Finite Group Databasebeta

← #15873313

Subgroups — #15873313

Order 3328.

830 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718498C211yesmaximal, characteristic
2#13117240C211yesmaximal, characteristic
3#13117229C211yesmaximal, characteristic
4#13115606C211yesmaximal, characteristic, complemented
5#13115604C211yesmaximal, characteristic, complemented
6#13115602C211yesmaximal, characteristic, complemented
7#13115598C211yesmaximal, characteristic, complemented
8#13115596C211yesmaximal, characteristic, complemented
9#13115593C211yesmaximal, characteristic, complemented
10#13115591C211yesmaximal, characteristic, complemented
11#13114870C211yesmaximal, characteristic
12#13114868C211yesmaximal, characteristic
13#13114868C211yesmaximal, characteristic
14#13114866C211yesmaximal, characteristic
15C13×#1949C211yesmaximal, characteristic, complemented
16C22×#6094C2211yescharacteristic
17C22×#6094C2211yescharacteristic
18C2×#88126—21no—
19C2×C42×D13C2211yescharacteristic
20C2×C13×#107C2211yescharacteristic
21C2×C13×#106C2211yescharacteristic
22C2×C13×C4⋊C8C2211yescharacteristic
23C13×#382C2211yescharacteristic
24C4×C13×C4⋊C4C2211yescharacteristic
25C2×#88058C2211yescharacteristic
26C2×#88057C2211yescharacteristic
27C2×#88054C2211yescharacteristic
28C2×#88054C2211yescharacteristic
29#11718648C2211yescharacteristic
30#11718648C2211yescharacteristic
31#11718648C2211yescharacteristic
32#11718648C2211yescharacteristic
33#11718637C2211yescharacteristic
34#11718637C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88048C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41#11718597C2211yescharacteristic
42C4×C4⋊Dic13C2211yescharacteristic
43C4×#6101—21no—
44C2×C4⋊#1055C2211yescharacteristic
45#11718498C418yescomplemented
46#11718328—21no—
47#11718327—21no—
48#11718325—21no—
49#11718312—21no—
50#11718311—21no—
51#11718309—21no—
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×C4×D13D411yescharacteristic
61C22×C4×D13D411yescharacteristic
62C22×C4×D13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C2×C13×C4⋊C4C2311yescharacteristic
66C2×C13×C4⋊C4C2311yescharacteristic
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C42×C13—21no—
70C2×C42×C13C2311yescharacteristic
71C2×#6103—21no—
72C2×#6103—21no—
73C2×#6103—21no—
74C2×#6103—21no—
75C2×C4⋊Dic13C2311yescharacteristic
76C2×C4⋊Dic13C2311yescharacteristic
77C2×C4⋊Dic13C2311yescharacteristic
78C2×C4⋊Dic13C2311yescharacteristic
79C2×#6101—21no—
80C2×#6101—21no—
81C2×#6101—21no—
82C2×#6101—21no—
83C2×C4×Dic13—21no—
84C2×C4×Dic13—21no—
85C2×C4×Dic13—21no—
86C2×C4×Dic13D411yescharacteristic
87C2×C4×Dic13D411yescharacteristic
88C2×C4×Dic13C2311yescharacteristic
89C22×#1055C2311yescharacteristic
90C22×#1055C2311yescharacteristic
91C2×#6094—41no—
92C2×#6094—41no—
93C2×#6094—41no—
94C2×#6094—41no—
95C2×#6094C2×C414yes—
96C2×#6094C2×C414yes—
97#88126—44no—
98C42×D13C2×C414yes—
99C42×D13Q814yes—
100C13×#107—41no—
101C13×#106—41no—
102C13×C4⋊C8C2×C414yes—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88058—41no—
116#88057—41no—
117#88054C2×C414yes—
118#88054C2×C414yes—
119#88048—41no—
120#88048—41no—
121#88047—41no—
122#88047—41no—
123C4⋊#1055C2×C414yes—
124#21963—131nomaximal, Sylow
125C23×D13C2×D411yescharacteristic
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13C2×D411yescharacteristic
131C22×C4×C13C2×D411yescharacteristic
132C22×C4×C13C2411yescharacteristic
133C22×Dic13—21no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13—21no—
137C22×Dic13C2×D411yescharacteristic
138C22×Dic13C2×D411yescharacteristic
139C22×Dic13C2×D411yescharacteristic
140C2×C4×D13—21no—
141C2×C4×D13—21no—
142C2×C4×D13—21no—
143C2×C4×D13—21no—
144C2×C4×D13—22no—
145C2×C4×D13—22no—
146C2×C4×D13—41no—
147C2×C4×D13—41no—
148C2×C4×D13C4⋊C414yes—
149C2×C4×D13C4⋊C414yes—
150C2×C4×D13C22×C412yes—
151C2×C4×D13C2×Q812yes—
152C2×C8×C13—21no—
153C2×C8×C13—21no—
154C2×C8×C13—21no—
155C2×C8×C13—21no—
156C2×C8×C13C22×C412yes—
157C2×C8×C13C22×C412yes—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C42×C13—42no—
167C42×C13C22×C412yes—
168C42×C13C2×Q812yes—
169#6103—42no—
170#6103—42no—
171#6103—42no—
172#6103—42no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181#6101—42no—
182#6101—42no—
183#6101—42no—
184#6101—42no—
185C4×Dic13—42no—
186C4×Dic13—42no—
187C4×Dic13—42no—
188C4×Dic13C4⋊C414yes—
189C4×Dic13C4⋊C414yes—
190C4×Dic13C22×C412yes—
191C4×Dic13C2×Q812yes—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055C22×C412yes—
197C2×#1055C22×C412yes—
198#6094—44no—
199#6094—44no—
200C2×#423—131no—
201#2285—131no—
202#2283—131no—
203#1949—131no—
204#1949—131no—
205#1949—131no—
206#1949D1311yescharacteristic, complemented
207#1948—131no—
208#1948—131no—
209#1947—131no—
210#1947—131no—
211#1809—131no—
212#1809—131no—
213#1809—131no—
214#1809—131no—
215C23×C13C22×D411yescharacteristic
216C22×D13—21no—
217C22×D13—21no—
218C22×D13—21no—
219C22×D13—21no—
220C22×D13—21no—
221C22×D13—21no—
222C22×D13C2×C4⋊C412yes—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13C2×C4⋊C412yes—
240C2×C4×C13C2×C4⋊C412yes—
241C2×C4×C13C23×C411yescharacteristic
242C2×C4×C13C22×Q811yescharacteristic
243C2×C4×C13H(5,𝔽2)11yescharacteristic
244C2×C4×C13H(5,𝔽2)11yescharacteristic
245C2×C4×C132−1+411yescharacteristic
246C2×C4×C132−1+411yescharacteristic
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—41no—
252C2×Dic13—41no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13C2×C4⋊C412yes—
264C2×Dic13C2×C4⋊C412yes—
265C2×Dic13C2×C4⋊C412yes—
266C2×Dic13AGL(1,ℤ/8)12yes—
267C2×Dic13#13712yes—
268C4×D13—22no—
269C4×D13—22no—
270C4×D13—22no—
271C4×D13—22no—
272C4×D13—28no—
273C4×D13—44no—
274C8×C13—22no—
275C8×C13—22no—
276#1055—22no—
277#1055—22no—
278C22×M16—131no—
279C22×M16—131no—
280C2×#117—261no—
281C22×C42—131no—
282C2×#107—131no—
283C2×#107—131no—
284C2×#107—131no—
285C2×#107C2×D1311yescharacteristic
286C2×#106—131no—
287C2×#106—131no—
288C2×#106—131no—
289C2×#106C2×D1311yescharacteristic
290#423—138no—
291C2×C4⋊C8—131no—
292C2×C4⋊C8—131no—
293C2×C4⋊C8—131no—
294C2×C4⋊C8C2×D1311yescharacteristic
295#384—131no—
296#384—131no—
297#383—261no—
298#382—131no—
299#382—261no—
300#382C2×D1311yescharacteristic
301#381—261no—
302#380—131no—
303#380—131no—
304#380—131no—
305#380—131no—
306#380—261no—
307C4×C4⋊C4—131no—
308C4×C4⋊C4—131no—
309C4×C4⋊C4—131no—
310C4×C4⋊C4—261no—
311C4×C4⋊C4C2×D1311yescharacteristic
312C4×C22⋊C4—261no—
313#376—261no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#340—131no—
319#340—131no—
320#340C2×D1311yescharacteristic
321#340C2×D1311yescharacteristic
322C22×C13C22×C4⋊C411yescharacteristic
323C22×C13#53711yescharacteristic
324C22×C13#53711yescharacteristic
325C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
326C22×C13C2×#13711yescharacteristic
327C22×C13#57511yescharacteristic
328C22×C13#57511yescharacteristic
329C2×D13—22no—
330C2×D13—22no—
331C2×D13—22no—
332C2×D13—41no—
333C2×D13—41no—
334C2×D13—41no—
335C2×D13—41no—
336C4×C13—22no—
337C4×C13—22no—
338C4×C13—81no—
339C4×C13—81no—
340C4×C13—81no—
341C4×C13—81no—
342C4×C13#51811yescharacteristic
343C4×C13#51911yescharacteristic
344C4×C13#54411yescharacteristic
345C4×C13#54411yescharacteristic
346Dic13—22no—
347Dic13—22no—
348Dic13—81no—
349Dic13—81no—
350Dic13—81no—
351Dic13—81no—
352Dic13#42814yes—
353C23×C4—131no—
354C23×C4—131no—
355C23×C4—131no—
356C2×M16—134no—
357C2×M16—134no—
358C2×M16—521no—
359C2×M16—521no—
360C2×M16—521no—
361C2×M16—521no—
362C22×C8—131no—
363C22×C8—131no—
364C22×C8C22×D1311yescharacteristic
365C22×C8C22×D1311yescharacteristic
366#117—524no—
367C2×C4⋊C4—131no—
368C2×C4⋊C4—131no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—131no—
371C2×C4⋊C4—261no—
372C2×C4⋊C4—261no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4—261no—
375C2×C4⋊C4C22×D1311yescharacteristic
376C2×C4⋊C4C22×D1311yescharacteristic
377C2×C4⋊C4C22×D1311yescharacteristic
378C2×C4⋊C4C22×D1311yescharacteristic
379C2×C22⋊C4—261no—
380C2×C22⋊C4—261no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C42—21no—
384C2×C42—131no—
385C2×C42—131no—
386C2×C42—131no—
387C2×C42—134no—
388C2×C42—134no—
389C2×C42—261no—
390C2×C42—261no—
391C2×C42—261no—
392C2×C42C22×D1311yescharacteristic
393#107—41no—
394#107—521no—
395#107—521no—
396#107—521no—
397#106—41no—
398#106—521no—
399#106—521no—
400#106—521no—
401C4⋊C8—134no—
402C4⋊C8—134no—
403C4⋊C8—134no—
404C4⋊C8C4×D1314yes—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412#95—261no—
413#95—261no—
414#95—261no—
415#95—261no—
416#95—261no—
417C2×C13#231611yescharacteristic
418C2×C13C2×#42811yescharacteristic
419C2×C13#291111yescharacteristic
420C2×C13#325111yescharacteristic
421C2×C13#325111yescharacteristic
422C2×C13#325211yescharacteristic
423C2×C13#325211yescharacteristic
424D13—42no—
425C24—131no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—131no—
431C22×C4—131no—
432C22×C4—131no—
433C22×C4—134no—
434C22×C4—132no—
435C22×C4—132no—
436C22×C4—134no—
437C22×C4—262no—
438C22×C4—262no—
439C22×C4—261no—
440C22×C4—261no—
441C22×C4—261no—
442C22×C4—261no—
443C22×C4—261no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—521no—
448C22×C4—521no—
449C22×C4D4×D1311yescharacteristic
450C22×C4D4×D1311yescharacteristic
451C22×C4C23×D1311yescharacteristic
452M16—524no—
453M16—524no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—132no—
459C2×C8—132no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8C2×C4×D1312yes—
465C2×C8C2×C4×D1312yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—261no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—261no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—522no—
483C4⋊C4—522no—
484C4⋊C4—522no—
485C4⋊C4—522no—
486C22⋊C4—522no—
487C22⋊C4—522no—
488C22⋊C4—522no—
489C22⋊C4—522no—
490C42—42no—
491C42—134no—
492C42—134no—
493C42—132no—
494C42—132no—
495C42—522no—
496C42—522no—
497C42—522no—
498C42C2×C4×D1312yes—
499C42Q8×D1312yes—
500C13#2196311yescharacteristic, complemented, Sylow
501C23—132no—
502C23—261no—
503C23—261no—
504C23—261no—
505C23—261no—
506C23—261no—
507C23—261no—
508C23C2×D4×D1311yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—132no—
526C2×C4—132no—
527C2×C4—132no—
528C2×C4—132no—
529C2×C4—132no—
530C2×C4—268no—
531C2×C4—261no—
532C2×C4—261no—
533C2×C4—261no—
534C2×C4—261no—
535C2×C4—262no—
536C2×C4—262no—
537C2×C4—262no—
538C2×C4—262no—
539C2×C4—524no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4C4⋊C4×D1312yes—
553C2×C4C4⋊C4×D1312yes—
554C2×C4C22×C4×D1311yescharacteristic
555C2×C4#8825111yescharacteristic
556C2×C4C2×Q8×D1311yescharacteristic
557C2×C4#8825411yescharacteristic
558C2×C4#8825611yescharacteristic
559C2×C4#8825711yescharacteristic
560C8—22no—
561C8—22no—
562C8—262no—
563C8—262no—
564C22—262no—
565C22—262no—
566C22—262no—
567C22—521no—
568C22—521no—
569C22—521no—
570C22—521no—
571C22C2×C4⋊C4×D1311yescharacteristic
572C22#1171931311yescharacteristic
573C22#1171933411yescharacteristic
574C22#1171947311yescharacteristic
575C22#1171947311yescharacteristic
576C22D13×AGL(1,ℤ/8)11yescharacteristic
577C22D13×#13711yescharacteristic
578C4—22no—
579C4—22no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—134no—
585C4—262no—
586C4—262no—
587C4—1041no—
588C4—1041no—
589C4—1041no—
590C4—1041no—
591C4#1171925011yescharacteristic
592C4#1171927711yescharacteristic
593C4#1171939411yescharacteristic
594C4#1171943411yescharacteristic
595C2—522no—
596C2#1311745111yescharacteristic
597C2D13×#42811yescharacteristic
598C2#1312342411yescharacteristic
599C2#1312619511yescharacteristic
600C2#1312619711yescharacteristic
601C2#1312620611yescharacteristic
602C2#1312620811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1949C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1949D13
hypocenterC13#21963
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #13115606C2semidirect
5. #13115604C2semidirect
6. #13115602C2semidirect
7. #13115598C2semidirect
8. #13115596C2semidirect
9. #13115593C2semidirect
10. #13115591C2semidirect
15. C13×#1949C2semidirect
45. #11718498C4semidirect
206. #1949D13semidirect
500. C13#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#117184982
#131172402
#131172292
#131156062
#131156042
#131156022
#131155982
#131155962
#131155932
#131155912
#131148702
#131148682
#131148662
C13×#19492
#2196313