Finite Group Databasebeta

← #15864288

Subgroups — #15864288

Order 3328.

1802 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88196C211yesmaximal, characteristic
2C2×#11718627C211yesmaximal, characteristic
3C2×#11718886C211yesmaximal, characteristic, complemented
4#13117326C211yesmaximal, characteristic
5#13115191C214yesmaximal, complemented
6#13115177C214yesmaximal, complemented
7#13114555C211yesmaximal, characteristic, complemented
8#13114530C211yesmaximal, characteristic, complemented
9C22×C4×Q8×C13C211yesmaximal, characteristic, complemented
10C23×Q8×C13C2211yescharacteristic
11C22×#6127C2211yescharacteristic
12C2×C4×Q8×C13—24no—
13C2×C4×Q8×C13—24no—
14C2×C4×Q8×C13—22no—
15C2×C4×Q8×C13C2214yes—
16C22×C13×C4⋊C4—21no—
17C22×C13×C4⋊C4C2211yescharacteristic
18C22×C42×C13—21no—
19C22×C42×C13C2211yescharacteristic
20C22×#6107C2212yes—
21C2×#88196—24no—
22C2×#88196—24no—
23C2×#88196C418yescomplemented
24C22×#6106C2212yes—
25C2×#88180C2211yescharacteristic
26C22×#6099C2211yescharacteristic
27C22×#6099C2211yescharacteristic
28#11718886—24no—
29C2×#88075C2214yes—
30#11718874—21no—
31#11718874—21no—
32C2×#88050C2212yes—
33#11718627—24no—
34C2×#88049C2212yes—
35#11718600C2212yes—
36C4×#6103—22no—
37#11718590C2212yes—
38#11718121C2214yes—
39#11718116C2214yes—
40C22×Q8×C13—22no—
41C22×Q8×C13—21no—
42C22×Q8×C13—22no—
43C22×Q8×C13—22no—
44C22×Q8×C13C2×C414yes—
45C22×Q8×C13D414yes—
46C22×Q8×C13D414yes—
47C22×Q8×C13C2312yes—
48C23×C4×C13—21no—
49C23×C4×C13—21no—
50C23×C4×C13D411yescharacteristic
51C23×C4×C13D411yescharacteristic
52C23×C4×C13C2311yescharacteristic
53C22×C22⋊D13—21no—
54C2×#6127—22no—
55C2×#6127—22no—
56C2×#6127—22no—
57C2×#6127—22no—
58C2×#6127—22no—
59C2×#6127C2×C414yes—
60C22×D52C2311yescharacteristic
61C22×C4×D13—21no—
62C22×Dic26C2311yescharacteristic
63C4×Q8×C13—216no—
64C4×Q8×C13—28no—
65C4×Q8×C13—28no—
66C2×C13×C4⋊C4—28no—
67C2×C13×C4⋊C4—24no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—22no—
70C2×C13×C4⋊C4D414yes—
71C2×C13×C4⋊C4C2312yes—
72C2×C42×C13—28no—
73C2×C42×C13—24no—
74C2×C42×C13—22no—
75C2×C42×C13—22no—
76C2×C42×C13D412yes—
77C2×C42×C13C2312yes—
78C2×C22⋊Dic13—21no—
79C2×#6107—24no—
80C2×#6107—24no—
81C2×#6107C2×C418yescomplemented
82#88196—232no—
83C2×#6106—24no—
84C2×#6106—24no—
85C2×#6106C2×C418yescomplemented
86C2×#6103—22no—
87C2×#6103—22no—
88#88180—22no—
89#88180—22no—
90C2×C4⋊Dic13C2312yes—
91C2×C4×Dic13—21no—
92C2×#6099—22no—
93C2×#6099—22no—
94C2×#6099—22no—
95C2×#6099—22no—
96C2×#6099—22no—
97C2×#6099C2×C414yes—
98C22×#1055C2312yes—
99C22×#1055C2312yes—
100#88075—28no—
101#88071—22no—
102#88071—22no—
103#88050—24no—
104#88049—24no—
105#20926—131nomaximal, Sylow
106C24×C13C2×D411yescharacteristic
107C23×D13—21no—
108C2×Q8×C13—24no—
109C2×Q8×C13—28no—
110C2×Q8×C13—24no—
111C2×Q8×C13—24no—
112C2×Q8×C13—28no—
113C2×Q8×C13—28no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13C22⋊C4116yescomplemented
118C2×Q8×C13C22×C414yes—
119C2×Q8×C13C2×D414yes—
120C2×Q8×C13C2×D414yes—
121C22×C4×C13—24no—
122C22×C4×C13—22no—
123C22×C4×C13—24no—
124C22×C4×C13—24no—
125C22×C4×C13—22no—
126C22×C4×C13—24no—
127C22×C4×C13—22no—
128C22×C4×C13—24no—
129C22×C4×C13—22no—
130C22×C4×C13—42no—
131C22×C4×C13—44no—
132C22×C4×C13C22×C412yes—
133C22×C4×C13C2×D412yes—
134C22×C4×C13C2×D412yes—
135C22×C4×C13C2×D412yes—
136C22×C4×C13C2×D412yes—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D412yes—
139C22×C4×C13#4112yes—
140C22×C4×C13#4112yes—
141C22×C4×C13C2411yescharacteristic
142C2×C22⋊D13—24no—
143C2×C22⋊D13—42no—
144C2×C22⋊D13—42no—
145C22×Dic13—21no—
146C22×Dic13—22no—
147#6127—28no—
148#6127—28no—
149#6127—44no—
150#6127—44no—
151C2×D52—21no—
152C2×D52—21no—
153C2×D52—21no—
154C2×D52—21no—
155C2×D52—21no—
156C2×D52C22×C412yes—
157C2×C4×D13—24no—
158C2×C4×D13—42no—
159C2×C4×D13—42no—
160C2×Dic26—21no—
161C2×Dic26—21no—
162C2×Dic26—21no—
163C2×Dic26—21no—
164C2×Dic26—21no—
165C2×Dic26C22×C412yes—
166C13×C4⋊C4—216no—
167C13×C4⋊C4—24no—
168C13×C4⋊C4—22no—
169C13×C4⋊C4C2×D414yes—
170C42×C13—216no—
171C42×C13—24no—
172C42×C13—22no—
173C42×C13—22no—
174C22⋊Dic13—42no—
175#6107—216no—
176#6106—216no—
177#6103—44no—
178#6103—44no—
179C4⋊Dic13—22no—
180C4⋊Dic13—22no—
181C4×Dic13—42no—
182#6099—28no—
183#6099—44no—
184C2×#1055—22no—
185C2×#1055—22no—
186C2×#1055—22no—
187C2×#1055—22no—
188C2×#1055—22no—
189C2×#1055C22×C414yes—
190C22×#137—131no—
191C22×C4×Q8D1311yescharacteristic, complemented
192C2×#417—131no—
193C2×#419—131no—
194#2297—131no—
195#1880—134no—
196#1878—134no—
197#1736—131no—
198#1728—131no—
199C23×C13—22no—
200C23×C13—22no—
201C23×C13C4×D412yes—
202C23×C13#12012yes—
203C23×C13#12112yes—
204C23×C13C22×D411yescharacteristic
205C22×D13—22no—
206C22×D13—41no—
207C22×D13—41no—
208C22×D13—42no—
209C22×D13—42no—
210Q8×C13—24no—
211Q8×C13—24no—
212Q8×C13—24no—
213Q8×C13—216no—
214Q8×C13C2×C22⋊C418yes—
215C2×C4×C13—28no—
216C2×C4×C13—24no—
217C2×C4×C13—22no—
218C2×C4×C13—24no—
219C2×C4×C13—24no—
220C2×C4×C13—22no—
221C2×C4×C13—216no—
222C2×C4×C13—22no—
223C2×C4×C13—24no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—44no—
227C2×C4×C13—48no—
228C2×C4×C13—44no—
229C2×C4×C13C2×C22⋊C414yes—
230C2×C4×C13C4×D414yes—
231C2×C4×C13C4×D412yes—
232C2×C4×C13#12012yes—
233C2×C4×C13#12114yes—
234C2×C4×C13#12112yes—
235C2×C4×C13#12314yes—
236C2×C4×C13#13714yescomplemented
237C2×C4×C13C23×C411yescharacteristic
238C2×C4×C13C22×D411yescharacteristic
239C2×C4×C13C22×D411yescharacteristic
240C2×C4×C13C22×D411yescharacteristic
241C2×C4×C13C2×#4111yescharacteristic
242C2×C4×C13C2×#4111yescharacteristic
243C22⋊D13—48no—
244C2×Dic13—22no—
245C2×Dic13—41no—
246C2×Dic13—41no—
247C2×Dic13—42no—
248C2×Dic13—42no—
249C2×Dic13—42no—
250D52—22no—
251D52—22no—
252D52—41no—
253D52—41no—
254C4×D13—48no—
255Dic26—22no—
256Dic26—22no—
257Dic26—41no—
258Dic26—41no—
259#1055—24no—
260#1055—42no—
261C22×#41—131no—
262C23×Q8C2×D1311yescharacteristic
263C2×#137—138no—
264C2×#137—264no—
265C2×#137—264no—
266C22×Q16—132no—
267C22×SD16—132no—
268C22×M16—131no—
269C22×M16—131no—
270C2×C4×Q8—24no—
271C2×C4×Q8—24no—
272C2×C4×Q8—22no—
273C2×C4×Q8C2×D1314yes—
274C2×#117—131no—
275C22×C4⋊C4—21no—
276C22×C4⋊C4C2×D1311yescharacteristic
277C22×C42—21no—
278C22×C42C2×D1311yescharacteristic
279#419—264no—
280#417—264no—
281C2×#103—134no—
282C2×#103—132no—
283C2×#102—132no—
284#391—132no—
285#390—132no—
286#386—261no—
287#386—261no—
288C4×C22⋊C4—262no—
289#340—134no—
290#336—134no—
291C22×C13—24no—
292C22×C13—22no—
293C22×C13—22no—
294C22×C13—24no—
295C22×C13#38614yes—
296C22×C13C2×C4×D411yescharacteristic
297C22×C13C2×#12011yescharacteristic
298C22×C13C2×#12111yescharacteristic
299C22×C13C2×#13712yes—
300C22×C13#57512yes—
301C2×D13—44no—
302C2×D13—42no—
303C2×D13—81no—
304C2×D13—81no—
305C2×D13—82no—
306C4×C13—24no—
307C4×C13—28no—
308C4×C13—42no—
309C4×C13—44no—
310C4×C13#38614yes—
311C4×C13C22×C22⋊C411yescharacteristic
312C4×C13C2×C4×D411yescharacteristic
313C4×C13C2×#12111yescharacteristic
314C4×C13C2×#12311yescharacteristic
315Dic13—42no—
316Dic13—82no—
317C2×#41—134no—
318C2×#41—262no—
319C2×#41—262no—
320C2×#41—262no—
321C2×#41—262no—
322C2×#41—262no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—21no—
327C22×Q8—131no—
328C22×Q8C4×D1314yes—
329C22×Q8D5214yes—
330C22×Q8C22⋊D1314yes—
331C22×Q8C22×D1312yes—
332C22×D4—131no—
333C22×D4—261no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—261no—
337C23×C4C22⋊D1311yescharacteristic
338C23×C4C22⋊D1311yescharacteristic
339C23×C4C22×D1311yescharacteristic
340#137—2632no—
341C2×Q16—138no—
342C2×Q16—264no—
343C2×Q16—264no—
344C2×SD16—138no—
345C2×SD16—264no—
346C2×SD16—264no—
347C2×M16—134no—
348C2×M16—262no—
349C2×M16—262no—
350C2×M16—262no—
351C2×M16—262no—
352C2×M16—262no—
353C22×C8—132no—
354C22×C8—132no—
355C4×Q8—216no—
356C4×Q8—28no—
357C4×Q8—28no—
358#117—262no—
359#117—262no—
360C2×C4⋊C4—28no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—132no—
365C2×C4⋊C4C22⋊D1314yes—
366C2×C4⋊C4C22×D1312yes—
367C2×C22⋊C4—261no—
368C2×C22⋊C4—262no—
369C2×C22⋊C4—262no—
370C2×C42—28no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42—261no—
375C2×C42C22⋊D1312yes—
376C2×C42C22×D1312yes—
377#103—268no—
378#103—264no—
379#102—264no—
380#95—262no—
381#95—262no—
382C2×C13—24no—
383C2×C13C2×#38611yescharacteristic
384C2×C13C4×#13712yes—
385C2×C13#300411yescharacteristic
386C2×C13#300511yescharacteristic
387C2×C13#303712yes—
388D13—82no—
389C24—261no—
390C24C2×C22⋊D1311yescharacteristic
391#41—268no—
392#41—268no—
393#41—524no—
394#41—524no—
395C2×Q8—24no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—132no—
405C2×Q8—261no—
406C2×Q8—261no—
407C2×Q8—261no—
408C2×Q8—261no—
409C2×Q8—261no—
410C2×Q8#6103116yescomplemented
411C2×Q8C2×C4×D1314yes—
412C2×Q8C2×D5214yes—
413C2×Q8C2×C22⋊D1314yes—
414C2×D4—132no—
415C2×D4—264no—
416C2×D4—261no—
417C2×D4—261no—
418C2×D4—261no—
419C2×D4—261no—
420C2×D4—261no—
421C2×D4—522no—
422C2×D4—522no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—261no—
435C22×C4—264no—
436C22×C4—262no—
437C22×C4—522no—
438C22×C4—522no—
439C22×C4C2×C4×D1312yes—
440C22×C4C2×D5212yes—
441C22×C4#612712yes—
442C22×C4#612712yes—
443C22×C4C2×C22⋊D1312yes—
444C22×C4C2×C22⋊D1312yes—
445C22×C4C2×C22⋊D1312yes—
446C22×C4C2×C22⋊D1311yescharacteristic
447C22×C4C2×C22⋊D1312yes—
448C22×C4C23×D1311yescharacteristic
449Q16—2616no—
450SD16—2616no—
451M16—268no—
452M16—524no—
453C2×C8—134no—
454C2×C8—262no—
455C2×C8—262no—
456C2×C8—262no—
457C2×C8—262no—
458C2×C8—262no—
459C4⋊C4—216no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—262no—
463C4⋊C4—262no—
464C4⋊C4C2×C22⋊D1314yes—
465C22⋊C4—524no—
466C22⋊C4—522no—
467C22⋊C4—524no—
468C42—216no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—522no—
473C13#2092611yescharacteristic, complemented, Sylow
474C23—22no—
475C23—22no—
476C23—262no—
477C23—521no—
478C23—522no—
479C23—522no—
480C23—521no—
481C23C4×C22⋊D1312yes—
482C23#8818412yes—
483C23#8820712yes—
484C23C22×C22⋊D1311yescharacteristic
485Q8—24no—
486Q8—24no—
487Q8—24no—
488Q8—216no—
489Q8—262no—
490Q8—262no—
491Q8—521no—
492Q8—521no—
493Q8C2×#610318yes—
494D4—262no—
495D4—262no—
496D4—528no—
497D4—521no—
498D4—521no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—216no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—262no—
514C2×C4—521no—
515C2×C4—521no—
516C2×C4—528no—
517C2×C4—522no—
518C2×C4—522no—
519C2×C4—522no—
520C2×C4C2×#610314yes—
521C2×C4C4×C22⋊D1314yes—
522C2×C4C4×C22⋊D1312yes—
523C2×C4#8818314yes—
524C2×C4#8818414yes—
525C2×C4#8818412yes—
526C2×C4#8819614yescomplemented
527C2×C4#8820712yes—
528C2×C4C22×C4×D1311yescharacteristic
529C2×C4C22×D5211yescharacteristic
530C2×C4C2×#612711yescharacteristic
531C2×C4C2×#612711yescharacteristic
532C2×C4C22×C22⋊D1311yescharacteristic
533C2×C4C22×C22⋊D1311yescharacteristic
534C8—264no—
535C8—522no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—522no—
541C22—524no—
542C22—1042no—
543C22—1041no—
544C22—1041no—
545C22#1171887414yes—
546C22C2×C4×C22⋊D1311yescharacteristic
547C22C2×#8818411yescharacteristic
548C22C2×#8819612yes—
549C22#1171952712yes—
550C22C2×#8820711yescharacteristic
551C4—28no—
552C4—24no—
553C4—42no—
554C4—44no—
555C4—522no—
556C4—1042no—
557C4#1171887414yes—
558C4C22×#610311yescharacteristic
559C4C2×C4×C22⋊D1311yescharacteristic
560C4C2×#8818311yescharacteristic
561C4C2×#8818411yescharacteristic
562C2—24no—
563C2—1042no—
564C2C2×#1171887411yescharacteristic
565C2C4×#8819612yes—
566C2#1312407011yescharacteristic
567C2#1312407811yescharacteristic
568C2#1312430812yes—
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC22×C4×Q8×C13C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC22×C4×Q8D13
hypocenterC13#20926
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#11718886C2semidirect
5. #13115191C2semidirect
6. #13115177C2semidirect
7. #13114555C2semidirect
8. #13114530C2semidirect
9. C22×C4×Q8×C13C2semidirect
23. C2×#88196C4semidirect
81. C2×#6107C2×C4semidirect
85. C2×#6106C2×C4semidirect
117. C2×Q8×C13C22⋊C4semidirect
191. C22×C4×Q8D13semidirect
236. C2×C4×C13#137semidirect
410. C2×Q8#6103semidirect
473. C13#20926semidirect
526. C2×C4#88196semidirect

Maximal subgroups

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

subgroupindex
C22×#881962
C2×#117186272
C2×#117188862
#131173262
#131151912
#131151772
#131145552
#131145302
C22×C4×Q8×C132
#2092613