Finite Group Databasebeta

← #15755582

Subgroups — #15755582

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×Dic13C211yesmaximal, characteristic
2#13114814C212yesmaximal, complemented
3C2×#11718117C211yesmaximal, characteristic
4#13112678C212yesmaximal, complemented
5C2×C13×#323C211yesmaximal, characteristic
6C24×Dic13C411yescharacteristic
7C22×C13×C22⋊C4C2211yescharacteristic
8C22×C22⋊Dic13C411yescharacteristic
9C22×C4×Dic13C411yescharacteristic
10C22×C4×Dic13C2211yescharacteristic
11C2×C13×C22⋊C8C2212yes—
12C2×#88071C411yescharacteristic
13C2×#88071C2211yescharacteristic
14C2×C22⋊#1055C2212yes—
15C2×#88054—22no—
16C22⋊C4×Dic13—22no—
17C22⋊C4×Dic13—22no—
18C13×#323—22no—
19#11718142—22no—
20#11718117—22no—
21C25×C13C2×C411yescharacteristic
22C23×C4×C13C2×C411yescharacteristic
23C23×C4×C13C2311yescharacteristic
24C23×Dic13—22no—
25C23×Dic13—21no—
26C23×Dic13—21no—
27C23×Dic13—22no—
28C23×Dic13—42no—
29C23×Dic13—42no—
30C23×Dic13C2×C411yescharacteristic
31C23×Dic13C2×C411yescharacteristic
32C23×Dic13C2×C411yescharacteristic
33C23×Dic13C2×C411yescharacteristic
34C22×C8×C13—22no—
35C2×C13×C22⋊C4—22no—
36C2×C13×C22⋊C4—22no—
37C2×C13×C22⋊C4D412yes—
38C2×C13×C22⋊C4Q812yes—
39C2×C22⋊Dic13—22no—
40C2×C22⋊Dic13—22no—
41C2×C22⋊Dic13C814yescomplemented
42C2×C4×Dic13—21no—
43C2×C4×Dic13—21no—
44C2×C4×Dic13—42no—
45C2×C4×Dic13—42no—
46C2×C4×Dic13D412yes—
47C2×C4×Dic13D412yes—
48C22×#1055—22no—
49C13×C22⋊C8—22no—
50C13×C22⋊C8—22no—
51#88071—22no—
52#88071—22no—
53C22⋊#1055—22no—
54C22⋊#1055—22no—
55#88054—44no—
56#8559—131nomaximal, Sylow
57C24×C13—21no—
58C24×C13—21no—
59C24×C13—21no—
60C24×C13—21no—
61C24×C13—41no—
62C24×C13—41no—
63C24×C13—41no—
64C24×C13—41no—
65C24×C13C4⋊C412yes—
66C24×C13C2×C812yes—
67C24×C13M1612yes—
68C24×C13C22×C411yescharacteristic
69C22×C4×C13—21no—
70C22×C4×C13—21no—
71C22×C4×C13—41no—
72C22×C4×C13—41no—
73C22×C4×C13—41no—
74C22×C4×C13—41no—
75C22×C4×C13C4⋊C412yes—
76C22×C4×C13C2×D412yes—
77C22×C4×C13C2×D411yescharacteristic
78C22×C4×C13C2×Q811yescharacteristic
79C22×C4×C13#4112yes—
80C22×Dic13—21no—
81C22×Dic13—21no—
82C22×Dic13—22no—
83C22×Dic13—22no—
84C22×Dic13—24no—
85C22×Dic13—24no—
86C22×Dic13—21no—
87C22×Dic13—21no—
88C22×Dic13—21no—
89C22×Dic13—21no—
90C22×Dic13—41no—
91C22×Dic13—41no—
92C22×Dic13—48no—
93C22×Dic13—48no—
94C22×Dic13—42no—
95C22×Dic13—42no—
96C22×Dic13—42no—
97C22×Dic13—42no—
98C22×Dic13—41no—
99C22×Dic13—41no—
100C22×Dic13—41no—
101C22×Dic13—41no—
102C22×Dic13C22⋊C412yes—
103C22×Dic13C22⋊C411yescharacteristic
104C22×Dic13C22⋊C411yescharacteristic
105C22×Dic13C2×C812yes—
106C22×Dic13C2×C812yes—
107C22×Dic13M1612yes—
108C22×Dic13M1612yes—
109C2×C8×C13—42no—
110C2×C8×C13—42no—
111C2×C8×C13—42no—
112C13×C22⋊C4—24no—
113C13×C22⋊C4—24no—
114C22⋊Dic13—28no—
115C4×Dic13—24no—
116C4×Dic13—42no—
117C4×Dic13—42no—
118C4×Dic13—42no—
119C2×#1055—42no—
120C2×#1055—42no—
121C2×#1055—42no—
122C2×C4×C22⋊C4—131no—
123#1797—132no—
124C2×#323—131no—
125C2×#323D1311yescharacteristic
126#1280—132no—
127C23×C13—21no—
128C23×C13—21no—
129C23×C13—21no—
130C23×C13—21no—
131C23×C13—22no—
132C23×C13—22no—
133C23×C13—22no—
134C23×C13—22no—
135C23×C13—22no—
136C23×C13—22no—
137C23×C13—42no—
138C23×C13—42no—
139C23×C13—42no—
140C23×C13—42no—
141C23×C13—42no—
142C23×C13—42no—
143C23×C13—42no—
144C23×C13—42no—
145C23×C13—42no—
146C23×C13—42no—
147C23×C13—42no—
148C23×C13—42no—
149C23×C13—41no—
150C23×C13—41no—
151C23×C13—41no—
152C23×C13—41no—
153C23×C13C4⋊C814yescomplemented
154C23×C13C2×C22⋊C411yescharacteristic
155C23×C13C2×C4⋊C411yescharacteristic
156C23×C13#11711yescharacteristic
157C23×C13C22×C811yescharacteristic
158C23×C13C2×M1611yescharacteristic
159C23×C13C2×M1611yescharacteristic
160C23×C13C2×M1611yescharacteristic
161C2×C4×C13—22no—
162C2×C4×C13—22no—
163C2×C4×C13—22no—
164C2×C4×C13—41no—
165C2×C4×C13—41no—
166C2×C4×C13—42no—
167C2×C4×C13—42no—
168C2×C4×C13—41no—
169C2×C4×C13—41no—
170C2×C4×C13—42no—
171C2×C4×C13#12112yes—
172C2×C4×C13C22⋊Q812yes—
173C2×Dic13—24no—
174C2×Dic13—24no—
175C2×Dic13—21no—
176C2×Dic13—21no—
177C2×Dic13—22no—
178C2×Dic13—21no—
179C2×Dic13—21no—
180C2×Dic13—22no—
181C2×Dic13—22no—
182C2×Dic13—22no—
183C2×Dic13—42no—
184C2×Dic13—42no—
185C2×Dic13—48no—
186C2×Dic13—48no—
187C2×Dic13—42no—
188C2×Dic13—42no—
189C2×Dic13—41no—
190C2×Dic13—41no—
191C2×Dic13—41no—
192C2×Dic13—41no—
193C2×Dic13—42no—
194C2×Dic13—42no—
195C2×Dic13C22⋊C814yes—
196C2×Dic13#9912yes—
197C2×Dic13#10012yes—
198C8×C13—82no—
199#1055—82no—
200C24×C4—131no—
201C22×C22⋊C4—131no—
202C22×C22⋊C4C2×D1311yescharacteristic
203C22×C42—131no—
204C22×C42—131no—
205C2×C4⋊C8—262no—
206C2×C22⋊C8—132no—
207C2×C22⋊C8C2×D1312yes—
208C4×C22⋊C4—262no—
209C4×C22⋊C4—262no—
210C2×#95—131no—
211C2×#95—131no—
212#336—262no—
213#323—22no—
214#323—262no—
215C22×C13—21no—
216C22×C13—21no—
217C22×C13—21no—
218C22×C13—21no—
219C22×C13—21no—
220C22×C13—21no—
221C22×C13—21no—
222C22×C13—21no—
223C22×C13—21no—
224C22×C13—21no—
225C22×C13—21no—
226C22×C13—21no—
227C22×C13—24no—
228C22×C13—44no—
229C22×C13—44no—
230C22×C13—44no—
231C22×C13—44no—
232C22×C13—42no—
233C22×C13—42no—
234C22×C13—42no—
235C22×C13—42no—
236C22×C13—42no—
237C22×C13—42no—
238C22×C13#38011yescharacteristic
239C22×C13C2×C22⋊C811yescharacteristic
240C22×C13#40711yescharacteristic
241C22×C13C2×#9911yescharacteristic
242C22×C13#41011yescharacteristic
243C22×C13C2×#10011yescharacteristic
244C22×C13#41311yescharacteristic
245C22×C13C2×C4⋊C811yescharacteristic
246C22×C13#42311yescharacteristic
247C22×C13#43111yescharacteristic
248C22×C13#43211yescharacteristic
249C4×C13—22no—
250C4×C13—22no—
251C4×C13—42no—
252Dic13—24no—
253Dic13—22no—
254Dic13—22no—
255Dic13—42no—
256Dic13—42no—
257C25C4×D1311yescharacteristic
258C23×C4—131no—
259C23×C4—131no—
260C23×C4—131no—
261C23×C4—131no—
262C23×C4—261no—
263C23×C4—261no—
264C23×C4—262no—
265C23×C4—262no—
266C23×C4—522no—
267C23×C4—522no—
268C23×C4C4×D1311yescharacteristic
269C23×C4C22×D1311yescharacteristic
270C22×C8—22no—
271C22×C8—262no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—134no—
275C2×C22⋊C4—262no—
276C2×C22⋊C4—262no—
277C2×C22⋊C4Dic2612yes—
278C2×C22⋊C4C22⋊D1312yes—
279C2×C42—132no—
280C2×C42—132no—
281C2×C42—261no—
282C2×C42—261no—
283C2×C42—522no—
284C2×C42—522no—
285C4⋊C8—524no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—262no—
289C22⋊C8—262no—
290#95—262no—
291#95—262no—
292C2×C13—21no—
293C2×C13—21no—
294C2×C13—21no—
295C2×C13—21no—
296C2×C13—44no—
297C2×C13#145911yescharacteristic
298C2×C13#146011yescharacteristic
299C2×C13#146211yescharacteristic
300C2×C13#146411yescharacteristic
301C2×C13#179711yescharacteristic
302C2×C13#179911yescharacteristic
303C2×C13#180011yescharacteristic
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24C8×D1312yes—
313C24#609412yes—
314C24#610112yes—
315C24C2×C4×D1311yescharacteristic
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—132no—
323C22×C4—131no—
324C22×C4—131no—
325C22×C4—132no—
326C22×C4—132no—
327C22×C4—132no—
328C22×C4—132no—
329C22×C4—262no—
330C22×C4—262no—
331C22×C4—261no—
332C22×C4—261no—
333C22×C4—264no—
334C22×C4—264no—
335C22×C4—261no—
336C22×C4—261no—
337C22×C4—261no—
338C22×C4—261no—
339C22×C4—522no—
340C22×C4—522no—
341C22×C4—521no—
342C22×C4—521no—
343C22×C4—528no—
344C22×C4—528no—
345C22×C4—522no—
346C22×C4—522no—
347C22×C4—521no—
348C22×C4—521no—
349C22×C4—521no—
350C22×C4—521no—
351C22×C4#610112yes—
352C22×C4C2×Dic2611yescharacteristic
353C22×C4D4×D1312yes—
354C22×C4#612912yes—
355C22×C4C2×C22⋊D1311yescharacteristic
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—522no—
360C2×C8—522no—
361C2×C8—522no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—268no—
365C42—264no—
366C42—522no—
367C42—522no—
368C42—522no—
369C13#855911yescharacteristic, complemented, Sylow
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#8805414yescomplemented
397C23#8813111yescharacteristic
398C23C22⋊C4×D1311yescharacteristic
399C23C2×C8×D1311yescharacteristic
400C23C2×#609411yescharacteristic
401C23M16×D1311yescharacteristic
402C23M16×D1311yescharacteristic
403C23C2×#610111yescharacteristic
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—132no—
415C2×C4—132no—
416C2×C4—134no—
417C2×C4—264no—
418C2×C4—264no—
419C2×C4—262no—
420C2×C4—261no—
421C2×C4—261no—
422C2×C4—262no—
423C2×C4—261no—
424C2×C4—261no—
425C2×C4—262no—
426C2×C4—262no—
427C2×C4—522no—
428C2×C4—522no—
429C2×C4—528no—
430C2×C4—528no—
431C2×C4—522no—
432C2×C4—522no—
433C2×C4—521no—
434C2×C4—521no—
435C2×C4—521no—
436C2×C4—521no—
437C2×C4—522no—
438C2×C4—522no—
439C2×C4#8813212yes—
440C2×C4#8819312yes—
441C8—82no—
442C8—1042no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—24no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22#1171837911yescharacteristic
467C22#1171838011yescharacteristic
468C22D13×C22⋊C811yescharacteristic
469C22#1171838711yescharacteristic
470C22#1171840111yescharacteristic
471C22D13×#9911yescharacteristic
472C22D13×#10011yescharacteristic
473C22#1171840611yescharacteristic
474C22#1171860311yescharacteristic
475C22C2×#8805411yescharacteristic
476C22#1171877911yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—264no—
481C4—262no—
482C4—262no—
483C4—522no—
484C4—522no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—44no—
490C2#1311315111yescharacteristic
491C2#1311315511yescharacteristic
492C2#1311316111yescharacteristic
493C2#1311316711yescharacteristic
494C2#1311481411yescharacteristic
495C2#1311482211yescharacteristic
496C2#1311482611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D13
commutatorC23×C13C22×C8
FrattiniC23×C4C22×D13
FittingC2×C13×#323C2
socleC23×C13C2×C22⋊C4
prime radicalC25×C13C2×C4
cosocleC23×C4×C13C23
hypercenterC2×#323D13
hypocenterC13#8559
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #13114814C2semidirect
4. #13112678C2semidirect
41. C2×C22⋊Dic13C8semidirect
153. C23×C13C4⋊C8semidirect
369. C13#8559semidirect
396. C23#88054semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×Dic132
#131148142
C2×#117181172
#131126782
C2×C13×#3232
#855913