Finite Group Databasebeta

← #15753808

Subgroups — #15753808

Order 3328.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13114598C211yesmaximal, characteristic
2C2×#11718130C212yesmaximal
3#13113063C211yesmaximal, characteristic
4#13112714C212yesmaximal
5C4×C13×C22⋊Q8C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8×C13C411yescharacteristic, complemented
8C22×C42×C13C2211yescharacteristic
9C13×#391C411yescharacteristic, complemented
10C13×#389C411yescharacteristic, complemented
11C13×#387—21no—
12C13×#385—21no—
13C13×#380C2211yescharacteristic
14C4×C13×C4⋊C4—21no—
15C4×C13×C4⋊C4C411yescharacteristic, complemented
16C4×C13×C22⋊C4—21no—
17C2×#88071—21no—
18C2×#88071—21no—
19C2×#88071C2212yes—
20C2×C22⋊#1055C2212yes—
21C2×#88043—21no—
22#11718130C418yescomplemented
23#11718116—21no—
24C22×Q8×C13C2×C411yescharacteristic
25C23×C4×C13D411yescharacteristic
26C23×C4×C13D411yescharacteristic
27C23×C4×C13C2311yescharacteristic
28C23×Dic13—22no—
29C13×C22⋊Q8—22no—
30C13×C22⋊Q8C2×C414yes—
31C4×Q8×C13—42no—
32C2×C13×C4⋊C4—21no—
33C2×C13×C4⋊C4—21no—
34C2×C13×C4⋊C4—21no—
35C2×C13×C4⋊C4C2×C411yescharacteristic
36C2×C13×C4⋊C4C2×C411yescharacteristic
37C2×C13×C4⋊C4C2×C411yescharacteristic
38C2×C13×C22⋊C4—21no—
39C2×C13×C22⋊C4—21no—
40C2×C42×C13—21no—
41C2×C42×C13—21no—
42C2×C42×C13—42no—
43C2×C42×C13C2×C411yescharacteristic
44C2×C42×C13C2×C411yescharacteristic
45C2×C42×C13D411yescharacteristic
46C2×C42×C13D411yescharacteristic
47C22×#1055—22no—
48C13×#95—21no—
49C13×#95—21no—
50C13×#95—21no—
51C13×#95—21no—
52C13×#95—21no—
53#88071—22no—
54#88071—42no—
55#88071—42no—
56#88071C2×C414yes—
57C22⋊#1055—22no—
58C22⋊#1055C2×C414yes—
59#88043—42no—
60#8217—131nomaximal, Sylow
61C24×C13C2×D411yescharacteristic
62C2×Q8×C13—21no—
63C2×Q8×C13—41no—
64C2×Q8×C13—41no—
65C2×Q8×C13C4212yes—
66C22×C4×C13—21no—
67C22×C4×C13—21no—
68C22×C4×C13—21no—
69C22×C4×C13—22no—
70C22×C4×C13—21no—
71C22×C4×C13—22no—
72C22×C4×C13—21no—
73C22×C4×C13—21no—
74C22×C4×C13—41no—
75C22×C4×C13—41no—
76C22×C4×C13—41no—
77C22×C4×C13—41no—
78C22×C4×C13—41no—
79C22×C4×C13C22⋊C411yescharacteristic
80C22×C4×C13C22⋊C411yescharacteristic
81C22×C4×C13C22⋊C411yescharacteristic
82C22×C4×C13C22⋊C411yescharacteristic
83C22×C4×C13C22×C412yes—
84C22×C4×C13C22×C411yescharacteristic
85C22×C4×C13C2×D411yescharacteristic
86C22×C4×C13#4112yes—
87C22×Dic13—24no—
88C22×Dic13—42no—
89C22×Dic13—42no—
90C22×Dic13—42no—
91C22×Dic13—42no—
92C22×Dic13—42no—
93C22×Dic13—42no—
94C13×C4⋊C4—21no—
95C13×C4⋊C4—24no—
96C13×C4⋊C4—41no—
97C13×C4⋊C4—41no—
98C13×C4⋊C4—42no—
99C13×C4⋊C4—42no—
100C13×C4⋊C4C4212yes—
101C13×C22⋊C4—24no—
102C13×C22⋊C4—41no—
103C13×C22⋊C4—41no—
104C42×C13—22no—
105C42×C13—22no—
106C42×C13—41no—
107C42×C13—41no—
108C42×C13—42no—
109C42×C13—42no—
110C2×#1055—24no—
111C2×#1055—42no—
112C2×#1055—42no—
113C4×C22⋊Q8D1311yescharacteristic
114#1749—131no—
115C2×#328—132no—
116#1434—131no—
117#1294—132no—
118C23×C13—21no—
119C23×C13—21no—
120C23×C13—21no—
121C23×C13—21no—
122C23×C13C2×C22⋊C411yescharacteristic
123C23×C13C4×D412yes—
124C23×C13AGL(1,ℤ/8)11yescharacteristic
125C23×C13#13711yescharacteristic
126C23×C13#13712yes—
127Q8×C13—42no—
128Q8×C13—42no—
129C2×C4×C13—21no—
130C2×C4×C13—21no—
131C2×C4×C13—21no—
132C2×C4×C13—21no—
133C2×C4×C13—21no—
134C2×C4×C13—21no—
135C2×C4×C13—22no—
136C2×C4×C13—22no—
137C2×C4×C13—21no—
138C2×C4×C13—21no—
139C2×C4×C13—21no—
140C2×C4×C13—21no—
141C2×C4×C13—21no—
142C2×C4×C13—21no—
143C2×C4×C13—21no—
144C2×C4×C13—21no—
145C2×C4×C13—21no—
146C2×C4×C13—42no—
147C2×C4×C13—42no—
148C2×C4×C13—44no—
149C2×C4×C13—42no—
150C2×C4×C13—42no—
151C2×C4×C13—41no—
152C2×C4×C13—41no—
153C2×C4×C13—41no—
154C2×C4×C13—41no—
155C2×C4×C13—41no—
156C2×C4×C13—41no—
157C2×C4×C13—41no—
158C2×C4×C13—41no—
159C2×C4×C13—41no—
160C2×C4×C13—41no—
161C2×C4×C13#9912yes—
162C2×C4×C13C2×C4211yescharacteristic
163C2×C4×C13#11711yescharacteristic
164C2×C4×C13C4×D412yes—
165C2×Dic13—44no—
166C2×Dic13—44no—
167C2×Dic13—44no—
168C2×Dic13—82no—
169C2×Dic13—82no—
170C2×Dic13—82no—
171C2×Dic13—82no—
172#1055—44no—
173C2×C22⋊Q8C2×D1311yescharacteristic
174C2×C4×Q8Dic1311yescharacteristic, complemented
175C22×C42C2×D1311yescharacteristic
176C2×C22⋊C8—132no—
177C2×#97—261no—
178#391Dic1311yescharacteristic, complemented
179#389Dic1311yescharacteristic, complemented
180#387—21no—
181#385—21no—
182#380C2×D1311yescharacteristic
183C4×C4⋊C4—21no—
184C4×C4⋊C4Dic1311yescharacteristic, complemented
185C4×C22⋊C4—21no—
186C2×#95—132no—
187C2×#95—261no—
188C2×#95—261no—
189#336—261no—
190#328—138no—
191C22×C13—22no—
192C22×C13—22no—
193C22×C13—22no—
194C22×C13—21no—
195C22×C13—21no—
196C22×C13—21no—
197C22×C13—21no—
198C22×C13—22no—
199C22×C13C4×C22⋊C411yescharacteristic
200C22×C13C2×#9911yescharacteristic
201C22×C13#41011yescharacteristic
202C22×C13#41711yescharacteristic
203C22×C13#41911yescharacteristic
204C22×C13#42111yescharacteristic
205C22×C13#42111yescharacteristic
206C22×C13#44012yes—
207C22×C13#44112yes—
208C4×C13—22no—
209C4×C13—21no—
210C4×C13—21no—
211C4×C13—22no—
212C4×C13—42no—
213C4×C13—42no—
214C4×C13—41no—
215C4×C13—41no—
216C4×C13—41no—
217C4×C13—81no—
218C4×C13—81no—
219Dic13—84no—
220C22×Q8C2×Dic1311yescharacteristic
221C23×C4—262no—
222C23×C4C22⋊D1311yescharacteristic
223C23×C4C22⋊D1311yescharacteristic
224C23×C4C22×D1311yescharacteristic
225C22×C8—262no—
226C22⋊Q8—22no—
227C22⋊Q8C4×D1314yes—
228C4×Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C2×Dic1311yescharacteristic
233C2×C4⋊C4C2×Dic1311yescharacteristic
234C2×C4⋊C4C2×Dic1311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42C2×Dic1311yescharacteristic
241C2×C42C2×Dic1311yescharacteristic
242C2×C42C22⋊D1311yescharacteristic
243C2×C42C22⋊D1311yescharacteristic
244C22⋊C8—134no—
245C22⋊C8—262no—
246#97—522no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—134no—
253#95—262no—
254#95—522no—
255#95—522no—
256C2×C13—22no—
257C2×C13—22no—
258C2×C13#151111yescharacteristic
259C2×C13#151211yescharacteristic
260C2×C13#151311yescharacteristic
261C2×C13#151511yescharacteristic
262C2×C13C4×#9911yescharacteristic
263C2×C13#175911yescharacteristic
264C2×C13#176311yescharacteristic
265C24C2×C22⋊D1311yescharacteristic
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8C4×Dic1312yes—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—264no—
284C22×C4—522no—
285C22×C4—522no—
286C22×C4—522no—
287C22×C4—522no—
288C22×C4—522no—
289C22×C4—522no—
290C22×C4C22⋊Dic1311yescharacteristic
291C22×C4C22⋊Dic1311yescharacteristic
292C22×C4C22⋊Dic1311yescharacteristic
293C22×C4C22⋊Dic1311yescharacteristic
294C22×C4C2×C4×D1312yes—
295C22×C4#612712yes—
296C22×C4C22×Dic1311yescharacteristic
297C22×C4C2×C22⋊D1311yescharacteristic
298C2×C8—264no—
299C2×C8—522no—
300C2×C8—522no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4C4×Dic1312yes—
308C22⋊C4—24no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—22no—
312C42—22no—
313C42—41no—
314C42—41no—
315C42—42no—
316C42—42no—
317C13#821711yescharacteristic, complemented, Sylow
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23C4×C22⋊D1312yes—
323C23#8819612yes—
324C23#8820311yescharacteristic
325C23#8820511yescharacteristic
326C23C2×C22⋊Dic1311yescharacteristic
327Q8—42no—
328Q8—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—524no—
362C2×C4—524no—
363C2×C4—524no—
364C2×C4—1042no—
365C2×C4—1042no—
366C2×C4—1042no—
367C2×C4—1042no—
368C2×C4#8807412yes—
369C2×C4C2×C4×Dic1311yescharacteristic
370C2×C4#8818011yescharacteristic
371C2×C4C4×C22⋊D1312yes—
372C8—524no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22#1171868712yes—
382C22#1171869112yes—
383C22C4×C22⋊Dic1311yescharacteristic
384C22C2×#8807411yescharacteristic
385C22#1171888611yescharacteristic
386C22#1171889311yescharacteristic
387C22#1171889711yescharacteristic
388C22#1171889711yescharacteristic
389C22#1171890111yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—1044no—
402C2—22no—
403C2—22no—
404C2#1311336911yescharacteristic
405C2#1311337411yescharacteristic
406C2#1311337911yescharacteristic
407C2#1311338711yescharacteristic
408C2C4×#8807411yescharacteristic
409C2#1311464611yescharacteristic
410C2#1311466611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic13
commutatorC2×C4×C13C2×C42
FrattiniC23×C4C22×D13
FittingC4×C13×C22⋊Q8C2
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenterC4×C22⋊Q8D13
hypocenterC13#8217
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8×C13C4semidirect
9. C13×#391C4semidirect
10. C13×#389C4semidirect
15. C4×C13×C4⋊C4C4semidirect
22. #11718130C4semidirect
174. C2×C4×Q8Dic13semidirect
178. #391Dic13semidirect
179. #389Dic13semidirect
184. C4×C4⋊C4Dic13semidirect
317. C13#8217semidirect

Maximal subgroups

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

subgroupindex
#131145982
C2×#117181302
#131130632
#131127142
C4×C13×C22⋊Q82
#821713