Finite Group Databasebeta

← #29652935

Subgroups — #29652935

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic, complemented
2C2×#16839913C212yesmaximal, complemented
3#21444758C214yesmaximal, complemented
4#21444741C214yesmaximal, complemented
5#21444390C212yesmaximal, complemented
6#21444137C211yesmaximal, characteristic, complemented
7C61×#2408C211yesmaximal, characteristic
8C22×#11758200—22no—
9#16840471—24no—
10#16840469—22no—
11D4×Dic122—22no—
12#16840393C2212yescomplemented
13#16840388—21no—
14#16840388C2212yescomplemented
15#16840319—21no—
16C2×#14519756—21no—
17C2×#14519756C2212yes—
18#16840237—21no—
19#16839913—24no—
20#16839913C2218yes—
21C2×#14519698—22no—
22C2×C22⋊#92418C2212yes—
23#16839498C2218yescomplemented
24#16839488C2212yescomplemented
25#16839486C2212yescomplemented
26C23×D4×C61C2211yescharacteristic
27C2×#121×C61—21no—
28C2×#120×C61—21no—
29C2×C4×D4×C61C2211yescharacteristic
30C61×#394C2212yes—
31C61×#392C2212yes—
32C61×#390—21no—
33C61×#380C2211yescharacteristic
34C61×#379—21no—
35C22×Dic122—21no—
36C22×Dic122D412yescomplemented
37C2×#11758200—28no—
38C2×#11758200—48no—
39C22×#92418—22no—
40#14519777—21no—
41#14519777—21no—
42#14519776—24no—
43#14519769—22no—
44#14519768—24no—
45#14519756—22no—
46#14519756—22no—
47#14519756—22no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51#14519756D414yescomplemented
52#14519756C2314yes—
53#14519754—24no—
54#14519754—24no—
55#14519754—24no—
56#14519754—24no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59D4×Dic61—24no—
60D4×Dic61—24no—
61#14519736—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D412yescomplemented
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—22no—
68#14519698—28no—
69C22⋊#92418—22no—
70C22⋊#92418C2314yes—
71C25×C61—21no—
72C22×D4×C61—22no—
73C22×D4×C61—22no—
74C22×D4×C61—21no—
75C22×D4×C61—22no—
76C22×D4×C61—44no—
77C22×D4×C61D411yescharacteristic
78C22×D4×C61C2312yes—
79C23×C4×C61D411yescharacteristic
80C23×C4×C61C2311yescharacteristic
81#121×C61—44no—
82#120×C61—44no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C22⋊C4×C61C2311yescharacteristic
93C2×C42×C61D411yescharacteristic
94#95×C61C2312yes—
95C24×C61—22no—
96C24×C61—44no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—82no—
101C24×C61C2×D411yescharacteristic
102C24×C61C2×D411yescharacteristic
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—28no—
106C2×D4×C61—24no—
107C2×D4×C61—24no—
108C2×D4×C61—44no—
109C2×D4×C61—42no—
110C2×D4×C61—44no—
111C2×D4×C61—48no—
112C2×D4×C61—44no—
113C2×D4×C61—44no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—82no—
117C2×D4×C61SD1614yescomplemented
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—44no—
138C2×Dic122—82no—
139C2×Dic122C2×D414yes—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—42no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—44no—
145C22⋊C4×C61—42no—
146C22⋊C4×C61—44no—
147C22⋊C4×C61—82no—
148C42×C61—21no—
149C42×C61—21no—
150C22⋊Dic61—24no—
151C22⋊Dic61—24no—
152C22⋊Dic61—24no—
153#11758200—416no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C2×D414yes—
160#11758196—24no—
161#11758196—24no—
162#11758196—24no—
163#11758196—24no—
164C4×Dic61—24no—
165C2×#92418—24no—
166C2×#92418—44no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—24no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—42no—
175C23×C61—48no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—84no—
182C23×C61—88no—
183C23×C61—88no—
184C23×C61—84no—
185C23×C61#12012yes—
186C23×C61C2×SD1612yes—
187C23×C61C2×SD1612yescomplemented
188C23×C61#13712yes—
189C23×C61C22×D411yescharacteristic
190D4×C61—28no—
191D4×C61—28no—
192D4×C61—44no—
193D4×C61—44no—
194D4×C61—44no—
195D4×C61—88no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—81no—
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C2×SD1612yescomplemented
214C2×C4×C61C22×D412yes—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—82no—
218C2×Dic61—82no—
219C2×Dic61—82no—
220C2×Dic61—82no—
221Dic122—44no—
222Dic122—44no—
223Dic122—82no—
224Dic122—82no—
225#92418—44no—
226#18196—611nomaximal, Sylow
227C22×C61—22no—
228C22×C61—21no—
229C22×C61—22no—
230C22×C61—21no—
231C22×C61—22no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—44no—
236C22×C61—44no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—44no—
240C22×C61—81no—
241C22×C61—82no—
242C22×C61—82no—
243C22×C61—816no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61#44814yes—
247C22×C61C2×#12011yescharacteristic
248C22×C61C22×SD1611yescharacteristic
249C22×C61C2×#13711yescharacteristic
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—21no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—82no—
256C4×C61B(4,𝔽2)12yes—
257Dic61—44no—
258Dic61—44no—
259#3501—611no—
260C2×#450—612no—
261#2408D6111yescharacteristic
262#1599—614no—
263#1596—614no—
264#1520—612no—
265#1459—611no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—24no—
269C2×C61—42no—
270C2×C61—84no—
271C2×C61C2×#44811yescharacteristic
272C2×C61#302111yescharacteristic
273C2×C61C2×B(4,𝔽2)11yescharacteristic
274C23×D4C2×D6111yescharacteristic
275C22×SD16—1222no—
276#554—1222no—
277#554—1224no—
278D4×Q8—1222no—
279#544—612no—
280#543—612no—
281#543—1221no—
282#533—1221no—
283C2×C22⋊Q8—612no—
284C2×C22⋊Q8—1221no—
285C2×#121—21no—
286C2×#120—21no—
287#518—1221no—
288C2×C4×D4C2×D6111yescharacteristic
289#450—618no—
290#450—1224no—
291C2×#102—1222no—
292C2×C22⋊C8—612no—
293#394C2×D6112yes—
294#392C2×D6112yes—
295#390—21no—
296#380C2×D6111yescharacteristic
297#379—21no—
298#328—618no—
299#324—612no—
300#323—612no—
301C61#1819611yescharacteristic, complemented, Sylow
302C25—21no—
303C22×Q8—612no—
304C22×Q8—1221no—
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—44no—
310C22×D4C22⋊D6111yescharacteristic
311C22×D4C22×D6112yes—
312C23×C4C22⋊D6111yescharacteristic
313C23×C4C22×D6111yescharacteristic
314C2×SD16—1228no—
315C2×SD16—2448no—
316C22×C8—1222no—
317C4⋊Q8—1221no—
318C4⋊Q8—1221no—
319C4⋊Q8—1224no—
320#125—1222no—
321#125—1224no—
322C22⋊Q8—614no—
323C22⋊Q8—614no—
324C22⋊Q8—1224no—
325C22⋊Q8—1224no—
326C22⋊Q8—1224no—
327C22⋊Q8—1224no—
328C22⋊Q8—1222no—
329C22⋊Q8—1222no—
330C22⋊Q8—1222no—
331C22⋊Q8—1222no—
332C22⋊Q8—1222no—
333C22⋊Q8—1222no—
334#121—44no—
335#120—44no—
336C4×Q8—1222no—
337C4×Q8—1222no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—1224no—
341C4×D4—1224no—
342#117—1222no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4C22⋊D6111yescharacteristic
356C2×C22⋊C4C22×D6111yescharacteristic
357C2×C42C22⋊D6111yescharacteristic
358#102—1228no—
359C22⋊C8—614no—
360C22⋊C8—1222no—
361#95C22×D6112yes—
362C24—22no—
363C24—44no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—82no—
368C24C2×C22⋊D6111yescharacteristic
369C24C2×C22⋊D6111yescharacteristic
370C2×Q8—614no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—4882no—
375C2×Q8—2444no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—82no—
390C2×D4#1175820014yescomplemented
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4D4×D6112yes—
401C22×C4D4×D6112yes—
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407SD16—24416no—
408C2×C8—1224no—
409C2×C8—2444no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—614no—
413C4⋊C4—1224no—
414C4⋊C4—1222no—
415C4⋊C4—1224no—
416C4⋊C4—1222no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1224no—
420C4⋊C4—1222no—
421C4⋊C4—1224no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—82no—
428C22⋊C4—1224no—
429C22⋊C4—1224no—
430C22⋊C4—1224no—
431C42—21no—
432C42—21no—
433C42—1224no—
434C23—21no—
435C23—21no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—48no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—84no—
449C23—88no—
450C23—88no—
451C23—84no—
452C23#1451974512yes—
453C23C2×#1175820012yes—
454C23C2×#1175820012yescomplemented
455C23#1451980712yes—
456C23C22×C22⋊D6111yescharacteristic
457Q8—4882no—
458Q8—4882no—
459Q8—2444no—
460Q8—2444no—
461D4—28no—
462D4—28no—
463D4—44no—
464D4—44no—
465D4—44no—
466D4—88no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—1224no—
483C2×C4—1224no—
484C2×C4—4882no—
485C2×C4—4882no—
486C2×C4—4882no—
487C2×C4—4882no—
488C2×C4#1451974511yescharacteristic
489C2×C4#1451974511yescharacteristic
490C2×C4C2×#1175820012yescomplemented
491C2×C4C2×D4×D6112yes—
492C8—2444no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—44no—
502C22—44no—
503C22—41no—
504C22—41no—
505C22—44no—
506C22—81no—
507C22—82no—
508C22—82no—
509C22—816no—
510C22—82no—
511C22—82no—
512C22#1683990014yes—
513C22C2×#1451974511yescharacteristic
514C22C22×#1175820011yescharacteristic
515C22C2×#1451980711yescharacteristic
516C4—22no—
517C4—21no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—82no—
522C4—2444no—
523C4—2444no—
524C4#1683994812yes—
525C2—21no—
526C2—21no—
527C2—24no—
528C2—42no—
529C2—84no—
530C2C2×#1683990011yescharacteristic
531C2#2145519011yescharacteristic
532C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2408C2
socleC22×C61C2×#120
hypercenter#2408D61
hypocenterC61#18196
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #21459622C2semidirect
2. C2×#16839913C2semidirect
3. #21444758C2semidirect
4. #21444741C2semidirect
5. #21444390C2semidirect
6. #21444137C2semidirect
12. #16840393C22semidirect
14. #16840388C22semidirect
23. #16839498C22semidirect
24. #16839488C22semidirect
25. #16839486C22semidirect
36. C22×Dic122D4semidirect
51. #14519756D4semidirect
63. C2×C4⋊Dic61D4semidirect
117. C2×D4×C61SD16semidirect
187. C23×C61C2×SD16semidirect
213. C2×C4×C61C2×SD16semidirect
301. C61#18196semidirect
390. C2×D4#11758200semidirect
454. C23C2×#11758200semidirect
490. C2×C4C2×#11758200semidirect

Maximal subgroups

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

subgroupindex
#214596222
C2×#168399132
#214447582
#214447412
#214443902
#214441372
C61×#24082
#1819661