Finite Group Databasebeta

← #13150710

Subgroups — #13150710

Order 1728.

1116 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88671C212yesmaximal, complemented
2C4×#6235C211yesmaximal, characteristic, complemented
3#64139⋊C3C211yesmaximal, characteristic, complemented
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5#11721306C212yesmaximal, complemented
6#11721159C211yesmaximal, characteristic, complemented
7#11721152C211yesmaximal, characteristic, complemented
8#11721136C211yesmaximal, characteristic, complemented
9#11721134C212yesmaximal, complemented
10#11721117C211yesmaximal, characteristic, complemented
11#11721102C212yesmaximal, complemented
12#10594907—31nomaximal
13#10594906—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2212yes—
19C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
20C2×#6200C2212yes—
21C2×#6199C2212yes—
22#88671—28no—
23C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
24C2×#6196C2212yes—
25#88552C2211yescharacteristic
26#88542C2211yescharacteristic
27#88540C2211yescharacteristic
28C4×#1084C2211yescharacteristic
29#3752⋊C3C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
33C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
34#88450C2212yes—
35#88450C2212yes—
36#88449C2212yes—
37#88449C2212yes—
38C2×#3805—32no—
39C2×#3804—32no—
40#64139D311yescharacteristic, complemented
41C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
42C3×#930—31no—
43C3×C4×#67—31no—
44C3×C4×#67—31no—
45C3×C4×#67—61no—
46C2×C3×C4⋊Dic3—31no—
47#63966—32no—
48#63962—32no—
49#63918—31no—
50#63918—31no—
51#63909—31no—
52#63908—31no—
53#63895—32no—
54#63894—31no—
55#63893—32no—
56Dic3×D12—31no—
57#63891—32no—
58#63890—32no—
59D3×C4⋊Dic3—31no—
60D3×C4⋊Dic3—31no—
61C23×H(3,𝔽3)C2311yescharacteristic
62C22×#272C2311yescharacteristic
63C22×(C3⋊D3)⋊C3C2312yes—
64C2×C4×H(3,𝔽3)—22no—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67#6235D414yescomplemented
68C2×#1084C2311yescharacteristic
69C2×#1084C2311yescharacteristic
70C2×#1084C2311yescharacteristic
71C4×#272—22no—
72#638⋊C3—24no—
73C2×(C3⋊Dic3)⋊C3—24no—
74C2×(C3⋊Dic3)⋊C3C2312yes—
75C2×(C3⋊Dic3)⋊C3C2312yes—
76C2×(C3⋊Dic3)⋊C3C2312yes—
77C4⋊((C3⋊D3)⋊C3)—22no—
78#6200—24no—
79#6199—24no—
80((C3⋊D3)⋊C3)⋊C4—24no—
81#6196—24no—
82#5611—91no—
83#5555—91no—
84#5507—91no—
85C22×C32×C4—21no—
86C22×C32×C4C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×#638C2×D312yes—
89C22×C3⋊Dic3C2×D312yes—
90C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
91C2×C3×#67—31no—
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C2×C3×#67—61no—
95C22×C3×Dic3—32no—
96C2×C3×D12—31no—
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3—61no—
100C2×#627—32no—
101C2×#627—32no—
102C2×#626—32no—
103C2×#625—32no—
104C2×#624—32no—
105#3805—68no—
106#3804—68no—
107C2×D3×Dic3—32no—
108C2×D3×Dic3—32no—
109C2×D3×Dic3—32no—
110#3752C2×D312yes—
111C4⋊(C3⋊Dic3)C2×D311yescharacteristic
112C4⋊(C3⋊Dic3)C2×D312yes—
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115C3×#217—31no—
116C3×#217—31no—
117C3×#217—61no—
118C3×(C2×D3)⋊C4—31no—
119C3×(C2×D3)⋊C4—31no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—61no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—31no—
125C3×C4⋊Dic3—31no—
126C3×Dic3⋊C4—31no—
127C3×Dic3⋊C4—31no—
128C3×Dic3⋊C4—61no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—61no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136#3722—32no—
137#3722—32no—
138(C2×D3)⋊Dic3—32no—
139(C2×D3)⋊Dic3—32no—
140C22×H(3,𝔽3)—21no—
141C22×H(3,𝔽3)—21no—
142C22×H(3,𝔽3)C2×D412yes—
143C22×H(3,𝔽3)C2411yescharacteristic
144C2×#272—21no—
145C2×#272—21no—
146C2×#272C2×D412yes—
147C2×(C3⋊D3)⋊C3—22no—
148C2×(C3⋊D3)⋊C3—22no—
149C2×(C3⋊D3)⋊C3—22no—
150C4×H(3,𝔽3)—21no—
151C4×H(3,𝔽3)#4112yescomplemented
152#1084—21no—
153#1084—21no—
154#1084C2×D412yes—
155(C3⋊Dic3)⋊C3—22no—
156(C3⋊Dic3)⋊C3—22no—
157(C3⋊Dic3)⋊C3—22no—
158C2×#237—92no—
159C2×#237—92no—
160C2×#235—92no—
161C3×#121—91no—
162C3×C4×D4—91no—
163C2×C3×C4⋊C4—91no—
164#938—91no—
165#935—92no—
166D4×Dic3—91no—
167#930—31no—
168#930—91no—
169#930—91no—
170#929—92no—
171C4×#67—31no—
172C4×#67—31no—
173C4×#67—61no—
174C2×C4⋊Dic3—31no—
175C2×C4⋊Dic3—91no—
176C2×C4⋊Dic3—91no—
177#897—92no—
178#896—92no—
179#895—91no—
180#894—92no—
181#893—91no—
182D3×C4⋊C4—91no—
183D3×C4⋊C4—91no—
184#888—91no—
185#888—91no—
186#886—92no—
187#886—92no—
188#883—92no—
189#878—92no—
190C4×D12—91no—
191#869—91no—
192C23×C32—21no—
193C23×C32C22×D311yescharacteristic
194C23×C32C22×D311yescharacteristic
195C22×C3⋊D3C22×D312yes—
196C22×C3×D3—31no—
197C22×C3×D3—31no—
198C22×C3×D3—32no—
199C22×C3×D3—61no—
200C2×C32×C4—22no—
201C2×C32×C4—22no—
202C2×C32×C4—21no—
203C2×C32×C4—21no—
204C2×C32×C4—42no—
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D311yescharacteristic
207C2×C32×C4C22×D311yescharacteristic
208C2×C32×C4C22×D311yescharacteristic
209#638—24no—
210C2×C3⋊Dic3—24no—
211C2×C3⋊Dic3C22×D312yes—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D312yes—
214C4⋊(C3⋊D3)—22no—
215C3×#67—34no—
216C3×#67—34no—
217C3×#67—64no—
218C3×#67—64no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—31no—
225C2×C3×Dic3—31no—
226C2×C3×Dic3—31no—
227C2×C3×Dic3—31no—
228C2×C3×Dic3—61no—
229C2×C3×Dic3—61no—
230C2×C3×Dic3—61no—
231C2×C3×Dic3—64no—
232C3×D12—62no—
233C3×C4×D3—62no—
234C3×C4×D3—62no—
235C3×C4×D3—122no—
236#627—64no—
237#627—64no—
238#626—64no—
239#625—64no—
240#624—64no—
241D3×Dic3—64no—
242D3×Dic3—64no—
243D3×Dic3—64no—
244#547—271noSylow
245C2×H(3,𝔽3)—22no—
246C2×H(3,𝔽3)C22×D411yescharacteristic
247C2×H(3,𝔽3)C2×#4111yescharacteristic
248C2×H(3,𝔽3)2−1+411yescharacteristic
249#272—22no—
250(C3⋊D3)⋊C3—42no—
251C2×C3×D4—91no—
252C2×C3×D4—91no—
253C2×C3×D4—92no—
254C22×C3×C4—31no—
255C22×C3×C4—31no—
256C22×C3×C4—61no—
257C22×C3×C4—92no—
258C22×C3×C4—91no—
259C22×C3×C4D3211yescharacteristic
260C2×#67—31no—
261C2×#67—31no—
262C2×#67—32no—
263C2×#67—61no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C22×Dic3—32no—
268C22×Dic3—92no—
269C22×Dic3—92no—
270C22×Dic3—92no—
271#237—188no—
272#237—188no—
273#235—188no—
274C2×D12—31no—
275C2×D12—91no—
276C2×D12—92no—
277C2×C4×D3—31no—
278C2×C4×D3—31no—
279C2×C4×D3—61no—
280C2×C4×D3—92no—
281C2×C4×D3—92no—
282C2×C4×D3—92no—
283C2×C4×D3—92no—
284C2×Dic6—92no—
285C2×Dic6—92no—
286C2×Dic6—92no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—91no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—92no—
291C3×C4⋊C4—91no—
292C3×C22⋊C4—92no—
293C3×C22⋊C4—91no—
294C3×C22⋊C4—91no—
295C3×C42—91no—
296#217—31no—
297#217—31no—
298#217—61no—
299#217—92no—
300#217—92no—
301(C2×D3)⋊C4—31no—
302(C2×D3)⋊C4—31no—
303(C2×D3)⋊C4—32no—
304(C2×D3)⋊C4—61no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—92no—
310C4⋊Dic3—31no—
311C4⋊Dic3—31no—
312C4⋊Dic3—31no—
313C4⋊Dic3—32no—
314C4⋊Dic3—92no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—91no—
318C4⋊Dic3—92no—
319C4⋊Dic3—92no—
320Dic3⋊C4—31no—
321Dic3⋊C4—31no—
322Dic3⋊C4—61no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327C4×Dic3—31no—
328C4×Dic3—31no—
329C4×Dic3—61no—
330C22×C32—21no—
331C22×C32—21no—
332C22×C32—21no—
333C22×C32—21no—
334C22×C32—21no—
335C22×C32—22no—
336C22×C32—41no—
337C22×C32—41no—
338C22×C32D3×D412yes—
339C22×C32D3×D412yes—
340C22×C32C23×D311yescharacteristic
341C22×C32C23×D311yescharacteristic
342C2×C3⋊D3—22no—
343C2×C3⋊D3—22no—
344C2×C3⋊D3—22no—
345C2×C3×D3—32no—
346C2×C3×D3—32no—
347C2×C3×D3—62no—
348C2×C3×D3—61no—
349C2×C3×D3—61no—
350C2×C3×D3—61no—
351C2×C3×D3—61no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—62no—
355C2×C3×D3—121no—
356C2×C3×D3—121no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—22no—
360C32×C4—41no—
361C32×C4#23512yescomplemented
362C32×C4#23712yescomplemented
363C3⋊Dic3—22no—
364C3⋊Dic3—22no—
365C3⋊Dic3—22no—
366C3×Dic3—32no—
367C3×Dic3—32no—
368C3×Dic3—62no—
369C3×Dic3—61no—
370C3×Dic3—62no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—61no—
374C3×Dic3—61no—
375C3×Dic3—61no—
376C3×Dic3—121no—
377C3×Dic3—121no—
378C2×#41—272no—
379C4⋊Q8—271no—
380#123—272no—
381#123—272no—
382C22⋊Q8—272no—
383#121—91no—
384#121—271no—
385C4×D4—91no—
386C4×D4—271no—
387C2×C4⋊C4—91no—
388C2×C4⋊C4—271no—
389H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
390C23×C3—31no—
391C23×C3—31no—
392C23×C3—61no—
393C23×C3—92no—
394C23×C3—91no—
395C23×C3C2×D3211yescharacteristic
396C22×D3—31no—
397C22×D3—31no—
398C22×D3—32no—
399C22×D3—61no—
400C22×D3—92no—
401C3×D4—94no—
402C3×D4—182no—
403C3×D4—184no—
404C2×C3×C4—22no—
405C2×C3×C4—31no—
406C2×C3×C4—31no—
407C2×C3×C4—31no—
408C2×C3×C4—31no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—91no—
414C2×C3×C4—92no—
415C2×C3×C4—92no—
416C2×C3×C4—92no—
417C2×C3×C4—91no—
418C2×C3×C4—91no—
419C2×C3×C4—122no—
420C2×C3×C4—184no—
421C2×C3×C4—182no—
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C2×D3211yescharacteristic
424#67—34no—
425#67—34no—
426#67—64no—
427#67—64no—
428#67—184no—
429#67—184no—
430#67—184no—
431C2×Dic3—31no—
432C2×Dic3—31no—
433C2×Dic3—31no—
434C2×Dic3—31no—
435C2×Dic3—31no—
436C2×Dic3—31no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—61no—
441C2×Dic3—61no—
442C2×Dic3—61no—
443C2×Dic3—64no—
444C2×Dic3—92no—
445C2×Dic3—92no—
446C2×Dic3—92no—
447C2×Dic3—184no—
448C2×Dic3—184no—
449C2×Dic3—184no—
450D12—62no—
451D12—182no—
452D12—184no—
453C4×D3—62no—
454C4×D3—62no—
455C4×D3—122no—
456C4×D3—184no—
457C4×D3—184no—
458C4×D3—184no—
459C4×D3—184no—
460Dic6—184no—
461Dic6—184no—
462Dic6—184no—
463C2×C32—22no—
464C2×C32—22no—
465C2×C32—21no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—42no—
469C2×C32C2×#23511yescharacteristic
470C2×C32C2×D3×D411yescharacteristic
471C2×C32C2×D3×D411yescharacteristic
472C2×C32C2×#23711yescharacteristic
473C2×C32#100711yescharacteristic
474C2×C32#101011yescharacteristic
475C3⋊D3—42no—
476C3×D3—62no—
477C3×D3—62no—
478C3×D3—122no—
479C3×D3—122no—
480#41—548no—
481C2×Q8—272no—
482C2×D4—91no—
483C2×D4—92no—
484C2×D4—91no—
485C2×D4—272no—
486C22×C4—92no—
487C22×C4—91no—
488C22×C4—272no—
489C22×C4—272no—
490C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
491C4⋊C4—91no—
492C4⋊C4—91no—
493C4⋊C4—92no—
494C4⋊C4—91no—
495C4⋊C4—91no—
496C4⋊C4—272no—
497C4⋊C4—272no—
498C22⋊C4—91no—
499C22⋊C4—91no—
500C22⋊C4—92no—
501C22⋊C4—272no—
502C22⋊C4—272no—
503C42—91no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—31no—
507C22×C3—31no—
508C22×C3—32no—
509C22×C3—32no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—61no—
513C22×C3—61no—
514C22×C3—61no—
515C22×C3—62no—
516C22×C3—92no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3—182no—
520C22×C3—182no—
521C22×C3—182no—
522C22×C3—181no—
523C22×C3—181no—
524C22×C3#380212yes—
525C22×C3C22×D3211yescharacteristic
526C2×D3—32no—
527C2×D3—32no—
528C2×D3—62no—
529C2×D3—61no—
530C2×D3—61no—
531C2×D3—61no—
532C2×D3—61no—
533C2×D3—62no—
534C2×D3—62no—
535C2×D3—62no—
536C2×D3—121no—
537C2×D3—121no—
538C2×D3—182no—
539C2×D3—182no—
540C2×D3—182no—
541C3×C4—21no—
542C3×C4—32no—
543C3×C4—32no—
544C3×C4—61no—
545C3×C4—61no—
546C3×C4—62no—
547C3×C4—92no—
548C3×C4—121no—
549C3×C4—182no—
550C3×C4—182no—
551C3×C4—182no—
552C3×C4—181no—
553C3×C4—181no—
554C3×C4#380412yes—
555Dic3—32no—
556Dic3—32no—
557Dic3—62no—
558Dic3—61no—
559Dic3—61no—
560Dic3—61no—
561Dic3—61no—
562Dic3—62no—
563Dic3—62no—
564Dic3—62no—
565Dic3—121no—
566Dic3—121no—
567Dic3—182no—
568Dic3—182no—
569Dic3—182no—
570C32—21no—
571C32#550711yescharacteristic, complemented
572C32#561111yescharacteristic, complemented
573C23—91no—
574C23—92no—
575C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
576Q8—544no—
577D4—94no—
578D4—184no—
579D4—182no—
580D4—544no—
581C2×C4—22no—
582C2×C4—91no—
583C2×C4—91no—
584C2×C4—91no—
585C2×C4—92no—
586C2×C4—92no—
587C2×C4—92no—
588C2×C4—184no—
589C2×C4—182no—
590C2×C4—544no—
591C2×C4—544no—
592C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
593C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
594C2×C3—22no—
595C2×C3—31no—
596C2×C3—31no—
597C2×C3—31no—
598C2×C3—31no—
599C2×C3—31no—
600C2×C3—31no—
601C2×C3—62no—
602C2×C3—62no—
603C2×C3—61no—
604C2×C3—61no—
605C2×C3—61no—
606C2×C3—122no—
607C2×C3—182no—
608C2×C3—362no—
609C2×C3#6429711yescharacteristic
610C2×C3C2×#380211yescharacteristic
611C2×C3C2×#380411yescharacteristic
612D3—62no—
613D3—62no—
614D3—122no—
615D3—122no—
616D3—362no—
617C22—21no—
618C22—21no—
619C22—92no—
620C22—181no—
621C22—181no—
622C22—182no—
623C22—182no—
624C22—182no—
625C22#8865512yes—
626C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
627C4—21no—
628C4—92no—
629C4—181no—
630C4—181no—
631C4—182no—
632C4—182no—
633C4—182no—
634C4#8867112yescomplemented
635C3—31no—
636C3—31no—
637C3—61no—
638C3#1059492811yescharacteristic
639C2—22no—
640C2—182no—
641C2—362no—
642C2#1172248211yescharacteristic
643C2C2×#8865511yescharacteristic
644C2C2×#8867111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#547
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
1. C2×#88671C2semidirect
2. C4×#6235C2semidirect
3. #64139⋊C3C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #11721306C2semidirect
6. #11721159C2semidirect
7. #11721152C2semidirect
8. #11721136C2semidirect
9. #11721134C2semidirect
10. #11721117C2semidirect
11. #11721102C2semidirect
40. #64139D3semidirect
41. C2×C4⋊(C3⋊Dic3)D3semidirect
67. #6235D4semidirect
151. C4×H(3,𝔽3)#41semidirect
361. C32×C4#235semidirect
362. C32×C4#237semidirect
389. H(3,𝔽3)#547semidirect
571. C32#5507semidirect
572. C32#5611semidirect
634. C4#88671semidirect

Maximal subgroups

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

subgroupindex
C2×#886712
C4×#62352
#64139⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#117213062
#117211592
#117211522
#117211362
#117211342
#117211172
#117211022
#105949073
#105949063