Finite Group Databasebeta

← #15846061

Subgroups — #15846061

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125751C211yesmaximal, characteristic
2C2×#11718749C211yesmaximal, characteristic, complemented
3#13123548C211yesmaximal, characteristic, complemented
4#13114371C214yesmaximal, complemented
5#13114247C214yesmaximal, complemented
6#13113582C211yesmaximal, characteristic, complemented
7#13113517C211yesmaximal, characteristic, complemented
8#13113503C211yesmaximal, characteristic, complemented
9C13×#3227C211yesmaximal, characteristic, complemented
10C13×#532C2211yescharacteristic
11C2×C13×C4⋊Q8C2211yescharacteristic
12C2×#88202—21no—
13C2×#88128C2211yescharacteristic
14C13×#498—21no—
15C13×#486—21no—
16C13×#481—22no—
17C13×#476C2214yescomplemented
18C13×#431C2211yescharacteristic
19#11718895—21no—
20#11718749—21no—
21#11718749—21no—
22#11718749—21no—
23#11718749C2212yes—
24#11718749C2212yes—
25#11718734—21no—
26#11718697—22no—
27#11718676C2214yes—
28#11718673—21no—
29#11718673C2212yescomplemented
30D4×#1055—21no—
31D4×#1055C2212yescomplemented
32C2×#88049—21no—
33C2×#88049—21no—
34#11718586C2211yescharacteristic
35#11718585C2211yescharacteristic
36C2×C4×#1055C2211yescharacteristic
37#11718138C2214yescomplemented
38#11718077C2214yes—
39#11718072C2212yescomplemented
40#11718071C2212yescomplemented
41C2×C13×#41—21no—
42C22×Q8×C13—21no—
43C22×D52—21no—
44C22×D52—21no—
45C22×D52—21no—
46C13×C4⋊Q8—21no—
47C13×C4⋊Q8—21no—
48C13×C4⋊Q8—21no—
49C13×C4⋊Q8D411yescharacteristic
50C13×C4⋊Q8C2312yes—
51C13×#127D411yescharacteristic
52C13×#124—21no—
53C13×#121—21no—
54C13×#121D412yes—
55C4×D4×C13—21no—
56C4×D4×C13C2312yes—
57C2×C13×C4⋊C4—21no—
58C2×C13×C4⋊C4—21no—
59C2×C42×C13C2311yescharacteristic
60#88202—44no—
61C2×#6104—24no—
62C2×#6099—21no—
63C2×#6099—22no—
64C22×#1055—22no—
65C22×#1055D411yescharacteristic
66C22×#1055D411yescharacteristic
67#88128—21no—
68#88128—21no—
69#88128—21no—
70#88128C2312yes—
71C13×#106—24no—
72C13×C4⋊C8C2312yes—
73C13×#102—24no—
74C13×C22⋊C8—21no—
75C4×C8×C13—21no—
76C22⋊#1055—21no—
77C22⋊#1055—22no—
78C22⋊#1055—22no—
79#88049—24no—
80#88049—22no—
81#88049—22no—
82#88049—22no—
83#88049—22no—
84C4⋊#1055—21no—
85C4⋊#1055C2312yes—
86C4⋊#1055C2312yes—
87C4×#1055—21no—
88C4×#1055D411yescharacteristic
89C4×#1055D411yescharacteristic
90C4×#1055C2311yescharacteristic
91C4×#1055C2311yescharacteristic
92#19196—131nomaximal, Sylow
93C23×D13—41no—
94C13×#41—42no—
95C13×#41—42no—
96C2×Q8×C13—21no—
97C2×Q8×C13—22no—
98C2×Q8×C13—41no—
99C2×Q8×C13—82no—
100C2×D4×C13—22no—
101C2×D4×C13—21no—
102C2×D4×C13—21no—
103C2×D4×C13C2×D412yes—
104C22×C4×C13—22no—
105C22×C4×C13—41no—
106C22×C4×C13C2×D411yescharacteristic
107C22×C4×C13C2×D411yescharacteristic
108C22×C4×C13#4111yescharacteristic
109C2×D52—22no—
110C2×D52—22no—
111C2×D52—22no—
112C2×D52—42no—
113C2×D52—42no—
114C2×D52—41no—
115C2×D52—41no—
116C2×D52—41no—
117C2×D52—82no—
118C2×D52—82no—
119C2×C8×C13—22no—
120C13×C4⋊C4—21no—
121C13×C4⋊C4—22no—
122C13×C4⋊C4—22no—
123C13×C4⋊C4—41no—
124C13×C4⋊C4—41no—
125C13×C4⋊C4C2×D412yes—
126C13×C22⋊C4—22no—
127C13×C22⋊C4—22no—
128C42×C13C2×D411yescharacteristic
129C42×C13C2×D411yescharacteristic
130C42×C13#4111yescharacteristic
131C42×C13C2411yescharacteristic
132#6104—84no—
133#6099—44no—
134#6099—42no—
135C2×#1055—22no—
136C2×#1055—22no—
137C2×#1055—22no—
138C2×#1055—22no—
139C2×#1055—44no—
140C2×#1055C2×D411yescharacteristic
141C2×#1055C2×D411yescharacteristic
142C2×#1055C2×D411yescharacteristic
143C2×#1055C2×D411yescharacteristic
144#3227D1311yescharacteristic, complemented
145#3205—131no—
146C2×#486—131no—
147#2931—131no—
148#1697—134no—
149#1677—134no—
150#1564—131no—
151#1548—131no—
152#1545—131no—
153C23×C13—22no—
154C23×C13#12111yescharacteristic
155C22×D13—42no—
156C22×D13—82no—
157C22×D13—82no—
158C22×D13—82no—
159Q8×C13—42no—
160Q8×C13—82no—
161Q8×C13—82no—
162D4×C13—24no—
163D4×C13—44no—
164D4×C13—42no—
165D4×C13—42no—
166C2×C4×C13—22no—
167C2×C4×C13—22no—
168C2×C4×C13—22no—
169C2×C4×C13—42no—
170C2×C4×C13—42no—
171C2×C4×C13—42no—
172C2×C4×C13—82no—
173C2×C4×C13#12111yescharacteristic
174C2×C4×C13#12111yescharacteristic
175C2×C4×C13#12111yescharacteristic
176C2×C4×C13#13512yes—
177C2×C4×C13AGL(1,ℤ/8)12yes—
178C2×C4×C13C22×D411yescharacteristic
179C2×C4×C13C22×D411yescharacteristic
180C2×C4×C13C2×#4111yescharacteristic
181D52—42no—
182D52—42no—
183D52—44no—
184D52—82no—
185D52—82no—
186D52—82no—
187D52—82no—
188C8×C13—42no—
189#1055—21no—
190#1055—21no—
191#1055—21no—
192#1055—21no—
193#1055—42no—
194#1055—42no—
195C2×#131—261no—
196#532C2×D1311yescharacteristic
197C2×C4⋊Q8C2×D1311yescharacteristic
198C2×#127—131no—
199#498—21no—
200#493—261no—
201#486—21no—
202#486—132no—
203#486—132no—
204#486—261no—
205#486—261no—
206#486—261no—
207#481—22no—
208#476C2×D1314yescomplemented
209#459—134no—
210#447—262no—
211#436—132no—
212#436—261no—
213C8×D4—132no—
214C8×D4—261no—
215#431—131no—
216#431C2×D1311yescharacteristic
217#423—131no—
218C2×#102—261no—
219C2×#102—261no—
220#408—261no—
221C2×C4×C8—131no—
222#335—132no—
223#334—132no—
224#333—134no—
225#331—134no—
226C22×C13—21no—
227C22×C13—21no—
228C22×C13—42no—
229C22×C13—42no—
230C22×C13—42no—
231C22×C13#46312yes—
232C22×C13C2×#12111yescharacteristic
233C2×D13—82no—
234C2×D13—82no—
235C2×D13—82no—
236C2×D13—164no—
237C4×C13—21no—
238C4×C13—21no—
239C4×C13—42no—
240C4×C13—82no—
241C4×C13#46312yes—
242C4×C13#47212yes—
243C4×C13C2×#13511yescharacteristic
244C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
245C4×C13#57712yes—
246C2×#41—21no—
247C22×Q8—21no—
248C22×D4—261no—
249C22×D4—261no—
250C22×D4—261no—
251C2×D8—264no—
252#131—524no—
253C2×M16—262no—
254C2×M16—261no—
255C22×C8—131no—
256C22×C8—131no—
257C22×C8—262no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8C22⋊D1311yescharacteristic
262C4⋊Q8C22×D1312yes—
263#127—132no—
264#127—261no—
265#127—261no—
266#127—261no—
267#127C22⋊D1311yescharacteristic
268#124—21no—
269#121—21no—
270#121C22⋊D1312yes—
271C4×D4—21no—
272C4×D4C22×D1312yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C42C22×D1311yescharacteristic
276#106—24no—
277C4⋊C8—132no—
278C4⋊C8—132no—
279C4⋊C8—261no—
280C4⋊C8C22×D1312yes—
281#102—24no—
282#102—262no—
283#102—262no—
284#102—262no—
285#102—262no—
286#102—264no—
287C22⋊C8—21no—
288C22⋊C8—262no—
289C22⋊C8—262no—
290C22⋊C8—261no—
291C4×C8—21no—
292C4×C8—131no—
293C4×C8—131no—
294C4×C8—131no—
295C4×C8—131no—
296C4×C8—261no—
297C2×C13—22no—
298C2×C13—82no—
299C2×C13C2×#46311yescharacteristic
300C2×C13#305711yescharacteristic
301C2×C13C2×#47211yescharacteristic
302D13—162no—
303C24—521no—
304#41—42no—
305#41—42no—
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—262no—
314C2×D4—262no—
315C2×D4—262no—
316C2×D4—522no—
317C2×D4—522no—
318C2×D4—521no—
319C2×D4—521no—
320C2×D4—521no—
321C2×D4—1042no—
322C2×D4—1042no—
323C2×D4C2×C22⋊D1312yes—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4D4×D1311yescharacteristic
327C22×C4#612911yescharacteristic
328C22×C4C2×C22⋊D1311yescharacteristic
329D8—1044no—
330M16—524no—
331M16—522no—
332C2×C8—22no—
333C2×C8—131no—
334C2×C8—131no—
335C2×C8—131no—
336C2×C8—131no—
337C2×C8—262no—
338C2×C8—262no—
339C2×C8—262no—
340C2×C8—262no—
341C2×C8—524no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4C2×C22⋊D1312yes—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C42D4×D1311yescharacteristic
351C42#612911yescharacteristic
352C42C2×C22⋊D1311yescharacteristic
353C42C23×D1311yescharacteristic
354C13#1919611yescharacteristic, complemented, Sylow
355C23—22no—
356C23—522no—
357C23—1042no—
358C23—1042no—
359C23—1042no—
360C23#8819311yescharacteristic
361Q8—42no—
362Q8—82no—
363Q8—82no—
364D4—24no—
365D4—44no—
366D4—42no—
367D4—42no—
368D4—522no—
369D4—522no—
370D4—524no—
371D4—1042no—
372D4—1042no—
373D4—1042no—
374D4—1042no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4#8816812yes—
383C2×C4#8817012yes—
384C2×C4#8819311yescharacteristic
385C2×C4#8819311yescharacteristic
386C2×C4#8819311yescharacteristic
387C2×C4C2×D4×D1311yescharacteristic
388C2×C4C2×#612911yescharacteristic
389C2×C4C22×C22⋊D1311yescharacteristic
390C8—42no—
391C8—261no—
392C8—261no—
393C8—261no—
394C8—261no—
395C8—522no—
396C8—522no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—1042no—
403C22—1042no—
404C22—1042no—
405C22—2084no—
406C22#1171882612yes—
407C22C2×#8819311yescharacteristic
408C4—21no—
409C4—21no—
410C4—42no—
411C4—82no—
412C4#1171882612yes—
413C4#1171886012yes—
414C4C2×#8816811yescharacteristic
415C4C2×#8817011yescharacteristic
416C4#1171954212yes—
417C2—22no—
418C2—82no—
419C2—2082no—
420C2C2×#1171882611yescharacteristic
421C2#1312447711yescharacteristic
422C2C2×#1171886011yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3227C2
socleC22×C13C2×#121
prime radical#13125751C2
hypercenter#3227D13
hypocenterC13#19196
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. C2×#11718749C2semidirect
3. #13123548C2semidirect
4. #13114371C2semidirect
5. #13114247C2semidirect
6. #13113582C2semidirect
7. #13113517C2semidirect
8. #13113503C2semidirect
9. C13×#3227C2semidirect
17. C13×#476C22semidirect
29. #11718673C22semidirect
31. D4×#1055C22semidirect
37. #11718138C22semidirect
39. #11718072C22semidirect
40. #11718071C22semidirect
144. #3227D13semidirect
208. #476C2×D13semidirect
354. C13#19196semidirect

Maximal subgroups

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

subgroupindex
#131257512
C2×#117187492
#131235482
#131143712
#131142472
#131135822
#131135172
#131135032
C13×#32272
#1919613