Finite Group Databasebeta

← #15762063

Subgroups — #15762063

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123137C211yesmaximal, characteristic
2#13114816C211yesmaximal, characteristic, complemented
3C2×#11718044C212yesmaximal, complemented
4C2×#11718043C212yesmaximal
5C13×#1434C211yesmaximal, characteristic, complemented
6C22×C42×C13C2211yescharacteristic
7C2×#88184C2211yescharacteristic
8C2×#88178C2211yescharacteristic
9C2×#88129C411yescharacteristic, complemented
10C2×C13×#95—21no—
11C2×C13×#95—21no—
12C2×C13×#95C2212yes—
13#11718874—21no—
14#11718874—21no—
15C2×C22⋊#1055C2212yes—
16#11718601—21no—
17#11718601—21no—
18#11718600C411yescharacteristic, complemented
19#11718597C411yescharacteristic, complemented
20#11718590C411yescharacteristic, complemented
21C2×C4⋊#1055—21no—
22#11718116—21no—
23#11718044—24no—
24#11718043—24no—
25C23×C4×C13—22no—
26C23×C4×C13D411yescharacteristic
27C23×C4×C13D411yescharacteristic
28C23×C4×C13C2311yescharacteristic
29C22×C22⋊D13—21no—
30C22×D52C2×C411yescharacteristic
31C22×Dic26C2×C411yescharacteristic
32C2×C42×C13—42no—
33C2×C42×C13C2×C411yescharacteristic
34C2×C42×C13C2×C411yescharacteristic
35C2×C42×C13D411yescharacteristic
36C2×C42×C13D411yescharacteristic
37C2×C22⋊Dic13—21no—
38#88184—22no—
39#88184—22no—
40C2×#6103—21no—
41C2×#6103—21no—
42C2×#6103—21no—
43C2×#6103—21no—
44C2×#6103—21no—
45C2×C4⋊Dic13C2×C411yescharacteristic
46C2×C4⋊Dic13C2×C411yescharacteristic
47#88178—22no—
48#88178—22no—
49C2×#6101—21no—
50C22×#1055—22no—
51#88129—42no—
52C13×#95—22no—
53C13×#95—42no—
54C13×#95—42no—
55C13×#95D414yes—
56#88071—21no—
57#88071—21no—
58#88071—21no—
59#88071—21no—
60C22⋊#1055—22no—
61C22⋊#1055—22no—
62C4⋊#1055—42no—
63#9716—131nomaximal, Sylow
64C24×C13C2×D411yescharacteristic
65C23×D13—21no—
66C22×C4×C13—22no—
67C22×C4×C13—21no—
68C22×C4×C13—21no—
69C22×C4×C13—22no—
70C22×C4×C13—24no—
71C22×C4×C13—41no—
72C22×C4×C13—41no—
73C22×C4×C13—41no—
74C22×C4×C13—41no—
75C22×C4×C13—41no—
76C22×C4×C13—42no—
77C22×C4×C13—42no—
78C22×C4×C13—42no—
79C22×C4×C13—42no—
80C22×C4×C13—42no—
81C22×C4×C13—42no—
82C22×C4×C13C22⋊C411yescharacteristic
83C22×C4×C13C22⋊C411yescharacteristic
84C22×C4×C13C22⋊C411yescharacteristic
85C22×C4×C13C22⋊C411yescharacteristic
86C22×C4×C13C22×C411yescharacteristic
87C22×C4×C13C2×D411yescharacteristic
88C22×C4×C13C2×D412yes—
89C22×C4×C13#4112yes—
90C2×C22⋊D13—42no—
91C2×C22⋊D13—42no—
92C2×C22⋊D13—41no—
93C2×C22⋊D13—41no—
94C22×Dic13—21no—
95C22×Dic13—21no—
96C22×Dic13—21no—
97C2×D52—22no—
98C2×D52—41no—
99C2×D52—41no—
100C2×Dic26—22no—
101C2×Dic26—41no—
102C2×Dic26—41no—
103C42×C13—22no—
104C42×C13—22no—
105C42×C13—41no—
106C42×C13—41no—
107C22⋊Dic13—42no—
108#6103—42no—
109#6103—42no—
110#6103—42no—
111#6103—42no—
112#6103—41no—
113#6103—41no—
114C4⋊Dic13—22no—
115C4⋊Dic13—22no—
116#6101—41no—
117#6101—41no—
118C2×#1055—42no—
119C2×#1055—42no—
120C2×#1055—42no—
121#2850—131no—
122#1797—131no—
123C2×#328—132no—
124C2×#327—132no—
125#1434D1311yescharacteristic, complemented
126C23×C13—21no—
127C23×C13—21no—
128C23×C13—21no—
129C23×C13—21no—
130C23×C13C2×C22⋊C411yescharacteristic
131C23×C13#12112yes—
132C23×C13#13512yes—
133C23×C13AGL(1,ℤ/8)11yescharacteristic
134C23×C13#13711yescharacteristic
135C22×D13—41no—
136C22×D13—41no—
137C22×D13—41no—
138C22×D13—41no—
139C22×D13—41no—
140C22×D13—41no—
141C22×D13—41no—
142C2×C4×C13—21no—
143C2×C4×C13—21no—
144C2×C4×C13—21no—
145C2×C4×C13—21no—
146C2×C4×C13—21no—
147C2×C4×C13—21no—
148C2×C4×C13—21no—
149C2×C4×C13—21no—
150C2×C4×C13—21no—
151C2×C4×C13—21no—
152C2×C4×C13—21no—
153C2×C4×C13—21no—
154C2×C4×C13—21no—
155C2×C4×C13—21no—
156C2×C4×C13—21no—
157C2×C4×C13—42no—
158C2×C4×C13—42no—
159C2×C4×C13—44no—
160C2×C4×C13—42no—
161C2×C4×C13—42no—
162C2×C4×C13—44no—
163C2×C4×C13—44no—
164C2×C4×C13—44no—
165C2×C4×C13—82no—
166C2×C4×C13—82no—
167C2×C4×C13—82no—
168C2×C4×C13—82no—
169C2×C4×C13#9912yes—
170C2×C4×C13C4×D412yes—
171C2×C4×C13#12411yescharacteristic
172C2×C4×C13#12711yescharacteristic
173C22⋊D13—82no—
174C22⋊D13—82no—
175C2×Dic13—41no—
176C2×Dic13—41no—
177C2×Dic13—41no—
178C2×Dic13—41no—
179C2×Dic13—41no—
180C2×Dic13—41no—
181C2×Dic13—41no—
182C2×Dic13—41no—
183C2×Dic13—41no—
184D52—82no—
185Dic26—82no—
186#1055—82no—
187C2×#124—131no—
188C2×C22⋊Q8—131no—
189C2×#121—131no—
190C22×C42C2×D1311yescharacteristic
191C2×C4⋊C8—261no—
192C2×C22⋊C8—132no—
193#391—131no—
194#390—131no—
195#388—261no—
196#388—261no—
197#386—261no—
198#386—261no—
199#382—131no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95C2×D1312yes—
203#336—261no—
204#328—264no—
205#327—264no—
206C22×C13—22no—
207C22×C13—22no—
208C22×C13—21no—
209C22×C13—21no—
210C22×C13—21no—
211C22×C13—21no—
212C22×C13—22no—
213C22×C13—22no—
214C22×C13#39011yescharacteristic
215C22×C13C2×#9911yescharacteristic
216C22×C13#41011yescharacteristic
217C22×C13#41611yescharacteristic
218C22×C13#41711yescharacteristic
219C22×C13#42111yescharacteristic
220C22×C13#42111yescharacteristic
221C22×C13#46312yes—
222C22×C13#46412yes—
223C2×D13—81no—
224C2×D13—81no—
225C2×D13—81no—
226C2×D13—81no—
227C2×D13—81no—
228C2×D13—81no—
229C2×D13—81no—
230C4×C13—21no—
231C4×C13—22no—
232C4×C13—21no—
233C4×C13—22no—
234C4×C13—41no—
235C4×C13—41no—
236C4×C13—41no—
237C4×C13—84no—
238Dic13—81no—
239Dic13—81no—
240Dic13—81no—
241C22×Q8—131no—
242C22×D4—131no—
243C22×D4—261no—
244C23×C4—22no—
245C23×C4C22⋊D1311yescharacteristic
246C23×C4C22⋊D1311yescharacteristic
247C23×C4C22×D1311yescharacteristic
248C22×C8—262no—
249#124—522no—
250C22⋊Q8—262no—
251C22⋊Q8—262no—
252#121—262no—
253#121—262no—
254C2×C4⋊C4—131no—
255C2×C4⋊C4—131no—
256C2×C4⋊C4—261no—
257C2×C22⋊C4—261no—
258C2×C22⋊C4—261no—
259C2×C22⋊C4—261no—
260C2×C22⋊C4—261no—
261C2×C22⋊C4—261no—
262C2×C22⋊C4—261no—
263C2×C42—42no—
264C2×C42C4×D1311yescharacteristic
265C2×C42C4×D1311yescharacteristic
266C2×C42D5211yescharacteristic
267C2×C42D5211yescharacteristic
268C4⋊C8—522no—
269C22⋊C8—262no—
270C22⋊C8—262no—
271#95—22no—
272#95—42no—
273#95—42no—
274#95—261no—
275#95—261no—
276#95—261no—
277#95—261no—
278#95D5214yes—
279C2×C13—22no—
280C2×C13—22no—
281C2×C13#151211yescharacteristic
282C2×C13#151311yescharacteristic
283C2×C13#151611yescharacteristic
284C2×C13#152111yescharacteristic
285C2×C13#196311yescharacteristic
286C2×C13#196411yescharacteristic
287C2×C13#198311yescharacteristic
288D13—161no—
289C24—261no—
290C24C2×C22⋊D1311yescharacteristic
291C2×Q8—262no—
292C2×Q8—521no—
293C2×Q8—521no—
294C2×D4—262no—
295C2×D4—521no—
296C2×D4—521no—
297C2×D4—522no—
298C2×D4—521no—
299C2×D4—521no—
300C2×D4—522no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—261no—
318C22×C4—261no—
319C22×C4—261no—
320C22×C4#610311yescharacteristic
321C22×C4#610311yescharacteristic
322C22×C4#610311yescharacteristic
323C22×C4#610311yescharacteristic
324C22×C4C2×C4×D1311yescharacteristic
325C22×C4C2×D5211yescharacteristic
326C22×C4C2×D5212yes—
327C22×C4#612712yes—
328C2×C8—522no—
329C2×C8—522no—
330C2×C8—522no—
331C4⋊C4—262no—
332C4⋊C4—262no—
333C4⋊C4—521no—
334C4⋊C4—521no—
335C22⋊C4—522no—
336C22⋊C4—522no—
337C22⋊C4—522no—
338C22⋊C4—522no—
339C22⋊C4—521no—
340C22⋊C4—521no—
341C22⋊C4—522no—
342C42—22no—
343C42—22no—
344C42—41no—
345C42—41no—
346C13#971611yescharacteristic, complemented, Sylow
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—521no—
352C23—521no—
353C23—521no—
354C23—521no—
355C23—521no—
356C23—521no—
357C23—521no—
358C23C2×#610311yescharacteristic
359C23#8818412yes—
360C23#8818611yescharacteristic
361C23#8819611yescharacteristic
362C23#8820412yes—
363Q8—1042no—
364D4—1042no—
365D4—1042no—
366D4—1042no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—521no—
395C2×C4—521no—
396C2×C4—521no—
397C2×C4—521no—
398C2×C4—521no—
399C2×C4—521no—
400C2×C4—521no—
401C2×C4—521no—
402C2×C4—521no—
403C2×C4#8804612yes—
404C2×C4C4×D5212yes—
405C2×C4#8812811yescharacteristic
406C2×C4#8812911yescharacteristic
407C8—1042no—
408C22—22no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—1041no—
417C22—1041no—
418C22—1041no—
419C22—1041no—
420C22—1041no—
421C22—1041no—
422C22—1041no—
423C22#1171860011yescharacteristic
424C22C2×#8804611yescharacteristic
425C22#1171862711yescharacteristic
426C22#1171866311yescharacteristic
427C22#1171866611yescharacteristic
428C22#1171866611yescharacteristic
429C22#1171866711yescharacteristic
430C22#1171867712yes—
431C22#1171868912yes—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—84no—
440C4—1041no—
441C4—1041no—
442C4—1041no—
443C2—22no—
444C2—22no—
445C2—2081no—
446C2#1311337211yescharacteristic
447C2#1311337711yescharacteristic
448C2#1311339011yescharacteristic
449C2#1311341111yescharacteristic
450C2#1311566811yescharacteristic
451C2#1311567211yescharacteristic
452C2#1311576811yescharacteristic
subgroupG/H
centerC23C2×#6103
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#1434C2
socleC23×C13C2×C22⋊C4
prime radicalC2×#88184C22
cosocleC23×C4×C13C23
hypercenter#1434D13
hypocenterC13#9716
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #13114816C2semidirect
3. C2×#11718044C2semidirect
5. C13×#1434C2semidirect
9. C2×#88129C4semidirect
18. #11718600C4semidirect
19. #11718597C4semidirect
20. #11718590C4semidirect
125. #1434D13semidirect
346. C13#9716semidirect

Maximal subgroups

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

subgroupindex
#131231372
#131148162
C2×#117180442
C2×#117180432
C13×#14342
#971613