Finite Group Databasebeta

← #28304268

Subgroups — #28304268

Order 13568.

964 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊#88324C211yesmaximal, characteristic
2#20290537C211yesmaximal, characteristic
3C2×#16811468C212yesmaximal, complemented
4#20289553C212yesmaximal, complemented
5C2×#16810498C211yesmaximal, characteristic
6C23×#88324—21no—
7C2×C22⋊#88324—24no—
8C2×C22⋊#88324—21no—
9C2×C22⋊#88324—21no—
10C2×C22⋊#88324—21no—
11C2×C22⋊#88324C2212yes—
12C2×C22⋊#88324C2212yes—
13#16811468—24no—
14C24×Dic53C2211yescharacteristic
15C22×C4⋊Dic53C411yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C22⋊Dic53C411yescharacteristic
18#16810498—21no—
19#16810498—21no—
20#16810498—21no—
21#16810498—21no—
22C2×#13133124C411yescharacteristic
23C2×#13133124C2211yescharacteristic
24C22×C22⋊C4×C53C411yescharacteristic
25C22×#88324—24no—
26C22×#88324—22no—
27C22×#88324—22no—
28C22×#88324—42no—
29C22×#88324—42no—
30C22⋊#88324—24no—
31C22⋊#88324—22no—
32C22⋊#88324—22no—
33C22⋊#88324—22no—
34C22⋊#88324—42no—
35C22⋊#88324—42no—
36C22⋊#88324D414yes—
37C23×Dic53—22no—
38C23×Dic53—22no—
39C23×Dic53—42no—
40C23×Dic53—42no—
41C23×Dic53C2×C411yescharacteristic
42C23×Dic53C2×C411yescharacteristic
43C23×Dic53C2×C411yescharacteristic
44C23×Dic53D411yescharacteristic
45C23×Dic53D411yescharacteristic
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C23×Dic53C2311yescharacteristic
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—42no—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—42no—
59#13133124—21no—
60#13133124—21no—
61#13133124—21no—
62#13133124—21no—
63C25×C53C2×C411yescharacteristic
64C23×C4×C53C2×C411yescharacteristic
65C23×C4×C53C2×C411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53C814yescomplemented
70C24×C53—22no—
71C24×C53—22no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—41no—
75C24×C53—41no—
76C24×C53C22⋊C411yescharacteristic
77C24×C53C22⋊C411yescharacteristic
78C24×C53C2×C812yes—
79C24×C53M1612yes—
80C24×C53C22×C411yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C2×C812yescomplemented
90C22×C4×C53C2×C812yescomplemented
91C22×Dic53—22no—
92C22×Dic53—22no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—22no—
99C22×Dic53—21no—
100C22×Dic53—22no—
101C22×Dic53—22no—
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—22no—
106C22×Dic53—22no—
107C22×Dic53—41no—
108C22×Dic53—41no—
109C22×Dic53—41no—
110C22×Dic53—41no—
111C22×Dic53—41no—
112C22×Dic53—41no—
113C22×Dic53—41no—
114C22×Dic53—48no—
115C22×Dic53—48no—
116C22×Dic53—41no—
117C22×Dic53—41no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C22×Dic53C22⋊C411yescharacteristic
121C22×Dic53C22⋊C411yescharacteristic
122C22×Dic53C2×D411yescharacteristic
123C22×Dic53C2×D411yescharacteristic
124C22×Dic53C2×D412yes—
125C22×Dic53#4112yes—
126C2×#88324—24no—
127C2×#88324—48no—
128C2×#88324—42no—
129C2×#88324—42no—
130C2×#88324—42no—
131C2×#88324—42no—
132C2×#88324—42no—
133C22⋊C4×C53—24no—
134C22⋊C4×C53—24no—
135C22⋊Dic53—42no—
136C22⋊Dic53—42no—
137C4⋊Dic53—42no—
138C4⋊Dic53—42no—
139C4⋊Dic53—41no—
140C4⋊Dic53—42no—
141C4⋊Dic53—41no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—24no—
151C23×C53—24no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—24no—
159C23×C53—24no—
160C23×C53—42no—
161C23×C53—42no—
162C23×C53—44no—
163C23×C53—44no—
164C23×C53—44no—
165C23×C53—44no—
166C23×C53C22⋊C812yes—
167C23×C53C22⋊C812yes—
168C23×C53#9912yes—
169C23×C53#10112yes—
170C23×C53C2×C22⋊C411yescharacteristic
171C23×C53C4×D412yes—
172C23×C53C22×C811yescharacteristic
173C23×C53C2×M1611yescharacteristic
174C23×C53#13112yes—
175C2×C4×C53—22no—
176C2×C4×C53—22no—
177C2×C4×C53—24no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×Dic53—24no—
186C2×Dic53—24no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—44no—
199C2×Dic53—44no—
200C2×Dic53—44no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—82no—
210C2×Dic53—82no—
211C2×Dic53#9912yes—
212C2×Dic53#12011yescharacteristic
213C2×Dic53#12112yes—
214C2×Dic53#12311yescharacteristic
215#88324—44no—
216#88324—82no—
217#9081—531nomaximal, Sylow
218C22×C53—22no—
219C22×C53—22no—
220C22×C53—22no—
221C22×C53—22no—
222C22×C53—22no—
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53—22no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—48no—
236C22×C53—48no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53#32414yescomplemented
242C22×C53#38611yescharacteristic
243C22×C53C2×C22⋊C811yescharacteristic
244C22×C53#40811yescharacteristic
245C22×C53C2×#9911yescharacteristic
246C22×C53C2×#9911yescharacteristic
247C22×C53C2×#10111yescharacteristic
248C22×C53#41311yescharacteristic
249C22×C53C8×D412yes—
250C22×C53#43512yes—
251C4×C53—42no—
252C4×C53—42no—
253Dic53—21no—
254Dic53—22no—
255Dic53—22no—
256Dic53—21no—
257Dic53—41no—
258Dic53—81no—
259Dic53—81no—
260C22×C22⋊C8—531no—
261C2×#385—531no—
262#1779—531no—
263C2×#324—532no—
264#1280—532no—
265C2×C53—24no—
266C2×C53—22no—
267C2×C53—22no—
268C2×C53—42no—
269C2×C53C2×#32411yescharacteristic
270C2×C53#145911yescharacteristic
271C2×C53#146811yescharacteristic
272C2×C53#147211yescharacteristic
273C2×C53#186911yescharacteristic
274C2×C53#189811yescharacteristic
275C2×C53#190111yescharacteristic
276C24×C4—531no—
277C23×C8—1061no—
278C22×C4⋊C4—531no—
279C22×C4⋊C4—531no—
280C22×C22⋊C4—531no—
281C22×C22⋊C4#615711yescharacteristic
282C2×C22⋊C8—532no—
283C2×C22⋊C8—532no—
284C2×C22⋊C8—1064no—
285C2×C22⋊C8—1061no—
286C2×C22⋊C8—1061no—
287C2×C22⋊C8—1061no—
288#385—1061no—
289#385—1061no—
290#385—1061no—
291#385—1061no—
292C2×#95—531no—
293C2×#95—531no—
294#324—1064no—
295C53#908111yescharacteristic, complemented, Sylow
296C25C2×#615711yescharacteristic
297C23×C4—531no—
298C23×C4—531no—
299C23×C4—531no—
300C23×C4—531no—
301C23×C4—531no—
302C23×C4—531no—
303C23×C4—531no—
304C23×C4—531no—
305C23×C4—1062no—
306C23×C4—1062no—
307C23×C4—2122no—
308C23×C4—2122no—
309C23×C4C2×#615711yescharacteristic
310C23×C4C2×#615711yescharacteristic
311C22×C8—1062no—
312C22×C8—1064no—
313C22×C8—1062no—
314C22×C8—2122no—
315C22×C8—2122no—
316C2×C4⋊C4—1062no—
317C2×C4⋊C4—1062no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4—1061no—
320C2×C4⋊C4—1061no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—2122no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—1061no—
327C2×C22⋊C4—1061no—
328C2×C22⋊C4—2122no—
329C2×C22⋊C4#8832414yescomplemented
330C22⋊C8—534no—
331C22⋊C8—1062no—
332C22⋊C8—1064no—
333C22⋊C8—1062no—
334C22⋊C8—1062no—
335C22⋊C8—2122no—
336C22⋊C8—2122no—
337#95—1061no—
338#95—1061no—
339#95—1061no—
340#95—1061no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24C2×#8832412yes—
348C24#1171994012yes—
349C24C22⋊#615711yescharacteristic
350C24C22⋊#615711yescharacteristic
351C24C22×#615711yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—531no—
361C22×C4—531no—
362C22×C4—531no—
363C22×C4—532no—
364C22×C4—532no—
365C22×C4—531no—
366C22×C4—1061no—
367C22×C4—1062no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1062no—
371C22×C4—1062no—
372C22×C4—1062no—
373C22×C4—1062no—
374C22×C4—1061no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1061no—
381C22×C4—1061no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4—2121no—
385C22×C4—2121no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4—2121no—
389C22×C4—2128no—
390C22×C4—2128no—
391C22×C4—2121no—
392C22×C4—2121no—
393C22×C4—2121no—
394C22×C4—2121no—
395C22×C4C2×#8832412yescomplemented
396C22×C4C2×#8832412yescomplemented
397C2×C8—1064no—
398C2×C8—2122no—
399C2×C8—2122no—
400C2×C8—2122no—
401C2×C8—2128no—
402C2×C8—2122no—
403C2×C8—2122no—
404C4⋊C4—2121no—
405C4⋊C4—2122no—
406C4⋊C4—2121no—
407C4⋊C4—2122no—
408C4⋊C4—2122no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—2122no—
412C22⋊C4—2122no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23C22⋊#8832412yes—
438C23C22⋊#8832412yes—
439C23#1313327612yes—
440C23#1313327812yes—
441C23C2×C22⋊#615711yescharacteristic
442C23D4×#615712yes—
443C23C22×#8832411yescharacteristic
444C23C2×#1171994011yescharacteristic
445C23#1313330412yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—531no—
457C2×C4—532no—
458C2×C4—531no—
459C2×C4—532no—
460C2×C4—1064no—
461C2×C4—1064no—
462C2×C4—1061no—
463C2×C4—1061no—
464C2×C4—1061no—
465C2×C4—1061no—
466C2×C4—1061no—
467C2×C4—1061no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—4242no—
472C2×C4—4242no—
473C2×C4—2122no—
474C2×C4—2122no—
475C2×C4—2124no—
476C2×C4—2124no—
477C2×C4—2124no—
478C2×C4—2122no—
479C2×C4—2122no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4—2121no—
484C2×C4—2121no—
485C2×C4—2121no—
486C8—4242no—
487C8—2124no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—48no—
506C22—48no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22#1681146814yescomplemented
512C22#1681157111yescharacteristic
513C22C2×C22⋊#8832411yescharacteristic
514C22#1681159611yescharacteristic
515C22D53⋊#9911yescharacteristic
516C22C2×#1313327611yescharacteristic
517C22C2×#1313327811yescharacteristic
518C22#1681161011yescharacteristic
519C22D4×#8832412yes—
520C22#1681165412yes—
521C4—42no—
522C4—42no—
523C4—1062no—
524C4—1061no—
525C4—1061no—
526C4—1062no—
527C4—4241no—
528C4—4241no—
529C4—2121no—
530C2—22no—
531C2—22no—
532C2—24no—
533C2—42no—
534C2C2×#1681146811yescharacteristic
535C2#2028995111yescharacteristic
536C2#2028998211yescharacteristic
537C2#2028999411yescharacteristic
538C2#2029068811yescharacteristic
539C2#2029073711yescharacteristic
540C2#2029074211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#9081
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16811468C2semidirect
4. #20289553C2semidirect
69. C2×C22⋊C4×C53C8semidirect
89. C22×C4×C53C2×C8semidirect
90. C22×C4×C53C2×C8semidirect
241. C22×C53#324semidirect
295. C53#9081semidirect
329. C2×C22⋊C4#88324semidirect
395. C22×C4C2×#88324semidirect
396. C22×C4C2×#88324semidirect
511. C22#16811468semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#883242
#202905372
C2×#168114682
#202895532
C2×#168104982
#908153