Finite Group Databasebeta

← #16799409

Subgroups — #16799409

Order 3328.

1114 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719129C211yesmaximal, characteristic
2C2×C4×#6122C211yesmaximal, characteristic
3#13131030C214yesmaximal, complemented
4#13131029C212yesmaximal, complemented
5#13131028C212yesmaximal, complemented
6#13130542C214yesmaximal
7C2×#11719327C211yesmaximal, characteristic
8C22×#6122C2212yes—
9C22×#6122C2212yes—
10C2×C13×C22⋊Q8C411yescharacteristic
11C2×Q8×Dic13C411yescharacteristic
12C2×#88198C2211yescharacteristic
13C2×#88178C411yescharacteristic
14C22×C4×Dic13C2211yescharacteristic
15#11719327—21no—
16#11719327—21no—
17#11719327—21no—
18#11719327—21no—
19#11719327—21no—
20#11719327—21no—
21#11719327—21no—
22#11719327—21no—
23C2×#88143C2212yes—
24C2×#88142C2211yescharacteristic
25C2×#88141C411yescharacteristic
26C2×#88141C412yes—
27C2×#88132C412yes—
28C2×#88131C2212yes—
29#11719129—21no—
30#11719129—21no—
31#11719129C2212yes—
32#11719129C2212yes—
33C2×#88112—21no—
34C4×#6122—21no—
35C4×#6122—21no—
36C4×#6122C2212yes—
37C4×#6122C2212yes—
38C2×#88110—21no—
39C2×C4⋊#1057—21no—
40C2×C4×#1057—21no—
41#11718291C2218yes—
42#11718291C2218yes—
43C22×Q8×C13C2×C411yescharacteristic
44C23×C4×C13C2×C411yescharacteristic
45C23×Dic13C2×C411yescharacteristic
46C23×Dic13C2311yescharacteristic
47C22×Dic26C2×C412yes—
48C22×Dic26C2×C411yescharacteristic
49C2×#6122—22no—
50C2×#6122—24no—
51C2×#6122—22no—
52C2×#6122—28no—
53C2×#6122Q818yescomplemented
54C2×#6122C2314yes—
55C2×#6122C2314yes—
56C22×#1057—22no—
57C22×#1057—22no—
58C13×C22⋊Q8—21no—
59C13×C22⋊Q8—21no—
60C13×C22⋊Q8—21no—
61C13×C22⋊Q8—21no—
62C2×C13×C4⋊C4C2×C411yescharacteristic
63C2×C13×C4⋊C4C2×C412yes—
64C2×C13×C22⋊C4C2×C412yes—
65C2×C22⋊Dic13C2×C412yes—
66Q8×Dic13—21no—
67Q8×Dic13—21no—
68Q8×Dic13—21no—
69Q8×Dic13—21no—
70#88198—21no—
71#88198—21no—
72#88198—21no—
73#88198—21no—
74C2×C4⋊Dic13C2×C412yes—
75C2×C4⋊Dic13C2×C411yescharacteristic
76#88178—21no—
77#88178—21no—
78#88178—21no—
79#88178—21no—
80C2×#6101C2×C412yes—
81C2×#6101C2×C412yes—
82C2×#6101C2×C412yes—
83C2×#6101C2312yes—
84C2×#6101C2312yes—
85C2×C4×Dic13C2×C412yes—
86C2×C4×Dic13C2×C412yes—
87C2×C4×Dic13C2×C411yescharacteristic
88C2×C4×Dic13C2×C411yescharacteristic
89C2×C4×Dic13D411yescharacteristic
90C2×C4×Dic13D411yescharacteristic
91C2×C4×Dic13D411yescharacteristic
92C2×C4×Dic13D411yescharacteristic
93C2×C4×Dic13D411yescharacteristic
94C2×C4×Dic13D411yescharacteristic
95C2×C4×Dic13D411yescharacteristic
96C2×C4×Dic13D411yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99#88143—22no—
100#88143—22no—
101#88143—22no—
102#88143—22no—
103#88142—21no—
104#88142—21no—
105#88142—21no—
106#88142—21no—
107#88141—21no—
108#88141—21no—
109#88141—21no—
110#88141—21no—
111#88141—22no—
112#88141—22no—
113#88141—22no—
114#88141—22no—
115#88132—22no—
116#88132—22no—
117#88132—22no—
118#88132—22no—
119#88131—22no—
120#88131—22no—
121#88131—22no—
122#88131—22no—
123#88112—22no—
124#88112—22no—
125#88110—22no—
126#88110—22no—
127C4⋊#1057—22no—
128C4⋊#1057—22no—
129C4×#1057—22no—
130C4×#1057—22no—
131#21337—131nomaximal, Sylow
132C24×C13C22×C411yescharacteristic
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—42no—
136C22×C4×C13—42no—
137C22×C4×C13C22⋊C411yescharacteristic
138C22×C4×C13C22⋊C411yescharacteristic
139C22×C4×C13C22⋊C411yescharacteristic
140C22×C4×C13C22⋊C411yescharacteristic
141C22×C4×C13C22×C411yescharacteristic
142C22×C4×C13C22×C412yes—
143C22×C4×C13C22×C412yes—
144C22×C4×C13C22×C411yescharacteristic
145C22×Dic13—42no—
146C22×Dic13C22⋊C411yescharacteristic
147C22×Dic13C22⋊C411yescharacteristic
148C22×Dic13C22⋊C411yescharacteristic
149C22×Dic13C22⋊C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C412yes—
152C22×Dic13C22×C412yes—
153C22×Dic13C22×C411yescharacteristic
154C22×Dic13C22×C411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C22×Dic13C2×D411yescharacteristic
157C22×Dic13C2×D411yescharacteristic
158C22×Dic13C2×D411yescharacteristic
159C22×Dic13C2×Q814yes—
160C22×Dic13#4114yes—
161C22×Dic13C2411yescharacteristic
162C2×Dic26—21no—
163C2×Dic26—21no—
164C2×Dic26—22no—
165C2×Dic26—22no—
166C2×Dic26—44no—
167C2×Dic26—42no—
168#6122—28no—
169#6122—24no—
170#6122—28no—
171#6122—44no—
172#6122C2×Q818yes—
173C2×#1057—28no—
174C2×#1057—24no—
175C2×#1057—24no—
176C2×#1057—44no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4—22no—
180C13×C4⋊C4—22no—
181C13×C22⋊C4—22no—
182C13×C22⋊C4—22no—
183C22⋊Dic13—22no—
184C22⋊Dic13—22no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—22no—
188C4⋊Dic13—22no—
189#6101—22no—
190#6101—22no—
191#6101—22no—
192#6101—22no—
193#6101—22no—
194#6101—22no—
195#6101—22no—
196#6101—22no—
197#6101—22no—
198#6101—22no—
199C4×Dic13—22no—
200C4×Dic13—22no—
201C4×Dic13—22no—
202C4×Dic13—22no—
203C4×Dic13C22⋊C411yescharacteristic
204C4×Dic13C22⋊C411yescharacteristic
205C4×Dic13C22⋊C411yescharacteristic
206C4×Dic13C22⋊C411yescharacteristic
207C4×Dic13C22⋊C411yescharacteristic
208C4×Dic13C22⋊C411yescharacteristic
209C4×Dic13C22⋊C411yescharacteristic
210C4×Dic13C22⋊C411yescharacteristic
211C4×Dic13C2×D411yescharacteristic
212C4×Dic13C2×D411yescharacteristic
213C4×Dic13C2×D411yescharacteristic
214C4×Dic13C2×D411yescharacteristic
215C4×Dic13C2×D411yescharacteristic
216C4×Dic13C2×D411yescharacteristic
217C4×Dic13C2×D411yescharacteristic
218C4×Dic13C2×D411yescharacteristic
219C2×#533—131no—
220C2×#423—131no—
221C2×C4×M16—131no—
222#1998—134no—
223#1997—132no—
224#1997—132no—
225#1742—134no—
226C23×C13—41no—
227C23×C13—41no—
228C23×C13C2×C22⋊C411yescharacteristic
229C23×C13C2×C22⋊C411yescharacteristic
230C23×C13C4×Q814yes—
231C23×C13C23×C411yescharacteristic
232Q8×C13—42no—
233Q8×C13—42no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13C2×C22⋊C411yescharacteristic
245C2×C4×C13C2×C22⋊C411yescharacteristic
246C2×C4×C13C2×C22⋊C411yescharacteristic
247C2×C4×C13C2×C22⋊C411yescharacteristic
248C2×Dic13—22no—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—42no—
257C2×Dic13—42no—
258C2×Dic13C2×C22⋊C411yescharacteristic
259C2×Dic13C2×C22⋊C411yescharacteristic
260C2×Dic13C2×C22⋊C411yescharacteristic
261C2×Dic13C2×C22⋊C411yescharacteristic
262C2×Dic13C2×C22⋊C411yescharacteristic
263C2×Dic13C2×C22⋊C411yescharacteristic
264C2×Dic13C4×Q814yes—
265C2×Dic13C22⋊Q814yes—
266C2×Dic13C22⋊Q814yes—
267C2×Dic13#12412yes—
268C2×Dic13#12412yes—
269C2×Dic13C4⋊Q812yes—
270C2×Dic13C4⋊Q812yes—
271C2×Dic13C22×D411yescharacteristic
272C2×Dic13C22×D411yescharacteristic
273C2×Dic13C22×Q812yes—
274C2×Dic13C2×#4112yes—
275Dic26—44no—
276Dic26—42no—
277Dic26—42no—
278Dic26—44no—
279#1057—28no—
280#1057—44no—
281C22×M16—132no—
282C22×M16—132no—
283#533—261no—
284#533—261no—
285#533—261no—
286#533—261no—
287#533—261no—
288#533—261no—
289#533—261no—
290#533—261no—
291C2×C4⋊Q8—131no—
292C2×C4⋊Q8—131no—
293C2×#125—132no—
294C2×C22⋊Q8—131no—
295C2×C22⋊Q8—132no—
296C2×C22⋊Q8#26111yescharacteristic
297C2×C4×Q8—131no—
298C2×C4×Q8—131no—
299C2×C4×Q8—132no—
300C2×#117—132no—
301C22×C42—131no—
302#423—132no—
303#423—132no—
304#423—261no—
305#423—261no—
306C2×C4⋊C8—261no—
307C2×C4⋊C8—261no—
308C4×M16—132no—
309C4×M16—132no—
310C4×M16—261no—
311C4×M16—261no—
312C2×#97—261no—
313C2×C4×C8—261no—
314#343—138no—
315#343—138no—
316C22×C13—21no—
317C22×C13—21no—
318C22×C13—21no—
319C22×C13—21no—
320C22×C13—21no—
321C22×C13—21no—
322C22×C13—44no—
323C22×C13#39112yes—
324C22×C13#39112yes—
325C22×C13C22×C22⋊C411yescharacteristic
326C22×C13C2×C4×Q812yes—
327C4×C13—21no—
328C4×C13—21no—
329C4×C13—21no—
330C4×C13—42no—
331C4×C13—42no—
332C4×C13#41312yes—
333Dic13—21no—
334Dic13—21no—
335Dic13—21no—
336Dic13—42no—
337Dic13—42no—
338Dic13#39112yes—
339Dic13#39112yes—
340Dic13#41312yes—
341Dic13C2×C22⋊Q812yes—
342Dic13C2×#12411yescharacteristic
343Dic13C2×C4⋊Q811yescharacteristic
344C22×Q8—132no—
345C22×Q8—131no—
346C22×Q8C2×#26111yescharacteristic
347C23×C4—131no—
348C23×C4—131no—
349C23×C4C2×#26111yescharacteristic
350C2×M16—134no—
351C2×M16—138no—
352C2×M16—134no—
353C2×M16—262no—
354C2×M16—264no—
355C2×M16—262no—
356C2×M16—268no—
357C22×C8—262no—
358C22×C8—262no—
359C4⋊Q8—261no—
360C4⋊Q8—261no—
361C4⋊Q8—261no—
362C4⋊Q8—261no—
363C4⋊Q8—261no—
364C4⋊Q8—261no—
365C4⋊Q8—261no—
366C4⋊Q8—261no—
367#125—262no—
368#125—262no—
369#125—262no—
370#125—262no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—262no—
376C22⋊Q8—262no—
377C22⋊Q8—262no—
378C22⋊Q8—262no—
379C22⋊Q8—261no—
380C22⋊Q8—261no—
381C22⋊Q8—261no—
382C22⋊Q8—261no—
383C4×Q8—262no—
384C4×Q8—262no—
385C4×Q8—262no—
386C4×Q8—262no—
387C4×Q8—261no—
388C4×Q8—261no—
389C4×Q8—261no—
390C4×Q8—261no—
391C4×Q8—261no—
392C4×Q8—261no—
393C4×Q8—261no—
394C4×Q8—261no—
395#117—262no—
396#117—262no—
397#117—262no—
398#117—262no—
399C2×C4⋊C4—132no—
400C2×C4⋊C4—132no—
401C2×C4⋊C4—132no—
402C2×C4⋊C4—132no—
403C2×C4⋊C4—132no—
404C2×C4⋊C4—132no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4C2×#26112yes—
407C2×C4⋊C4C2×#26111yescharacteristic
408C2×C22⋊C4—132no—
409C2×C22⋊C4C2×#26112yes—
410C2×C42—131no—
411C2×C42—131no—
412C2×C42—131no—
413C2×C42—131no—
414C2×C42—131no—
415C2×C42—131no—
416C2×C42—131no—
417C2×C42—131no—
418C2×C42—131no—
419C2×C42—131no—
420C2×C42—131no—
421C2×C42—131no—
422C2×C42—132no—
423C2×C42—132no—
424C4⋊C8—262no—
425C4⋊C8—262no—
426C4⋊C8—262no—
427C4⋊C8—262no—
428#97—262no—
429#97—262no—
430C4×C8—262no—
431C4×C8—262no—
432C2×C13—22no—
433C2×C13—22no—
434C2×C13—22no—
435C2×C13C2×#39111yescharacteristic
436C2×C13C2×#41311yescharacteristic
437C2×C13C2×#41311yescharacteristic
438C24C22×#26111yescharacteristic
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—42no—
442C2×Q8—262no—
443C2×Q8—262no—
444C2×Q8—261no—
445C2×Q8—261no—
446C2×Q8—522no—
447C2×Q8—524no—
448C22×C4—42no—
449C22×C4—134no—
450C22×C4—134no—
451C22×C4—131no—
452C22×C4—131no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—131no—
459C22×C4—131no—
460C22×C4—131no—
461C22×C4—131no—
462C22×C4—131no—
463C22×C4—132no—
464C22×C4—132no—
465C22×C4—522no—
466C22×C4C22⋊#26111yescharacteristic
467C22×C4C22⋊#26111yescharacteristic
468C22×C4C22⋊#26111yescharacteristic
469C22×C4C22⋊#26111yescharacteristic
470C22×C4C22×#26111yescharacteristic
471C22×C4C22×#26112yes—
472C22×C4C22×#26112yes—
473C22×C4C22×#26111yescharacteristic
474M16—138no—
475M16—264no—
476M16—268no—
477M16—268no—
478M16—524no—
479C2×C8—264no—
480C2×C8—268no—
481C2×C8—264no—
482C2×C8—524no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—262no—
488C4⋊C4—262no—
489C4⋊C4—262no—
490C4⋊C4—262no—
491C4⋊C4—262no—
492C4⋊C4—262no—
493C4⋊C4—262no—
494C4⋊C4—262no—
495C4⋊C4—262no—
496C4⋊C4—262no—
497C4⋊C4—262no—
498C4⋊C4—262no—
499C4⋊C4—261no—
500C4⋊C4—261no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—262no—
504C22⋊C4—262no—
505C42—131no—
506C42—131no—
507C42—131no—
508C42—131no—
509C42—131no—
510C42—131no—
511C42—131no—
512C42—131no—
513C42—131no—
514C42—131no—
515C42—131no—
516C42—131no—
517C42—131no—
518C42—131no—
519C42—131no—
520C42—131no—
521C42—262no—
522C42—262no—
523C42—262no—
524C42—262no—
525C13#2133711yescharacteristic, complemented, Sylow
526C23—41no—
527C23—41no—
528C23Q8×#26114yes—
529C23C2×C22⋊#26111yescharacteristic
530C23C2×C22⋊#26111yescharacteristic
531C23C23×#26111yescharacteristic
532Q8—42no—
533Q8—42no—
534Q8—522no—
535Q8—524no—
536Q8—524no—
537Q8—522no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—132no—
549C2×C4—132no—
550C2×C4—134no—
551C2×C4—131no—
552C2×C4—131no—
553C2×C4—131no—
554C2×C4—131no—
555C2×C4—131no—
556C2×C4—134no—
557C2×C4—132no—
558C2×C4—134no—
559C2×C4—132no—
560C2×C4—132no—
561C2×C4—132no—
562C2×C4—131no—
563C2×C4—131no—
564C2×C4—131no—
565C2×C4—262no—
566C2×C4—262no—
567C2×C4—262no—
568C2×C4—262no—
569C2×C4—262no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4C2×C22⋊#26111yescharacteristic
576C2×C4C2×C22⋊#26111yescharacteristic
577C2×C4C2×C22⋊#26111yescharacteristic
578C2×C4C2×C22⋊#26111yescharacteristic
579C8—268no—
580C8—524no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—44no—
588C22(C2×#261)⋊Q812yes—
589C22(C2×#261)⋊Q812yes—
590C22C2×Q8×#26112yes—
591C22C22×C22⋊#26111yescharacteristic
592C4—21no—
593C4—21no—
594C4—21no—
595C4—42no—
596C4—42no—
597C4—132no—
598C4—131no—
599C4—131no—
600C4—132no—
601C4—132no—
602C4—132no—
603C4—261no—
604C4—261no—
605C4—261no—
606C4—522no—
607C4—522no—
608C4#1171914812yes—
609C2—22no—
610C2—22no—
611C2—22no—
612C2C2×(C2×#261)⋊Q811yescharacteristic
613C2D13⋊#41311yescharacteristic
614C2C2×#1171914811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#261
commutatorC23×C13C23×C4
FrattiniC23C23×#261
FittingC2×C13×C22⋊Q8C4
socleC22×C13C22×C22⋊C4
prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenterC2×C22⋊Q8#261
hypocenterC13#21337
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13131030C2semidirect
4. #13131029C2semidirect
5. #13131028C2semidirect
53. C2×#6122Q8semidirect
525. C13#21337semidirect

Maximal subgroups

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

subgroupindex
C2×#117191292
C2×C4×#61222
#131310302
#131310292
#131310282
#131305422
C2×#117193272
#2133713