
586C22—91no—
587C22—91no—
588C22—91no—
589C22—91no—
590C22—181no—
591C22—181no—
592C22—181no—
593C22—361no—
594C22—361no—
595C22—361no—
596C22—361no—
597C22—541no—
598C22—541no—
599C22—541no—
600C22—1081no—
601C22—1081no—
602C22—1081no—
603C22—1081no—
604C22C2×D3×#15711yescharacteristic
605C4—32no—
606C4—182no—
607C4—272no—
608C4—361no—
609C4—542no—
610C4—1081no—
611C3—22no—
612C3—42no—
613C3#1059382711yescharacteristic, complemented
614C2—41no—
615C2—91no—
616C2—91no—
617C2—91no—
618C2—91no—
619C2—121no—
620C2—361no—
621C2—1081no—
622C2D3×C22⋊#15711yescharacteristic
623C2#67×#15711yescharacteristic
624C2#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#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C2×Dic3⋊#157C2semidirect
5. C2×(C2×D3)⋊#157C2semidirect
6. C2×(C2×D3)⋊#157C2semidirect
7. C2×Dic3×#157C2semidirect
15. C2×#6312C4semidirect
16. C2×C32×#67C4semidirect
19. C2×#6292C4semidirect
20. C2×#6290C4semidirect
33. C2×C22⋊#157D3semidirect
58. C2×C3⋊#157D4semidirect
64. C2×C3×#157D4semidirect
134. C3⋊(C3⋊Dic3)C22⋊C4semidirect
135. C32×Dic3C22⋊C4semidirect
191. C2×#157#67semidirect
237. C2×#67#157semidirect
328. C33#390semidirect
510. Dic3C22⋊#157semidirect
513. C32#4957semidirect
613. C3#10593827semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×#672
C2×#889942
C2×C3×C22⋊#1572
C2×Dic3⋊#1572
C2×(C2×D3)⋊#1572
C2×Dic3×#1572
#105938273
#49579