Finite Group Databasebeta

← #15846789

Subgroups — #15846789

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125415C211yesmaximal, characteristic
2#13124958C212yesmaximal, complemented
3#13114252C214yesmaximal, complemented
4#13114240C212yesmaximal, complemented
5#13114196C212yesmaximal, complemented
6#13113599C211yesmaximal, characteristic, complemented
7#13113529C212yesmaximal, complemented
8C2×C8×Q8×C13C211yesmaximal, characteristic, complemented
9C2×C4×Q8×C13C2211yescharacteristic
10#11719182C2212yes—
11C8×Q8×C13—21no—
12C8×Q8×C13—21no—
13C8×Q8×C13C2212yes—
14C8×Q8×C13C2212yes—
15C2×C13×C4⋊C8—21no—
16C2×C13×C4⋊C8C2211yescharacteristic
17C2×C4×C8×C13—21no—
18C2×C4×C8×C13C2211yescharacteristic
19#11718888—22no—
20#11718776—21no—
21#11718776—21no—
22#11718688C2214yes—
23#11718687—24no—
24#11718682C2214yes—
25#11718628—22no—
26#11718587C2212yes—
27#11718356C2211yescharacteristic
28#11718355C2212yes—
29C4×D104—21no—
30C4×D104—21no—
31#11718343C2211yescharacteristic
32#11718077C2214yes—
33#11718073C2214yes—
34#11718050C2212yes—
35#11718049C2212yes—
36C22×Q8×C13D411yescharacteristic
37C2×#6127—22no—
38C22×C8×C13—21no—
39C22×C8×C13D411yescharacteristic
40C22×C8×C13D411yescharacteristic
41C4×Q8×C13—21no—
42C4×Q8×C13—21no—
43C4×Q8×C13D412yes—
44C4×Q8×C13C2312yes—
45C2×C13×C4⋊C4—21no—
46C2×C13×C4⋊C4D411yescharacteristic
47C2×C42×C13—21no—
48C2×C42×C13C2311yescharacteristic
49C2×#6106—24no—
50#88184—22no—
51#88184—22no—
52C2×D104—21no—
53C2×D104—21no—
54#88129—22no—
55#88128C2312yes—
56C4×D52—22no—
57C4×D52—22no—
58#88123C2312yes—
59C13×C4⋊C8—22no—
60C13×C4⋊C8—22no—
61C13×C4⋊C8—21no—
62C13×C4⋊C8C2312yes—
63C4×C8×C13—22no—
64C4×C8×C13—22no—
65C4×C8×C13D411yescharacteristic
66C4×C8×C13D411yescharacteristic
67C4×C8×C13C2311yescharacteristic
68C4×C8×C13C2311yescharacteristic
69#88075—24no—
70C22⋊#1055—22no—
71#88061—22no—
72#88061—22no—
73#88058—21no—
74#88058—21no—
75#88049—24no—
76#88047—24no—
77C4⋊#1055C2314yes—
78#88043—22no—
79#19264—131nomaximal, Sylow
80C2×Q8×C13—24no—
81C2×Q8×C13—21no—
82C2×Q8×C13C2×D412yes—
83C22×C4×C13—21no—
84C22×C4×C13—21no—
85C22×C4×C13C2×D411yescharacteristic
86C22×C4×C13C2×D411yescharacteristic
87C22×C4×C13#4111yescharacteristic
88C2×C22⋊D13—42no—
89#6127—84no—
90C2×D52—22no—
91C2×D52—22no—
92C2×D52—22no—
93C2×C4×D13—42no—
94C2×Dic26—22no—
95C2×C8×C13—22no—
96C2×C8×C13—22no—
97C2×C8×C13—41no—
98C2×C8×C13—42no—
99C2×C8×C13C2×D411yescharacteristic
100C2×C8×C13C2×D411yescharacteristic
101C2×C8×C13C2×D411yescharacteristic
102C2×C8×C13C2×D411yescharacteristic
103C13×C4⋊C4—22no—
104C13×C4⋊C4—21no—
105C13×C4⋊C4—22no—
106C13×C4⋊C4C2×D412yes—
107C42×C13—22no—
108C42×C13—22no—
109C42×C13C2×D411yescharacteristic
110C42×C13C2×D411yescharacteristic
111C42×C13#4111yescharacteristic
112C42×C13C2411yescharacteristic
113#6106—48no—
114#6103—42no—
115#6103—42no—
116C4⋊Dic13—22no—
117C4⋊Dic13—22no—
118C2×#1055—24no—
119D104—42no—
120D104—42no—
121#3169—131no—
122#3112—132no—
123C2×C8×Q8D1311yescharacteristic, complemented
124#1678—134no—
125#1676—132no—
126#1668—132no—
127#1568—131no—
128#1551—132no—
129C23×C13#12111yescharacteristic
130C22×D13—42no—
131Q8×C13—24no—
132Q8×C13—22no—
133Q8×C13#13514yes—
134C2×C4×C13—24no—
135C2×C4×C13—22no—
136C2×C4×C13—22no—
137C2×C4×C13—22no—
138C2×C4×C13—21no—
139C2×C4×C13—21no—
140C2×C4×C13—22no—
141C2×C4×C13—42no—
142C2×C4×C13#12111yescharacteristic
143C2×C4×C13#12111yescharacteristic
144C2×C4×C13#12111yescharacteristic
145C2×C4×C13C22×D411yescharacteristic
146C2×C4×C13C22×D411yescharacteristic
147C2×C4×C13C2×#4111yescharacteristic
148C22⋊D13—84no—
149C2×Dic13—42no—
150D52—44no—
151D52—42no—
152D52—42no—
153D52—82no—
154C4×D13—84no—
155Dic26—82no—
156C8×C13—21no—
157C8×C13—21no—
158C8×C13—21no—
159C8×C13—42no—
160C8×C13#13712yes—
161#1055—44no—
162#532—132no—
163C2×C4×Q8C2×D1311yescharacteristic
164#500—131no—
165#493—131no—
166#486—132no—
167#480—262no—
168#477—134no—
169#466—261no—
170#466—261no—
171#460—134no—
172#448—262no—
173C8×Q8—21no—
174C8×Q8—21no—
175C8×Q8C2×D1312yes—
176C8×Q8C2×D1312yes—
177#440—264no—
178C4×D8—261no—
179C4×D8—261no—
180#432—132no—
181C2×C4⋊C8—21no—
182C2×C4⋊C8C2×D1311yescharacteristic
183C2×C4×C8—21no—
184C2×C4×C8C2×D1311yescharacteristic
185#335—132no—
186#334—132no—
187#333—134no—
188#331—134no—
189C22×C13—21no—
190C22×C13—21no—
191C22×C13#47212yes—
192C22×C13C2×#12111yescharacteristic
193C2×D13—82no—
194C2×D13—82no—
195C2×D13—82no—
196C4×C13—21no—
197C4×C13—24no—
198C4×C13—21no—
199C4×C13—21no—
200C4×C13—21no—
201C4×C13—42no—
202C4×C13C2×#13512yes—
203C4×C13C2×#13711yescharacteristic
204C4×C13#57511yescharacteristic
205Dic13—82no—
206C2×#41—262no—
207C22×Q8D5211yescharacteristic
208C2×SD16—264no—
209C2×D8—261no—
210C2×D8—261no—
211C22×C8—21no—
212C22×C8C22⋊D1311yescharacteristic
213C22×C8C22⋊D1311yescharacteristic
214C4⋊Q8—132no—
215#127—132no—
216#124—262no—
217#121—262no—
218#121—262no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8D5212yes—
222C4×Q8C22×D1312yes—
223C4×D4—262no—
224C4×D4—262no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4D5211yescharacteristic
227C2×C42—21no—
228C2×C42C22×D1311yescharacteristic
229#107—261no—
230#107—261no—
231#107—264no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—21no—
235C4⋊C8—134no—
236C4⋊C8C22×D1312yes—
237#103—264no—
238#102—262no—
239#102—262no—
240#102—264no—
241C22⋊C8—262no—
242#97—262no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8C22⋊D1311yescharacteristic
246C4×C8C22⋊D1311yescharacteristic
247C4×C8C22×D1311yescharacteristic
248C4×C8C22×D1311yescharacteristic
249C2×C13—22no—
250C2×C13#303711yescharacteristic
251C2×C13#305011yescharacteristic
252C2×C13C2×#47211yescharacteristic
253D13—162no—
254#41—1044no—
255C2×Q8—24no—
256C2×Q8—21no—
257C2×Q8—262no—
258C2×Q8C2×D5212yes—
259C2×D4—262no—
260C2×D4—262no—
261C2×D4—262no—
262C2×D4—522no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—522no—
266C22×C4C2×D5211yescharacteristic
267C22×C4#612711yescharacteristic
268C22×C4C2×C22⋊D1311yescharacteristic
269SD16—528no—
270D8—522no—
271D8—522no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—264no—
277C2×C8C2×C22⋊D1311yescharacteristic
278C2×C8C2×C22⋊D1311yescharacteristic
279C2×C8C2×C22⋊D1311yescharacteristic
280C2×C8C2×C22⋊D1311yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—262no—
285C4⋊C4—262no—
286C4⋊C4C2×D5212yes—
287C22⋊C4—522no—
288C22⋊C4—522no—
289C42—22no—
290C42—22no—
291C42C2×D5211yescharacteristic
292C42#612711yescharacteristic
293C42C2×C22⋊D1311yescharacteristic
294C42C23×D1311yescharacteristic
295C13#1926411yescharacteristic, complemented, Sylow
296C23—522no—
297C23#8818411yescharacteristic
298Q8—24no—
299Q8—22no—
300Q8—1042no—
301Q8#8815814yes—
302D4—524no—
303D4—522no—
304D4—522no—
305D4—1042no—
306D4—1044no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—522no—
316C2×C4—1044no—
317C2×C4#8818411yescharacteristic
318C2×C4#8818411yescharacteristic
319C2×C4#8818411yescharacteristic
320C2×C4C22×D5211yescharacteristic
321C2×C4C2×#612711yescharacteristic
322C2×C4C22×C22⋊D1311yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—42no—
327C8—524no—
328C8#8819612yes—
329C22—21no—
330C22—21no—
331C22—1042no—
332C22—1042no—
333C22—1042no—
334C22#1171880312yes—
335C22C2×#8818411yescharacteristic
336C4—24no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—1042no—
343C4C2×#8815812yes—
344C4C2×#8819611yescharacteristic
345C4#1171952711yescharacteristic
346C2—22no—
347C2—2082no—
348C2#1312430811yescharacteristic
349C2#1312441711yescharacteristic
350C2C2×#1171880311yescharacteristic
subgroupG/H
centerC22C2×#88184
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×C8×Q8×C13C2
socleC22×C13C2×#121
prime radical#13125415C2
hypercenterC2×C8×Q8D13
hypocenterC13#19264
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #13124958C2semidirect
3. #13114252C2semidirect
4. #13114240C2semidirect
5. #13114196C2semidirect
6. #13113599C2semidirect
7. #13113529C2semidirect
8. C2×C8×Q8×C13C2semidirect
123. C2×C8×Q8D13semidirect
295. C13#19264semidirect

Maximal subgroups

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

subgroupindex
#131254152
#131249582
#131142522
#131142402
#131141962
#131135992
#131135292
C2×C8×Q8×C132
#1926413