Finite Group Databasebeta

← #15880612

Subgroups — #15880612

Order 3328.

866 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88047C211yesmaximal, characteristic
2#13117707C211yesmaximal, characteristic
3#13117705C211yesmaximal, characteristic
4#13115573C212yesmaximal, complemented
5#13115431C212yesmaximal, complemented
6#13115426C212yesmaximal, complemented
7#13114923C211yesmaximal, characteristic
8#13114859C211yesmaximal, characteristic
9#13114732C212yesmaximal, complemented
10#13114615C211yesmaximal, characteristic
11C13×#1823C211yesmaximal, characteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C2×#88180—21no—
14C22×C4⋊Dic13C2211yescharacteristic
15C23×#1055C2211yescharacteristic
16C2×#88131—21no—
17C2×C13×C22⋊C8C2211yescharacteristic
18C13×#380C2211yescharacteristic
19#11718873C2211yescharacteristic
20#11718872—21no—
21C4×C22⋊Dic13C2211yescharacteristic
22C2×#88055C2212yes—
23#11718646C2212yes—
24#11718642—22no—
25C4⋊C4×Dic13C2212yes—
26C4⋊C4×Dic13C2212yes—
27#11718634C2212yes—
28C2×#88048C2212yes—
29C2×#88047C418yescomplemented
30C2×#88047C2211yescharacteristic
31C2×#88047C2211yescharacteristic
32C2×#88047C2211yescharacteristic
33C2×#88047C2211yescharacteristic
34#11718604—21no—
35#11718315—22no—
36C13×#340C2212yes—
37C13×#340C2212yes—
38#11718143C2212yes—
39#11718143C2212yes—
40#11718142C2212yes—
41#11718121C2212yes—
42C23×C4×C13—21no—
43C23×C4×C13C2311yescharacteristic
44C23×Dic13—21no—
45C22×C8×C13C2312yes—
46C2×C13×C4⋊C4—21no—
47C2×C13×C4⋊C4—22no—
48C2×C13×C4⋊C4C2×C414yes—
49C2×C13×C4⋊C4C2312yes—
50C2×C13×C4⋊C4C2311yescharacteristic
51C2×C13×C4⋊C4C2311yescharacteristic
52C2×C13×C22⋊C4—21no—
53C2×C22⋊Dic13—21no—
54C2×C22⋊Dic13D411yescharacteristic
55C2×C22⋊Dic13D411yescharacteristic
56#88180—44no—
57C2×C4⋊Dic13—21no—
58C2×C4⋊Dic13—22no—
59C2×C4⋊Dic13—22no—
60C2×C4⋊Dic13C2×C414yes—
61C2×C4⋊Dic13C2312yes—
62C2×C4⋊Dic13C2311yescharacteristic
63C2×C4⋊Dic13C2311yescharacteristic
64C2×#6101—22no—
65C2×C4×Dic13—22no—
66C2×C4×Dic13—22no—
67C2×C4×Dic13C2312yes—
68C22×#1055—24no—
69C22×#1055C2×C414yes—
70C22×#1055C2311yescharacteristic
71C22×#1055C2311yescharacteristic
72#88131—44no—
73C13×C22⋊C8—22no—
74C13×#95—21no—
75#88071—21no—
76#88071—22no—
77#88071—22no—
78#88071—22no—
79#88071—22no—
80#88071D412yes—
81#88055—24no—
82#88048—24no—
83#88047C2×C414yes—
84#88047C2×C414yes—
85#88047C2×C414yes—
86#88047C2×C414yes—
87#22617—131nomaximal, Sylow
88C24×C13C2×D411yescharacteristic
89C22×C4×C13—22no—
90C22×C4×C13—22no—
91C22×C4×C13—24no—
92C22×C4×C13—21no—
93C22×C4×C13—21no—
94C22×C4×C13—22no—
95C22×C4×C13—42no—
96C22×C4×C13—42no—
97C22×C4×C13C22×C412yes—
98C22×C4×C13C2×D411yescharacteristic
99C22×C4×C13#4112yes—
100C22×C4×C13C2411yescharacteristic
101C22×Dic13—24no—
102C22×Dic13—21no—
103C22×Dic13—21no—
104C22×Dic13—22no—
105C22×Dic13—42no—
106C22×Dic13—42no—
107C22×Dic13C2×D412yes—
108C22×Dic13C2×D412yes—
109C2×C8×C13—22no—
110C2×C8×C13—22no—
111C2×C8×C13—22no—
112C13×C4⋊C4—24no—
113C13×C4⋊C4—21no—
114C13×C4⋊C4—21no—
115C13×C4⋊C4—22no—
116C13×C4⋊C4—22no—
117C13×C4⋊C4C22×C412yes—
118C13×C4⋊C4C22×C412yes—
119C13×C22⋊C4—42no—
120C22⋊Dic13—22no—
121C22⋊Dic13—22no—
122C22⋊Dic13—42no—
123C4⋊Dic13—24no—
124C4⋊Dic13—21no—
125C4⋊Dic13—21no—
126C4⋊Dic13—22no—
127C4⋊Dic13—22no—
128C4⋊Dic13—44no—
129C4⋊Dic13C22×C412yes—
130C4⋊Dic13C22×C412yes—
131#6101—44no—
132C4×Dic13—22no—
133C4×Dic13—22no—
134C4×Dic13—44no—
135C4×Dic13—44no—
136C2×#1055—22no—
137C2×#1055—28no—
138C2×#1055C22×C412yes—
139C2×#1055C22×C412yes—
140C2×#1055#4114yes—
141C22×#107—131no—
142#2344—131no—
143#2344—131no—
144#1943—132no—
145#1921—132no—
146#1921—132no—
147#1823—131no—
148#1823D1311yescharacteristic
149#1807—131no—
150#1776—132no—
151#1752—131no—
152C23×C13—22no—
153C23×C13—22no—
154C23×C13C4×D412yes—
155C23×C13AGL(1,ℤ/8)12yes—
156C23×C13AGL(1,ℤ/8)12yes—
157C23×C13C22×D411yescharacteristic
158C2×C4×C13—22no—
159C2×C4×C13—22no—
160C2×C4×C13—22no—
161C2×C4×C13—24no—
162C2×C4×C13—22no—
163C2×C4×C13—22no—
164C2×C4×C13—41no—
165C2×C4×C13—41no—
166C2×C4×C13—41no—
167C2×C4×C13—41no—
168C2×C4×C13—48no—
169C2×C4×C13—42no—
170C2×C4×C13—42no—
171C2×C4×C13—42no—
172C2×C4×C13C4×D412yes—
173C2×C4×C13C23×C411yescharacteristic
174C2×C4×C13C2×#4112yes—
175C2×C4×C13C2×#4111yescharacteristic
176C2×C4×C132−1+412yes—
177C2×Dic13—22no—
178C2×Dic13—22no—
179C2×Dic13—22no—
180C2×Dic13—22no—
181C2×Dic13—22no—
182C2×Dic13—22no—
183C2×Dic13—22no—
184C2×Dic13—22no—
185C2×Dic13—41no—
186C2×Dic13—41no—
187C2×Dic13—41no—
188C2×Dic13—41no—
189C2×Dic13—48no—
190C2×Dic13—42no—
191C2×Dic13—42no—
192C2×Dic13—42no—
193C8×C13—42no—
194#1055—22no—
195#1055#11714yescomplemented
196C23×C8—131no—
197C2×#117—261no—
198C2×#117—261no—
199C22×C4⋊C4—131no—
200C22×C4⋊C4C2×D1311yescharacteristic
201C2×#107—138no—
202C2×#107—131no—
203C2×#107—131no—
204C2×#107—131no—
205C2×#107—131no—
206C2×#106—132no—
207C2×C22⋊C8C2×D1311yescharacteristic
208C2×#97—132no—
209#389—132no—
210#389—132no—
211#387—262no—
212#387—262no—
213#381—261no—
214#381—261no—
215#380—131no—
216#380C2×D1311yescharacteristic
217C4×C4⋊C4—132no—
218C4×C4⋊C4—132no—
219C4×C22⋊C4—131no—
220#340—132no—
221#340—132no—
222#340—132no—
223#340C2×D1312yes—
224#340C2×D1312yes—
225#336—132no—
226C22×C13—24no—
227C22×C13—24no—
228C22×C13—22no—
229C22×C13—22no—
230C22×C13#44214yes—
231C22×C13C2×C4×D411yescharacteristic
232C22×C13#54712yes—
233C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
234C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
235C22×C13#57812yes—
236C4×C13—22no—
237C4×C13—42no—
238C4×C13—42no—
239C4×C13—82no—
240C4×C13C2×#11711yescharacteristic
241C4×C13#51911yescharacteristic
242C4×C13#55212yes—
243Dic13—42no—
244Dic13—42no—
245Dic13—42no—
246Dic13—42no—
247Dic13—82no—
248C23×C4—21no—
249C23×C4—261no—
250C23×C4C22×D1311yescharacteristic
251C22×C8—131no—
252C22×C8—134no—
253C22×C8—131no—
254C22×C8—264no—
255C22×C8C22×D1312yes—
256#117—524no—
257#117—524no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—134no—
261C2×C4⋊C4—131no—
262C2×C4⋊C4—131no—
263C2×C4⋊C4—132no—
264C2×C4⋊C4—261no—
265C2×C4⋊C4—262no—
266C2×C4⋊C4—262no—
267C2×C4⋊C4—262no—
268C2×C4⋊C4C4×D1314yes—
269C2×C4⋊C4C22×D1312yes—
270C2×C4⋊C4C22×D1311yescharacteristic
271C2×C4⋊C4C22×D1311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—131no—
274C2×C22⋊C4—131no—
275C2×C22⋊C4—261no—
276C2×C42—132no—
277C2×C42—262no—
278C2×C42—262no—
279#107—134no—
280#107—134no—
281#107—134no—
282#107—134no—
283#106—264no—
284C22⋊C8—22no—
285#97—264no—
286#95—21no—
287#95—132no—
288#95—262no—
289#95—262no—
290#95—262no—
291#95—262no—
292#95—261no—
293C2×C13—24no—
294C2×C13#234411yescharacteristic
295C2×C13C2×#44211yescharacteristic
296C2×C13#295011yescharacteristic
297C2×C13#334312yes—
298C2×C13#334312yes—
299C24D4×D1311yescharacteristic
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—132no—
309C22×C4—132no—
310C22×C4—264no—
311C22×C4—261no—
312C22×C4—261no—
313C22×C4—262no—
314C22×C4—522no—
315C22×C4—522no—
316C22×C4C2×C4×D1312yes—
317C22×C4D4×D1311yescharacteristic
318C22×C4#612912yes—
319C22×C4C23×D1311yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—132no—
324C2×C8—134no—
325C2×C8—132no—
326C2×C8—268no—
327C2×C8—262no—
328C4⋊C4—24no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—132no—
334C4⋊C4—132no—
335C4⋊C4—264no—
336C4⋊C4—261no—
337C4⋊C4—261no—
338C4⋊C4—262no—
339C4⋊C4—262no—
340C4⋊C4—524no—
341C4⋊C4—524no—
342C4⋊C4C2×C4×D1312yes—
343C4⋊C4C2×C4×D1312yes—
344C22⋊C4—42no—
345C22⋊C4—262no—
346C22⋊C4—262no—
347C22⋊C4—522no—
348C42—262no—
349C42—262no—
350C42—524no—
351C42—524no—
352C13#2261711yescharacteristic, complemented, Sylow
353C23—22no—
354C23—22no—
355C23#8813512yes—
356C23#8816512yes—
357C23#8816812yes—
358C23C2×D4×D1311yescharacteristic
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—48no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—262no—
374C2×C4—262no—
375C2×C4—262no—
376C2×C4—262no—
377C2×C4—262no—
378C2×C4—262no—
379C2×C4—262no—
380C2×C4—262no—
381C2×C4—521no—
382C2×C4—521no—
383C2×C4—521no—
384C2×C4—521no—
385C2×C4—528no—
386C2×C4—522no—
387C2×C4—522no—
388C2×C4—522no—
389C2×C4#8813512yes—
390C2×C4C22×C4×D1311yescharacteristic
391C2×C4C2×#612911yescharacteristic
392C2×C4#41×D1312yes—
393C2×C4#8825712yes—
394C8—42no—
395C8—134no—
396C8—262no—
397C22—22no—
398C22—24no—
399C22—24no—
400C22—22no—
401C22#1171844614yes—
402C22C2×#8813511yescharacteristic
403C22#1171936012yes—
404C22C2×#8816511yescharacteristic
405C22C2×#8816811yescharacteristic
406C22#1171948412yes—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—82no—
411C4—522no—
412C4—522no—
413C4—522no—
414C4—522no—
415C4—1042no—
416C4#1171922111yescharacteristic
417C4D13×#11711yescharacteristic
418C4#1171935512yes—
419C2—24no—
420C2#1311770011yescharacteristic
421C2C2×#1171844611yescharacteristic
422C2#1312370011yescharacteristic
423C2#1312727412yes—
424C2#1312728012yes—
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1823C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1823D13
hypocenterC13#22617
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #13115573C2semidirect
5. #13115431C2semidirect
6. #13115426C2semidirect
9. #13114732C2semidirect
29. C2×#88047C4semidirect
195. #1055#117semidirect
352. C13#22617semidirect

Maximal subgroups

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

subgroupindex
C22×#880472
#131177072
#131177052
#131155732
#131154312
#131154262
#131149232
#131148592
#131147322
#131146152
C13×#18232
#2261713