Finite Group Databasebeta

← #15754365

Subgroups — #15754365

Order 3328.

1276 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88178C211yesmaximal, characteristic
2C2×#11718059C214yesmaximal, complemented
3#13113059C211yesmaximal, characteristic
4C13×#1779C211yesmaximal, characteristic
5C24×C4×C13C2211yescharacteristic
6C23×Dic26C411yescharacteristic, complemented
7C22×C22⋊Dic13—21no—
8C22×C22⋊Dic13—21no—
9C22×C4⋊Dic13C411yescharacteristic, complemented
10C2×#88178—21no—
11C2×#88178—21no—
12C2×#88178—28no—
13C2×#88178C412yescomplemented
14C2×#88178C2212yes—
15C22×#6101—21no—
16C2×C13×C22⋊C8—21no—
17C2×C13×C22⋊C8—21no—
18C2×C13×C22⋊C8C2212yes—
19C2×#88071—21no—
20C2×#88071C2212yes—
21#11718059—28no—
22C25×C13D411yescharacteristic
23C23×C4×C13—24no—
24C23×C4×C13—42no—
25C23×C4×C13—42no—
26C23×C4×C13C2×C411yescharacteristic
27C23×C4×C13C2×C411yescharacteristic
28C23×C4×C13D411yescharacteristic
29C23×C4×C13D411yescharacteristic
30C23×C4×C13D411yescharacteristic
31C23×C4×C13C2311yescharacteristic
32C23×Dic13—21no—
33C23×Dic13—21no—
34C23×Dic13—22no—
35C22×Dic26—21no—
36C22×Dic26—24no—
37C22×Dic26—42no—
38C22×Dic26—42no—
39C22×Dic26C2×C412yes—
40C22×C8×C13—22no—
41C2×C22⋊Dic13—28no—
42C2×C22⋊Dic13—22no—
43C2×C22⋊Dic13—22no—
44C2×C22⋊Dic13—24no—
45C2×C22⋊Dic13—42no—
46C2×C22⋊Dic13—42no—
47C2×C4⋊Dic13—21no—
48C2×C4⋊Dic13—24no—
49C2×C4⋊Dic13C2×C412yes—
50#88178—28no—
51#88178—28no—
52#88178—24no—
53#88178—24no—
54#88178—24no—
55#88178D418yes—
56C2×#6101—28no—
57C2×#6101—22no—
58C2×#6101—22no—
59C13×C22⋊C8—24no—
60C13×C22⋊C8—42no—
61C13×C22⋊C8—42no—
62#88071—24no—
63#88071—42no—
64#8329—131nomaximal, Sylow
65C24×C13—22no—
66C24×C13—22no—
67C24×C13—22no—
68C24×C13—22no—
69C24×C13—41no—
70C24×C13—41no—
71C24×C13C22⋊C411yescharacteristic
72C24×C13C22⋊C411yescharacteristic
73C24×C13SD1612yes—
74C24×C13Q1612yes—
75C24×C13C2×D411yescharacteristic
76C22×C4×C13—22no—
77C22×C4×C13—22no—
78C22×C4×C13—22no—
79C22×C4×C13—22no—
80C22×C4×C13—22no—
81C22×C4×C13—22no—
82C22×C4×C13—22no—
83C22×C4×C13—22no—
84C22×C4×C13—22no—
85C22×C4×C13—22no—
86C22×C4×C13—41no—
87C22×C4×C13—41no—
88C22×C4×C13—41no—
89C22×C4×C13—41no—
90C22×C4×C13—41no—
91C22×C4×C13—41no—
92C22×C4×C13—41no—
93C22×C4×C13—416no—
94C22×C4×C13C22⋊C411yescharacteristic
95C22×C4×C13C22⋊C411yescharacteristic
96C22×C4×C13C22×C411yescharacteristic
97C22×C4×C13C2×D414yes—
98C22×C4×C13C2×D411yescharacteristic
99C22×Dic13—24no—
100C22×Dic13—24no—
101C22×Dic13—22no—
102C22×Dic13—22no—
103C22×Dic13—24no—
104C22×Dic13—42no—
105C22×Dic13—42no—
106C22×Dic13—42no—
107C22×Dic13—42no—
108C22×Dic13—42no—
109C22×Dic13—42no—
110C22×Dic13—42no—
111C22×Dic13—42no—
112C22×Dic13—42no—
113C22×Dic13—42no—
114C2×Dic26—24no—
115C2×Dic26—22no—
116C2×Dic26—42no—
117C2×Dic26—416no—
118C2×Dic26—42no—
119C2×Dic26—42no—
120C2×Dic26—42no—
121C2×Dic26C22⋊C414yescomplemented
122C2×C8×C13—42no—
123C2×C8×C13—42no—
124C2×C8×C13—42no—
125C22⋊Dic13—28no—
126C22⋊Dic13—28no—
127C22⋊Dic13—48no—
128C4⋊Dic13—24no—
129C4⋊Dic13—22no—
130C4⋊Dic13C22⋊C414yescomplemented
131#6101—28no—
132#6101—28no—
133C22×C22⋊Q8—131no—
134#1779D1311yescharacteristic
135C2×#328—134no—
136C22⋊#95—131no—
137C23×C13—21no—
138C23×C13—21no—
139C23×C13—21no—
140C23×C13—21no—
141C23×C13—21no—
142C23×C13—21no—
143C23×C13—21no—
144C23×C13—21no—
145C23×C13—24no—
146C23×C13—24no—
147C23×C13—21no—
148C23×C13—21no—
149C23×C13—21no—
150C23×C13—21no—
151C23×C13—21no—
152C23×C13—21no—
153C23×C13—24no—
154C23×C13—24no—
155C23×C13—42no—
156C23×C13—42no—
157C23×C13—44no—
158C23×C13—44no—
159C23×C13—44no—
160C23×C13—44no—
161C23×C13#9912yes—
162C23×C13#10312yes—
163C23×C13#10312yes—
164C23×C13#10412yes—
165C23×C13C2×C22⋊C411yescharacteristic
166C23×C13#12012yes—
167C23×C13C2×SD1611yescharacteristic
168C23×C13C2×Q1611yescharacteristic
169C23×C13#13712yes—
170C2×C4×C13—28no—
171C2×C4×C13—21no—
172C2×C4×C13—21no—
173C2×C4×C13—21no—
174C2×C4×C13—21no—
175C2×C4×C13—21no—
176C2×C4×C13—21no—
177C2×C4×C13—21no—
178C2×C4×C13—21no—
179C2×C4×C13—21no—
180C2×C4×C13—44no—
181C2×C4×C13—44no—
182C2×C4×C13—44no—
183C2×C4×C13—44no—
184C2×C4×C13—44no—
185C2×C4×C13#9912yes—
186C2×C4×C13C2×C22⋊C412yes—
187C2×C4×C13#12012yes—
188C2×Dic13—24no—
189C2×Dic13—24no—
190C2×Dic13—48no—
191C2×Dic13—48no—
192C2×Dic13—42no—
193C2×Dic13—42no—
194C2×Dic13—42no—
195C2×Dic13—42no—
196C2×Dic13—44no—
197C2×Dic13—44no—
198C2×Dic13—44no—
199C2×Dic13—82no—
200C2×Dic13—82no—
201C2×Dic13—82no—
202C2×Dic13—82no—
203Dic26—48no—
204Dic26—44no—
205Dic26—44no—
206C8×C13—82no—
207C23×Q8—131no—
208C24×C4C2×D1311yescharacteristic
209C2×C22⋊Q8—132no—
210C2×C22⋊Q8—132no—
211C2×C22⋊Q8—268no—
212C2×C22⋊Q8—261no—
213C2×C22⋊Q8—261no—
214C22×C4⋊C4—131no—
215C22×C4⋊C4—261no—
216C22×C22⋊C4—261no—
217C22×C22⋊C4—261no—
218C2×C22⋊C8—21no—
219C2×C22⋊C8—21no—
220C2×C22⋊C8C2×D1312yes—
221C2×#95—132no—
222C2×#95—261no—
223#328—268no—
224C22×C13—21no—
225C22×C13—21no—
226C22×C13—22no—
227C22×C13—22no—
228C22×C13—22no—
229C22×C13—22no—
230C22×C13—21no—
231C22×C13—22no—
232C22×C13—21no—
233C22×C13—22no—
234C22×C13—22no—
235C22×C13—22no—
236C22×C13—22no—
237C22×C13—22no—
238C22×C13—22no—
239C22×C13—22no—
240C22×C13—22no—
241C22×C13—48no—
242C22×C13—48no—
243C22×C13—41no—
244C22×C13—41no—
245C22×C13—41no—
246C22×C13—41no—
247C22×C13#32814yescomplemented
248C22×C13#37911yescharacteristic
249C22×C13C2×#9911yescharacteristic
250C22×C13C2×#9911yescharacteristic
251C22×C13C2×#10311yescharacteristic
252C22×C13#41911yescharacteristic
253C22×C13C2×#10411yescharacteristic
254C22×C13#42111yescharacteristic
255C22×C13#44812yes—
256C22×C13#45112yes—
257C4×C13—22no—
258C4×C13—22no—
259C4×C13—22no—
260C4×C13—41no—
261Dic13—44no—
262Dic13—44no—
263Dic13—84no—
264C25D5211yescharacteristic
265C22×Q8—132no—
266C22×Q8—264no—
267C22×Q8—261no—
268C22×Q8—522no—
269C22×Q8—522no—
270C23×C4—24no—
271C23×C4—42no—
272C23×C4—42no—
273C23×C4—261no—
274C23×C4—261no—
275C23×C4—262no—
276C23×C4C4×D1311yescharacteristic
277C23×C4C4×D1311yescharacteristic
278C23×C4D5211yescharacteristic
279C23×C4C22⋊D1311yescharacteristic
280C23×C4C22⋊D1311yescharacteristic
281C23×C4C22×D1311yescharacteristic
282C22×C8—22no—
283C22⋊Q8—138no—
284C22⋊Q8—264no—
285C22⋊Q8—264no—
286C22⋊Q8—268no—
287C22⋊Q8—268no—
288C22⋊Q8—264no—
289C2×C4⋊C4—132no—
290C2×C4⋊C4—264no—
291C2×C4⋊C4—268no—
292C2×C4⋊C4—261no—
293C2×C4⋊C4—262no—
294C2×C4⋊C4—262no—
295C2×C22⋊C4—268no—
296C2×C22⋊C4—262no—
297C2×C22⋊C4—262no—
298C2×C22⋊C4—264no—
299C2×C22⋊C4—522no—
300C2×C22⋊C4—522no—
301C22⋊C8—24no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#95—264no—
305#95—522no—
306C2×C13—22no—
307C2×C13—24no—
308C2×C13—22no—
309C2×C13—42no—
310C2×C13C2×#32811yescharacteristic
311C2×C13#150811yescharacteristic
312C2×C13#151111yescharacteristic
313C2×C13#152011yescharacteristic
314C2×C13#178111yescharacteristic
315C2×C13#178711yescharacteristic
316C2×C13#178911yescharacteristic
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—41no—
322C24—41no—
323C24#609512yes—
324C24Dic5212yes—
325C24#610311yescharacteristic
326C24#610311yescharacteristic
327C24C2×D5211yescharacteristic
328C2×Q8—134no—
329C2×Q8—264no—
330C2×Q8—262no—
331C2×Q8—5216no—
332C2×Q8—522no—
333C2×Q8—522no—
334C2×Q8—522no—
335C2×Q8—522no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—416no—
354C22×C4—264no—
355C22×C4—264no—
356C22×C4—262no—
357C22×C4—262no—
358C22×C4—264no—
359C22×C4—522no—
360C22×C4—522no—
361C22×C4—522no—
362C22×C4—522no—
363C22×C4—522no—
364C22×C4—522no—
365C22×C4—522no—
366C22×C4—522no—
367C22×C4—522no—
368C22×C4—522no—
369C22×C4#610311yescharacteristic
370C22×C4#610311yescharacteristic
371C22×C4C2×C4×D1311yescharacteristic
372C22×C4D4×D1314yes—
373C22×C4C2×C22⋊D1311yescharacteristic
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—134no—
378C4⋊C4—268no—
379C4⋊C4—264no—
380C4⋊C4—262no—
381C4⋊C4—268no—
382C22⋊C4—268no—
383C22⋊C4—268no—
384C22⋊C4—528no—
385C13#832911yescharacteristic, complemented, Sylow
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—24no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—24no—
399C23—24no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23#8804612yes—
411C23#8805612yes—
412C23#8805612yes—
413C23#8806512yes—
414C23#8813612yes—
415C23C2×#609511yescharacteristic
416C23C2×Dic5211yescharacteristic
417C23#8816312yes—
418C23C2×#610311yescharacteristic
419Q8—524no—
420Q8—524no—
421Q8—528no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—264no—
438C2×C4—264no—
439C2×C4—528no—
440C2×C4—528no—
441C2×C4—522no—
442C2×C4—522no—
443C2×C4—522no—
444C2×C4—522no—
445C2×C4—524no—
446C2×C4—524no—
447C2×C4—524no—
448C2×C4—1042no—
449C2×C4—1042no—
450C2×C4—1042no—
451C2×C4—1042no—
452C2×C4#8804612yes—
453C2×C4C22⋊C4×D1312yes—
454C2×C4#8819112yes—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—48no—
474C22—48no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#1171805914yescomplemented
480C22#1171839912yes—
481C22#1171840012yes—
482C22C2×#8804611yescharacteristic
483C22C2×#8804611yescharacteristic
484C22#1171861611yescharacteristic
485C22C2×#8805611yescharacteristic
486C22#1171878111yescharacteristic
487C22C2×#8806511yescharacteristic
488C22#1171880011yescharacteristic
489C4—22no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—524no—
494C4—524no—
495C4—1044no—
496C2—22no—
497C2—22no—
498C2—24no—
499C2—42no—
500C2C2×#1171805911yescharacteristic
501C2#1311335711yescharacteristic
502C2#1311337011yescharacteristic
503C2#1311340911yescharacteristic
504C2#1311475011yescharacteristic
505C2#1311477211yescharacteristic
506C2#1311477911yescharacteristic
subgroupG/H
centerC23C2×#6103
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#1779C2
socleC23×C13C2×C22⋊C4
prime radicalC25×C13D4
cosocleC23×C4×C13C23
hypercenter#1779D13
hypocenterC13#8329
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#11718059C2semidirect
6. C23×Dic26C4semidirect
9. C22×C4⋊Dic13C4semidirect
13. C2×#88178C4semidirect
121. C2×Dic26C22⋊C4semidirect
130. C4⋊Dic13C22⋊C4semidirect
247. C22×C13#328semidirect
385. C13#8329semidirect
479. C22#11718059semidirect

Maximal subgroups

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

subgroupindex
C22×#881782
C2×#117180592
#131130592
C13×#17792
#832913