Finite Group Databasebeta

← #16780664

Subgroups — #16780664

Order 3328.

720 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130563C211yesmaximal, characteristic
2C2×#11718241C212yesmaximal
3C4×(C2×#261)⋊C4C211yesmaximal, characteristic
4#13129709C212yesmaximal
5C4×#88151C211yesmaximal, characteristic
6C22×C4×#261—21no—
7C2×#88151C2211yescharacteristic
8C2×C42×D13C2211yescharacteristic
9C2×C4×Dic26C411yescharacteristic, complemented
10C2×(C2×#261)⋊C4—21no—
11C2×(C2×#261)⋊C4C2212yes—
12C2×#88111C2212yes—
13C2×#88110—21no—
14C4×C13×C4⋊C4C411yescharacteristic, complemented
15#11718648C2211yescharacteristic
16#11718646C411yescharacteristic, complemented
17#11718635C411yescharacteristic, complemented
18C4×#6103—21no—
19C4×C4⋊Dic13—21no—
20#11718329—21no—
21#11718314—21no—
22#11718290—21no—
23#11718241C418yescomplemented
24C23×#261—22no—
25C22×C4×D13D411yescharacteristic
26C22×C4×D13D411yescharacteristic
27C22×C4×D13C2311yescharacteristic
28C22×Dic26C2×C411yescharacteristic
29C22×#1057—22no—
30C2×C4×#261—24no—
31C2×C4×#261—42no—
32C2×C4×#261—42no—
33C2×C13×C4⋊C4C2×C411yescharacteristic
34C2×C13×C4⋊C4C2×C411yescharacteristic
35C2×C42×C13—21no—
36C2×C42×C13C2×C411yescharacteristic
37C2×#6103—21no—
38C2×#6103—21no—
39C2×C4⋊Dic13—21no—
40C2×C4⋊Dic13—21no—
41C2×C4⋊Dic13C2×C411yescharacteristic
42C2×#6101—21no—
43C2×C4×Dic13—21no—
44C2×C4×Dic13C2×C411yescharacteristic
45C2×C4×Dic13D411yescharacteristic
46C2×C4×Dic13D411yescharacteristic
47#88151—22no—
48#88151C2×C414yes—
49C42×D13—42no—
50C4×Dic26—42no—
51(C2×#261)⋊C4—24no—
52(C2×#261)⋊C4—22no—
53(C2×#261)⋊C4C2×C414yes—
54#88111—22no—
55#88111C2×C414yes—
56#88110—42no—
57C13×#95—21no—
58#88071—21no—
59#88071—21no—
60#88071—21no—
61#88071—21no—
62#8219—131nomaximal, Sylow
63C23×D13C2×D411yescharacteristic
64C22×#261—24no—
65C22×#261—24no—
66C22×#261—24no—
67C22×#261—42no—
68C22×#261—42no—
69C22×#261—42no—
70C22×#261—42no—
71C22×C4×C13—21no—
72C22×C4×C13—21no—
73C22×C4×C13C22⋊C411yescharacteristic
74C22×C4×C13C22⋊C411yescharacteristic
75C22×C4×C13C22×C411yescharacteristic
76C22×Dic13—21no—
77C22×Dic13—21no—
78C22×Dic13C22⋊C411yescharacteristic
79C22×Dic13C22⋊C411yescharacteristic
80C22×Dic13C2×D411yescharacteristic
81C2×C4×D13—21no—
82C2×C4×D13—21no—
83C2×C4×D13—22no—
84C2×C4×D13—22no—
85C2×C4×D13—22no—
86C2×C4×D13—22no—
87C2×C4×D13—41no—
88C2×C4×D13—41no—
89C2×C4×D13—41no—
90C2×C4×D13C22×C412yes—
91C2×C4×D13#4112yes—
92C2×Dic26—21no—
93C2×Dic26—41no—
94C2×Dic26—41no—
95C2×Dic26C4212yescomplemented
96C2×#1057—24no—
97C2×#1057—42no—
98C2×#1057—42no—
99C4×#261—48no—
100C13×C4⋊C4—21no—
101C13×C4⋊C4—41no—
102C13×C4⋊C4C4212yescomplemented
103C42×C13—22no—
104C42×C13—42no—
105#6103—24no—
106#6103—41no—
107#6103—41no—
108C4⋊Dic13—24no—
109C4⋊Dic13—42no—
110C4⋊Dic13—41no—
111#6101—42no—
112C4×Dic13—22no—
113C4×Dic13—41no—
114C4×Dic13—41no—
115C4×Dic13—42no—
116C4×C22⋊Q8—131no—
117#1749—131no—
118C2×#328—132no—
119C4×#95—131no—
120#1294—132no—
121C23×C13C2×C22⋊C411yescharacteristic
122C22×D13—21no—
123C22×D13—21no—
124C22×D13—21no—
125C22×D13—21no—
126C22×D13C4×D412yes—
127C22×D13AGL(1,ℤ/8)11yescharacteristic
128C22×D13#13712yes—
129C22×D13#13711yescharacteristic
130C2×#261—28no—
131C2×#261—44no—
132C2×#261—44no—
133C2×#261—44no—
134C2×#261—44no—
135C2×#261—44no—
136C2×#261—44no—
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—22no—
144C2×C4×C13—41no—
145C2×C4×C13—41no—
146C2×C4×C13—41no—
147C2×C4×C13—41no—
148C2×C4×C13—41no—
149C2×C4×C13#9912yes—
150C2×C4×C13#10412yes—
151C2×C4×C13C2×C4211yescharacteristic
152C2×C4×C13#11711yescharacteristic
153C2×Dic13—21no—
154C2×Dic13—21no—
155C2×Dic13—21no—
156C2×Dic13—21no—
157C2×Dic13—22no—
158C2×Dic13—21no—
159C2×Dic13—21no—
160C2×Dic13—21no—
161C2×Dic13—41no—
162C2×Dic13—41no—
163C2×Dic13—41no—
164C2×Dic13—41no—
165C2×Dic13—41no—
166C2×Dic13#10412yes—
167C2×Dic13C4×D412yes—
168C4×D13—24no—
169C4×D13—24no—
170C4×D13—42no—
171C4×D13—42no—
172C4×D13—44no—
173Dic26—42no—
174Dic26—42no—
175#1057—44no—
176C2×C22⋊Q8—131no—
177C2×C4×Q8—131no—
178C22×C42—131no—
179C22×C42—261no—
180C2×C22⋊C8—132no—
181C2×#97—261no—
182#391—131no—
183#389—131no—
184#387—261no—
185#385—261no—
186#380—131no—
187C4×C4⋊C4—261no—
188C4×C4⋊C4#26111yescharacteristic, complemented
189C4×C22⋊C4—261no—
190C2×#95—132no—
191C2×#95—261no—
192#336—261no—
193#328—138no—
194C22×C13C4×C22⋊C411yescharacteristic
195C22×C13C2×#9911yescharacteristic
196C22×C13#41011yescharacteristic
197C22×C13#41711yescharacteristic
198C22×C13#41911yescharacteristic
199C22×C13C2×#10411yescharacteristic
200C22×C13C2×#10411yescharacteristic
201C2×D13—22no—
202C2×D13—22no—
203C2×D13—21no—
204C2×D13—21no—
205C2×D13—22no—
206C2×D13—22no—
207C2×D13—21no—
208C2×D13—21no—
209C2×D13#44012yes—
210C2×D13#44112yes—
211#261—48no—
212C4×C13—21no—
213C4×C13—21no—
214C4×C13—22no—
215C4×C13—22no—
216C4×C13—42no—
217C4×C13—81no—
218Dic13—22no—
219Dic13—22no—
220Dic13—42no—
221Dic13—41no—
222Dic13—81no—
223C22×Q8—131no—
224C23×C4—131no—
225C23×C4—131no—
226C23×C4—131no—
227C23×C4—262no—
228C22×C8—262no—
229C22⋊Q8—134no—
230C22⋊Q8—262no—
231C4×Q8—522no—
232C2×C4⋊C4—131no—
233C2×C4⋊C4—261no—
234C2×C4⋊C4—261no—
235C2×C4⋊C4—261no—
236C2×C4⋊C4C2×#26111yescharacteristic
237C2×C4⋊C4C2×#26111yescharacteristic
238C2×C22⋊C4—261no—
239C2×C22⋊C4—261no—
240C2×C42—21no—
241C2×C42—131no—
242C2×C42—131no—
243C2×C42—131no—
244C2×C42—264no—
245C2×C42—261no—
246C2×C42—522no—
247C2×C42—522no—
248C2×C42—522no—
249C2×C42C2×#26111yescharacteristic
250C22⋊C8—134no—
251C22⋊C8—262no—
252#97—522no—
253#95—21no—
254#95—134no—
255#95—261no—
256#95—261no—
257#95—261no—
258#95—261no—
259#95—262no—
260#95—264no—
261C2×C13#150411yescharacteristic
262C2×C13#150611yescharacteristic
263C2×C13#150811yescharacteristic
264C2×C13#150911yescharacteristic
265C2×C13C4×#9911yescharacteristic
266C2×C13C4×#10411yescharacteristic
267C2×C13#176311yescharacteristic
268D13—22no—
269D13—22no—
270C24—131no—
271C2×Q8—132no—
272C2×Q8—261no—
273C2×Q8—521no—
274C2×Q8—521no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—131no—
278C22×C4—131no—
279C22×C4—132no—
280C22×C4—132no—
281C22×C4—131no—
282C22×C4—262no—
283C22×C4—261no—
284C22×C4—261no—
285C22×C4—262no—
286C22×C4—262no—
287C22×C4—264no—
288C22×C4—264no—
289C22×C4—264no—
290C22×C4—261no—
291C22×C4—262no—
292C22×C4—261no—
293C22×C4—521no—
294C22×C4—521no—
295C22×C4—521no—
296C22×C4—522no—
297C22×C4—522no—
298C22×C4—522no—
299C22×C4—522no—
300C22×C4C22⋊#26111yescharacteristic
301C22×C4C22⋊#26111yescharacteristic
302C22×C4C22×#26111yescharacteristic
303C2×C8—264no—
304C2×C8—522no—
305C2×C8—522no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—264no—
309C4⋊C4—521no—
310C4⋊C4—522no—
311C4⋊C4—522no—
312C4⋊C4C4×#26112yescomplemented
313C22⋊C4—264no—
314C22⋊C4—521no—
315C22⋊C4—521no—
316C42—22no—
317C42—42no—
318C42—262no—
319C42—521no—
320C42—521no—
321C42—522no—
322C42—528no—
323C13#821911yescharacteristic, complemented, Sylow
324C23—132no—
325C23—132no—
326C23—131no—
327C23—131no—
328C23—261no—
329C23—261no—
330C23—261no—
331C23—261no—
332C23C2×C22⋊#26111yescharacteristic
333Q8—522no—
334Q8—522no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—132no—
348C2×C4—132no—
349C2×C4—264no—
350C2×C4—264no—
351C2×C4—261no—
352C2×C4—261no—
353C2×C4—261no—
354C2×C4—261no—
355C2×C4—268no—
356C2×C4—261no—
357C2×C4—262no—
358C2×C4—261no—
359C2×C4—261no—
360C2×C4—522no—
361C2×C4—522no—
362C2×C4—524no—
363C2×C4—524no—
364C2×C4—524no—
365C2×C4—524no—
366C2×C4—524no—
367C2×C4—524no—
368C2×C4—524no—
369C2×C4—521no—
370C2×C4—521no—
371C2×C4—521no—
372C2×C4—521no—
373C2×C4—521no—
374C2×C4#8810712yes—
375C2×C4#8811612yes—
376C2×C4C2×C4×#26111yescharacteristic
377C2×C4#8823711yescharacteristic
378C8—524no—
379C22—132no—
380C22—132no—
381C22—262no—
382C22—262no—
383C22—261no—
384C22—261no—
385C22—262no—
386C22—262no—
387C22—261no—
388C22—261no—
389C22C2×#8810711yescharacteristic
390C22#1171911611yescharacteristic
391C22C4×C22⋊#26111yescharacteristic
392C22C2×#8811611yescharacteristic
393C22#1171915311yescharacteristic
394C22D13⋊#10411yescharacteristic
395C22#1171916411yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—42no—
401C4—81no—
402C4—262no—
403C4—262no—
404C4—528no—
405C4—522no—
406C4—521no—
407C4—1041no—
408C2—262no—
409C2—262no—
410C2#1313022411yescharacteristic
411C2#1313022911yescharacteristic
412C2#1313023511yescharacteristic
413C2#1313023811yescharacteristic
414C2C4×#8810711yescharacteristic
415C2C4×#8811611yescharacteristic
416C2#1313060811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC2×C4×C13C2×C42
FrattiniC22×C4C22×#261
FittingC4×C13×C4⋊C4C4
socleC23×C13C2×C22⋊C4
prime radicalC23×D13C2×D4
cosocleC22×C4×D13C23
hypercenterC4×C4⋊C4#261
hypocenterC13#8219
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. C2×C4×Dic26C4semidirect
14. C4×C13×C4⋊C4C4semidirect
16. #11718646C4semidirect
17. #11718635C4semidirect
23. #11718241C4semidirect
95. C2×Dic26C42semidirect
102. C13×C4⋊C4C42semidirect
188. C4×C4⋊C4#261semidirect
312. C4⋊C4C4×#261semidirect
323. C13#8219semidirect

Maximal subgroups

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

subgroupindex
#131305632
C2×#117182412
C4×(C2×#261)⋊C42
#131297092
C4×#881512
#821913