Finite Group Databasebeta

← #15918732

Subgroups — #15918732

Order 3328.

1078 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88144C211yesmaximal, characteristic
2C2×#11718517C212yesmaximal, complemented
3#13123186C211yesmaximal, characteristic
4#13116177C212yesmaximal, complemented
5#13116176C212yesmaximal, complemented
6#13115803C212yesmaximal, complemented
7#13115029C211yesmaximal, characteristic
8#13115026C211yesmaximal, characteristic
9#13115023C211yesmaximal, characteristic
10#13114600C211yesmaximal, characteristic
11C13×#1853C211yesmaximal, characteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic13—21no—
14C22×C4⋊Dic13C2211yescharacteristic
15C22×#6101—21no—
16C22×C4×Dic13C2211yescharacteristic
17C2×#88144—21no—
18C2×#88144—21no—
19C2×#88144—28no—
20C2×#88144C2212yes—
21C2×#88144C2212yes—
22C2×#88142C2211yescharacteristic
23C2×#88132—21no—
24C2×C13×#106C2212yes—
25C2×C13×C22⋊C8C2211yescharacteristic
26C13×#380C2211yescharacteristic
27#11718873C2211yescharacteristic
28C2×C22⋊#1055C2211yescharacteristic
29C2×#88058C2212yes—
30C2×#88055C2212yes—
31#11718646C2211yescharacteristic
32#11718637C2211yescharacteristic
33#11718634C2211yescharacteristic
34C2×#88048C2212yes—
35C2×#88047C2212yes—
36#11718603—21no—
37#11718517—28no—
38#11718308—21no—
39#11718307—21no—
40C13×#340C2212yes—
41#11718143C2212yes—
42#11718142C2212yes—
43#11718121C2212yes—
44#11718121C2212yes—
45C23×C4×C13—21no—
46C23×C4×C13C2311yescharacteristic
47C23×Dic13—21no—
48C23×Dic13D411yescharacteristic
49C23×Dic13D411yescharacteristic
50C22×Dic26—21no—
51C22×C8×C13C2312yes—
52C2×C13×C4⋊C4—21no—
53C2×C13×C4⋊C4—24no—
54C2×C13×C4⋊C4C2311yescharacteristic
55C2×C13×C4⋊C4C2311yescharacteristic
56C2×C13×C4⋊C4C2312yes—
57C2×C13×C22⋊C4—21no—
58C2×C22⋊Dic13—21no—
59C2×C4⋊Dic13—28no—
60C2×C4⋊Dic13—21no—
61C2×C4⋊Dic13—22no—
62C2×C4⋊Dic13—22no—
63C2×C4⋊Dic13—24no—
64C2×C4⋊Dic13—21no—
65C2×C4⋊Dic13C2311yescharacteristic
66C2×C4⋊Dic13C2311yescharacteristic
67C2×C4⋊Dic13C2312yes—
68C2×#6101—28no—
69C2×#6101—22no—
70C2×#6101—22no—
71C2×#6101—21no—
72C2×#6101—21no—
73C2×#6101—21no—
74C2×#6101—21no—
75C2×C4×Dic13—24no—
76C2×C4×Dic13D411yescharacteristic
77C2×C4×Dic13D411yescharacteristic
78C2×C4×Dic13C2311yescharacteristic
79C2×C4×Dic13C2311yescharacteristic
80C22×#1055C2312yes—
81#88144—28no—
82#88144—28no—
83#88144—24no—
84#88144—24no—
85#88144—24no—
86#88144D418yes—
87#88142—24no—
88#88132—44no—
89C13×#106—24no—
90C13×C22⋊C8Q814yescomplemented
91C13×#95—21no—
92#88071—21no—
93#88071—21no—
94#88071—21no—
95C22⋊#1055Q814yescomplemented
96#88058—24no—
97#88055—24no—
98#88048—24no—
99#88047—24no—
100#25819—131nomaximal, Sylow
101C24×C13C2×D411yescharacteristic
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—24no—
105C22×C4×C13—22no—
106C22×C4×C13—21no—
107C22×C4×C13—21no—
108C22×C4×C13—42no—
109C22×C4×C13—42no—
110C22×C4×C13C2×D411yescharacteristic
111C22×C4×C13C2×Q812yes—
112C22×C4×C13C2×Q812yes—
113C22×C4×C13C2411yescharacteristic
114C22×Dic13—24no—
115C22×Dic13—22no—
116C22×Dic13—22no—
117C22×Dic13—22no—
118C22×Dic13—22no—
119C22×Dic13—22no—
120C22×Dic13—21no—
121C22×Dic13—21no—
122C22×Dic13—41no—
123C22×Dic13—41no—
124C22×Dic13—42no—
125C22×Dic13—42no—
126C22×Dic13C2×D411yescharacteristic
127C22×Dic13C2×D411yescharacteristic
128C22×Dic13C2×D411yescharacteristic
129C22×Dic13C2×D411yescharacteristic
130C2×Dic26—42no—
131C2×Dic26—42no—
132C2×Dic26—42no—
133C2×C8×C13—22no—
134C2×C8×C13—22no—
135C2×C8×C13C2×Q814yes—
136C13×C4⋊C4—24no—
137C13×C4⋊C4—22no—
138C13×C4⋊C4—22no—
139C13×C4⋊C4—22no—
140C13×C4⋊C4C2×D414yes—
141C13×C22⋊C4—42no—
142C22⋊Dic13—42no—
143C4⋊Dic13—28no—
144C4⋊Dic13—24no—
145C4⋊Dic13—22no—
146C4⋊Dic13—28no—
147C4⋊Dic13—22no—
148C4⋊Dic13—22no—
149C4⋊Dic13—42no—
150C4⋊Dic13C2×D414yes—
151#6101—28no—
152#6101—28no—
153#6101—42no—
154#6101—42no—
155#6101—42no—
156#6101—42no—
157C4×Dic13—22no—
158C4×Dic13—22no—
159C4×Dic13—22no—
160C4×Dic13C2×D414yes—
161C2×#1055—22no—
162C2×#1055—22no—
163C2×#1055C2×Q814yes—
164C22×#125—131no—
165C2×#501—132no—
166#2857—131no—
167#2059—132no—
168#2059—132no—
169#1992—132no—
170#1853—131no—
171#1853D1311yescharacteristic
172#1852—131no—
173#1852—131no—
174#1749—131no—
175C23×C13—21no—
176C23×C13—21no—
177C23×C13—21no—
178C23×C13—21no—
179C23×C13C4⋊Q812yes—
180C23×C13AGL(1,ℤ/8)12yes—
181C23×C13#13712yes—
182C23×C13C22×D411yescharacteristic
183C2×C4×C13—21no—
184C2×C4×C13—21no—
185C2×C4×C13—24no—
186C2×C4×C13—24no—
187C2×C4×C13—24no—
188C2×C4×C13—48no—
189C2×C4×C13—42no—
190C2×C4×C13—42no—
191C2×C4×C13—41no—
192C2×C4×C13—41no—
193C2×C4×C13—41no—
194C2×C4×C13—41no—
195C2×C4×C13—41no—
196C2×C4×C13—41no—
197C2×C4×C13C4⋊Q812yes—
198C2×C4×C13C22×D412yes—
199C2×C4×C13C22×Q811yescharacteristic
200C2×C4×C13C22×Q811yescharacteristic
201C2×C4×C132−1+412yes—
202C2×Dic13—21no—
203C2×Dic13—21no—
204C2×Dic13—21no—
205C2×Dic13—21no—
206C2×Dic13—21no—
207C2×Dic13—21no—
208C2×Dic13—24no—
209C2×Dic13—21no—
210C2×Dic13—21no—
211C2×Dic13—21no—
212C2×Dic13—21no—
213C2×Dic13—44no—
214C2×Dic13—44no—
215C2×Dic13—48no—
216C2×Dic13—42no—
217C2×Dic13—42no—
218C2×Dic13—41no—
219C2×Dic13—41no—
220C2×Dic13—41no—
221C2×Dic13—41no—
222C2×Dic13—41no—
223C2×Dic13—41no—
224C2×Dic13#12012yes—
225C2×Dic13#12012yes—
226Dic26—84no—
227C8×C13—24no—
228#1055—24no—
229C2×C4⋊Q8—131no—
230C2×#125—132no—
231C2×#125—132no—
232C2×#125—261no—
233C2×#125—261no—
234C2×#125—268no—
235C2×C22⋊Q8—261no—
236C22×C4⋊C4—131no—
237C22×C4⋊C4—261no—
238C22×C4⋊C4—261no—
239C22×C4⋊C4C2×D1311yescharacteristic
240C22×C42—131no—
241#501—268no—
242C2×#107—132no—
243C2×#107—132no—
244C2×#106—132no—
245C2×#106C2×D1312yes—
246C2×C22⋊C8—131no—
247C2×C22⋊C8C2×D1311yescharacteristic
248C2×#97—132no—
249#391—261no—
250#389—131no—
251#389—131no—
252#389—261no—
253#384—131no—
254#380—131no—
255#380—261no—
256#380C2×D1311yescharacteristic
257#340—132no—
258#340—132no—
259#340—132no—
260#340C2×D1312yes—
261#336—132no—
262C22×C13—22no—
263C22×C13—22no—
264C22×C13—22no—
265C22×C13—22no—
266C22×C13—22no—
267C22×C13—22no—
268C22×C13#50114yescomplemented
269C22×C13C2×#12011yescharacteristic
270C22×C13C2×C4⋊Q811yescharacteristic
271C22×C13#53611yescharacteristic
272C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C13C2×#13711yescharacteristic
274C22×C13#57511yescharacteristic
275C22×C13#57511yescharacteristic
276C4×C13—22no—
277C4×C13—44no—
278C4×C13—81no—
279C4×C13—81no—
280C4×C13D4×Q812yes—
281C4×C13D4×Q812yes—
282Dic13—22no—
283Dic13—22no—
284Dic13—41no—
285Dic13—44no—
286Dic13—41no—
287Dic13—81no—
288Dic13—81no—
289C22×Q8—261no—
290C23×C4—21no—
291C23×C4—131no—
292C23×C4—131no—
293C23×C4—261no—
294C23×C4C22×D1311yescharacteristic
295C22×C8—132no—
296C22×C8C22×D1312yes—
297C4⋊Q8—264no—
298#125—138no—
299#125—264no—
300#125—264no—
301#125—268no—
302#125—268no—
303#125—264no—
304C22⋊Q8—524no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—132no—
308C2×C4⋊C4—131no—
309C2×C4⋊C4—131no—
310C2×C4⋊C4—264no—
311C2×C4⋊C4—262no—
312C2×C4⋊C4—262no—
313C2×C4⋊C4—262no—
314C2×C4⋊C4—262no—
315C2×C4⋊C4—268no—
316C2×C4⋊C4—268no—
317C2×C4⋊C4—261no—
318C2×C4⋊C4—261no—
319C2×C4⋊C4—261no—
320C2×C4⋊C4—261no—
321C2×C4⋊C4—261no—
322C2×C4⋊C4—261no—
323C2×C4⋊C4C22×D1311yescharacteristic
324C2×C4⋊C4C22×D1311yescharacteristic
325C2×C4⋊C4C22×D1312yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—261no—
328C2×C42—131no—
329C2×C42—131no—
330C2×C42—131no—
331C2×C42—131no—
332C2×C42—264no—
333#107—264no—
334#107—264no—
335#106—24no—
336#106—264no—
337C22⋊C8—134no—
338C22⋊C8Dic2614yescomplemented
339#97—264no—
340#95—21no—
341#95—261no—
342#95—261no—
343#95—261no—
344C2×C13—24no—
345C2×C13#243411yescharacteristic
346C2×C13#300611yescharacteristic
347C2×C13#300711yescharacteristic
348C2×C13C2×#50111yescharacteristic
349C2×C13#316611yescharacteristic
350C2×C13#323411yescharacteristic
351C2×C13#323611yescharacteristic
352C24D4×D1311yescharacteristic
353C2×Q8—522no—
354C2×Q8—522no—
355C2×Q8—522no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—131no—
365C22×C4—131no—
366C22×C4—131no—
367C22×C4—131no—
368C22×C4—262no—
369C22×C4—262no—
370C22×C4—264no—
371C22×C4—262no—
372C22×C4—262no—
373C22×C4—262no—
374C22×C4—261no—
375C22×C4—261no—
376C22×C4—521no—
377C22×C4—521no—
378C22×C4—522no—
379C22×C4—522no—
380C22×C4C2×Dic2612yes—
381C22×C4D4×D1311yescharacteristic
382C22×C4Q8×D1312yes—
383C22×C4C23×D1311yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—134no—
387C2×C8—262no—
388C2×C8—262no—
389C2×C8C2×Dic2614yes—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—134no—
395C4⋊C4—268no—
396C4⋊C4—268no—
397C4⋊C4—268no—
398C4⋊C4—264no—
399C4⋊C4—268no—
400C4⋊C4—262no—
401C4⋊C4—262no—
402C4⋊C4—262no—
403C4⋊C4—522no—
404C4⋊C4—522no—
405C4⋊C4—522no—
406C4⋊C4—522no—
407C4⋊C4—522no—
408C4⋊C4D4×D1314yes—
409C22⋊C4—42no—
410C22⋊C4—522no—
411C42—134no—
412C42—262no—
413C42—262no—
414C42—262no—
415C13#2581911yescharacteristic, complemented, Sylow
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23#8814212yes—
421C23#8816812yes—
422C23#8816912yes—
423C23C2×D4×D1311yescharacteristic
424Q8—1044no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—132no—
440C2×C4—132no—
441C2×C4—261no—
442C2×C4—261no—
443C2×C4—261no—
444C2×C4—261no—
445C2×C4—261no—
446C2×C4—261no—
447C2×C4—261no—
448C2×C4—261no—
449C2×C4—261no—
450C2×C4—261no—
451C2×C4—264no—
452C2×C4—524no—
453C2×C4—524no—
454C2×C4—528no—
455C2×C4—522no—
456C2×C4—522no—
457C2×C4—521no—
458C2×C4—521no—
459C2×C4—521no—
460C2×C4—521no—
461C2×C4—521no—
462C2×C4—521no—
463C2×C4#8814212yes—
464C2×C4C22×Dic2611yescharacteristic
465C2×C4C2×D4×D1312yes—
466C2×C4C2×Q8×D1311yescharacteristic
467C2×C4#8825712yes—
468C8—24no—
469C8—264no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22#1171851714yescomplemented
477C22C2×#8814211yescharacteristic
478C22D13×#12011yescharacteristic
479C22#1171935311yescharacteristic
480C22C2×#8816811yescharacteristic
481C22C2×#8816911yescharacteristic
482C22#1171947811yescharacteristic
483C22#1171948211yescharacteristic
484C4—22no—
485C4—44no—
486C4—81no—
487C4—81no—
488C4—262no—
489C4—262no—
490C4—521no—
491C4—524no—
492C4—521no—
493C4—1041no—
494C4—1041no—
495C4D4×Dic2612yes—
496C4#1171933712yes—
497C2—24no—
498C2#1311847811yescharacteristic
499C2#1312408711yescharacteristic
500C2#1312409111yescharacteristic
501C2C2×#1171851711yescharacteristic
502C2#1312538611yescharacteristic
503C2#1312602511yescharacteristic
504C2#1312603911yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1853C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1853D13
hypocenterC13#25819
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#11718517C2semidirect
4. #13116177C2semidirect
5. #13116176C2semidirect
6. #13115803C2semidirect
90. C13×C22⋊C8Q8semidirect
95. C22⋊#1055Q8semidirect
268. C22×C13#501semidirect
338. C22⋊C8Dic26semidirect
415. C13#25819semidirect
476. C22#11718517semidirect

Maximal subgroups

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

subgroupindex
C22×#881442
C2×#117185172
#131231862
#131161772
#131161762
#131158032
#131150292
#131150262
#131150232
#131146002
C13×#18532
#2581913