Finite Group Databasebeta

← #15799630

Subgroups — #15799630

Order 3328.

1934 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88142C211yesmaximal, characteristic
2C2×#11718317C212yesmaximal
3C2×#11718640C212yesmaximal
4C2×#11718308C211yesmaximal, characteristic
5C2×#11718637C212yesmaximal
6#13113101C211yesmaximal, characteristic
7#13113100C211yesmaximal, characteristic
8#13113099C211yesmaximal, characteristic
9#13113064C212yesmaximal
10#95×Dic13C211yesmaximal, characteristic
11C13×#1444C211yesmaximal, characteristic
12C23×Dic26C2212yes—
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C13×C4⋊C4C2212yes—
15C22×C4⋊Dic13C2211yescharacteristic
16C22×C4⋊Dic13C2211yescharacteristic
17C22×C4⋊Dic13C2212yes—
18C22×#6101C2211yescharacteristic
19C22×#6101C2211yescharacteristic
20C22×#6101C2212yes—
21C22×#6101C2212yes—
22C22×#6101C2212yes—
23C22×#6101C2211yescharacteristic
24C22×C4×Dic13C2211yescharacteristic
25C22×C4×Dic13C2212yes—
26C2×#88142—28no—
27C2×#88142—24no—
28C2×C13×#95C2211yescharacteristic
29C2×C13×#95C2212yes—
30C2×C13×#95C2211yescharacteristic
31C2×#88071C2211yescharacteristic
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2211yescharacteristic
37#11718640—28no—
38#11718637—28no—
39#11718317—28no—
40#11718308—24no—
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2312yes—
43C23×C4×C13C2311yescharacteristic
44C23×C4×C13C2311yescharacteristic
45C23×C4×C13C2312yes—
46C23×Dic13C2311yescharacteristic
47C23×Dic13C2312yes—
48C23×Dic13C2311yescharacteristic
49C23×Dic13C2311yescharacteristic
50C23×Dic13C2312yes—
51C23×Dic13C2311yescharacteristic
52C22×Dic26—24no—
53C22×Dic26—24no—
54C22×Dic26—24no—
55C22×Dic26—24no—
56C22×Dic26—24no—
57C22×Dic26—42no—
58C22×Dic26—42no—
59C2×C13×C4⋊C4—24no—
60C2×C13×C4⋊C4—22no—
61C2×C13×C4⋊C4—24no—
62C2×C13×C4⋊C4—24no—
63C2×C13×C4⋊C4—42no—
64C2×C4⋊Dic13—22no—
65C2×C4⋊Dic13—24no—
66C2×C4⋊Dic13—24no—
67C2×C4⋊Dic13—22no—
68C2×C4⋊Dic13—24no—
69C2×C4⋊Dic13—24no—
70C2×C4⋊Dic13—42no—
71C2×#6101—24no—
72C2×#6101—24no—
73C2×#6101—22no—
74C2×#6101—22no—
75C2×#6101—24no—
76C2×#6101—22no—
77C2×#6101—22no—
78C2×#6101—24no—
79C2×#6101—24no—
80C2×#6101—24no—
81C2×#6101—24no—
82C2×#6101—24no—
83C2×#6101—42no—
84C2×#6101—42no—
85C2×#6101D418yes—
86C2×#6101Q814yescomplemented
87C2×C4×Dic13—24no—
88C2×C4×Dic13—24no—
89C2×C4×Dic13—24no—
90C2×C4×Dic13D414yes—
91C2×C4×Dic13Q818yescomplemented
92#88142—232no—
93C13×#95—22no—
94C13×#95—24no—
95C13×#95Q814yescomplemented
96#88071—22no—
97#88071—24no—
98#88071—24no—
99#88071—24no—
100#88071D418yes—
101#88071Q814yescomplemented
102#15143—131nomaximal, Sylow
103C24×C13C2411yescharacteristic
104C22×C4×C13—22no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—21no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—21no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—21no—
124C22×C4×C13—21no—
125C22×C4×C13C2×D414yes—
126C22×C4×C13C2×D412yes—
127C22×C4×C13C2×D414yes—
128C22×C4×C13C2×Q812yes—
129C22×C4×C13C2×Q814yes—
130C22×C4×C13C2×Q812yes—
131C22×C4×C13C2×Q812yes—
132C22×C4×C13C2×Q814yes—
133C22×Dic13—22no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—21no—
142C22×Dic13—22no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—21no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—21no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13C2×D414yes—
156C22×Dic13C2×D414yes—
157C22×Dic13C2×D412yes—
158C22×Dic13C2×D414yes—
159C22×Dic13C2×D414yes—
160C22×Dic13C2×D412yes—
161C22×Dic13C2×Q812yes—
162C22×Dic13C2×Q814yes—
163C22×Dic13C2×Q814yes—
164C22×Dic13C2×Q814yes—
165C22×Dic13C2×Q812yes—
166C22×Dic13#4112yes—
167C2×Dic26—216no—
168C2×Dic26—48no—
169C2×Dic26—48no—
170C2×Dic26—48no—
171C2×Dic26—48no—
172C2×Dic26—48no—
173C2×Dic26—48no—
174C13×C4⋊C4—28no—
175C13×C4⋊C4—48no—
176C4⋊Dic13—28no—
177C4⋊Dic13—48no—
178C4⋊Dic13—44no—
179#6101—216no—
180#6101—28no—
181#6101—48no—
182#6101—48no—
183#6101—44no—
184#6101—44no—
185C4×Dic13—28no—
186C4×Dic13—216no—
187C22×C4⋊Q8—131no—
188C2×#395—132no—
189C2×#395—132no—
190C2×#389—131no—
191C2×#384—132no—
192#1444—131no—
193#1444—131no—
194#1444—131no—
195#1444D1311yescharacteristic
196C4⋊#95—132no—
197C4×#95—131no—
198C23×C13C22×D411yescharacteristic
199C23×C13C22×D412yes—
200C23×C13C22×D412yes—
201C23×C13C22×Q812yes—
202C23×C13C22×Q811yescharacteristic
203C23×C13C22×Q811yescharacteristic
204C23×C13C22×Q811yescharacteristic
205C23×C13C2×#4111yescharacteristic
206C23×C13H(5,𝔽2)11yescharacteristic
207C23×C13H(5,𝔽2)11yescharacteristic
208C23×C132−1+412yes—
209C2×C4×C13—22no—
210C2×C4×C13—24no—
211C2×C4×C13—22no—
212C2×C4×C13—24no—
213C2×C4×C13—24no—
214C2×C4×C13—24no—
215C2×C4×C13—24no—
216C2×C4×C13—24no—
217C2×C4×C13—24no—
218C2×C4×C13—24no—
219C2×C4×C13—24no—
220C2×C4×C13—22no—
221C2×C4×C13—24no—
222C2×C4×C13—24no—
223C2×C4×C13—24no—
224C2×C4×C13—41no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—41no—
228C2×C4×C13—42no—
229C2×C4×C13—42no—
230C2×C4×C13—42no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13C4⋊Q814yes—
237C2×C4×C13C4⋊Q818yescomplemented
238C2×Dic13—22no—
239C2×Dic13—24no—
240C2×Dic13—22no—
241C2×Dic13—24no—
242C2×Dic13—24no—
243C2×Dic13—24no—
244C2×Dic13—24no—
245C2×Dic13—24no—
246C2×Dic13—24no—
247C2×Dic13—24no—
248C2×Dic13—24no—
249C2×Dic13—22no—
250C2×Dic13—24no—
251C2×Dic13—24no—
252C2×Dic13—24no—
253C2×Dic13—41no—
254C2×Dic13—42no—
255C2×Dic13—42no—
256C2×Dic13—41no—
257C2×Dic13—42no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13#12014yes—
266C2×Dic13C22⋊Q818yes—
267C2×Dic13C22⋊Q814yes—
268C2×Dic13#12714yes—
269C2×Dic13C4⋊Q814yes—
270C2×Dic13C4⋊Q818yes—
271C2×Dic13C4⋊Q818yescomplemented
272Dic26—432no—
273C23×Q8—132no—
274C2×C4⋊Q8—264no—
275C2×C4⋊Q8—268no—
276C22×C4⋊C4—132no—
277C22×C4⋊C4—131no—
278C22×C4⋊C4—131no—
279C22×C4⋊C4—131no—
280C22×C4⋊C4—131no—
281C22×C4⋊C4—131no—
282C22×C4⋊C4—132no—
283C22×C4⋊C4—132no—
284C22×C4⋊C4—132no—
285C22×C4⋊C4C2×D1311yescharacteristic
286C22×C4⋊C4C2×D1312yes—
287C22×C42—131no—
288C22×C42—132no—
289#395—268no—
290#395—268no—
291#389—264no—
292#384—268no—
293C2×#95—132no—
294C2×#95—131no—
295C2×#95—132no—
296C2×#95—132no—
297C2×#95—132no—
298C2×#95—131no—
299C2×#95C2×D1312yes—
300C2×#95C2×D1311yescharacteristic
301C2×#95C2×D1311yescharacteristic
302C22×C13C2×#12011yescharacteristic
303C22×C13C2×C22⋊Q811yescharacteristic
304C22×C13C2×C22⋊Q812yes—
305C22×C13C2×#12711yescharacteristic
306C22×C13C2×C4⋊Q811yescharacteristic
307C22×C13C2×C4⋊Q812yes—
308C22×C13C2×C4⋊Q812yes—
309C22×C13C2×C4⋊Q811yescharacteristic
310C22×C13C2×C4⋊Q812yes—
311C22×C13#53511yescharacteristic
312C22×C13#53611yescharacteristic
313C22×C13#54412yes—
314C22×C13#54412yes—
315C22×C13D4212yes—
316C22×C13D4212yes—
317C22×C13#54712yes—
318C22×C13D4×Q812yes—
319C22×C13D4×Q812yes—
320C22×C13D4×Q812yes—
321C22×C13#55411yescharacteristic
322C22×C13#55411yescharacteristic
323C22×C13Q8211yescharacteristic
324C22×C13Q8211yescharacteristic
325C4×C13—24no—
326C4×C13—28no—
327C4×C13—44no—
328C4×C13—42no—
329C4×C13—81no—
330Dic13—24no—
331Dic13—28no—
332Dic13—44no—
333Dic13—42no—
334Dic13—81no—
335Dic13#39518yescomplemented
336C22×Q8—264no—
337C22×Q8—264no—
338C22×Q8—264no—
339C22×Q8—264no—
340C22×Q8—264no—
341C22×Q8—522no—
342C22×Q8—522no—
343C23×C4—131no—
344C23×C4—131no—
345C23×C4—132no—
346C23×C4—132no—
347C23×C4—131no—
348C23×C4—131no—
349C23×C4C22×D1311yescharacteristic
350C23×C4C22×D1312yes—
351C23×C4C22×D1311yescharacteristic
352C23×C4C22×D1311yescharacteristic
353C23×C4C22×D1312yes—
354C4⋊Q8—2632no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—138no—
361C2×C4⋊C4—134no—
362C2×C4⋊C4—264no—
363C2×C4⋊C4—264no—
364C2×C4⋊C4—264no—
365C2×C4⋊C4—262no—
366C2×C4⋊C4—262no—
367C2×C4⋊C4—262no—
368C2×C4⋊C4—264no—
369C2×C4⋊C4—264no—
370C2×C4⋊C4—262no—
371C2×C4⋊C4—262no—
372C2×C4⋊C4—264no—
373C2×C4⋊C4—264no—
374C2×C4⋊C4—264no—
375C2×C4⋊C4—264no—
376C2×C4⋊C4—262no—
377C2×C4⋊C4—264no—
378C2×C4⋊C4—264no—
379C2×C4⋊C4—264no—
380C2×C4⋊C4—522no—
381C2×C4⋊C4—522no—
382C2×C4⋊C4—522no—
383C2×C42—134no—
384C2×C42—138no—
385C2×C42—264no—
386C2×C42—264no—
387C2×C42—264no—
388#95—22no—
389#95—24no—
390#95—138no—
391#95—134no—
392#95—264no—
393#95—264no—
394#95—262no—
395#95—264no—
396#95Dic2614yescomplemented
397C2×C13C2×#39511yescharacteristic
398C2×C13#243311yescharacteristic
399C2×C13#243411yescharacteristic
400C2×C13#244012yes—
401C2×C13#244012yes—
402C2×C13#254211yescharacteristic
403C2×C13#255112yes—
404C2×C13#255112yes—
405C2×C13#256711yescharacteristic
406C2×C13#266011yescharacteristic
407C2×C13#286811yescharacteristic
408C24C23×D1311yescharacteristic
409C2×Q8—2616no—
410C2×Q8—528no—
411C2×Q8—528no—
412C2×Q8—528no—
413C2×Q8—528no—
414C2×Q8—528no—
415C2×Q8—528no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—132no—
438C22×C4—134no—
439C22×C4—134no—
440C22×C4—132no—
441C22×C4—132no—
442C22×C4—134no—
443C22×C4—134no—
444C22×C4—132no—
445C22×C4—134no—
446C22×C4—134no—
447C22×C4—134no—
448C22×C4—132no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—261no—
452C22×C4—262no—
453C22×C4—262no—
454C22×C4—261no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—261no—
458C22×C4—262no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—261no—
464C22×C4—262no—
465C22×C4—262no—
466C22×C4—261no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4—261no—
470C22×C4—261no—
471C22×C4C2×Dic2612yes—
472C22×C4C2×Dic2614yes—
473C22×C4D4×D1314yes—
474C22×C4D4×D1312yes—
475C22×C4D4×D1314yes—
476C22×C4Q8×D1312yes—
477C22×C4Q8×D1312yes—
478C22×C4Q8×D1314yes—
479C4⋊C4—28no—
480C4⋊C4—48no—
481C4⋊C4—2616no—
482C4⋊C4—268no—
483C4⋊C4—268no—
484C4⋊C4—528no—
485C4⋊C4—528no—
486C4⋊C4—524no—
487C4⋊C4—524no—
488C4⋊C4—524no—
489C4⋊C4—528no—
490C42—268no—
491C42—2616no—
492C13#1514311yescharacteristic, complemented, Sylow
493C23C22×Dic2611yescharacteristic
494C23C2×D4×D1311yescharacteristic
495C23C2×D4×D1312yes—
496C23C2×D4×D1312yes—
497C23#8825111yescharacteristic
498C23C2×Q8×D1311yescharacteristic
499C23C2×Q8×D1311yescharacteristic
500C23C2×Q8×D1312yes—
501C23#41×D1311yescharacteristic
502C23#8825611yescharacteristic
503C23#8825712yes—
504Q8—5232no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—138no—
533C2×C4—134no—
534C2×C4—134no—
535C2×C4—138no—
536C2×C4—134no—
537C2×C4—138no—
538C2×C4—134no—
539C2×C4—262no—
540C2×C4—264no—
541C2×C4—262no—
542C2×C4—264no—
543C2×C4—264no—
544C2×C4—264no—
545C2×C4—264no—
546C2×C4—264no—
547C2×C4—264no—
548C2×C4—264no—
549C2×C4—264no—
550C2×C4—264no—
551C2×C4—264no—
552C2×C4—262no—
553C2×C4—264no—
554C2×C4—522no—
555C2×C4—522no—
556C2×C4—522no—
557C2×C4—521no—
558C2×C4—522no—
559C2×C4—521no—
560C2×C4—522no—
561C2×C4—521no—
562C2×C4—522no—
563C2×C4—521no—
564C2×C4—522no—
565C2×C4—521no—
566C2×C4#8814214yes—
567C2×C4#8814218yescomplemented
568C22C2×#8814211yescharacteristic
569C22C2×#8814212yes—
570C22D4×Dic2612yes—
571C22#1171924411yescharacteristic
572C22Q8×Dic2611yescharacteristic
573C22D13×#12011yescharacteristic
574C22#1171929512yes—
575C22#1171931211yescharacteristic
576C22#1171931412yes—
577C22#1171932812yes—
578C22D13×C22⋊Q811yescharacteristic
579C22D13×C22⋊Q812yes—
580C22#1171933712yes—
581C22#1171933712yes—
582C22#1171935311yescharacteristic
583C22#1171936012yes—
584C22#1171939412yes—
585C22#1171939511yescharacteristic
586C22D13×#12711yescharacteristic
587C22#1171942711yescharacteristic
588C22D13×C4⋊Q812yes—
589C22D13×C4⋊Q812yes—
590C22D13×C4⋊Q811yescharacteristic
591C4—24no—
592C4—28no—
593C4—44no—
594C4—42no—
595C4—81no—
596C4—138no—
597C4—264no—
598C4—268no—
599C4—524no—
600C4—522no—
601C4—1041no—
602C2D13×#39511yescharacteristic
603C2#1311846811yescharacteristic
604C2#1311847811yescharacteristic
605C2#1311853312yes—
606C2#1311853712yes—
607C2#1311978411yescharacteristic
608C2#1311986112yes—
609C2#1311986412yes—
610C2#1312004311yescharacteristic
611C2#1312095911yescharacteristic
612C2#1312323311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, prime radical, cosocleC24×C13C24
FittingC13×#1444C2
hypercenter#1444D13
hypocenterC13#15143
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
86. C2×#6101Q8semidirect
91. C2×C4×Dic13Q8semidirect
95. C13×#95Q8semidirect
101. #88071Q8semidirect
237. C2×C4×C13C4⋊Q8semidirect
271. C2×Dic13C4⋊Q8semidirect
335. Dic13#395semidirect
396. #95Dic26semidirect
492. C13#15143semidirect
567. C2×C4#88142semidirect

Maximal subgroups

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

subgroupindex
C22×#881422
C2×#117183172
C2×#117186402
C2×#117183082
C2×#117186372
#131131012
#131131002
#131130992
#131130642
#95×Dic132
C13×#14442
#1514313