Finite Group Databasebeta

← #13166521

Subgroups — #13166521

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×D12C211yesmaximal, characteristic
3C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
6#11723106C211yesmaximal, characteristic, complemented
7C3×#63781C211yesmaximal, characteristic, complemented
8#10593826—31nomaximal
9C22×D3×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×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C32×D12C411yescharacteristic, complemented
21C2×#6291C411yescharacteristic, complemented
22C2×#6291C411yescharacteristic, complemented
23C3⋊D3×D12—22no—
24C3⋊D3×D12—22no—
25(C2×D3)⋊#157—22no—
26(C2×D3)⋊#157—22no—
27(C2×D3)⋊#157—22no—
28C23×#157—31no—
29C2×D4×C3⋊D3—31no—
30C2×D3×D12—62no—
31C2×C22⋊#157—31no—
32C2×C22⋊#157—31no—
33C2×C22⋊#157—31no—
34#63781—31no—
35#63781D311yescharacteristic, complemented
36C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
37C22×C3⋊(C3⋊D3)C2×C411yescharacteristic, complemented
38C22×C3×C3⋊D3C2311yescharacteristic
39C22×C32×D3C2×C411yescharacteristic
40C22×C32×D3C2×C411yescharacteristic, complemented
41C2×D3×C3⋊D3—21no—
42C2×D3×C3⋊D3—21no—
43C2×D3×C3⋊D3—21no—
44C2×D3×C3⋊D3—21no—
45C2×D3×C3⋊D3—21no—
46C2×D3×C3⋊D3—21no—
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—
66C2×C33×C4C2×C411yescharacteristic, complemented
67C4⋊(C3⋊(C3⋊D3))—22no—
68C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
69C3×C4×C3⋊D3—21no—
70C3×C4×C3⋊D3—21no—
71C32×D12—22no—
72#6291—22no—
73#6291—22no—
74#4665—91nomaximal
75C23×C3⋊D3—31no—
76C23×C3⋊D3—31no—
77C22×D32—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—
89D4×C3⋊D3—62no—
90D4×C3⋊D3—62no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C4⋊(C3⋊D3)—62no—
93C2×C4⋊(C3⋊D3)—122no—
94C2×C4×C3⋊D3C2×D311yescharacteristic
95C2×C3×D12—22no—
96C2×C3×C4×D3—62no—
97C2×#626—62no—
98C2×#626—62no—
99D3×D12—68no—
100D3×D12—68no—
101C22⋊#157—62no—
102C22⋊#157—62no—
103C22⋊#157—62no—
104C22×C33C22×C411yescharacteristic
105C2×C3⋊(C3⋊D3)—21no—
106C2×C3⋊(C3⋊D3)—21no—
107C2×C3⋊(C3⋊D3)—21no—
108C2×C3⋊(C3⋊D3)—21no—
109C2×C3⋊(C3⋊D3)—21no—
110C2×C3⋊(C3⋊D3)C22⋊C412yescomplemented
111C2×C3×C3⋊D3C2×D411yescharacteristic
112C2×C3×C3⋊D3C2×D411yescharacteristic
113C2×C3×C3⋊D3C2×D411yescharacteristic
114C2×C3×C3⋊D3C2×D411yescharacteristic
115C2×C3×C3⋊D3#4111yescharacteristic
116C2×C3×C3⋊D3#4111yescharacteristic
117C2×C32×D3—21no—
118C2×C32×D3—21no—
119C2×C32×D3—21no—
120C2×C32×D3—21no—
121C2×C32×D3—21no—
122C2×C32×D3C22⋊C412yescomplemented
123D3×C3⋊D3—22no—
124D3×C3⋊D3—22no—
125D3×C3⋊D3—22no—
126D3×C3⋊D3—22no—
127D3×C3⋊D3—41no—
128D3×C3⋊D3—41no—
129D3×C3⋊D3—41no—
130D3×C3⋊D3—41no—
131C3⋊#157—22no—
132C3⋊#157—41no—
133C3×#157—22no—
134C3×#157—41no—
135C33×C4—21no—
136C3×C3⋊Dic3—21no—
137C2×D3×D4—182no—
138C22×D12—91no—
139C22×C4×D3—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4—91no—
143C3×#95—91no—
144C3⋊#95—91no—
145C23×C32—31no—
146C23×C32—31no—
147C22×C3⋊D3—31no—
148C22×C3⋊D3—32no—
149C22×C3⋊D3—32no—
150C22×C3⋊D3—31no—
151C22×C3⋊D3—62no—
152C22×C3⋊D3—62no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
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—61no—
163C22×C3⋊D3—122no—
164C22×C3⋊D3—122no—
165C22×C3⋊D3C22×D311yescharacteristic
166C22×C3×D3—22no—
167C22×C3×D3—22no—
168C22×C3×D3—62no—
169C2×D32—64no—
170C2×D32—64no—
171C2×D32—64no—
172C2×D32—64no—
173C2×D32—122no—
174C2×D32—122no—
175C2×D32—122no—
176C2×D32—122no—
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×#157D1212yescomplemented
194C32×D4—62no—
195C2×C32×C4—22no—
196C2×C32×C4—42no—
197C2×C32×C4C4×D311yescharacteristic, complemented
198#638—62no—
199#638—62no—
200C2×C3⋊Dic3C4×D311yescharacteristic, complemented
201C4⋊(C3⋊D3)—62no—
202C4⋊(C3⋊D3)—68no—
203C4⋊(C3⋊D3)—128no—
204C4×C3⋊D3—21no—
205C4×C3⋊D3—21no—
206C2×C3×Dic3—62no—
207C3×D12—28no—
208C3×C4×D3—64no—
209C3×C4×D3—64no—
210#626—68no—
211#626—68no—
212#386—271noSylow
213C2×C33C2×C22⋊C411yescharacteristic
214C2×C33C4×D411yescharacteristic
215C2×C33C4×D411yescharacteristic
216C3⋊(C3⋊D3)—22no—
217C3⋊(C3⋊D3)—41no—
218C3×C3⋊D3#12011yescharacteristic
219C3×C3⋊D3#12111yescharacteristic
220C3×C3⋊D3#12111yescharacteristic
221C3×C3⋊D3#12311yescharacteristic
222C32×D3—22no—
223C32×D3—41no—
224C23×D3—91no—
225C23×D3—91no—
226C23×D3—182no—
227C23×D3—182no—
228C2×C3×D4—62no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C2×#67—182no—
233C2×#67—182no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236D3×D4—188no—
237D3×D4—188no—
238C2×D12—91no—
239C2×D12—91no—
240C2×D12—91no—
241C2×D12—182no—
242C2×D12—182no—
243C2×D12—182no—
244C2×D12—362no—
245C2×D12#15711yescharacteristic, complemented
246C2×C4×D3—62no—
247C2×C4×D3—182no—
248C2×C4×D3—182no—
249C2×C4×D3—182no—
250(C2×D3)⋊C4—182no—
251(C2×D3)⋊C4—182no—
252(C2×D3)⋊C4—182no—
253C22×C32—22no—
254C22×C32—32no—
255C22×C32—42no—
256C22×C32—61no—
257C22×C32—61no—
258C22×C32—61no—
259C22×C32—61no—
260C22×C32—61no—
261C22×C32C2×C4×D311yescharacteristic
262C2×C3⋊D3—32no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—61no—
265C2×C3⋊D3—64no—
266C2×C3⋊D3—61no—
267C2×C3⋊D3—61no—
268C2×C3⋊D3—62no—
269C2×C3⋊D3—62no—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—61no—
273C2×C3⋊D3—61no—
274C2×C3⋊D3—124no—
275C2×C3⋊D3—124no—
276C2×C3⋊D3—122no—
277C2×C3⋊D3—122no—
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—242no—
287C2×C3⋊D3—242no—
288C2×C3⋊D3C2×D1211yescharacteristic
289C2×C3⋊D3#23511yescharacteristic
290C2×C3⋊D3D3×D411yescharacteristic
291C2×C3⋊D3D3×D411yescharacteristic
292C2×C3⋊D3D3×D411yescharacteristic
293C2×C3⋊D3#23911yescharacteristic
294C2×C3×D3—24no—
295C2×C3×D3—24no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306D32—124no—
307D32—124no—
308D32—124no—
309D32—124no—
310D32—124no—
311D32—124no—
312D32—124no—
313D32—124no—
314#157—22no—
315#157—41no—
316#157—62no—
317#157—121no—
318C32×C4—21no—
319C32×C4—24no—
320C32×C4—44no—
321C3⋊Dic3—21no—
322C3×Dic3—64no—
323C22×D4—271no—
324C23×C4—271no—
325C2×C22⋊C4—271no—
326C2×C22⋊C4—271no—
327C2×C22⋊C4—271no—
328#95—91no—
329#95—271no—
330C33#38611yescharacteristic, complemented, Sylow
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—91no—
334C22×D3—62no—
335C22×D3—91no—
336C22×D3—92no—
337C22×D3—92no—
338C22×D3—91no—
339C22×D3—184no—
340C22×D3—184no—
341C22×D3—182no—
342C22×D3—182no—
343C22×D3—181no—
344C22×D3—181no—
345C22×D3—181no—
346C22×D3—181no—
347C22×D3—181no—
348C22×D3—181no—
349C22×D3—181no—
350C22×D3—181no—
351C22×D3—181no—
352C22×D3—181no—
353C22×D3—184no—
354C22×D3—184no—
355C22×D3—362no—
356C22×D3—362no—
357C22×D3—362no—
358C22×D3—362no—
359C22×D3—362no—
360C22×D3—362no—
361C22×D3—362no—
362C22×D3—362no—
363C22×D3—362no—
364C22×D3—362no—
365C22×D3C2×#15711yescharacteristic
366C22×D3C2×#15711yescharacteristic, complemented
367C3×D4—68no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—92no—
371C2×C3×C4—91no—
372C2×C3×C4—181no—
373C2×C3×C4—181no—
374C2×C3×C4—181no—
375C2×C3×C4—181no—
376C2×C3×C4—181no—
377C2×C3×C4—181no—
378C2×C3×C4—181no—
379C2×C3×C4C2×#15711yescharacteristic, complemented
380#67—188no—
381#67—188no—
382C2×Dic3—62no—
383C2×Dic3—92no—
384C2×Dic3—181no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389D12—22no—
390D12—182no—
391D12—188no—
392D12—182no—
393D12—182no—
394D12—368no—
395C4×D3—64no—
396C4×D3—64no—
397C4×D3—188no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—42no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—62no—
405C2×C32—121no—
406C2×C32C4×D1211yescharacteristic
407C2×C32D3×C22⋊C411yescharacteristic
408C2×C32#89311yescharacteristic
409C3⋊D3—62no—
410C3⋊D3—124no—
411C3⋊D3—121no—
412C3⋊D3—124no—
413C3⋊D3—244no—
414C3⋊D3—244no—
415C3⋊D3#88211yescharacteristic
416C3⋊D3#88411yescharacteristic
417C3⋊D3#89411yescharacteristic
418C3⋊D3#89511yescharacteristic
419C3×D3—44no—
420C3×D3—44no—
421C3×D3—62no—
422C3×D3—62no—
423C3×D3—62no—
424C3×D3—62no—
425C24—271no—
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—91no—
436C22×C4—271no—
437C22×C4—271no—
438C22×C4—542no—
439C22×C4—542no—
440C22×C4—542no—
441C22⋊C4—542no—
442C22⋊C4—542no—
443C22⋊C4—542no—
444C22×C3—22no—
445C22×C3—42no—
446C22×C3—64no—
447C22×C3—64no—
448C22×C3—91no—
449C22×C3—91no—
450C22×C3—91no—
451C22×C3—91no—
452C22×C3—91no—
453C22×C3—91no—
454C22×C3—122no—
455C22×C3—122no—
456C22×C3—122no—
457C22×C3—122no—
458C22×C3C22×#15711yescharacteristic
459C2×D3—21no—
460C2×D3—21no—
461C2×D3—21no—
462C2×D3—21no—
463C2×D3—21no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—62no—
467C2×D3—62no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—92no—
471C2×D3—184no—
472C2×D3—182no—
473C2×D3—182no—
474C2×D3—182no—
475C2×D3—182no—
476C2×D3—181no—
477C2×D3—181no—
478C2×D3—184no—
479C2×D3—181no—
480C2×D3—181no—
481C2×D3—181no—
482C2×D3—364no—
483C2×D3—364no—
484C2×D3—364no—
485C2×D3—364no—
486C2×D3—364no—
487C2×D3—362no—
488C2×D3—362no—
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—
502C2×D3—722no—
503C2×D3—722no—
504C2×D3C22⋊#15712yescomplemented
505C3×C4—24no—
506C3×C4—21no—
507C3×C4—44no—
508C3×C4—182no—
509C3×C4—181no—
510C3×C4—361no—
511Dic3—64no—
512Dic3—182no—
513Dic3—361no—
514C32—22no—
515C32—42no—
516C32#466511yescharacteristic, complemented
517C23—31no—
518C23—31no—
519C23—91no—
520C23—272no—
521C23—272no—
522C23—271no—
523C23—271no—
524C23—541no—
525C23—541no—
526C23—541no—
527C23—541no—
528C23—541no—
529C23—541no—
530C23—541no—
531C23—541no—
532C23—541no—
533C23—541no—
534D4—62no—
535D4—542no—
536D4—542no—
537D4—542no—
538C2×C4—92no—
539C2×C4—91no—
540C2×C4—181no—
541C2×C4—181no—
542C2×C4—181no—
543C2×C4—181no—
544C2×C4—181no—
545C2×C4—181no—
546C2×C4—181no—
547C2×C4—272no—
548C2×C4—548no—
549C2×C4—541no—
550C2×C4—541no—
551C2×C4—541no—
552C2×C4—541no—
553C2×C4—541no—
554C2×C4D3×#15711yescharacteristic, complemented
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—22no—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3—42no—
561C2×C3—91no—
562C2×C3—91no—
563C2×C3—91no—
564C2×C3—91no—
565C2×C3—124no—
566C2×C3—124no—
567C2×C3D4×#15711yescharacteristic
568C2×C3D4×#15711yescharacteristic
569C2×C3C2×C22⋊#15711yescharacteristic
570D3—22no—
571D3—41no—
572D3—62no—
573D3—62no—
574D3—62no—
575D3—62no—
576D3—182no—
577D3—364no—
578D3—364no—
579D3—361no—
580D3—724no—
581D3—724no—
582C22—32no—
583C22—61no—
584C22—61no—
585C22—61no—
586C22—61no—
587C22—61no—
588C22—91no—
589C22—91no—
590C22—91no—
591C22—91no—
592C22—91no—
593C22—91no—
594C22—272no—
595C22—542no—
596C22—542no—
597C22—542no—
598C22—542no—
599C22—541no—
600C22—541no—
601C22—541no—
602C22—541no—
603C22—541no—
604C22—1081no—
605C22—1081no—
606C22—1081no—
607C22—1081no—
608C22C2×D3×#15711yescharacteristic
609C4—21no—
610C4—182no—
611C4—181no—
612C4—361no—
613C4—542no—
614C4—1081no—
615C3—22no—
616C3—42no—
617C3#1059382611yescharacteristic, complemented
618C2—62no—
619C2—91no—
620C2—91no—
621C2—91no—
622C2—91no—
623C2—121no—
624C2—542no—
625C2—1081no—
626C2D12×#15711yescharacteristic
627C2#1172425911yescharacteristic
628C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×C3⋊D3×D12C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D3×#157C2semidirect
3. C2×(C2×D3)⋊#157C2semidirect
4. C2×(C2×D3)⋊#157C2semidirect
5. C2×(C2×D3)⋊#157C2semidirect
6. #11723106C2semidirect
7. C3×#63781C2semidirect
18. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
20. C2×C32×D12C4semidirect
21. C2×#6291C4semidirect
22. C2×#6291C4semidirect
35. #63781D3semidirect
37. C22×C3⋊(C3⋊D3)C2×C4semidirect
40. C22×C32×D3C2×C4semidirect
58. C2×C3⋊#157D4semidirect
64. C2×C3×#157D4semidirect
66. C2×C33×C4C2×C4semidirect
68. C2×C3×C3⋊Dic3C2×C4semidirect
110. C2×C3⋊(C3⋊D3)C22⋊C4semidirect
122. C2×C32×D3C22⋊C4semidirect
193. C2×#157D12semidirect
197. C2×C32×C4C4×D3semidirect
200. C2×C3⋊Dic3C4×D3semidirect
245. C2×D12#157semidirect
330. C33#386semidirect
366. C22×D3C2×#157semidirect
379. C2×C3×C4C2×#157semidirect
504. C2×D3C22⋊#157semidirect
516. C32#4665semidirect
554. C2×C4D3×#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×D122
C2×(C2×D3)⋊#1572
#117231062
C3×#637812
#105938263
#46659