Finite Group Databasebeta

← #13148711

Subgroups — #13148711

Order 1728.

1329 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1329 conjugacy classes of subgroups — too many to draw. Only its 151 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×#88677C211yesmaximal, characteristic
2C2×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3C22⋊C4×#272C211yesmaximal, characteristic, complemented
4#64092⋊C3C212yesmaximal, complemented
5#11721320C211yesmaximal, characteristic, complemented
6#11721311C212yesmaximal, complemented
7#11721161C211yesmaximal, characteristic, complemented
8#11721152C211yesmaximal, characteristic, complemented
9#11721147C211yesmaximal, characteristic, complemented
10#11721130C212yesmaximal, complemented
11#11721100C212yesmaximal, complemented
12D3×#883—32nomaximal
13C23×#272C2211yescharacteristic
14C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
19#88677—24no—
20C2×#6200C2212yes—
21C2×#6199C2211yescharacteristic
22C2×#6199C2211yescharacteristic
23C2×#6199C2211yescharacteristic
24C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
25C4×((C3⋊D3)⋊C3)⋊C2—24no—
26C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
27#88552C2211yescharacteristic, complemented
28#88542C2211yescharacteristic, complemented
29#88542C2211yescharacteristic, complemented
30#3757⋊C3C2212yescomplemented
31#3752⋊C3C2212yescomplemented
32C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
33#3750⋊C3C2212yescomplemented
34#88451C2212yescomplemented
35#88450C2212yescomplemented
36#88449C2212yescomplemented
37#88448C2212yescomplemented
38C2×D3×#67—32no—
39C2×C4×D32—32no—
40#64092D312yescomplemented
41C3×#883—32no—
42C3×D3×C22⋊C4—32no—
43C3×D3×C22⋊C4—61no—
44#63975—32no—
45#63969—32no—
46D3×#217—32no—
47D3×(C2×D3)⋊C4—32no—
48#63905—32no—
49#63891—32no—
50D3×C4⋊Dic3—32no—
51D3×Dic3⋊C4—32no—
52#63854—32no—
53#63853—32no—
54C23×H(3,𝔽3)C2311yescharacteristic
55C22×#272—22no—
56C22×#272—22no—
57C22×#272—22no—
58C22×#272C2311yescharacteristic
59C22×#272C2311yescharacteristic
60C22×(C3⋊D3)⋊C3C2312yes—
61C2×((C3⋊D3)⋊C3)⋊C2—22no—
62C2×((C3⋊D3)⋊C3)⋊C2—22no—
63C2×((C3⋊D3)⋊C3)⋊C2D414yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68C4×#272—22no—
69C4×#272—22no—
70#638⋊C3—24no—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C4×(C3⋊D3)⋊C3—24no—
75#6200—24no—
76#6199—22no—
77#6199—22no—
78#6199—22no—
79((C3⋊D3)⋊C3)⋊C4—24no—
80D3×#123—92no—
81C2×#883—91no—
82C23×C3×D3—32no—
83C23×C3×D3—61no—
84C22×D32—32no—
85C2×#638C2×D312yes—
86C2×C4×C3⋊D3C2×D312yescomplemented
87C2×C3×#67—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—61no—
92C2×C3×C4×D3—61no—
93D3×#67—68no—
94C2×#626—32no—
95C2×#625—32no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C4×D32—68no—
102C32×C22⋊C4—21no—
103C32×C22⋊C4C2×D312yescomplemented
104#3757C2×D312yescomplemented
105#3752C2×D312yescomplemented
106C4⋊(C3⋊Dic3)C2×D312yescomplemented
107#3750C2×D312yescomplemented
108C3×#217—32no—
109C3×#217—32no—
110C3×#217—61no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—61no—
115C3×(C2×D3)⋊C4—61no—
116C3×C4⋊Dic3—32no—
117C3×Dic3⋊C4—32no—
118#3724—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123(C2×D3)⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126C22×H(3,𝔽3)—21no—
127C22×H(3,𝔽3)—21no—
128C22×H(3,𝔽3)—21no—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×#272—21no—
131C2×#272—21no—
132C2×#272—21no—
133C2×#272—44no—
134C2×#272C2×D412yes—
135C2×#272#4112yes—
136C2×#272#4112yes—
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3C2×D414yes—
140((C3⋊D3)⋊C3)⋊C2—28no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)—21no—
143#1084—21no—
144#1084—21no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148C22×#67—91no—
149C2×D3×D4—92no—
150C22×C4×D3—91no—
151C22×C4×D3—92no—
152C3×#123—92no—
153C2×C3×C22⋊C4—91no—
154C2×#217—91no—
155#935—92no—
156#929—92no—
157C2×(C2×D3)⋊C4—91no—
158C2×C4⋊Dic3—91no—
159C2×Dic3⋊C4—91no—
160#894—92no—
161#894—92no—
162D3×C4⋊C4—92no—
163D3×C4⋊C4—92no—
164#886—92no—
165#883—32no—
166#883—92no—
167#883—92no—
168#883—92no—
169#883—92no—
170#883—92no—
171D3×C22⋊C4—32no—
172D3×C22⋊C4—61no—
173D3×C22⋊C4—92no—
174D3×C22⋊C4—92no—
175C23×C32—21no—
176C23×C32C22×D312yes—
177C22×C3⋊D3C22×D312yes—
178C22×C3×D3—32no—
179C22×C3×D3—32no—
180C22×C3×D3—32no—
181C22×C3×D3—61no—
182C22×C3×D3—61no—
183C22×C3×D3—64no—
184C22×C3×D3—64no—
185C22×C3×D3—64no—
186C22×C3×D3—64no—
187C22×C3×D3—122no—
188C22×C3×D3—122no—
189C2×D32—38no—
190C2×D32—64no—
191C2×D32—64no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D312yes—
196#638—24no—
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C4×C3⋊D3—24no—
201C3×#67—64no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C3×C4×D3—64no—
210C3×C4×D3—64no—
211C3×C4×D3—64no—
212C3×C4×D3—122no—
213C3×C4×D3—122no—
214#626—64no—
215#625—64no—
216#624—64no—
217D3×Dic3—64no—
218D3×Dic3—64no—
219D3×Dic3—64no—
220D3×Dic3—64no—
221C2×#123—271noSylow
222C2×H(3,𝔽3)—41no—
223C2×H(3,𝔽3)C22×D411yescharacteristic
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×H(3,𝔽3)C2×#4111yescharacteristic
226#272—41no—
227#272#12314yescomplemented
228(C3⋊D3)⋊C3—24no—
229C24×C3—91no—
230C23×D3—32no—
231C23×D3—61no—
232C23×D3—91no—
233C23×D3—92no—
234C2×C3×D4—92no—
235C22×C3×C4—92no—
236C22×C3×C4—91no—
237C22×C3×C4—91no—
238C2×#67—32no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—92no—
242C2×#67—184no—
243C22×Dic3—91no—
244C22×Dic3—91no—
245C22×Dic3—91no—
246C22×Dic3—92no—
247D3×D4—188no—
248C2×D12—92no—
249C2×C4×D3—32no—
250C2×C4×D3—32no—
251C2×C4×D3—32no—
252C2×C4×D3—61no—
253C2×C4×D3—61no—
254C2×C4×D3—92no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—184no—
261C2×C4×D3—188no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4—92no—
264C3×C22⋊C4—32no—
265C3×C22⋊C4—61no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—91no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4D3211yescharacteristic
272#217—32no—
273#217—32no—
274#217—61no—
275#217—92no—
276#217—92no—
277#217—92no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—61no—
282(C2×D3)⋊C4—61no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288C4⋊Dic3—32no—
289C4⋊Dic3—92no—
290C4⋊Dic3—92no—
291C4⋊Dic3—92no—
292Dic3⋊C4—32no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32C23×D312yes—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3D3×D414yes—
309C2×C3×D3—34no—
310C2×C3×D3—34no—
311C2×C3×D3—34no—
312C2×C3×D3—34no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C2×C3×D3—128no—
325C2×C3×D3—128no—
326D32—616no—
327C32×C4—22no—
328C32×C4—22no—
329C32×C4—41no—
330C32×C4—41no—
331C3⋊Dic3—22no—
332C3⋊Dic3—22no—
333C3⋊Dic3—22no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C22×D4—271no—
342C23×C4—271no—
343#123—92no—
344#123—272no—
345#123—272no—
346#123—272no—
347C2×C4⋊C4—271no—
348C2×C4⋊C4—271no—
349C2×C22⋊C4—91no—
350C2×C22⋊C4—271no—
351C2×C22⋊C4—271no—
352H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
353C23×C3—32no—
354C23×C3—61no—
355C23×C3—92no—
356C23×C3—91no—
357C23×C3—91no—
358C23×C3—182no—
359C23×C3—182no—
360C23×C3—182no—
361C23×C3C2×D3211yescharacteristic
362C22×D3—32no—
363C22×D3—32no—
364C22×D3—32no—
365C22×D3—61no—
366C22×D3—61no—
367C22×D3—64no—
368C22×D3—64no—
369C22×D3—64no—
370C22×D3—64no—
371C22×D3—92no—
372C22×D3—94no—
373C22×D3—98no—
374C22×D3—122no—
375C22×D3—122no—
376C22×D3—184no—
377C22×D3—184no—
378C22×D3—182no—
379C22×D3—182no—
380C3×D4—184no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—92no—
386C2×C3×C4—92no—
387C2×C3×C4—92no—
388C2×C3×C4—91no—
389C2×C3×C4—91no—
390C2×C3×C4—182no—
391C2×C3×C4—182no—
392C2×C3×C4—184no—
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395#67—64no—
396#67—184no—
397#67—184no—
398#67—184no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—32no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—92no—
407C2×Dic3—92no—
408C2×Dic3—92no—
409C2×Dic3—182no—
410C2×Dic3—182no—
411C2×Dic3—182no—
412C2×Dic3—184no—
413D12—184no—
414C4×D3—64no—
415C4×D3—64no—
416C4×D3—64no—
417C4×D3—122no—
418C4×D3—122no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—184no—
422C4×D3—184no—
423C4×D3—184no—
424C4×D3—184no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—42no—
429C2×C32—42no—
430C2×C32C2×D3×D412yes—
431C2×C32D3×#4112yes—
432C2×C32D3×#4112yes—
433C3⋊D3—24no—
434C3×D3—38no—
435C3×D3—64no—
436C3×D3—64no—
437C3×D3—122no—
438C3×D3—122no—
439C24—91no—
440C24—271no—
441C2×D4—92no—
442C2×D4—272no—
443C2×D4—544no—
444C22×C4—91no—
445C22×C4—91no—
446C22×C4—92no—
447C22×C4—271no—
448C22×C4—271no—
449C22×C4—271no—
450C22×C4—272no—
451C22×C4—544no—
452C4⋊C4—92no—
453C4⋊C4—92no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C22⋊C4—91no—
457C22⋊C4—92no—
458C22⋊C4—91no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—32no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—62no—
471C22×C3—62no—
472C22×C3—61no—
473C22×C3—94no—
474C22×C3—92no—
475C22×C3—92no—
476C22×C3—92no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3—181no—
482C22×C3—181no—
483C22×C3—181no—
484C22×C3—364no—
485C22×C3C22×D3211yescharacteristic
486C2×D3—34no—
487C2×D3—34no—
488C2×D3—34no—
489C2×D3—34no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—94no—
500C2×D3—121no—
501C2×D3—121no—
502C2×D3—128no—
503C2×D3—128no—
504C2×D3—182no—
505C2×D3—182no—
506C2×D3—1816no—
507C2×D3—188no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—62no—
511C3×C4—62no—
512C3×C4—121no—
513C3×C4—121no—
514C3×C4—182no—
515C3×C4—182no—
516C3×C4—182no—
517C3×C4—181no—
518C3×C4—181no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—62no—
523Dic3—62no—
524Dic3—121no—
525Dic3—121no—
526Dic3—182no—
527Dic3—182no—
528Dic3—182no—
529C32—21no—
530C32D3×#12312yescomplemented
531C23—91no—
532C23—92no—
533C23—91no—
534C23—182no—
535C23—182no—
536C23—182no—
537C23—274no—
538C23—542no—
539C23—542no—
540C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
541D4—184no—
542D4—544no—
543C2×C4—92no—
544C2×C4—91no—
545C2×C4—91no—
546C2×C4—92no—
547C2×C4—92no—
548C2×C4—182no—
549C2×C4—184no—
550C2×C4—182no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4—542no—
554C2×C4—542no—
555C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
556C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
557C2×C3—32no—
558C2×C3—32no—
559C2×C3—32no—
560C2×C3—41no—
561C2×C3—61no—
562C2×C3—61no—
563C2×C3—61no—
564C2×C3—94no—
565C2×C3—122no—
566C2×C3—122no—
567C2×C3—184no—
568C2×C3—361no—
569C2×C3#6430611yescharacteristic
570C2×C3D32×D411yescharacteristic
571C2×C3#6431211yescharacteristic
572D3—38no—
573D3—64no—
574D3—64no—
575D3—122no—
576D3—122no—
577D3—184no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—92no—
582C22—92no—
583C22—92no—
584C22—94no—
585C22—182no—
586C22—182no—
587C22—181no—
588C22—181no—
589C22—181no—
590C22—364no—
591C22—548no—
592C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
593C4—21no—
594C4—21no—
595C4—181no—
596C4—182no—
597C4—182no—
598C4—181no—
599C4—182no—
600C3—32no—
601C3—61no—
602C3#1059426011yescharacteristic
603C2—41no—
604C2—94no—
605C2—184no—
606C2—361no—
607C2#1172249111yescharacteristic
608C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
609C2#1172250511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC2×#88677C2
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. C22⋊C4×#272C2semidirect
4. #64092⋊C3C2semidirect
5. #11721320C2semidirect
6. #11721311C2semidirect
7. #11721161C2semidirect
8. #11721152C2semidirect
9. #11721147C2semidirect
10. #11721130C2semidirect
11. #11721100C2semidirect
18. C2×C4×(C3⋊D3)⋊C3C22semidirect
24. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
26. C22⋊C4×H(3,𝔽3)C22semidirect
27. #88552C22semidirect
28. #88542C22semidirect
29. #88542C22semidirect
30. #3757⋊C3C22semidirect
31. #3752⋊C3C22semidirect
32. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
33. #3750⋊C3C22semidirect
34. #88451C22semidirect
35. #88450C22semidirect
36. #88449C22semidirect
37. #88448C22semidirect
40. #64092D3semidirect
86. C2×C4×C3⋊D3C2×D3semidirect
103. C32×C22⋊C4C2×D3semidirect
104. #3757C2×D3semidirect
105. #3752C2×D3semidirect
106. C4⋊(C3⋊Dic3)C2×D3semidirect
107. #3750C2×D3semidirect
227. #272#123semidirect
352. H(3,𝔽3)C2×#123semidirect
463. C22⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
530. C32D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#886772
C2×C4×((C3⋊D3)⋊C3)⋊C22
C22⋊C4×#2722
#64092⋊C32
#117213202
#117213112
#117211612
#117211522
#117211472
#117211302
#117211002
D3×#8833