Finite Group Databasebeta

← #15848204

Subgroups — #15848204

Order 3328.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125763C211yesmaximal, characteristic
2C2×#11718681C211yesmaximal, characteristic, complemented
3#13124308C211yesmaximal, characteristic
4#13114266C214yesmaximal, complemented
5#13114184C214yesmaximal, complemented
6#13113607C211yesmaximal, characteristic, complemented
7#13113520C211yesmaximal, characteristic, complemented
8#13113483C211yesmaximal, characteristic
9C13×#2982C211yesmaximal, characteristic, complemented
10C2×C4×Q8×C13C2211yescharacteristic
11C2×#88196—21no—
12#11719182C2211yescharacteristic
13C2×#88123C2211yescharacteristic
14C8×Q8×C13—22no—
15C8×Q8×C13C2214yes—
16C13×#431—22no—
17C13×#431C2211yescharacteristic
18C13×#431C2211yescharacteristic
19C2×#88075—21no—
20#11718692C2212yes—
21#11718689—22no—
22#11718688C2212yes—
23#11718681—22no—
24#11718681C2214yes—
25#11718627—21no—
26C2×#88048—21no—
27#11718585C2211yescharacteristic
28C2×C4⋊#1055C2211yescharacteristic
29#11718502—22no—
30#11718399—22no—
31#11718354C2212yes—
32#11718342C2212yes—
33#11718078C2212yes—
34#11718077C2212yes—
35#11718074C2214yes—
36#11718049C2214yes—
37C22×Q8×C13D411yescharacteristic
38C2×#6127—21no—
39C22×Dic26—21no—
40C4×Q8×C13—21no—
41C4×Q8×C13—21no—
42C4×Q8×C13D412yes—
43C4×Q8×C13C2312yes—
44C2×C13×C4⋊C4—21no—
45C2×C13×C4⋊C4D411yescharacteristic
46C2×C42×C13—21no—
47C2×C42×C13C2311yescharacteristic
48C2×#6107—22no—
49#88196—44no—
50C2×#6106—22no—
51#88184—21no—
52#88184—21no—
53C2×C4⋊Dic13—21no—
54C2×C4⋊Dic13—21no—
55C2×#6099—21no—
56C22×#1055—21no—
57C2×#6095—24no—
58#88129—21no—
59#88128C2311yescharacteristic
60C4×D52—21no—
61C4×D52—21no—
62#88123—21no—
63#88123—21no—
64#88123—21no—
65#88123C2312yes—
66#88123C2311yescharacteristic
67C13×C4⋊C8—24no—
68C13×C4⋊C8D412yes—
69C13×C4⋊C8C2312yes—
70C13×C22⋊C8—21no—
71C13×C22⋊C8D412yes—
72C4×C8×C13—24no—
73C4×C8×C13—21no—
74C4×C8×C13C2312yes—
75#88075—24no—
76#88056—24no—
77#88050—22no—
78#88049—22no—
79#88048—24no—
80C4⋊#1055—21no—
81C4⋊#1055—21no—
82C4⋊#1055C2312yes—
83C4⋊#1055C2312yes—
84#19397—131nomaximal, Sylow
85C2×Q8×C13—21no—
86C2×Q8×C13—22no—
87C2×Q8×C13SD1614yescomplemented
88C2×Q8×C13C2×D412yes—
89C22×C4×C13—21no—
90C22×C4×C13—21no—
91C22×C4×C13C2×D411yescharacteristic
92C22×C4×C13C2×D411yescharacteristic
93C22×C4×C13#4111yescharacteristic
94C2×C22⋊D13—41no—
95C22×Dic13—41no—
96#6127—42no—
97#6127—42no—
98C2×D52—21no—
99C2×D52—21no—
100C2×D52—21no—
101C2×C4×D13—41no—
102C2×Dic26—22no—
103C2×Dic26—21no—
104C2×Dic26—41no—
105C2×Dic26—82no—
106C2×C8×C13—22no—
107C2×C8×C13C2×D414yes—
108C13×C4⋊C4—22no—
109C13×C4⋊C4—21no—
110C13×C4⋊C4—22no—
111C13×C4⋊C4C2×D412yes—
112C42×C13—22no—
113C42×C13—22no—
114C42×C13C2×D411yescharacteristic
115C42×C13C2×D411yescharacteristic
116C42×C13#4111yescharacteristic
117C42×C13C2411yescharacteristic
118#6107—44no—
119#6106—44no—
120#6103—41no—
121#6103—41no—
122C4⋊Dic13—22no—
123C4⋊Dic13—21no—
124C4⋊Dic13—22no—
125C4⋊Dic13—21no—
126C4⋊Dic13—41no—
127C4⋊Dic13—41no—
128#6099—42no—
129C2×#1055—22no—
130C2×#1055—22no—
131C2×#1055—42no—
132#6095—84no—
133#3206—131no—
134C2×#475—131no—
135#3037—131no—
136#2982D1311yescharacteristic, complemented
137#1680—134no—
138#1666—134no—
139#1570—131no—
140#1549—131no—
141#1540—131no—
142C23×C13#12111yescharacteristic
143C22×D13—41no—
144Q8×C13—22no—
145Q8×C13—24no—
146Q8×C13C2×SD1614yes—
147C2×C4×C13—22no—
148C2×C4×C13—24no—
149C2×C4×C13—22no—
150C2×C4×C13—22no—
151C2×C4×C13—41no—
152C2×C4×C13—81no—
153C2×C4×C13#12111yescharacteristic
154C2×C4×C13#12111yescharacteristic
155C2×C4×C13#12111yescharacteristic
156C2×C4×C13C2×SD1612yes—
157C2×C4×C13#13512yes—
158C2×C4×C13C22×D411yescharacteristic
159C2×C4×C13C22×D411yescharacteristic
160C2×C4×C13C2×#4111yescharacteristic
161C22⋊D13—82no—
162C2×Dic13—42no—
163C2×Dic13—41no—
164C2×Dic13—82no—
165D52—41no—
166D52—41no—
167D52—42no—
168D52—81no—
169C4×D13—82no—
170Dic26—41no—
171Dic26—41no—
172Dic26—44no—
173Dic26—82no—
174C8×C13—24no—
175C8×C13—42no—
176#1055—42no—
177#1055—42no—
178C2×#137—261no—
179#532—131no—
180C2×C4⋊Q8—131no—
181C2×C4×Q8C2×D1311yescharacteristic
182#492—132no—
183#487—132no—
184#475—134no—
185#475—262no—
186#464—262no—
187#462—132no—
188#461—262no—
189#460—132no—
190#448—262no—
191C8×Q8—22no—
192C8×Q8C2×D1314yes—
193#431—22no—
194#431C2×D1311yescharacteristic
195#431C2×D1311yescharacteristic
196C2×#106—261no—
197#423—131no—
198C2×C4⋊C8—131no—
199#417—261no—
200C2×#103—261no—
201#334—134no—
202#332—132no—
203#332—134no—
204#331—132no—
205C22×C13—21no—
206C22×C13—21no—
207C22×C13#46512yes—
208C22×C13C2×#12111yescharacteristic
209C2×D13—81no—
210C2×D13—81no—
211C2×D13—81no—
212C4×C13—21no—
213C4×C13—24no—
214C4×C13—21no—
215C4×C13—21no—
216C4×C13—42no—
217C4×C13#46512yes—
218C4×C13C22×SD1611yescharacteristic
219C4×C13C2×#13511yescharacteristic
220C4×C13#57712yes—
221Dic13—82no—
222Dic13—81no—
223C2×#41—261no—
224C22×Q8—261no—
225C22×Q8D5211yescharacteristic
226#137—524no—
227C2×Q16—262no—
228C2×SD16—262no—
229C2×SD16—264no—
230C2×M16—261no—
231C22×C8—261no—
232C4⋊Q8—132no—
233C4⋊Q8—131no—
234C4⋊Q8—261no—
235C4⋊Q8—261no—
236C4⋊Q8—261no—
237#127—131no—
238#124—261no—
239#121—261no—
240#121—261no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8D5212yes—
244C4×Q8C22×D1312yes—
245C4×D4—261no—
246C4×D4—261no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—261no—
249C2×C4⋊C4—261no—
250C2×C4⋊C4D5211yescharacteristic
251C2×C42—21no—
252C2×C42C22×D1311yescharacteristic
253#106—264no—
254C4⋊C8—24no—
255C4⋊C8—132no—
256C4⋊C8—132no—
257C4⋊C8—261no—
258C4⋊C8—261no—
259C4⋊C8C22⋊D1312yes—
260C4⋊C8C22×D1312yes—
261#103—264no—
262#103—264no—
263#103—262no—
264#102—262no—
265C22⋊C8—21no—
266C22⋊C8C22⋊D1312yes—
267C4×C8—24no—
268C4×C8—21no—
269C4×C8C22×D1312yes—
270C2×C13—22no—
271C2×C13#304311yescharacteristic
272C2×C13C2×#46511yescharacteristic
273C2×C13#306911yescharacteristic
274D13—161no—
275#41—522no—
276#41—522no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—261no—
280C2×Q8—262no—
281C2×Q8—521no—
282C2×Q8—1042no—
283C2×Q8#609514yescomplemented
284C2×Q8C2×D5212yes—
285C2×D4—261no—
286C2×D4—261no—
287C2×D4—261no—
288C2×D4—521no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—521no—
292C22×C4—521no—
293C22×C4C2×D5211yescharacteristic
294C22×C4#612711yescharacteristic
295C22×C4C2×C22⋊D1311yescharacteristic
296Q16—524no—
297SD16—524no—
298SD16—1044no—
299M16—522no—
300C2×C8—22no—
301C2×C8—262no—
302C2×C8—262no—
303C2×C8—522no—
304C2×C8C2×C22⋊D1314yes—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—262no—
309C4⋊C4—262no—
310C4⋊C4—261no—
311C4⋊C4—261no—
312C4⋊C4—521no—
313C4⋊C4—521no—
314C4⋊C4C2×D5212yes—
315C22⋊C4—521no—
316C22⋊C4—521no—
317C42—22no—
318C42—22no—
319C42C2×D5211yescharacteristic
320C42#612711yescharacteristic
321C42C2×C22⋊D1311yescharacteristic
322C42C23×D1311yescharacteristic
323C13#1939711yescharacteristic, complemented, Sylow
324C23—521no—
325C23#8818411yescharacteristic
326Q8—22no—
327Q8—24no—
328Q8—521no—
329Q8—521no—
330Q8—524no—
331Q8—1042no—
332Q8C2×#609514yes—
333D4—521no—
334D4—521no—
335D4—522no—
336D4—1041no—
337D4—1042no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—521no—
345C2×C4—522no—
346C2×C4—1042no—
347C2×C4—1042no—
348C2×C4C2×#609512yes—
349C2×C4#8815812yes—
350C2×C4#8818411yescharacteristic
351C2×C4#8818411yescharacteristic
352C2×C4#8818411yescharacteristic
353C2×C4C22×D5211yescharacteristic
354C2×C4C2×#612711yescharacteristic
355C2×C4C22×C22⋊D1311yescharacteristic
356C8—24no—
357C8—42no—
358C8—522no—
359C8—522no—
360C22—21no—
361C22—21no—
362C22—1041no—
363C22—1041no—
364C22—1041no—
365C22#1171877512yes—
366C22C2×#8818411yescharacteristic
367C4—21no—
368C4—24no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—1042no—
373C4—1041no—
374C4#1171877512yes—
375C4C22×#609511yescharacteristic
376C4C2×#8815811yescharacteristic
377C4#1171954212yes—
378C2—22no—
379C2—2081no—
380C2#1312435711yescharacteristic
381C2C2×#1171877511yescharacteristic
382C2#1312457311yescharacteristic
subgroupG/H
centerC22C2×#88184
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#2982C2
socleC22×C13C2×#121
prime radical#11719182C22
hypercenter#2982D13
hypocenterC13#19397
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#11718681C2semidirect
4. #13114266C2semidirect
5. #13114184C2semidirect
6. #13113607C2semidirect
7. #13113520C2semidirect
9. C13×#2982C2semidirect
87. C2×Q8×C13SD16semidirect
136. #2982D13semidirect
283. C2×Q8#6095semidirect
323. C13#19397semidirect

Maximal subgroups

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

subgroupindex
#131257632
C2×#117186812
#131243082
#131142662
#131141842
#131136072
#131135202
#131134832
C13×#29822
#1939713