Finite Group Databasebeta

← #13150847

Subgroups — #13150847

Order 1728.

927 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721830C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721319C212yesmaximal, complemented
5#11721312C212yesmaximal, complemented
6#11721156C211yesmaximal, characteristic, complemented
7#11721149C211yesmaximal, characteristic, complemented
8#11721129C211yesmaximal, characteristic, complemented
9#11721123C212yesmaximal, complemented
10#11721093C211yesmaximal, characteristic, complemented
11#11721092C212yesmaximal, complemented
12#10594967—31nomaximal
13#10594945—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6235C2212yes—
16C2×C4⋊#272C2211yescharacteristic
17C2×#638⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6199C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88542C2212yes—
23C4⋊#1084C2211yescharacteristic
24#3757⋊C3C2212yes—
25#3757⋊C3C2211yescharacteristic
26#3752⋊C3C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32#88451C2212yes—
33#88451C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37C4×#638D311yescharacteristic, complemented
38#64135D311yescharacteristic, complemented
39C3×#930—31no—
40C3×#930—31no—
41C3×#930—61no—
42C3×C4×#67—31no—
43C3×#926—31no—
44#63970—32no—
45#63969—32no—
46#63969—32no—
47#63958—32no—
48#63916—31no—
49#63908—31no—
50#63902—32no—
51#63900—31no—
52#63898—31no—
53Dic3×D12—31no—
54D3×C4⋊Dic3—31no—
55#63880—32no—
56#63872—31no—
57#63869—31no—
58#63862—32no—
59#63859—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×#272C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66#6235—42no—
67C2×#1084C2312yes—
68C4⋊#272—22no—
69#638⋊C3—22no—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C4×(C3⋊D3)⋊C3—22no—
76#6200—24no—
77#6199—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79#5614—91no—
80#5565—91no—
81#5530—91no—
82C22×C32×C4—21no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C2×D311yescharacteristic
85C2×#638C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C3×#67—31no—
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67—62no—
91C2×C3×D12—31no—
92C2×C3×D12—31no—
93C2×C3×D12—61no—
94C2×C3×C4×D3—31no—
95C2×#626—32no—
96C2×#625—32no—
97C2×#624—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101#3757C2×D312yes—
102#3757C2×D311yescharacteristic
103#3752C2×D311yescharacteristic
104C4⋊(C3⋊Dic3)C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750C2×D311yescharacteristic
107C4×C3⋊Dic3C2×D312yes—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C3×#217—32no—
110C3×#217—31no—
111C3×(C2×D3)⋊C4—31no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—62no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—61no—
120C3×Dic3⋊C4—31no—
121C3×C4×Dic3—32no—
122C3×C4×Dic3—31no—
123#3724—32no—
124#3724—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×#272—22no—
138C2×#272—22no—
139C2×#272—22no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3#4112yes—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)#4112yes—
145#1084—22no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3—21no—
151C3×#121—91no—
152C3×C4×D4—91no—
153C3×#117—91no—
154#939—92no—
155#938—91no—
156#938—92no—
157#935—92no—
158D4×Dic3—91no—
159#930—31no—
160#930—31no—
161#930—61no—
162#930—91no—
163C4×#67—31no—
164C4×#67—91no—
165#926—31no—
166#926—91no—
167C2×C4⋊Dic3—91no—
168#898—92no—
169#895—91no—
170#893—91no—
171D3×C4⋊C4—91no—
172D3×C4⋊C4—91no—
173#890—91no—
174#890—91no—
175#884—92no—
176#883—92no—
177#883—92no—
178#67⋊C4—92no—
179#67⋊C4—92no—
180#879—92no—
181#876—92no—
182C4×D12—92no—
183C4×D12—91no—
184#870—91no—
185C23×C32—21no—
186C23×C32C22×D311yescharacteristic
187C23×C32C22×D311yescharacteristic
188C22×C3⋊D3C22×D311yescharacteristic
189C22×C3×D3—31no—
190C22×C3×D3—32no—
191C22×C3×D3—32no—
192C22×C3×D3—62no—
193C2×C32×C4—22no—
194C2×C32×C4—22no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—42no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202#638—22no—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D311yescharacteristic
208C4×C3⋊D3—22no—
209C3×#67—62no—
210C3×#67—122no—
211C3×#67—122no—
212C3×#67—124no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—62no—
221C3×D12—62no—
222C3×D12—62no—
223C3×D12—122no—
224C3×C4×D3—62no—
225#626—64no—
226#625—64no—
227#624—122no—
228D3×Dic3—122no—
229D3×Dic3—122no—
230D3×Dic3—122no—
231#553—271noSylow
232C2×H(3,𝔽3)—41no—
233C2×H(3,𝔽3)C2×#4111yescharacteristic
234C2×H(3,𝔽3)C2×#4111yescharacteristic
235C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
236#272—42no—
237(C3⋊D3)⋊C3—22no—
238C2×C3×D4—91no—
239C2×C3×D4—92no—
240C2×C3×D4—91no—
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—61no—
244C22×C3×C4—91no—
245C22×C3×C4D3211yescharacteristic
246C2×#67—31no—
247C2×#67—32no—
248C2×#67—32no—
249C2×#67—62no—
250C2×#67—91no—
251C2×#67—92no—
252C2×#67—92no—
253C22×Dic3—92no—
254C22×Dic3—92no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—61no—
258C2×D12—92no—
259C2×C4×D3—31no—
260C2×C4×D3—91no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C3×C4⋊C4—91no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4—91no—
268C3×C4⋊C4—91no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C42—91no—
274C3×C42—92no—
275#217—32no—
276#217—31no—
277#217—91no—
278#217—92no—
279#217—92no—
280#217—92no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—31no—
284(C2×D3)⋊C4—62no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—61no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—92no—
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300Dic3⋊C4—31no—
301Dic3⋊C4—91no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—32no—
309C4×Dic3—31no—
310C4×Dic3—92no—
311C4×Dic3—91no—
312C22×C32—22no—
313C22×C32—21no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32—41no—
321C22×C32—41no—
322C22×C32C23×D311yescharacteristic
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—21no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3#23512yes—
327C2×C3×D3—32no—
328C2×C3×D3—61no—
329C2×C3×D3—61no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—122no—
337C2×C3×D3—122no—
338C2×C3×D3—122no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—22no—
342C32×C4—41no—
343C32×C4#23512yes—
344C32×C4#23712yes—
345C3⋊Dic3—21no—
346C3⋊Dic3—21no—
347C3⋊Dic3—22no—
348C3⋊Dic3—22no—
349C3⋊Dic3—21no—
350C3×Dic3—62no—
351C3×Dic3—61no—
352C3×Dic3—61no—
353C3×Dic3—61no—
354C3×Dic3—62no—
355C3×Dic3—62no—
356C3×Dic3—62no—
357C3×Dic3—122no—
358#126—272no—
359#125—271no—
360#123—272no—
361#121—91no—
362#121—272no—
363#121—271no—
364C4×D4—91no—
365C4×D4—272no—
366C4×D4—271no—
367#117—91no—
368C2×C4⋊C4—271no—
369H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
370C23×C3—31no—
371C23×C3—31no—
372C23×C3—61no—
373C23×C3—91no—
374C23×C3—92no—
375C23×C3C2×D3211yescharacteristic
376C22×D3—31no—
377C22×D3—32no—
378C22×D3—32no—
379C22×D3—62no—
380C22×D3—91no—
381C3×D4—182no—
382C3×D4—182no—
383C3×D4—362no—
384C2×C3×C4—22no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—62no—
390C2×C3×C4—62no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—91no—
394C2×C3×C4—91no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397C2×C3×C4—91no—
398C2×C3×C4—92no—
399C2×C3×C4—122no—
400C2×C3×C4—182no—
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403#67—62no—
404#67—122no—
405#67—122no—
406#67—124no—
407#67—182no—
408#67—184no—
409#67—184no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—32no—
415C2×Dic3—32no—
416C2×Dic3—31no—
417C2×Dic3—62no—
418C2×Dic3—91no—
419C2×Dic3—92no—
420C2×Dic3—91no—
421C2×Dic3—92no—
422C2×Dic3—91no—
423C2×Dic3—362no—
424C2×Dic3—362no—
425D12—62no—
426D12—62no—
427D12—122no—
428D12—184no—
429C4×D3—62no—
430C4×D3—182no—
431C4×D3—362no—
432C4×D3—362no—
433C4×D3—362no—
434C4×D3—362no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—42no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32C2×#23511yescharacteristic
442C2×C32C2×#23511yescharacteristic
443C2×C32C2×#23711yescharacteristic
444C2×C32D3×#4111yescharacteristic
445C2×C32#100911yescharacteristic
446C2×C32#100911yescharacteristic
447C3⋊D3—22no—
448C3×D3—62no—
449C3×D3—122no—
450C3×D3—122no—
451C3×D3—242no—
452C2×D4—91no—
453C2×D4—92no—
454C2×D4—91no—
455C2×D4—272no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C4⋊C4—91no—
461C4⋊C4—91no—
462C4⋊C4—91no—
463C4⋊C4—91no—
464C4⋊C4—272no—
465C4⋊C4—272no—
466C4⋊C4—272no—
467C22⋊C4—92no—
468C22⋊C4—91no—
469C22⋊C4—91no—
470C22⋊C4—92no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C42—92no—
474C42—91no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—31no—
479C22×C3—31no—
480C22×C3—61no—
481C22×C3—61no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—62no—
487C22×C3—61no—
488C22×C3—92no—
489C22×C3—121no—
490C22×C3—121no—
491C22×C3—181no—
492C22×C3—181no—
493C22×C3—182no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3C22×D3211yescharacteristic
497C2×D3—32no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—62no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—92no—
507C2×D3—122no—
508C2×D3—122no—
509C2×D3—122no—
510C2×D3—181no—
511C2×D3—181no—
512C3×C4—21no—
513C3×C4—32no—
514C3×C4—32no—
515C3×C4—61no—
516C3×C4—61no—
517C3×C4—62no—
518C3×C4—121no—
519C3×C4—181no—
520C3×C4—182no—
521C3×C4—182no—
522C3×C4—181no—
523C3×C4—181no—
524C3×C4—182no—
525C3×C4#380412yes—
526Dic3—62no—
527Dic3—62no—
528Dic3—61no—
529Dic3—62no—
530Dic3—62no—
531Dic3—61no—
532Dic3—61no—
533Dic3—122no—
534Dic3—181no—
535Dic3—181no—
536Dic3—182no—
537Dic3—181no—
538Dic3—182no—
539C32—21no—
540C32#553011yescharacteristic, complemented
541C32#556511yescharacteristic, complemented
542C23—91no—
543C23—92no—
544C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
545D4—182no—
546D4—182no—
547D4—362no—
548D4—544no—
549C2×C4—22no—
550C2×C4—91no—
551C2×C4—92no—
552C2×C4—92no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—92no—
556C2×C4—182no—
557C2×C4—1082no—
558C2×C4—1082no—
559C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
560C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
561C2×C3—31no—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—41no—
568C2×C3—61no—
569C2×C3—61no—
570C2×C3—61no—
571C2×C3—121no—
572C2×C3—121no—
573C2×C3—122no—
574C2×C3—182no—
575C2×C3—362no—
576C2×C3D3×#23511yescharacteristic
577C2×C3#6430811yescharacteristic
578C2×C3C2×#380411yescharacteristic
579D3—62no—
580D3—122no—
581D3—122no—
582D3—182no—
583D3—242no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—92no—
588C22—181no—
589C22—181no—
590C22—182no—
591C22—182no—
592C22—182no—
593C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
594C4—21no—
595C4—181no—
596C4—181no—
597C4—182no—
598C4—181no—
599C4—182no—
600C4—182no—
601C4#8867112yes—
602C3—31no—
603C3—31no—
604C3—61no—
605C3#1059494611yescharacteristic
606C2—41no—
607C2—182no—
608C2—362no—
609C2#1172249011yescharacteristic
610C2#1172249311yescharacteristic
611C2C2×#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)#553
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. #11721830C2semidirect
2. C4×#638⋊C3C2semidirect
3. #64135⋊C3C2semidirect
4. #11721319C2semidirect
5. #11721312C2semidirect
6. #11721156C2semidirect
7. #11721149C2semidirect
8. #11721129C2semidirect
9. #11721123C2semidirect
10. #11721093C2semidirect
11. #11721092C2semidirect
37. C4×#638D3semidirect
38. #64135D3semidirect
369. H(3,𝔽3)#553semidirect
540. C32#5530semidirect
541. C32#5565semidirect

Maximal subgroups

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

subgroupindex
#117218302
C4×#638⋊C32
#64135⋊C32
#117213192
#117213122
#117211562
#117211492
#117211292
#117211232
#117210932
#117210922
#105949673
#105949453