Finite Group Databasebeta

← #15848197

Subgroups — #15848197

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125774C211yesmaximal, characteristic
2C2×#11718688C211yesmaximal, characteristic
3#13123917C211yesmaximal, characteristic, complemented
4#13114254C214yesmaximal, complemented
5#13114186C214yesmaximal, complemented
6#13113608C211yesmaximal, characteristic, complemented
7#13113505C211yesmaximal, characteristic
8#13113493C211yesmaximal, characteristic, complemented
9C13×#3078C211yesmaximal, characteristic, complemented
10C13×#533C2211yescharacteristic
11C2×C4×Q8×C13C2211yescharacteristic
12C22×#6106—21no—
13C2×#88128C2211yescharacteristic
14C13×#479—22no—
15C13×#477C2212yescomplemented
16C13×#475C2212yescomplemented
17C13×#428—21no—
18C13×#423C2211yescharacteristic
19C13×#419—21no—
20#11718749C2212yes—
21#11718746C2212yes—
22#11718707—22no—
23#11718688—22no—
24#11718688—22no—
25#11718688C2214yes—
26Q8×#1055—22no—
27Q8×#1055C2214yescomplemented
28C2×#88049—21no—
29#11718586—22no—
30#11718586C2211yescharacteristic
31#11718586C2211yescharacteristic
32C2×C4⋊#1055C2211yescharacteristic
33#11718139C2212yescomplemented
34#11718138C2212yescomplemented
35#11718077C2214yes—
36#11718072C2214yescomplemented
37C22×Q8×C13D411yescharacteristic
38C22×D52—21no—
39C22×D52—21no—
40C22×D52—21no—
41C2×C13×M16—21no—
42C13×C4⋊Q8C2311yescharacteristic
43C13×C4⋊Q8C2311yescharacteristic
44C13×#125—21no—
45C13×C22⋊Q8—21no—
46C13×C22⋊Q8—21no—
47C4×Q8×C13—21no—
48C4×Q8×C13—21no—
49C4×Q8×C13—21no—
50C4×Q8×C13—21no—
51C4×Q8×C13D412yes—
52C4×Q8×C13C2312yes—
53C13×#117—21no—
54C2×C13×C4⋊C4—21no—
55C2×C13×C4⋊C4D411yescharacteristic
56C2×C42×C13—21no—
57C2×C42×C13C2311yescharacteristic
58C2×#6106—24no—
59C2×#6106—24no—
60C2×#6106—44no—
61C22×#1055—21no—
62#88128—21no—
63#88128—21no—
64#88128—21no—
65#88128C2312yes—
66C13×#107—22no—
67C13×#106—22no—
68C13×C4⋊C8—21no—
69C13×C4⋊C8C2312yes—
70C13×#103—22no—
71C13×#103—22no—
72C22⋊#1055—21no—
73C22⋊#1055D412yes—
74#88049—24no—
75#88049—24no—
76C4⋊#1055—21no—
77C4⋊#1055—24no—
78C4⋊#1055D412yes—
79C4⋊#1055C2312yes—
80C4⋊#1055C2312yes—
81C4×#1055—21no—
82C4×#1055—24no—
83C4×#1055C2312yes—
84#19396—131nomaximal, Sylow
85C23×D13—41no—
86C2×Q8×C13—21no—
87C2×Q8×C13—22no—
88C2×Q8×C13—21no—
89C2×Q8×C13—21no—
90C2×Q8×C13SD1614yescomplemented
91C2×Q8×C13C2×D412yes—
92C22×C4×C13—21no—
93C22×C4×C13—21no—
94C22×C4×C13C2×D411yescharacteristic
95C22×C4×C13C2×D411yescharacteristic
96C22×C4×C13#4111yescharacteristic
97C2×D52—22no—
98C2×D52—22no—
99C2×D52—22no—
100C2×D52—42no—
101C2×D52—42no—
102C2×D52—41no—
103C2×D52—41no—
104C2×D52—41no—
105C2×D52—82no—
106C2×D52—82no—
107C13×M16—42no—
108C2×C8×C13—22no—
109C13×C4⋊C4—22no—
110C13×C4⋊C4—21no—
111C13×C4⋊C4—22no—
112C13×C4⋊C4—21no—
113C13×C4⋊C4—21no—
114C13×C4⋊C4—21no—
115C13×C4⋊C4—21no—
116C13×C4⋊C4—41no—
117C13×C4⋊C4—41no—
118C13×C4⋊C4C2×D412yes—
119C13×C22⋊C4—41no—
120C42×C13—22no—
121C42×C13—22no—
122C42×C13—41no—
123C42×C13C2×D411yescharacteristic
124C42×C13C2×D411yescharacteristic
125C42×C13#4111yescharacteristic
126C42×C13C2411yescharacteristic
127#6106—48no—
128#6106—84no—
129C2×#1055—22no—
130C2×#1055—22no—
131C2×#1055—42no—
132C2×#1055C2×D414yes—
133#3208—131no—
134#3078D1311yescharacteristic, complemented
135C2×#460—131no—
136#2982—131no—
137#1678—134no—
138#1666—134no—
139#1570—131no—
140#1545—131no—
141#1542—131no—
142C23×C13#12111yescharacteristic
143C22×D13—42no—
144C22×D13—82no—
145C22×D13—82no—
146C22×D13—82no—
147Q8×C13—22no—
148Q8×C13—24no—
149Q8×C13—42no—
150Q8×C13—81no—
151Q8×C13C2×SD1614yes—
152C2×C4×C13—22no—
153C2×C4×C13—24no—
154C2×C4×C13—22no—
155C2×C4×C13—22no—
156C2×C4×C13—41no—
157C2×C4×C13—41no—
158C2×C4×C13—41no—
159C2×C4×C13—81no—
160C2×C4×C13#12111yescharacteristic
161C2×C4×C13#12111yescharacteristic
162C2×C4×C13#12111yescharacteristic
163C2×C4×C13C2×SD1612yes—
164C2×C4×C13#13512yes—
165C2×C4×C13C22×D411yescharacteristic
166C2×C4×C13C22×D411yescharacteristic
167C2×C4×C13C2×#4111yescharacteristic
168D52—42no—
169D52—42no—
170D52—44no—
171D52—82no—
172D52—82no—
173D52—82no—
174D52—82no—
175C8×C13—42no—
176#1055—24no—
177#1055—42no—
178#1055—42no—
179C22×SD16—261no—
180#533C2×D1311yescharacteristic
181C2×#127—131no—
182C2×C4×Q8C2×D1311yescharacteristic
183#492—132no—
184#486—132no—
185#479—22no—
186#477C2×D1312yescomplemented
187#475C2×D1312yescomplemented
188#460—134no—
189#460—262no—
190#460—262no—
191#450—262no—
192C8×Q8—134no—
193C8×Q8—262no—
194#431—131no—
195#431—131no—
196#431—262no—
197#428—21no—
198#423C2×D1311yescharacteristic
199C2×C4⋊C8—131no—
200#419—21no—
201C2×#102—261no—
202#334—134no—
203#333—132no—
204#332—132no—
205#331—134no—
206C22×C13—21no—
207C22×C13—21no—
208C22×C13#46512yes—
209C22×C13C2×#12111yescharacteristic
210C2×D13—82no—
211C2×D13—82no—
212C2×D13—82no—
213C2×D13—164no—
214C4×C13—21no—
215C4×C13—24no—
216C4×C13—21no—
217C4×C13—21no—
218C4×C13—42no—
219C4×C13—81no—
220C4×C13—81no—
221C4×C13#46512yes—
222C4×C13C22×SD1611yescharacteristic
223C4×C13C2×#13511yescharacteristic
224C4×C13#57712yes—
225C22×Q8C22⋊D1311yescharacteristic
226C22×D4—261no—
227C22×D4—261no—
228C22×D4—261no—
229C2×SD16—264no—
230C2×SD16—264no—
231C2×SD16—524no—
232C2×M16—21no—
233C22×C8—261no—
234C4⋊Q8C22×D1311yescharacteristic
235C4⋊Q8C22×D1311yescharacteristic
236#127—132no—
237#127—261no—
238#127—261no—
239#127—261no—
240#125—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8C22⋊D1312yes—
248C4×Q8C22×D1312yes—
249#117—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22⋊D1311yescharacteristic
252C2×C42—21no—
253C2×C42C22×D1311yescharacteristic
254#107—22no—
255#106—22no—
256C4⋊C8—21no—
257C4⋊C8—132no—
258C4⋊C8—132no—
259C4⋊C8—132no—
260C4⋊C8—264no—
261C4⋊C8—261no—
262C4⋊C8C22×D1312yes—
263#103—22no—
264#103—22no—
265#102—264no—
266#102—264no—
267C22⋊C8—132no—
268C22⋊C8—261no—
269C4×C8—132no—
270C4×C8—264no—
271C4×C8—261no—
272C2×C13—22no—
273C2×C13#304311yescharacteristic
274C2×C13C2×#46511yescharacteristic
275C2×C13#306811yescharacteristic
276D13—162no—
277C24—521no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8#610614yescomplemented
283C2×Q8C2×C22⋊D1312yes—
284C2×D4—262no—
285C2×D4—262no—
286C2×D4—262no—
287C2×D4—521no—
288C2×D4—521no—
289C2×D4—521no—
290C2×D4—522no—
291C2×D4—522no—
292C2×D4—1042no—
293C2×D4—1042no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4D4×D1311yescharacteristic
297C22×C4#612911yescharacteristic
298C22×C4C2×C22⋊D1311yescharacteristic
299SD16—528no—
300SD16—1044no—
301M16—42no—
302C2×C8—22no—
303C2×C8—134no—
304C2×C8—262no—
305C2×C8—262no—
306C2×C8—522no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4C2×C22⋊D1312yes—
317C22⋊C4—41no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42D4×D1311yescharacteristic
322C42#612911yescharacteristic
323C42C2×C22⋊D1311yescharacteristic
324C42C23×D1311yescharacteristic
325C13#1939611yescharacteristic, complemented, Sylow
326C23—522no—
327C23—1042no—
328C23—1042no—
329C23—1042no—
330C23#8819311yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335Q8C2×#610614yes—
336D4—524no—
337D4—522no—
338D4—522no—
339D4—1042no—
340D4—1042no—
341D4—1042no—
342D4—1042no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#8819311yescharacteristic
352C2×C4#8819311yescharacteristic
353C2×C4#8819311yescharacteristic
354C2×C4C2×#610612yes—
355C2×C4#8820412yes—
356C2×C4C2×D4×D1311yescharacteristic
357C2×C4C2×#612911yescharacteristic
358C2×C4C22×C22⋊D1311yescharacteristic
359C8—42no—
360C8—264no—
361C8—522no—
362C8—522no—
363C22—21no—
364C22—21no—
365C22—1042no—
366C22—1042no—
367C22—1042no—
368C22—2084no—
369C22#1171870912yes—
370C22C2×#8819311yescharacteristic
371C4—21no—
372C4—24no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1171870912yes—
379C4#1171948312yes—
380C4C22×#610611yescharacteristic
381C4C2×#8820411yescharacteristic
382C2—22no—
383C2—2082no—
384C2#1312435911yescharacteristic
385C2C2×#1171870911yescharacteristic
386C2#1312456611yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3078C2
socleC22×C13C2×#121
prime radicalC2×#88128C22
hypercenter#3078D13
hypocenterC13#19396
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. #13123917C2semidirect
4. #13114254C2semidirect
5. #13114186C2semidirect
6. #13113608C2semidirect
8. #13113493C2semidirect
9. C13×#3078C2semidirect
15. C13×#477C22semidirect
16. C13×#475C22semidirect
27. Q8×#1055C22semidirect
33. #11718139C22semidirect
34. #11718138C22semidirect
36. #11718072C22semidirect
90. C2×Q8×C13SD16semidirect
134. #3078D13semidirect
186. #477C2×D13semidirect
187. #475C2×D13semidirect
282. C2×Q8#6106semidirect
325. C13#19396semidirect

Maximal subgroups

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

subgroupindex
#131257742
C2×#117186882
#131239172
#131142542
#131141862
#131136082
#131135052
#131134932
C13×#30782
#1939613