Finite Group Databasebeta

← #13147834

Subgroups — #13147834

Order 1728.

1179 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88650C212yesmaximal, complemented
2C4×C4⋊#272C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5#11721158C211yesmaximal, characteristic, complemented
6#11721156C211yesmaximal, characteristic, complemented
7C4×#6200C212yesmaximal, complemented
8#11721141C211yesmaximal, characteristic, complemented
9#11721140C212yesmaximal, complemented
10#11721116C211yesmaximal, characteristic, complemented
11#11721103C212yesmaximal, complemented
12#10593972—31nomaximal
13#10593968—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2212yes—
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C2×#6200C2212yes—
22C2×#6200C2212yes—
23C2×#6200C2212yes—
24C2×#6198C2212yes—
25C2×#6196C2212yes—
26#88650—28no—
27C42×H(3,𝔽3)C2211yescharacteristic
28#88542C2212yes—
29C4⋊#1084C2211yescharacteristic
30#3752⋊C3C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35#88451C2212yes—
36#88450C2212yes—
37#88448C2212yes—
38C2×#3798—32no—
39C2×#3798—32no—
40C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
41C42×C3⋊D3D311yescharacteristic, complemented
42C3×C4×D12—31no—
43C3×C4×D12—31no—
44C3×C4×D12—31no—
45C3×C4×D12—61no—
46C3×C42×D3—31no—
47#63919—31no—
48#63918—31no—
49#63916—31no—
50#63916—31no—
51#63915—32no—
52#63914—31no—
53#63912—32no—
54#63911—31no—
55#63909—31no—
56#63908—31no—
57#63906—32no—
58C4×#626—32no—
59C4×#625—32no—
60#63852—32no—
61C22×#272C2312yes—
62C22×(C3⋊D3)⋊C3C2312yes—
63C22×(C3⋊D3)⋊C3C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×#1084C2312yes—
68C4⋊#272—22no—
69C4×#272—24no—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4⋊((C3⋊D3)⋊C3)—22no—
75C4×(C3⋊D3)⋊C3—24no—
76C4×(C3⋊D3)⋊C3—22no—
77C4×(C3⋊D3)⋊C3—22no—
78C4×(C3⋊D3)⋊C3D414yescomplemented
79#6200—24no—
80#6200—24no—
81#6200—24no—
82#6198—24no—
83#6196—24no—
84#5548—91no—
85#5548—91no—
86#5477—91no—
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
88C2×C4×C3⋊D3C2×D312yes—
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×C4×C3⋊D3C2×D311yescharacteristic
92C2×C3×D12—31no—
93C2×C3×D12—31no—
94C2×C3×D12—31no—
95C2×C3×D12—61no—
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—32no—
100C2×C3×C4×D3—31no—
101C2×C3×C4×D3—31no—
102C2×C3×C4×D3—62no—
103C2×#627—32no—
104C2×#627—32no—
105C2×#626—32no—
106C2×#626—32no—
107C2×#626—32no—
108C2×#626—32no—
109C2×#626—32no—
110C2×#625—32no—
111C2×#625—32no—
112C2×#625—32no—
113#3798—68no—
114#3798—68no—
115C32×C42—21no—
116C32×C42C2×D311yescharacteristic
117C32×C42C2×D311yescharacteristic
118#3752C2×D312yes—
119C4⋊(C3⋊Dic3)C2×D311yescharacteristic
120C4×C3⋊Dic3C2×D311yescharacteristic
121C4×C3⋊Dic3C2×D311yescharacteristic
122C4×C3⋊Dic3C2×D311yescharacteristic
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—32no—
126C3×(C2×D3)⋊C4—62no—
127C3×C4⋊Dic3—31no—
128C3×C4⋊Dic3—31no—
129C3×C4⋊Dic3—31no—
130C3×C4⋊Dic3—61no—
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)C2411yescharacteristic
141C2×#272—22no—
142C2×#272—22no—
143C2×#272—22no—
144C2×(C3⋊D3)⋊C3—21no—
145C2×(C3⋊D3)⋊C3—21no—
146C2×(C3⋊D3)⋊C3—22no—
147C2×(C3⋊D3)⋊C3—22no—
148C2×(C3⋊D3)⋊C3—22no—
149C2×(C3⋊D3)⋊C3C2×D412yes—
150C4×H(3,𝔽3)—21no—
151C4×H(3,𝔽3)C2×D412yes—
152C4×H(3,𝔽3)#4112yescomplemented
153#1084—22no—
154(C3⋊Dic3)⋊C3—22no—
155(C3⋊Dic3)⋊C3—21no—
156(C3⋊Dic3)⋊C3C2×D412yes—
157(C3⋊Dic3)⋊C3#4112yescomplemented
158C2×#235—92no—
159C2×#235—92no—
160C2×#235—92no—
161C3×C4×D4—91no—
162C3×C4×D4—91no—
163C2×C3×C42—91no—
164#940—91no—
165#940—91no—
166#938—91no—
167#938—91no—
168#930—91no—
169#930—92no—
170#930—91no—
171C4×#67—92no—
172C4×#67—92no—
173#895—91no—
174#895—91no—
175#888—91no—
176#888—91no—
177#885—92no—
178#885—92no—
179#884—92no—
180#884—92no—
181#875—92no—
182#874—91no—
183C4×D12—31no—
184C4×D12—31no—
185C4×D12—31no—
186C4×D12—61no—
187C4×D12—91no—
188C4×D12—92no—
189C42×D3—31no—
190C42×D3—91no—
191#869—91no—
192C22×C3⋊D3C22×D312yes—
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3×D3—31no—
195C22×C3×D3—32no—
196C22×C3×D3—32no—
197C22×C3×D3—32no—
198C22×C3×D3—62no—
199C2×C32×C4—21no—
200C2×C32×C4—21no—
201C2×C32×C4—21no—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D311yescharacteristic
207C2×C32×C4C22×D311yescharacteristic
208C2×C3⋊Dic3C22×D311yescharacteristic
209C2×C3⋊Dic3C22×D312yes—
210C2×C3⋊Dic3C22×D311yescharacteristic
211C2×C3⋊Dic3C22×D311yescharacteristic
212C4⋊(C3⋊D3)—22no—
213C4×C3⋊D3—22no—
214C4×C3⋊D3—24no—
215C4×C3⋊D3—22no—
216C4×C3⋊D3D1214yescomplemented
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—62no—
224C3×D12—64no—
225C3×D12—62no—
226C3×D12—62no—
227C3×D12—62no—
228C3×C4×D3—34no—
229C3×C4×D3—62no—
230C3×C4×D3—64no—
231C3×C4×D3—64no—
232C3×C4×D3—62no—
233C3×C4×D3—64no—
234C3×C4×D3—124no—
235#627—64no—
236#627—64no—
237#626—64no—
238#626—64no—
239#626—64no—
240#626—64no—
241#626—64no—
242#625—64no—
243#625—64no—
244#625—64no—
245#532—271noSylow
246C2×H(3,𝔽3)C22×D411yescharacteristic
247C2×H(3,𝔽3)C2×#4111yescharacteristic
248C2×H(3,𝔽3)C2×#4111yescharacteristic
249#272—42no—
250(C3⋊D3)⋊C3—22no—
251(C3⋊D3)⋊C3—42no—
252C2×C3×D4—91no—
253C2×C3×D4—91no—
254C22×C3×C4—91no—
255C22×C3×C4—92no—
256C22×C3×C4—91no—
257C22×C3×C4—91no—
258C22×C3×C4—92no—
259C2×#67—92no—
260C2×#67—92no—
261C2×#67—92no—
262C2×#67—92no—
263C2×#67—92no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267#235—188no—
268#235—188no—
269#235—188no—
270C2×D12—31no—
271C2×D12—31no—
272C2×D12—31no—
273C2×D12—61no—
274C2×D12—91no—
275C2×D12—92no—
276C2×D12—92no—
277C2×D12—92no—
278C2×D12—92no—
279C2×C4×D3—31no—
280C2×C4×D3—32no—
281C2×C4×D3—32no—
282C2×C4×D3—31no—
283C2×C4×D3—31no—
284C2×C4×D3—32no—
285C2×C4×D3—62no—
286C2×C4×D3—91no—
287C2×C4×D3—91no—
288C2×C4×D3—91no—
289C2×C4×D3—92no—
290C2×Dic6—92no—
291C2×Dic6—92no—
292C2×Dic6—92no—
293C3×C4⋊C4—91no—
294C3×C4⋊C4—91no—
295C3×C22⋊C4—92no—
296C3×C22⋊C4—92no—
297C3×C42—31no—
298C3×C42—31no—
299C3×C42—61no—
300C3×C42—91no—
301C3×C42—91no—
302C3×C42—91no—
303C3×C42D3211yescharacteristic
304#217—92no—
305#217—92no—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—32no—
308(C2×D3)⋊C4—32no—
309(C2×D3)⋊C4—62no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315C4⋊Dic3—31no—
316C4⋊Dic3—31no—
317C4⋊Dic3—31no—
318C4⋊Dic3—61no—
319C4⋊Dic3—91no—
320C4⋊Dic3—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323C4×Dic3—31no—
324C4×Dic3—31no—
325C4×Dic3—31no—
326C4×Dic3—91no—
327C4×Dic3—91no—
328C4×Dic3—91no—
329C22×C32—21no—
330C22×C32C23×D311yescharacteristic
331C22×C32C23×D311yescharacteristic
332C2×C3⋊D3—21no—
333C2×C3⋊D3—21no—
334C2×C3⋊D3—22no—
335C2×C3⋊D3—22no—
336C2×C3⋊D3—22no—
337C2×C3⋊D3C2×D1212yes—
338C2×C3×D3—32no—
339C2×C3×D3—61no—
340C2×C3×D3—61no—
341C2×C3×D3—64no—
342C2×C3×D3—62no—
343C2×C3×D3—62no—
344C2×C3×D3—62no—
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—122no—
352C2×C3×D3—122no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—22no—
356C32×C4—22no—
357C32×C4—41no—
358C32×C4C2×D1212yes—
359C32×C4#23512yescomplemented
360C32×C4#23512yescomplemented
361C32×C4D3×D412yes—
362C3⋊Dic3—21no—
363C3⋊Dic3—22no—
364C3⋊Dic3C2×D1212yes—
365C3⋊Dic3#23512yescomplemented
366C3×Dic3—32no—
367C3×Dic3—32no—
368C3×Dic3—62no—
369C3×Dic3—62no—
370C3×Dic3—61no—
371C3×Dic3—62no—
372C3×Dic3—122no—
373C2×#41—272no—
374C4⋊Q8—271no—
375#127—271no—
376#124—272no—
377#121—272no—
378#121—271no—
379#121—271no—
380C4×D4—91no—
381C4×D4—91no—
382C4×D4—272no—
383C2×C42—91no—
384H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
385C23×C3—91no—
386C23×C3—92no—
387C23×C3—92no—
388C22×D3—31no—
389C22×D3—32no—
390C22×D3—32no—
391C22×D3—32no—
392C22×D3—62no—
393C22×D3—91no—
394C22×D3—92no—
395C3×D4—182no—
396C3×D4—182no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—31no—
402C2×C3×C4—31no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—94no—
407C2×C3×C4—91no—
408C2×C3×C4—91no—
409C2×C3×C4—92no—
410C2×C3×C4—91no—
411C2×C3×C4—92no—
412C2×C3×C4—184no—
413C2×C3×C4—182no—
414C2×C3×C4—182no—
415C2×C3×C4—184no—
416C2×C3×C4C2×D3211yescharacteristic
417C2×C3×C4C2×D3211yescharacteristic
418C2×C3×C4C2×D3211yescharacteristic
419#67—184no—
420#67—184no—
421#67—184no—
422#67—184no—
423#67—184no—
424#67—184no—
425#67—184no—
426#67—184no—
427C2×Dic3—31no—
428C2×Dic3—31no—
429C2×Dic3—32no—
430C2×Dic3—32no—
431C2×Dic3—31no—
432C2×Dic3—32no—
433C2×Dic3—62no—
434C2×Dic3—91no—
435C2×Dic3—91no—
436C2×Dic3—91no—
437C2×Dic3—92no—
438D12—62no—
439D12—62no—
440D12—64no—
441D12—62no—
442D12—182no—
443D12—184no—
444D12—184no—
445D12—184no—
446D12—184no—
447C4×D3—34no—
448C4×D3—62no—
449C4×D3—64no—
450C4×D3—64no—
451C4×D3—62no—
452C4×D3—64no—
453C4×D3—94no—
454C4×D3—124no—
455C4×D3—182no—
456C4×D3—182no—
457C4×D3—184no—
458Dic6—184no—
459Dic6—184no—
460Dic6—184no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32C22×D1211yescharacteristic
465C2×C32C2×#23511yescharacteristic
466C2×C32C2×#23511yescharacteristic
467C2×C32C2×#23511yescharacteristic
468C2×C32C2×D3×D411yescharacteristic
469C2×C32D3×#4111yescharacteristic
470C3⋊D3—22no—
471C3⋊D3—42no—
472C3×D3—62no—
473C3×D3—124no—
474C3×D3—122no—
475C3×D3—122no—
476C3×D3—122no—
477#41—548no—
478C2×Q8—272no—
479C2×D4—91no—
480C2×D4—91no—
481C2×D4—272no—
482C2×D4—272no—
483C2×D4—272no—
484C2×D4—272no—
485C22×C4—91no—
486C22×C4—91no—
487C22×C4—91no—
488C22×C4—92no—
489C22×C4—92no—
490C4⋊C4—91no—
491C4⋊C4—91no—
492C4⋊C4—272no—
493C22⋊C4—92no—
494C22⋊C4—92no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C42—91no—
498C42—91no—
499C42—91no—
500C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C22×C3—31no—
502C22×C3—31no—
503C22×C3—61no—
504C22×C3—92no—
505C22×C3—181no—
506C22×C3—181no—
507C22×C3—182no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3—182no—
511C22×C3—182no—
512C22×C3—182no—
513C22×C3C22×D3211yescharacteristic
514C2×D3—32no—
515C2×D3—61no—
516C2×D3—61no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—64no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—92no—
528C2×D3—122no—
529C2×D3—122no—
530C2×D3—181no—
531C2×D3—181no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—182no—
535C3×C4—21no—
536C3×C4—32no—
537C3×C4—32no—
538C3×C4—32no—
539C3×C4—32no—
540C3×C4—61no—
541C3×C4—61no—
542C3×C4—62no—
543C3×C4—62no—
544C3×C4—92no—
545C3×C4—92no—
546C3×C4—121no—
547C3×C4—182no—
548C3×C4—181no—
549C3×C4—182no—
550C3×C4#379812yes—
551C3×C4D3×D1212yes—
552Dic3—32no—
553Dic3—32no—
554Dic3—62no—
555Dic3—62no—
556Dic3—61no—
557Dic3—62no—
558Dic3—92no—
559Dic3—92no—
560Dic3—122no—
561Dic3—182no—
562Dic3—181no—
563C32—21no—
564C32#547711yescharacteristic, complemented
565C32#554811yescharacteristic, complemented
566C23—91no—
567C23—92no—
568C23—92no—
569Q8—544no—
570D4—182no—
571D4—182no—
572D4—544no—
573D4—544no—
574D4—544no—
575D4—544no—
576C2×C4—91no—
577C2×C4—91no—
578C2×C4—94no—
579C2×C4—91no—
580C2×C4—92no—
581C2×C4—92no—
582C2×C4—182no—
583C2×C4—182no—
584C2×C4—184no—
585C2×C4—184no—
586C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
587C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
588C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
589C2×C3—31no—
590C2×C3—31no—
591C2×C3—31no—
592C2×C3—31no—
593C2×C3—31no—
594C2×C3—31no—
595C2×C3—61no—
596C2×C3—61no—
597C2×C3—61no—
598C2×C3—182no—
599C2×C3—362no—
600C2×C3—362no—
601C2×C3C2×#379811yescharacteristic
602C2×C3C2×D3×D1211yescharacteristic
603C2×C3D3×#23511yescharacteristic
604D3—62no—
605D3—122no—
606D3—122no—
607D3—124no—
608D3—122no—
609D3—182no—
610D3—362no—
611C22—92no—
612C22—181no—
613C22—181no—
614C22—182no—
615C22—182no—
616C22—182no—
617C22—182no—
618C22—182no—
619C22—182no—
620C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
621C4—21no—
622C4—92no—
623C4—92no—
624C4—181no—
625C4—182no—
626C4—182no—
627C4#8865012yescomplemented
628C4#8865412yes—
629C3—31no—
630C3—31no—
631C3—61no—
632C3#1059396811yescharacteristic
633C2—182no—
634C2—362no—
635C2—362no—
636C2C2×#8865011yescharacteristic
637C2C2×#8865411yescharacteristic
638C2#1172249011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88650C2semidirect
2. C4×C4⋊#272C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C42×(C3⋊D3)⋊C3C2semidirect
5. #11721158C2semidirect
6. #11721156C2semidirect
7. C4×#6200C2semidirect
8. #11721141C2semidirect
9. #11721140C2semidirect
10. #11721116C2semidirect
11. #11721103C2semidirect
40. C4×C4⋊(C3⋊D3)D3semidirect
41. C42×C3⋊D3D3semidirect
78. C4×(C3⋊D3)⋊C3D4semidirect
152. C4×H(3,𝔽3)#41semidirect
157. (C3⋊Dic3)⋊C3#41semidirect
216. C4×C3⋊D3D12semidirect
359. C32×C4#235semidirect
360. C32×C4#235semidirect
365. C3⋊Dic3#235semidirect
384. H(3,𝔽3)#532semidirect
564. C32#5477semidirect
565. C32#5548semidirect
627. C4#88650semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C4×C4⋊#2722
C4×C4⋊((C3⋊D3)⋊C3)2
C42×(C3⋊D3)⋊C32
#117211582
#117211562
C4×#62002
#117211412
#117211402
#117211162
#117211032
#105939723
#105939683