Finite Group Databasebeta

← #15757692

Subgroups — #15757692

Order 3328.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718688C211yesmaximal, characteristic
2Q8×C4⋊Dic13C211yesmaximal, characteristic, complemented
3#13115762C211yesmaximal, characteristic
4#13115642C211yesmaximal, characteristic
5#13112720C211yesmaximal, characteristic, complemented
6#13112676C211yesmaximal, characteristic, complemented
7C2×C13×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×C4×Q8×C13C2211yescharacteristic
9C2×Q8×Dic13—21no—
10C22×#6106—21no—
11C2×#88128C2211yescharacteristic
12C2×C13×C4⋊C8C2211yescharacteristic
13C2×C4×C8×C13C2211yescharacteristic
14C2×#88061—21no—
15C2×#88061—21no—
16C2×#88061—22no—
17#11718688—22no—
18#11718688C414yescomplemented
19#11718646—21no—
20#11718646—22no—
21C2×#88049—21no—
22#11718596—21no—
23#11718596C2211yescharacteristic
24C4×C4⋊Dic13—21no—
25C4×C4⋊Dic13C2211yescharacteristic
26C2×C4⋊#1055C2211yescharacteristic
27C13×Q8⋊C8—22no—
28C22×Q8×C13D411yescharacteristic
29C22×D52—21no—
30C22×D52—21no—
31C22×D52—21no—
32C22×C8×C13—21no—
33C22×C8×C13—21no—
34C4×Q8×C13—21no—
35C4×Q8×C13—21no—
36C4×Q8×C13C2×C412yes—
37C4×Q8×C13D412yes—
38C2×C13×C4⋊C4—21no—
39C2×C13×C4⋊C4D411yescharacteristic
40C2×C42×C13—21no—
41C2×C42×C13C2311yescharacteristic
42Q8×Dic13—42no—
43Q8×Dic13—42no—
44C2×#6106—24no—
45C2×#6106—42no—
46C2×#6106—42no—
47C2×C4⋊Dic13—22no—
48C2×C4⋊Dic13—21no—
49C2×C4⋊Dic13—21no—
50C2×C4⋊Dic13—22no—
51C2×C4⋊Dic13D411yescharacteristic
52C2×C4⋊Dic13D411yescharacteristic
53C2×C4⋊Dic13D412yes—
54C2×C4×Dic13—22no—
55C2×C4×Dic13—21no—
56C22×#1055—21no—
57#88128—21no—
58#88128—21no—
59#88128—21no—
60#88128C2×C412yes—
61C13×C4⋊C8—21no—
62C13×C4⋊C8—21no—
63C4×C8×C13—21no—
64C4×C8×C13—21no—
65#88071—22no—
66#88071—21no—
67#88061—42no—
68#88061—44no—
69#88061—42no—
70#88049—24no—
71C4⋊#1055—21no—
72C4⋊#1055C2×C412yes—
73#8979—131nomaximal, Sylow
74C23×D13—41no—
75C2×Q8×C13—21no—
76C2×Q8×C13—21no—
77C2×Q8×C13—21no—
78C2×Q8×C13C22⋊C412yes—
79C2×Q8×C13D812yes—
80C2×Q8×C13SD1612yes—
81C22×C4×C13—21no—
82C22×C4×C13—21no—
83C22×C4×C13C2×D411yescharacteristic
84C22×C4×C13C2×D411yescharacteristic
85C22×C4×C13C2×D411yescharacteristic
86C22×Dic13—21no—
87C22×Dic13—21no—
88C22×Dic13—22no—
89C2×D52—22no—
90C2×D52—22no—
91C2×D52—22no—
92C2×D52—42no—
93C2×D52—42no—
94C2×D52—41no—
95C2×D52—41no—
96C2×D52—41no—
97C2×D52—41no—
98C2×D52—41no—
99C2×D52—41no—
100C2×D52—41no—
101C2×D52—41no—
102C2×D52—41no—
103C2×D52—41no—
104C2×D52—41no—
105C2×C8×C13—22no—
106C2×C8×C13—41no—
107C2×C8×C13—41no—
108C2×C8×C13—41no—
109C2×C8×C13—41no—
110C2×C8×C13—41no—
111C13×C4⋊C4—22no—
112C13×C4⋊C4—22no—
113C13×C4⋊C4—21no—
114C13×C4⋊C4C22⋊C412yes—
115C42×C13—22no—
116C42×C13—22no—
117C42×C13C22×C411yescharacteristic
118C42×C13C2×D411yescharacteristic
119C42×C13#4111yescharacteristic
120C42×C13#4111yescharacteristic
121#6106—48no—
122C4⋊Dic13—21no—
123C4⋊Dic13—21no—
124C4⋊Dic13—21no—
125C4⋊Dic13—44no—
126C4⋊Dic13—41no—
127C4⋊Dic13—41no—
128C4⋊Dic13—41no—
129C4⋊Dic13—41no—
130C4⋊Dic13—42no—
131C4⋊Dic13—82no—
132C4⋊Dic13SD1612yescomplemented
133C4×Dic13—44no—
134C4×Dic13—41no—
135C4×Dic13—41no—
136C2×#1055—22no—
137C2×#1055—41no—
138C2×#1055—41no—
139C2×#460—131no—
140Q8×C4⋊C4—131no—
141#1982—131no—
142#1957—131no—
143C2×Q8⋊C8D1311yescharacteristic, complemented
144#1296—131no—
145#1279—131no—
146C23×C13#12011yescharacteristic
147C22×D13—42no—
148C22×D13—81no—
149C22×D13—81no—
150C22×D13—81no—
151C22×D13—81no—
152C22×D13—81no—
153C22×D13—81no—
154Q8×C13—22no—
155Q8×C13—22no—
156Q8×C13—22no—
157Q8×C13#10214yescomplemented
158C2×C4×C13—22no—
159C2×C4×C13—22no—
160C2×C4×C13—22no—
161C2×C4×C13—21no—
162C2×C4×C13—21no—
163C2×C4×C13—22no—
164C2×C4×C13—41no—
165C2×C4×C13—41no—
166C2×C4×C13C2×C22⋊C411yescharacteristic
167C2×C4×C13C4×D411yescharacteristic
168C2×C4×C13C4×D411yescharacteristic
169C2×C4×C13#12111yescharacteristic
170C2×C4×C13#12111yescharacteristic
171C2×C4×C13#12311yescharacteristic
172C2×C4×C13C2×D811yescharacteristic
173C2×C4×C13C2×SD1611yescharacteristic
174C2×C4×C13C2×SD1611yescharacteristic
175C2×C4×C13#13511yescharacteristic
176C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C13#13711yescharacteristic
179C2×C4×C13#13711yescharacteristic
180C2×Dic13—22no—
181C2×Dic13—41no—
182C2×Dic13—41no—
183C2×Dic13—42no—
184C2×Dic13—41no—
185C2×Dic13—41no—
186C2×Dic13—42no—
187C2×Dic13—42no—
188C2×Dic13—41no—
189D52—44no—
190D52—42no—
191D52—42no—
192D52—42no—
193D52—42no—
194D52—82no—
195D52—81no—
196D52—81no—
197D52—81no—
198D52—81no—
199C8×C13—42no—
200C8×C13—81no—
201#1055—42no—
202C22×SD16—261no—
203C2×#127—131no—
204C2×C4×Q8—261no—
205C2×C4×Q8C2×D1311yescharacteristic
206#460—134no—
207#460—262no—
208C2×C4⋊C8—131no—
209C2×C4⋊C8C2×D1311yescharacteristic
210C2×#102—261no—
211C2×#102—262no—
212C2×#102—261no—
213C2×#102—261no—
214C2×C4×C8C2×D1311yescharacteristic
215#389—262no—
216#389—261no—
217#384—131no—
218#384—261no—
219C4×C4⋊C4—131no—
220C4×C4⋊C4—261no—
221Q8⋊C8—22no—
222C22×C13#38611yescharacteristic
223C22×C13#44711yescharacteristic
224C22×C13#44811yescharacteristic
225C22×C13#44811yescharacteristic
226C22×C13#44911yescharacteristic
227C22×C13#45311yescharacteristic
228C22×C13#45411yescharacteristic
229C2×D13—82no—
230C2×D13—82no—
231C2×D13—82no—
232C2×D13—161no—
233C2×D13—161no—
234C2×D13—161no—
235C2×D13—161no—
236C4×C13—24no—
237C4×C13—21no—
238C4×C13—21no—
239C4×C13—42no—
240C4×C13C2×#10211yescharacteristic
241C4×C13#41711yescharacteristic
242C4×C13C4×SD1611yescharacteristic
243C4×C13#44011yescharacteristic
244C4×C13#46011yescharacteristic
245C4×C13#46411yescharacteristic
246C4×C13#48011yescharacteristic
247C4×C13#48111yescharacteristic
248Dic13—42no—
249Dic13—81no—
250Dic13—82no—
251C22×Q8D5211yescharacteristic
252C22×D4—261no—
253C22×D4—261no—
254C22×D4—261no—
255C2×SD16—264no—
256C2×SD16—522no—
257C2×SD16—522no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—261no—
261#127—132no—
262#127—261no—
263#127—261no—
264#127—261no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—522no—
268C4×Q8—522no—
269C4×Q8C4×D1312yes—
270C4×Q8C22⋊D1312yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—131no—
273C2×C4⋊C4—131no—
274C2×C4⋊C4—132no—
275C2×C4⋊C4—262no—
276C2×C4⋊C4—261no—
277C2×C4⋊C4—261no—
278C2×C4⋊C4—262no—
279C2×C4⋊C4D5211yescharacteristic
280C2×C42—21no—
281C2×C42—262no—
282C2×C42—261no—
283C2×C42C22×D1311yescharacteristic
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—132no—
287C4⋊C8—261no—
288#102—264no—
289#102—522no—
290#102—522no—
291#102—524no—
292C4×C8—21no—
293C4×C8—21no—
294#95—262no—
295#95—261no—
296C2×C13#162111yescharacteristic
297C2×C13#162311yescharacteristic
298C2×C13#162611yescharacteristic
299C2×C13#162711yescharacteristic
300C2×C13#187411yescharacteristic
301C2×C13#188011yescharacteristic
302C2×C13#188311yescharacteristic
303D13—162no—
304C24—521no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8#609512yes—
309C2×Q8D10412yes—
310C2×Q8#610312yes—
311C2×D4—262no—
312C2×D4—262no—
313C2×D4—262no—
314C2×D4—522no—
315C2×D4—522no—
316C2×D4—521no—
317C2×D4—521no—
318C2×D4—521no—
319C2×D4—521no—
320C2×D4—521no—
321C2×D4—521no—
322C2×D4—521no—
323C2×D4—521no—
324C2×D4—521no—
325C2×D4—521no—
326C2×D4—521no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—261no—
330C22×C4—261no—
331C22×C4—262no—
332C22×C4C2×D5211yescharacteristic
333C22×C4D4×D1311yescharacteristic
334C22×C4D4×D1311yescharacteristic
335SD16—528no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—262no—
343C2×C8—521no—
344C2×C8—521no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—132no—
349C4⋊C4—261no—
350C4⋊C4—261no—
351C4⋊C4—261no—
352C4⋊C4—522no—
353C4⋊C4—524no—
354C4⋊C4—521no—
355C4⋊C4—521no—
356C4⋊C4—521no—
357C4⋊C4—521no—
358C4⋊C4—1042no—
359C4⋊C4#610312yes—
360C42—22no—
361C42—22no—
362C42—524no—
363C42—521no—
364C42—521no—
365C42C2×C4×D1311yescharacteristic
366C42#612911yescharacteristic
367C42#612911yescharacteristic
368C42C2×C22⋊D1311yescharacteristic
369C13#897911yescharacteristic, complemented, Sylow
370C23—522no—
371C23—1041no—
372C23—1041no—
373C23—1041no—
374C23—1041no—
375C23—1041no—
376C23—1041no—
377C23#8813611yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#8806114yescomplemented
382D4—524no—
383D4—522no—
384D4—522no—
385D4—522no—
386D4—522no—
387D4—1042no—
388D4—1041no—
389D4—1041no—
390D4—1041no—
391D4—1041no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—262no—
401C2×C4—521no—
402C2×C4—521no—
403C2×C4—521no—
404C2×C4—522no—
405C2×C4—522no—
406C2×C4—521no—
407C2×C4—522no—
408C2×C4—521no—
409C2×C4#8813511yescharacteristic
410C2×C4#8813511yescharacteristic
411C2×C4#8814011yescharacteristic
412C2×C4C2×#609511yescharacteristic
413C2×C4C2×D10411yescharacteristic
414C2×C4#8816211yescharacteristic
415C2×C4#8816311yescharacteristic
416C2×C4SD16×D1311yescharacteristic
417C2×C4#8816811yescharacteristic
418C2×C4#8817211yescharacteristic
419C2×C4#8817311yescharacteristic
420C2×C4C2×#610311yescharacteristic
421C2×C4#8819311yescharacteristic
422C2×C4#8819311yescharacteristic
423C8—42no—
424C8—81no—
425C8—522no—
426C22—1042no—
427C22—1042no—
428C22—1042no—
429C22—2081no—
430C22—2081no—
431C22—2081no—
432C22—2081no—
433C22#1171839311yescharacteristic
434C22#1171839911yescharacteristic
435C22#1171846711yescharacteristic
436C22#1171847211yescharacteristic
437C22#1171848311yescharacteristic
438C22#1171848811yescharacteristic
439C22#1171861911yescharacteristic
440C4—24no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—522no—
445C4—1041no—
446C4—1042no—
447C4#1171839611yescharacteristic
448C4#1171839711yescharacteristic
449C4#1171844911yescharacteristic
450C4#1171847811yescharacteristic
451C4C2×#8806111yescharacteristic
452C4#1171879411yescharacteristic
453C4#1171882411yescharacteristic
454C4#1171884611yescharacteristic
455C2—2082no—
456C2#1311390311yescharacteristic
457C2#1311391611yescharacteristic
458C2#1311393511yescharacteristic
459C2#1311394111yescharacteristic
460C2#1311515111yescharacteristic
461C2#1311518911yescharacteristic
462C2#1311521011yescharacteristic
subgroupG/H
centerC23#88136
commutatorC42×C13C22×C4
FrattiniC2×C42C22×D13
FittingC2×C13×Q8⋊C8C2
socleC23×C13#120
prime radicalC2×#88128C22
cosocleC2×C42×C13C23
hypercenterC2×Q8⋊C8D13
hypocenterC13#8979
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic13C2semidirect
5. #13112720C2semidirect
6. #13112676C2semidirect
7. C2×C13×Q8⋊C8C2semidirect
18. #11718688C4semidirect
132. C4⋊Dic13SD16semidirect
143. C2×Q8⋊C8D13semidirect
157. Q8×C13#102semidirect
369. C13#8979semidirect
381. Q8#88061semidirect

Maximal subgroups

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

subgroupindex
C2×#117186882
Q8×C4⋊Dic132
#131157622
#131156422
#131127202
#131126762
C2×C13×Q8⋊C82
#897913