Finite Group Databasebeta

← #13150697

Subgroups — #13150697

Order 1728.

1179 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88671C212yesmaximal, complemented
2#11721830C211yesmaximal, characteristic, complemented
3#64139⋊C3C211yesmaximal, characteristic, complemented
4C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#11721322C212yesmaximal, complemented
6#11721158C211yesmaximal, characteristic, complemented
7#11721149C211yesmaximal, characteristic, complemented
8#11721132C211yesmaximal, characteristic, complemented
9#11721123C212yesmaximal, complemented
10#11721116C211yesmaximal, characteristic, complemented
11#11721103C212yesmaximal, complemented
12#10594929—31nomaximal
13#10594910—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2212yes—
16C2×C4⋊#272C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
21C2×#6200C2212yes—
22C2×#6200C2212yes—
23C2×#6199C2212yes—
24#88671—28no—
25C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
26C2×#6196C2212yes—
27#88542C2212yes—
28C4⋊#1084C2211yescharacteristic
29#3752⋊C3C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35#88451C2212yes—
36#88450C2212yes—
37#88448C2212yes—
38C2×#3805—32no—
39C2×#3804—32no—
40#64139D311yescharacteristic, complemented
41C2×C4×C3⋊Dic3D311yescharacteristic, complemented
42C3×#930—31no—
43C3×#930—31no—
44C3×#930—31no—
45C3×#930—61no—
46C2×C3×C4×Dic3—31no—
47#63980—32no—
48#63978—32no—
49#63919—31no—
50#63918—31no—
51#63915—32no—
52#63914—31no—
53#63911—31no—
54#63902—32no—
55#63898—31no—
56#63898—31no—
57#63894—31no—
58Dic3×D12—31no—
59#63880—32no—
60#63852—32no—
61C23×H(3,𝔽3)C2311yescharacteristic
62C22×#272C2312yes—
63C22×(C3⋊D3)⋊C3C2312yes—
64C2×C4×H(3,𝔽3)—22no—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67#6235—24no—
68C2×#1084C2312yes—
69C4⋊#272—22no—
70#638⋊C3—24no—
71C2×(C3⋊Dic3)⋊C3—22no—
72C2×(C3⋊Dic3)⋊C3D414yescomplemented
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
76C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
77C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
78C4⋊((C3⋊D3)⋊C3)—22no—
79#6200—24no—
80#6200—24no—
81#6199—24no—
82((C3⋊D3)⋊C3)⋊C4—24no—
83#6196—24no—
84#5596—91no—
85#5596—91no—
86#5477—91no—
87C22×C32×C4—21no—
88C22×C32×C4C2×D311yescharacteristic
89C22×C32×C4C2×D311yescharacteristic
90C2×#638C2×D312yes—
91C22×C3⋊Dic3C2×D311yescharacteristic
92C22×C3⋊Dic3C2×D311yescharacteristic
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—32no—
97C2×C3×#67—62no—
98C22×C3×Dic3—31no—
99C22×C3×Dic3—31no—
100C2×C3×D12—31no—
101C2×C3×D12—31no—
102C2×C3×D12—31no—
103C2×C3×D12—61no—
104C2×#627—32no—
105C2×#627—32no—
106C2×#626—32no—
107C2×#626—32no—
108C2×#625—32no—
109C2×#625—32no—
110C2×#624—32no—
111#3805—68no—
112#3804—68no—
113C2×D3×Dic3—32no—
114C2×D3×Dic3—32no—
115C2×D3×Dic3—32no—
116#3752C2×D312yes—
117C4⋊(C3⋊Dic3)C2×D311yescharacteristic
118C4×C3⋊Dic3C2×D311yescharacteristic
119C4×C3⋊Dic3C2×D311yescharacteristic
120C4×C3⋊Dic3C2×D311yescharacteristic
121C4×C3⋊Dic3C2×D311yescharacteristic
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—62no—
126C3×C4⋊Dic3—31no—
127C3×C4⋊Dic3—31no—
128C3×C4⋊Dic3—31no—
129C3×C4⋊Dic3—61no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—31no—
132C3×C4×Dic3—31no—
133C3×C4×Dic3—31no—
134#3724—32no—
135Dic3⋊Dic3—32no—
136#3722—32no—
137(C2×D3)⋊Dic3—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—22no—
145C2×#272—22no—
146C2×#272—22no—
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—22no—
153(C3⋊Dic3)⋊C3—22no—
154(C3⋊Dic3)⋊C3—21no—
155(C3⋊Dic3)⋊C3C2×D412yes—
156(C3⋊Dic3)⋊C3C2×D412yes—
157(C3⋊Dic3)⋊C3#4112yescomplemented
158C2×#237—92no—
159C2×#237—92no—
160C2×#235—92no—
161C3×#121—91no—
162C3×#121—91no—
163C2×C3×C42—91no—
164#940—91no—
165#940—91no—
166#939—92no—
167#939—92no—
168D4×Dic3—91no—
169D4×Dic3—91no—
170#930—31no—
171#930—31no—
172#930—31no—
173#930—61no—
174#930—91no—
175#930—92no—
176C2×C4×Dic3—31no—
177C2×C4×Dic3—91no—
178#893—91no—
179#893—91no—
180#888—91no—
181#888—91no—
182#885—92no—
183#885—92no—
184#67⋊C4—92no—
185#67⋊C4—92no—
186#875—92no—
187#874—91no—
188C4×D12—91no—
189C4×D12—92no—
190C4×D12—91no—
191#869—91no—
192C23×C32—21no—
193C23×C32C22×D311yescharacteristic
194C23×C32C22×D311yescharacteristic
195C22×C3⋊D3C22×D312yes—
196C22×C3×D3—32no—
197C22×C3×D3—32no—
198C22×C3×D3—32no—
199C22×C3×D3—62no—
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—22no—
211C2×C3⋊Dic3D1214yescomplemented
212C2×C3⋊Dic3C22×D311yescharacteristic
213C2×C3⋊Dic3C22×D312yes—
214C2×C3⋊Dic3C22×D311yescharacteristic
215C2×C3⋊Dic3C22×D311yescharacteristic
216C2×C3⋊Dic3C22×D311yescharacteristic
217C4⋊(C3⋊D3)—22no—
218C3×#67—64no—
219C3×#67—64no—
220C3×#67—64no—
221C3×#67—124no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—31no—
224C2×C3×Dic3—31no—
225C2×C3×Dic3—31no—
226C2×C3×Dic3—31no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—62no—
231C2×C3×Dic3—62no—
232C3×D12—62no—
233C3×D12—62no—
234C3×D12—62no—
235C3×D12—122no—
236#627—64no—
237#627—64no—
238#626—64no—
239#626—64no—
240#625—64no—
241#625—64no—
242#624—64no—
243D3×Dic3—64no—
244D3×Dic3—64no—
245D3×Dic3—64no—
246#532—271noSylow
247C2×H(3,𝔽3)—22no—
248C2×H(3,𝔽3)C22×D411yescharacteristic
249C2×H(3,𝔽3)C2×#4111yescharacteristic
250C2×H(3,𝔽3)C2×#4111yescharacteristic
251#272—42no—
252(C3⋊D3)⋊C3—42no—
253C2×C3×D4—91no—
254C2×C3×D4—92no—
255C2×C3×D4—92no—
256C2×C3×D4—91no—
257C22×C3×C4—31no—
258C22×C3×C4—31no—
259C22×C3×C4—61no—
260C22×C3×C4—91no—
261C22×C3×C4—91no—
262C22×C3×C4D3211yescharacteristic
263C2×#67—32no—
264C2×#67—32no—
265C2×#67—32no—
266C2×#67—62no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—92no—
271C2×#67—92no—
272C22×Dic3—31no—
273C22×Dic3—31no—
274C22×Dic3—91no—
275C22×Dic3—91no—
276C22×Dic3—92no—
277C22×Dic3—92no—
278#237—188no—
279#237—188no—
280#235—188no—
281C2×D12—31no—
282C2×D12—31no—
283C2×D12—31no—
284C2×D12—61no—
285C2×D12—91no—
286C2×D12—92no—
287C2×D12—92no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—92no—
291C2×C4×D3—92no—
292C2×Dic6—92no—
293C2×Dic6—92no—
294C2×Dic6—92no—
295C3×C4⋊C4—91no—
296C3×C4⋊C4—91no—
297C3×C22⋊C4—92no—
298C3×C22⋊C4—92no—
299C3×C42—91no—
300C3×C42—91no—
301C3×C42—91no—
302C3×C42—91no—
303#217—92no—
304#217—92no—
305(C2×D3)⋊C4—32no—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—32no—
308(C2×D3)⋊C4—62no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314C4⋊Dic3—31no—
315C4⋊Dic3—31no—
316C4⋊Dic3—31no—
317C4⋊Dic3—61no—
318C4⋊Dic3—91no—
319C4⋊Dic3—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322C4×Dic3—31no—
323C4×Dic3—31no—
324C4×Dic3—31no—
325C4×Dic3—31no—
326C4×Dic3—91no—
327C4×Dic3—91no—
328C4×Dic3—91no—
329C4×Dic3—91no—
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×C32C2×D1212yes—
339C22×C32D3×D412yes—
340C22×C32C23×D311yescharacteristic
341C22×C32C23×D311yescharacteristic
342C2×C3⋊D3—22no—
343C2×C3⋊D3—22no—
344C2×C3⋊D3—22no—
345C2×C3×D3—62no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—62no—
349C2×C3×D3—62no—
350C2×C3×D3—62no—
351C2×C3×D3—62no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—122no—
355C2×C3×D3—122no—
356C2×C3×D3—122no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—22no—
360C32×C4—41no—
361C32×C4#23512yescomplemented
362C32×C4#23712yescomplemented
363C3⋊Dic3—21no—
364C3⋊Dic3—22no—
365C3⋊Dic3C2×D1212yes—
366C3⋊Dic3C2×D1212yes—
367C3⋊Dic3#23512yescomplemented
368C3×Dic3—32no—
369C3×Dic3—32no—
370C3×Dic3—32no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—61no—
375C3×Dic3—122no—
376C2×#41—272no—
377C4⋊Q8—271no—
378#127—271no—
379#124—272no—
380#121—91no—
381#121—91no—
382#121—272no—
383C4×D4—272no—
384C4×D4—271no—
385C4×D4—271no—
386C2×C42—91no—
387H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
388C23×C3—31no—
389C23×C3—31no—
390C23×C3—61no—
391C23×C3—92no—
392C23×C3—92no—
393C23×C3C2×D3211yescharacteristic
394C22×D3—32no—
395C22×D3—32no—
396C22×D3—32no—
397C22×D3—62no—
398C22×D3—92no—
399C3×D4—182no—
400C3×D4—184no—
401C3×D4—184no—
402C3×D4—182no—
403C2×C3×C4—22no—
404C2×C3×C4—31no—
405C2×C3×C4—31no—
406C2×C3×C4—31no—
407C2×C3×C4—31no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—91no—
413C2×C3×C4—91no—
414C2×C3×C4—91no—
415C2×C3×C4—94no—
416C2×C3×C4—92no—
417C2×C3×C4—91no—
418C2×C3×C4—92no—
419C2×C3×C4—122no—
420C2×C3×C4—182no—
421C2×C3×C4C2×D3211yescharacteristic
422C2×C3×C4C2×D3211yescharacteristic
423#67—64no—
424#67—64no—
425#67—64no—
426#67—124no—
427#67—184no—
428#67—184no—
429#67—184no—
430#67—184no—
431#67—184no—
432C2×Dic3—31no—
433C2×Dic3—31no—
434C2×Dic3—31no—
435C2×Dic3—34no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—31no—
439C2×Dic3—32no—
440C2×Dic3—62no—
441C2×Dic3—62no—
442C2×Dic3—94no—
443C2×Dic3—91no—
444C2×Dic3—91no—
445C2×Dic3—91no—
446C2×Dic3—91no—
447C2×Dic3—92no—
448C2×Dic3—182no—
449C2×Dic3—184no—
450C2×Dic3—184no—
451D12—62no—
452D12—62no—
453D12—62no—
454D12—122no—
455D12—182no—
456D12—184no—
457D12—184no—
458C4×D3—184no—
459C4×D3—184no—
460C4×D3—184no—
461C4×D3—184no—
462Dic6—184no—
463Dic6—184no—
464Dic6—184no—
465C2×C32—22no—
466C2×C32—22no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—42no—
471C2×C32C22×D1211yescharacteristic
472C2×C32C2×#23511yescharacteristic
473C2×C32C2×#23511yescharacteristic
474C2×C32C2×D3×D411yescharacteristic
475C2×C32C2×#23711yescharacteristic
476C2×C32D3×#4111yescharacteristic
477C3⋊D3—42no—
478C3×D3—122no—
479C3×D3—122no—
480C3×D3—122no—
481C3×D3—242no—
482#41—548no—
483C2×Q8—272no—
484C2×D4—92no—
485C2×D4—92no—
486C2×D4—91no—
487C2×D4—91no—
488C2×D4—272no—
489C2×D4—272no—
490C22×C4—91no—
491C22×C4—91no—
492C22×C4—272no—
493C22×C4—272no—
494C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
495C4⋊C4—91no—
496C4⋊C4—91no—
497C4⋊C4—272no—
498C22⋊C4—92no—
499C22⋊C4—92no—
500C22⋊C4—272no—
501C22⋊C4—272no—
502C42—91no—
503C42—91no—
504C42—91no—
505C42—91no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—31no—
509C22×C3—31no—
510C22×C3—32no—
511C22×C3—32no—
512C22×C3—61no—
513C22×C3—61no—
514C22×C3—61no—
515C22×C3—61no—
516C22×C3—61no—
517C22×C3—62no—
518C22×C3—121no—
519C22×C3—121no—
520C22×C3—182no—
521C22×C3—182no—
522C22×C3—182no—
523C22×C3—182no—
524C22×C3—182no—
525C22×C3—182no—
526C22×C3D3×D1212yes—
527C22×C3C22×D3211yescharacteristic
528C2×D3—62no—
529C2×D3—62no—
530C2×D3—62no—
531C2×D3—62no—
532C2×D3—62no—
533C2×D3—62no—
534C2×D3—62no—
535C2×D3—62no—
536C2×D3—62no—
537C2×D3—122no—
538C2×D3—122no—
539C2×D3—122no—
540C2×D3—182no—
541C2×D3—182no—
542C2×D3—182no—
543C3×C4—21no—
544C3×C4—32no—
545C3×C4—32no—
546C3×C4—61no—
547C3×C4—61no—
548C3×C4—62no—
549C3×C4—92no—
550C3×C4—92no—
551C3×C4—92no—
552C3×C4—121no—
553C3×C4—182no—
554C3×C4—181no—
555C3×C4—182no—
556C3×C4#380412yes—
557Dic3—32no—
558Dic3—32no—
559Dic3—32no—
560Dic3—62no—
561Dic3—62no—
562Dic3—62no—
563Dic3—61no—
564Dic3—92no—
565Dic3—92no—
566Dic3—92no—
567Dic3—122no—
568Dic3—182no—
569Dic3—181no—
570C32—21no—
571C32#547711yescharacteristic, complemented
572C32#559611yescharacteristic, complemented
573C23—92no—
574C23—92no—
575C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
576Q8—544no—
577D4—184no—
578D4—184no—
579D4—182no—
580D4—182no—
581D4—544no—
582D4—544no—
583C2×C4—22no—
584C2×C4—91no—
585C2×C4—91no—
586C2×C4—91no—
587C2×C4—94no—
588C2×C4—91no—
589C2×C4—92no—
590C2×C4—92no—
591C2×C4—182no—
592C2×C4—544no—
593C2×C4—544no—
594C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
595C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
596C2×C3—22no—
597C2×C3—31no—
598C2×C3—31no—
599C2×C3—31no—
600C2×C3—31no—
601C2×C3—31no—
602C2×C3—31no—
603C2×C3—62no—
604C2×C3—62no—
605C2×C3—61no—
606C2×C3—61no—
607C2×C3—61no—
608C2×C3—122no—
609C2×C3—362no—
610C2×C3—362no—
611C2×C3C2×D3×D1211yescharacteristic
612C2×C3D3×#23511yescharacteristic
613C2×C3C2×#380411yescharacteristic
614D3—122no—
615D3—122no—
616D3—122no—
617D3—242no—
618D3—362no—
619C22—21no—
620C22—21no—
621C22—182no—
622C22—182no—
623C22—182no—
624C22—182no—
625C22—182no—
626C22—182no—
627C22#8865412yes—
628C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
629C4—21no—
630C4—92no—
631C4—92no—
632C4—92no—
633C4—182no—
634C4—182no—
635C4—181no—
636C4#8867112yescomplemented
637C3—31no—
638C3—31no—
639C3—61no—
640C3#1059491011yescharacteristic
641C2—22no—
642C2—362no—
643C2—362no—
644C2C2×#8865411yescharacteristic
645C2#1172249011yescharacteristic
646C2C2×#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)#532
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. #11721830C2semidirect
3. #64139⋊C3C2semidirect
4. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
5. #11721322C2semidirect
6. #11721158C2semidirect
7. #11721149C2semidirect
8. #11721132C2semidirect
9. #11721123C2semidirect
10. #11721116C2semidirect
11. #11721103C2semidirect
40. #64139D3semidirect
41. C2×C4×C3⋊Dic3D3semidirect
72. C2×(C3⋊Dic3)⋊C3D4semidirect
151. C4×H(3,𝔽3)#41semidirect
157. (C3⋊Dic3)⋊C3#41semidirect
211. C2×C3⋊Dic3D12semidirect
361. C32×C4#235semidirect
362. C32×C4#237semidirect
367. C3⋊Dic3#235semidirect
387. H(3,𝔽3)#532semidirect
571. C32#5477semidirect
572. C32#5596semidirect
636. C4#88671semidirect

Maximal subgroups

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

subgroupindex
C2×#886712
#117218302
#64139⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#117213222
#117211582
#117211492
#117211322
#117211232
#117211162
#117211032
#105949293
#105949103