Finite Group Databasebeta

← #16791781

Subgroups — #16791781

Order 3328.

1794 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C2×#261)⋊Q8C212yesmaximal
2C2×#11719083C212yesmaximal
3#13130021C214yesmaximal
4#13130014C211yesmaximal, characteristic
5#13130014C212yesmaximal
6#13130012C212yesmaximal
7C4×(C2×#261)⋊C4C211yesmaximal, characteristic
8C2×C4×Q8×D13C211yesmaximal, characteristic
9C22×Q8×D13C2211yescharacteristic
10C22×C4⋊#261C2212yes—
11C22×C4⋊#261C2214yes—
12C22×C4×#261C2212yes—
13C2×C4×Q8×C13C411yescharacteristic, complemented
14C2×Q8×Dic13C411yescharacteristic, complemented
15C4×Q8×D13—28no—
16C2×C4⋊C4×D13C2212yes—
17C2×C4⋊C4×D13C2211yescharacteristic
18C2×#88141C411yescharacteristic, complemented
19C2×#88141C412yescomplemented
20C2×C42×D13C2211yescharacteristic
21C2×C42×D13C2212yes—
22C2×C4×Dic26C411yescharacteristic, complemented
23C2×C4×Dic26C412yescomplemented
24(C2×#261)⋊Q8—28no—
25C2×(C2×#261)⋊C4C2211yescharacteristic
26C2×(C2×#261)⋊C4C2212yes—
27C2×(C2×#261)⋊C4C2214yes—
28C2×(C2×#261)⋊C4C2212yes—
29C2×(C2×#261)⋊C4C2214yes—
30C2×(C2×#261)⋊C4C2214yes—
31C2×(C2×#261)⋊C4C2212yes—
32C2×(C2×#261)⋊C4C2211yescharacteristic
33#11719083—28no—
34C23×#261C2314yes—
35C23×#261C2312yes—
36C23×#261C2312yes—
37C22×Q8×C13C2×C411yescharacteristic
38C2×Q8×D13—24no—
39C2×Q8×D13—24no—
40C2×Q8×D13D414yes—
41C2×Q8×D13Q814yes—
42C22×C4×D13C2311yescharacteristic
43C22×C4×D13C2312yes—
44C22×C4×D13C2312yes—
45C22×C4×D13C2311yescharacteristic
46C22×C4×D13C2311yescharacteristic
47C22×Dic26C2×C411yescharacteristic
48C22×Dic26C2×C412yes—
49C2×C4⋊#261—28no—
50C2×C4⋊#261—28no—
51C2×C4⋊#261—24no—
52C2×C4⋊#261—24no—
53C2×C4⋊#261—24no—
54C2×C4⋊#261—44no—
55C2×C4×#261—24no—
56C2×C4×#261—24no—
57C2×C4×#261—24no—
58C4×Q8×C13—24no—
59C2×C13×C4⋊C4C2×C412yes—
60C2×C13×C4⋊C4C2×C411yescharacteristic
61C2×C42×C13C2×C411yescharacteristic
62C2×C42×C13C2×C412yes—
63Q8×Dic13—24no—
64C2×C4⋊Dic13C2×C412yes—
65C2×C4⋊Dic13C2×C411yescharacteristic
66C2×#6101C2×C412yes—
67C2×#6101C2×C412yes—
68C2×#6101C2×C412yes—
69C2×C4×Dic13C2×C411yescharacteristic
70C2×C4×Dic13C2×C412yes—
71C2×C4×Dic13C2×C412yes—
72C2×C4×Dic13C2×C411yescharacteristic
73C2×C4×Dic13C2×C411yescharacteristic
74C2×C4×Dic13C2×C412yes—
75C4⋊C4×D13—28no—
76C4⋊C4×D13D418yes—
77#88141—24no—
78#88141—28no—
79C42×D13—28no—
80C42×D13—24no—
81C4×Dic26—24no—
82C4×Dic26—28no—
83(C2×#261)⋊C4—24no—
84(C2×#261)⋊C4—28no—
85(C2×#261)⋊C4—28no—
86(C2×#261)⋊C4—28no—
87(C2×#261)⋊C4—22no—
88(C2×#261)⋊C4—24no—
89(C2×#261)⋊C4—42no—
90(C2×#261)⋊C4Q814yescomplemented
91#13935—131nomaximal, Sylow
92C23×D13C2411yescharacteristic
93C22×#261—22no—
94C22×#261—22no—
95C22×#261—22no—
96C22×#261—22no—
97C22×#261—22no—
98C22×#261—22no—
99C22×#261—22no—
100C22×#261—22no—
101C22×#261—22no—
102C22×#261—22no—
103C22×#261—22no—
104C22×#261—22no—
105C22×#261—24no—
106C22×#261—24no—
107C22×#261—24no—
108C22×#261—24no—
109C22×#261—24no—
110C22×#261—24no—
111C22×#261—24no—
112C22×#261C2×Q814yes—
113C22×#261#4114yes—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13C4⋊C414yescomplemented
117C22×C4×C13C22×C411yescharacteristic
118C22×C4×C13C22×C412yes—
119C22×C4×C13C22×C412yes—
120C22×C4×C13C22×C411yescharacteristic
121C22×C4×C13C22×C411yescharacteristic
122C22×Dic13C22×C411yescharacteristic
123C22×Dic13C22×C412yes—
124C22×Dic13C22×C412yes—
125C22×Dic13C22×C411yescharacteristic
126C22×Dic13C22×C411yescharacteristic
127Q8×D13—216no—
128Q8×D13—216no—
129C2×C4×D13—22no—
130C2×C4×D13—24no—
131C2×C4×D13—24no—
132C2×C4×D13—24no—
133C2×C4×D13—22no—
134C2×C4×D13—22no—
135C2×C4×D13—22no—
136C2×C4×D13—22no—
137C2×C4×D13—22no—
138C2×C4×D13—21no—
139C2×C4×D13—21no—
140C2×C4×D13C2×D418yes—
141C2×C4×D13C2×D414yes—
142C2×C4×D13C2×D412yes—
143C2×C4×D13C2×D414yes—
144C2×C4×D13C2×Q814yes—
145C2×C4×D13C2×Q812yes—
146C2×C4×D13C2×Q812yes—
147C2×C4×D13#4112yes—
148C2×C4×D13#4114yes—
149C2×Dic26—22no—
150C2×Dic26—22no—
151C2×Dic26—24no—
152C2×Dic26—24no—
153C2×Dic26C4⋊C414yescomplemented
154C2×Dic26C4⋊C418yescomplemented
155C4⋊#261—416no—
156C4⋊#261—48no—
157C4×#261—48no—
158C13×C4⋊C4—24no—
159C13×C4⋊C4C22⋊C414yes—
160C42×C13—24no—
161C42×C13—22no—
162C4⋊Dic13—24no—
163C4⋊Dic13C22⋊C414yes—
164#6101—24no—
165#6101—24no—
166#6101C22⋊C418yes—
167C4×Dic13—24no—
168C4×Dic13—22no—
169C4×Dic13—24no—
170C4×Dic13—22no—
171C4×Dic13—22no—
172C4×Dic13—24no—
173C22×C4×Q8—131no—
174C2×#391—132no—
175C2×#389—132no—
176#1441—134no—
177C4⋊#95—131no—
178C4⋊#95—132no—
179#1434—132no—
180C4×#95—131no—
181C23×C13C23×C411yescharacteristic
182C22×D13C22×D411yescharacteristic
183C22×D13C22×D412yes—
184C22×D13C22×Q811yescharacteristic
185C22×D13C22×Q811yescharacteristic
186C22×D13C2×#4112yes—
187C22×D13C2×#4111yescharacteristic
188C22×D13C2×#4112yes—
189C22×D13H(5,𝔽2)11yescharacteristic
190C22×D132−1+411yescharacteristic
191C22×D132−1+412yes—
192C2×#261—24no—
193C2×#261—24no—
194C2×#261—24no—
195C2×#261—24no—
196C2×#261—24no—
197C2×#261—24no—
198C2×#261—42no—
199C2×#261—42no—
200C2×#261—42no—
201C2×#261—42no—
202C2×#261—42no—
203C2×#261—42no—
204C2×#261—42no—
205C2×#261—42no—
206C2×#261—44no—
207C2×#261—44no—
208C2×#261—44no—
209C2×#261—44no—
210C2×#261—44no—
211C2×#261—44no—
212C2×#261—44no—
213Q8×C13—28no—
214C2×C4×C13—21no—
215C2×C4×C13—22no—
216C2×C4×C13—22no—
217C2×C4×C13—22no—
218C2×C4×C13—21no—
219C2×C4×C13—22no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13C2×C22⋊C414yes—
224C2×C4×C13C2×C22⋊C412yes—
225C2×C4×C13C2×C4⋊C414yes—
226C2×C4×C13C2×C4⋊C412yes—
227C2×C4×C13#11712yes—
228C2×C4×C13C4×Q812yes—
229C2×Dic13—21no—
230C2×Dic13—22no—
231C2×Dic13—22no—
232C2×Dic13—22no—
233C2×Dic13—21no—
234C2×Dic13—22no—
235C2×Dic13—21no—
236C2×Dic13—21no—
237C2×Dic13—21no—
238C2×Dic13C2×C22⋊C414yes—
239C2×Dic13C2×C22⋊C412yes—
240C2×Dic13C2×C4⋊C414yes—
241C2×Dic13C2×C4⋊C412yes—
242C2×Dic13#11712yes—
243C2×Dic13C4×Q812yes—
244C4×D13—28no—
245C4×D13—28no—
246C4×D13—28no—
247C4×D13—28no—
248C4×D13—24no—
249C4×D13—24no—
250C4×D13—42no—
251C4×D13—42no—
252C4×D13#12118yes—
253C4×D13C22⋊Q818yes—
254Dic26—28no—
255Dic26—216no—
256C23×Q8—131no—
257C2×C4×Q8—131no—
258C2×C4×Q8—132no—
259C2×C4×Q8—131no—
260C2×C4×Q8—132no—
261C2×C4×Q8—131no—
262C2×C4×Q8—268no—
263C2×C4×Q8#26111yescharacteristic, complemented
264C22×C4⋊C4—131no—
265C22×C4⋊C4—132no—
266C22×C4⋊C4—132no—
267C22×C4⋊C4—134no—
268C22×C42—131no—
269C22×C42—132no—
270C22×C42—132no—
271#391—268no—
272#389—268no—
273C2×#95—131no—
274C2×#95—131no—
275C2×#95—132no—
276C2×#95—132no—
277C2×#95—134no—
278C2×#95—134no—
279C2×#95—134no—
280C2×#95—132no—
281C22×C13C22×C22⋊C411yescharacteristic
282C22×C13C22×C4⋊C411yescharacteristic
283C22×C13C2×#11711yescharacteristic
284C22×C13C2×C4×Q811yescharacteristic
285C22×C13C4×#4111yescharacteristic
286C22×C13#51911yescharacteristic
287C22×C13#52011yescharacteristic
288C2×D13C2×#12112yes—
289C2×D13C2×C22⋊Q812yes—
290C2×D13#52912yes—
291C2×D13#53312yes—
292C2×D13#54112yes—
293C2×D13#54212yes—
294C2×D13#54612yes—
295C2×D13#54712yes—
296C2×D13#54812yes—
297C2×D13#54812yes—
298C2×D13D4×Q811yescharacteristic
299C2×D13D4×Q812yes—
300C2×D13#55011yescharacteristic
301C2×D13#55712yes—
302C2×D13#55711yescharacteristic
303C2×D13Q8211yescharacteristic
304#261—44no—
305#261—44no—
306#261—84no—
307C4×C13—24no—
308C4×C13—22no—
309C4×C13—24no—
310C4×C13—41no—
311C4×C13#38014yes—
312Dic13—24no—
313Dic13—22no—
314Dic13—24no—
315Dic13—41no—
316Dic13#38014yes—
317C22×Q8—134no—
318C22×Q8—134no—
319C22×Q8—132no—
320C22×Q8—131no—
321C22×Q8—264no—
322C22×Q8—264no—
323C22×Q8C2×#26111yescharacteristic
324C23×C4—131no—
325C23×C4—132no—
326C23×C4—132no—
327C23×C4—131no—
328C23×C4—132no—
329C23×C4—132no—
330C23×C4—131no—
331C23×C4—134no—
332C4×Q8—24no—
333C4×Q8—268no—
334C4×Q8—264no—
335C4×Q8—268no—
336C4×Q8—264no—
337C4×Q8—264no—
338C2×C4⋊C4—132no—
339C2×C4⋊C4—131no—
340C2×C4⋊C4—138no—
341C2×C4⋊C4—132no—
342C2×C4⋊C4—132no—
343C2×C4⋊C4—132no—
344C2×C4⋊C4—268no—
345C2×C4⋊C4—264no—
346C2×C4⋊C4—268no—
347C2×C4⋊C4—264no—
348C2×C4⋊C4—264no—
349C2×C4⋊C4—268no—
350C2×C4⋊C4—524no—
351C2×C4⋊C4C2×#26112yes—
352C2×C4⋊C4C2×#26111yescharacteristic
353C2×C42—131no—
354C2×C42—131no—
355C2×C42—132no—
356C2×C42—132no—
357C2×C42—132no—
358C2×C42—131no—
359C2×C42—264no—
360C2×C42—268no—
361C2×C42—264no—
362C2×C42—264no—
363C2×C42—264no—
364C2×C42C2×#26111yescharacteristic
365C2×C42C2×#26112yes—
366#95—134no—
367#95—262no—
368#95—264no—
369#95—264no—
370#95—268no—
371#95—268no—
372#95—268no—
373#95—522no—
374C2×C13C2×#38011yescharacteristic
375C2×C13#229511yescharacteristic
376C2×C13#232011yescharacteristic
377C2×C13#233911yescharacteristic
378C2×C13Q8×C22⋊C411yescharacteristic
379C2×C13Q8×C4⋊C411yescharacteristic
380C2×C13#235111yescharacteristic
381D13#242612yes—
382D13#242912yes—
383D13#245112yes—
384D13#245312yes—
385C24—131no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—138no—
389C2×Q8—134no—
390C2×Q8—264no—
391C2×Q8—264no—
392C2×Q8—2616no—
393C2×Q8—2616no—
394C2×Q8—262no—
395C2×Q8—262no—
396C2×Q8C4⋊#26114yescomplemented
397C22×C4—131no—
398C22×C4—132no—
399C22×C4—132no—
400C22×C4—134no—
401C22×C4—134no—
402C22×C4—131no—
403C22×C4—132no—
404C22×C4—132no—
405C22×C4—134no—
406C22×C4—134no—
407C22×C4—132no—
408C22×C4—132no—
409C22×C4—138no—
410C22×C4—134no—
411C22×C4—134no—
412C22×C4—131no—
413C22×C4—262no—
414C22×C4—264no—
415C22×C4—264no—
416C22×C4—262no—
417C22×C4—264no—
418C22×C4—262no—
419C22×C4—262no—
420C22×C4—262no—
421C22×C4—261no—
422C22×C4—262no—
423C22×C4—261no—
424C22×C4—262no—
425C22×C4—262no—
426C22×C4—262no—
427C22×C4—262no—
428C22×C4—262no—
429C22×C4—262no—
430C22×C4—262no—
431C22×C4—262no—
432C22×C4—262no—
433C22×C4—262no—
434C22×C4—262no—
435C22×C4—262no—
436C22×C4—264no—
437C22×C4—264no—
438C22×C4—264no—
439C22×C4—264no—
440C22×C4—264no—
441C22×C4—264no—
442C22×C4—264no—
443C22×C4C22×#26111yescharacteristic
444C22×C4C22×#26112yes—
445C22×C4C22×#26111yescharacteristic
446C22×C4C22×#26112yes—
447C22×C4C22×#26111yescharacteristic
448C4⋊C4—24no—
449C4⋊C4—138no—
450C4⋊C4—134no—
451C4⋊C4—264no—
452C4⋊C4—264no—
453C4⋊C4—264no—
454C4⋊C4—528no—
455C4⋊C4—5216no—
456C4⋊C4C22⋊#26114yes—
457C42—24no—
458C42—22no—
459C42—262no—
460C42—264no—
461C42—264no—
462C42—262no—
463C42—264no—
464C42—262no—
465C42—528no—
466C13#1393511yescharacteristic, complemented, Sylow
467C23—131no—
468C23—131no—
469C23—132no—
470C23—131no—
471C23—131no—
472C23—132no—
473C23—132no—
474C23—131no—
475C23—131no—
476C23—132no—
477C23C23×#26111yescharacteristic
478Q8—28no—
479Q8—2616no—
480Q8—268no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—132no—
491C2×C4—132no—
492C2×C4—132no—
493C2×C4—138no—
494C2×C4—138no—
495C2×C4—134no—
496C2×C4—132no—
497C2×C4—134no—
498C2×C4—268no—
499C2×C4—268no—
500C2×C4—264no—
501C2×C4—264no—
502C2×C4—268no—
503C2×C4—268no—
504C2×C4—262no—
505C2×C4—261no—
506C2×C4—262no—
507C2×C4—262no—
508C2×C4—261no—
509C2×C4—262no—
510C2×C4—264no—
511C2×C4—264no—
512C2×C4—264no—
513C2×C4—264no—
514C2×C4—264no—
515C2×C4—264no—
516C2×C4—261no—
517C2×C4—261no—
518C2×C4—261no—
519C2×C4—522no—
520C2×C4—522no—
521C2×C4—522no—
522C2×C4—522no—
523C2×C4—522no—
524C2×C4—522no—
525C2×C4—522no—
526C2×C4—522no—
527C2×C4—522no—
528C2×C4—522no—
529C2×C4—524no—
530C2×C4—524no—
531C2×C4—524no—
532C2×C4—524no—
533C2×C4—524no—
534C2×C4—524no—
535C2×C4—524no—
536C2×C4C2×C4⋊#26114yes—
537C2×C4C2×C4⋊#26112yes—
538C2×C4#8823712yes—
539C2×C4Q8×#26112yes—
540C2×C4C2×C22⋊#26114yes—
541C2×C4C2×C22⋊#26112yes—
542C22—132no—
543C22—132no—
544C22—132no—
545C22—132no—
546C22—132no—
547C22—132no—
548C22—132no—
549C22—132no—
550C22—132no—
551C22—132no—
552C22—131no—
553C22—131no—
554C22—131no—
555C22—131no—
556C22—132no—
557C22—132no—
558C22C22×C4⋊#26111yescharacteristic
559C22C2×#8823711yescharacteristic
560C22C2×Q8×#26111yescharacteristic
561C22#1171962711yescharacteristic
562C22#41×#26111yescharacteristic
563C22#1171963111yescharacteristic
564C22C22×C22⋊#26111yescharacteristic
565C4—24no—
566C4—24no—
567C4—22no—
568C4—41no—
569C4—134no—
570C4—264no—
571C4—264no—
572C4—262no—
573C4—524no—
574C4—524no—
575C4—521no—
576C4—1044no—
577C4#1171913614yes—
578C2—132no—
579C2—132no—
580C2—132no—
581C2—132no—
582C2C2×#1171913611yescharacteristic
583C2#1313129511yescharacteristic
584C2#1313132011yescharacteristic
585C2#1313133911yescharacteristic
586C2Q8×C22⋊#26111yescharacteristic
587C2Q8×C4⋊#26111yescharacteristic
588C2#1313135111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC2×C4×Q8×C13C4
prime radical, cosocleC23×D13C24
hypercenterC2×C4×Q8#261
hypocenterC13#13935
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
13. C2×C4×Q8×C13C4semidirect
14. C2×Q8×Dic13C4semidirect
18. C2×#88141C4semidirect
19. C2×#88141C4semidirect
22. C2×C4×Dic26C4semidirect
23. C2×C4×Dic26C4semidirect
90. (C2×#261)⋊C4Q8semidirect
116. C2×Q8×C13C4⋊C4semidirect
153. C2×Dic26C4⋊C4semidirect
154. C2×Dic26C4⋊C4semidirect
263. C2×C4×Q8#261semidirect
396. C2×Q8C4⋊#261semidirect
466. C13#13935semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×#261)⋊Q82
C2×#117190832
#131300212
#131300142
#131300122
C4×(C2×#261)⋊C42
C2×C4×Q8×D132
#1393513