Finite Group Databasebeta

← #15839031

Subgroups — #15839031

Order 3328.

962 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125216C211yesmaximal, characteristic, complemented
2#13124966C211yesmaximal, characteristic, complemented
3#13124935C211yesmaximal, characteristic
4#13114050C214yesmaximal, complemented
5#13114043C214yesmaximal, complemented
6#13113562C211yesmaximal, characteristic, complemented
7#13113283C211yesmaximal, characteristic, complemented
8#13113279C211yesmaximal, characteristic
9C2×C13×#554C211yesmaximal, characteristic
10C13×#554—22no—
11C13×#554—22no—
12C13×#554—22no—
13C13×#554C2214yes—
14C2×C13×C4⋊Q8C2211yescharacteristic
15C2×C13×#125—21no—
16C2×C13×C22⋊Q8—21no—
17C2×C13×C22⋊Q8—22no—
18C2×C4×Q8×C13C2211yescharacteristic
19C2×C4×D4×C13—21no—
20C2×C4×D4×C13C2211yescharacteristic
21C22×C13×C4⋊C4—21no—
22#11719175C2211yescharacteristic
23#11718889—21no—
24#11718886—22no—
25#11718879—21no—
26#11718743—22no—
27#11718742C2212yescomplemented
28#11718742C2212yescomplemented
29#11718692C2212yescomplemented
30#11718691—22no—
31#11718681C2212yescomplemented
32#11718680C2212yes—
33#11718679—22no—
34#11718668C2212yes—
35#11718632—21no—
36#11718631—21no—
37#11718587C2211yescharacteristic
38#11718587C2211yescharacteristic
39C4×#6099C2211yescharacteristic
40#11718138C2214yescomplemented
41Q8⋊#1055C2214yescomplemented
42D4⋊#1055C2214yes—
43C22×Q8×C13D411yescharacteristic
44C22×Q8×C13D412yes—
45C22×D4×C13D411yescharacteristic
46C23×C4×C13—22no—
47C23×C4×C13—21no—
48C13×C4⋊Q8—21no—
49C13×C4⋊Q8—21no—
50C13×C4⋊Q8D412yes—
51C13×C4⋊Q8C2312yes—
52C13×#125—24no—
53C13×#125—24no—
54C13×C22⋊Q8—24no—
55C13×C22⋊Q8—28no—
56C13×C22⋊Q8—24no—
57C13×C22⋊Q8—28no—
58C4×Q8×C13—21no—
59C4×Q8×C13—21no—
60C4×Q8×C13D412yes—
61C4×Q8×C13C2312yes—
62C4×D4×C13—24no—
63C4×D4×C13—24no—
64C4×D4×C13—21no—
65C4×D4×C13—21no—
66C4×D4×C13D412yes—
67C4×D4×C13C2312yes—
68C2×C13×C4⋊C4—22no—
69C2×C13×C4⋊C4—24no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—24no—
75C2×C13×C4⋊C4D411yescharacteristic
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C22⋊C4—22no—
78C2×C13×C22⋊C4—21no—
79C2×C42×C13—21no—
80C2×C42×C13C2311yescharacteristic
81C2×#6107—22no—
82C2×#6105—22no—
83#88180—21no—
84#88178—21no—
85#88178—21no—
86C2×#6099—21no—
87#88124—21no—
88#88123C2311yescharacteristic
89#88123C2311yescharacteristic
90C4×Dic26—21no—
91C4×Dic26—21no—
92#88075—22no—
93#88075—24no—
94#88075—24no—
95#88072—22no—
96C22⋊#1055—21no—
97C22⋊#1055—21no—
98#88050—22no—
99#88050—22no—
100#88048—22no—
101#88047—22no—
102C4⋊#1055C2312yes—
103C4⋊#1055C2312yes—
104#88043—21no—
105#88043—21no—
106#88043—21no—
107C4×#1055C2312yes—
108#18530—131nomaximal, Sylow
109C24×C13—21no—
110C2×Q8×C13—21no—
111C2×Q8×C13—22no—
112C2×Q8×C13—42no—
113C2×Q8×C13—44no—
114C2×Q8×C13C2×D412yes—
115C2×Q8×C13C2×D414yes—
116C2×D4×C13—24no—
117C2×D4×C13—21no—
118C2×D4×C13C2×D412yes—
119C22×C4×C13—21no—
120C22×C4×C13—21no—
121C22×C4×C13—21no—
122C22×C4×C13—22no—
123C22×C4×C13—28no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—44no—
127C22×C4×C13—84no—
128C22×C4×C13C2×D411yescharacteristic
129C22×C4×C13C2×D411yescharacteristic
130C22×C4×C13C2×D411yescharacteristic
131C2×Dic26—21no—
132C2×Dic26—21no—
133C13×C4⋊C4—28no—
134C13×C4⋊C4—28no—
135C13×C4⋊C4—21no—
136C13×C4⋊C4—22no—
137C13×C4⋊C4—22no—
138C13×C4⋊C4—22no—
139C13×C4⋊C4—21no—
140C13×C4⋊C4—22no—
141C13×C4⋊C4—22no—
142C13×C4⋊C4—22no—
143C13×C4⋊C4C2×D412yes—
144C13×C4⋊C4C2×D412yes—
145C13×C22⋊C4—28no—
146C13×C22⋊C4—22no—
147C13×C22⋊C4—22no—
148C42×C13—22no—
149C42×C13—22no—
150C42×C13C2×D411yescharacteristic
151C42×C13C2×D411yescharacteristic
152C42×C13C2×D411yescharacteristic
153C42×C13C2411yescharacteristic
154C22⋊Dic13—41no—
155#6107—44no—
156#6105—44no—
157C4⋊Dic13—21no—
158C4⋊Dic13—21no—
159C4⋊Dic13—21no—
160C4⋊Dic13—21no—
161#6101—41no—
162#6101—41no—
163C4×Dic13—41no—
164#6099—42no—
165C2×#1055—22no—
166C2×#1055—22no—
167C2×#1055—22no—
168C2×#554D1311yescharacteristic
169#3139—131no—
170#3113—131no—
171#3110—131no—
172#1644—134no—
173#1643—134no—
174#1558—131no—
175#1491—131no—
176#1490—131no—
177C23×C13—24no—
178C23×C13—22no—
179C23×C13—42no—
180C23×C13—42no—
181C23×C13#12011yescharacteristic
182Q8×C13—22no—
183Q8×C13—22no—
184Q8×C13—42no—
185Q8×C13—44no—
186Q8×C13—82no—
187D4×C13—24no—
188D4×C13—22no—
189D4×C13#13714yes—
190C2×C4×C13—22no—
191C2×C4×C13—22no—
192C2×C4×C13—22no—
193C2×C4×C13—22no—
194C2×C4×C13—22no—
195C2×C4×C13—22no—
196C2×C4×C13—24no—
197C2×C4×C13—44no—
198C2×C4×C13—42no—
199C2×C4×C13—42no—
200C2×C4×C13—42no—
201C2×C4×C13—44no—
202C2×C4×C13—44no—
203C2×C4×C13—88no—
204C2×C4×C13#12011yescharacteristic
205C2×C4×C13#12011yescharacteristic
206C2×C4×C13#12011yescharacteristic
207C2×C4×C13C22×D411yescharacteristic
208C2×C4×C13C22×D411yescharacteristic
209C2×C4×C13C22×D411yescharacteristic
210C2×Dic13—41no—
211C2×Dic13—41no—
212Dic26—42no—
213Dic26—42no—
214#1055—42no—
215#1055—42no—
216#1055—42no—
217#554—22no—
218#554—22no—
219#554—22no—
220#554C2×D1314yes—
221#533—131no—
222C2×C4⋊Q8C2×D1311yescharacteristic
223C2×#125—21no—
224C2×C22⋊Q8—21no—
225C2×C22⋊Q8—22no—
226C2×C4×Q8C2×D1311yescharacteristic
227C2×C4×D4—21no—
228C2×C4×D4C2×D1311yescharacteristic
229C22×C4⋊C4—21no—
230#491—262no—
231#487—132no—
232#487—132no—
233#484—261no—
234#483—261no—
235#475—132no—
236#474—132no—
237#462—132no—
238#461—132no—
239#451—261no—
240#450—261no—
241#441—262no—
242#440—262no—
243#432—131no—
244#432—131no—
245#419—262no—
246C4×M16—131no—
247#333—134no—
248Q8⋊C8—134no—
249D4⋊C8—134no—
250C22×C13—21no—
251C22×C13—21no—
252C22×C13—24no—
253C22×C13—48no—
254C22×C13—42no—
255C22×C13—42no—
256C22×C13#45512yes—
257C22×C13C2×#12011yescharacteristic
258C4×C13—21no—
259C4×C13—21no—
260C4×C13—21no—
261C4×C13—21no—
262C4×C13—42no—
263C4×C13—42no—
264C4×C13—42no—
265C4×C13—44no—
266C4×C13C2×#13712yes—
267C4×C13#57512yes—
268Dic13—81no—
269Dic13—81no—
270C22×Q8C22⋊D1311yescharacteristic
271C22×Q8C22⋊D1312yes—
272C22×D4C22⋊D1311yescharacteristic
273C23×C4—22no—
274C23×C4—21no—
275C2×Q16—262no—
276C2×SD16—262no—
277C2×M16—261no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—131no—
281C4⋊Q8—131no—
282C4⋊Q8C22⋊D1312yes—
283C4⋊Q8C22×D1312yes—
284#125—24no—
285#125—24no—
286#125—261no—
287C22⋊Q8—24no—
288C22⋊Q8—28no—
289C22⋊Q8—24no—
290C22⋊Q8—28no—
291C22⋊Q8—261no—
292C22⋊Q8—261no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—261no—
296C4×Q8—261no—
297C4×Q8C22⋊D1312yes—
298C4×Q8C22×D1312yes—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C22⋊D1312yes—
304C4×D4C22×D1312yes—
305#117—261no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4C22⋊D1311yescharacteristic
314C2×C4⋊C4C22⋊D1311yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C42—21no—
318C2×C42C22×D1311yescharacteristic
319#107—262no—
320#106—262no—
321C4⋊C8—132no—
322C4⋊C8—132no—
323#103—264no—
324#103—264no—
325#103—262no—
326#103—262no—
327#103—262no—
328#102—262no—
329C22⋊C8—261no—
330C22⋊C8—261no—
331#97—261no—
332#97—261no—
333#97—261no—
334C4×C8—132no—
335C2×C13—22no—
336C2×C13—44no—
337C2×C13#300411yescharacteristic
338C2×C13#300511yescharacteristic
339C2×C13C2×#45511yescharacteristic
340C24—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—44no—
345C2×Q8—261no—
346C2×Q8—261no—
347C2×Q8C2×C22⋊D1312yes—
348C2×Q8C2×C22⋊D1314yes—
349C2×D4—24no—
350C2×D4—21no—
351C2×D4C2×C22⋊D1312yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—44no—
360C22×C4—84no—
361C22×C4C2×C22⋊D1311yescharacteristic
362C22×C4C2×C22⋊D1311yescharacteristic
363C22×C4C2×C22⋊D1311yescharacteristic
364Q16—524no—
365SD16—524no—
366M16—522no—
367C2×C8—262no—
368C2×C8—262no—
369C2×C8—262no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—261no—
381C4⋊C4—261no—
382C4⋊C4—261no—
383C4⋊C4—261no—
384C4⋊C4—521no—
385C4⋊C4—521no—
386C4⋊C4C2×C22⋊D1312yes—
387C4⋊C4C2×C22⋊D1312yes—
388C22⋊C4—28no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—521no—
392C42—22no—
393C42—22no—
394C42—521no—
395C42C2×C22⋊D1311yescharacteristic
396C42C2×C22⋊D1311yescharacteristic
397C42C2×C22⋊D1311yescharacteristic
398C42C23×D1311yescharacteristic
399C13#1853011yescharacteristic, complemented, Sylow
400C23—24no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23#8820711yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—42no—
409Q8—82no—
410Q8—522no—
411Q8—522no—
412D4—24no—
413D4—22no—
414D4#8819614yes—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—88no—
429C2×C4—521no—
430C2×C4—521no—
431C2×C4#8820711yescharacteristic
432C2×C4#8820711yescharacteristic
433C2×C4#8820711yescharacteristic
434C2×C4C22×C22⋊D1311yescharacteristic
435C2×C4C22×C22⋊D1311yescharacteristic
436C2×C4C22×C22⋊D1311yescharacteristic
437C8—522no—
438C8—522no—
439C8—522no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22#1171890312yes—
447C22C2×#8820711yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—44no—
454C4—42no—
455C4—42no—
456C4—1041no—
457C4—1041no—
458C4C2×#8819612yes—
459C4#1171952712yes—
460C2—22no—
461C2—44no—
462C2#1312407011yescharacteristic
463C2#1312407811yescharacteristic
464C2C2×#1171890311yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×C13×#554C2
socleC22×C13C2×#120
prime radicalC22×D4×C13D4
hypercenterC2×#554D13
hypocenterC13#18530
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #13125216C2semidirect
2. #13124966C2semidirect
4. #13114050C2semidirect
5. #13114043C2semidirect
6. #13113562C2semidirect
7. #13113283C2semidirect
27. #11718742C22semidirect
28. #11718742C22semidirect
29. #11718692C22semidirect
31. #11718681C22semidirect
40. #11718138C22semidirect
41. Q8⋊#1055C22semidirect
399. C13#18530semidirect

Maximal subgroups

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

subgroupindex
#131252162
#131249662
#131249352
#131140502
#131140432
#131135622
#131132832
#131132792
C2×C13×#5542
#1853013