Finite Group Databasebeta

← #29652936

Subgroups — #29652936

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
1C2×#16839912C212yesmaximal, complemented
2#21449175C211yesmaximal, characteristic
3#21444759C214yesmaximal, complemented
4#21444742C214yesmaximal, complemented
5#21444391C212yesmaximal, complemented
6#21444137C211yesmaximal, characteristic, complemented
7C61×#3501C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9C22×#11758201—22no—
10C2×#14519751—21no—
11C2×#14519745—21no—
12#16839912—24no—
13#16839912C2218yes—
14C2×#14519697—22no—
15C2×C22⋊#92418C2212yes—
16#16839679C2212yes—
17#16839671C2212yes—
18#16839664—21no—
19#16839622C2211yescharacteristic
20#16839617—21no—
21#16839499C2218yescomplemented
22#16839488C2212yescomplemented
23#16839486C2212yescomplemented
24C61×#554—24no—
25C61×#554—22no—
26D4×Q8×C61—22no—
27C61×#544C2212yescomplemented
28C61×#543—21no—
29C61×#543C2212yescomplemented
30C61×#533—21no—
31C2×C22⋊Q8×C61—21no—
32C2×C22⋊Q8×C61C2212yes—
33C61×#518—21no—
34C2×C4×D4×C61C2211yescharacteristic
35C24×D61—21no—
36C22×D244—22no—
37C22×D244—21no—
38C22×D244—22no—
39C22×D244—44no—
40C22×D244C2312yes—
41C22×C22⋊D61—22no—
42C2×#11758201—28no—
43C2×#11758201—48no—
44C22×#92418—22no—
45#14519751—44no—
46#14519745—44no—
47C2×C4⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—22no—
50C2×#11758198—22no—
51C2×#11758198—22no—
52#14519697—28no—
53C22⋊#92418—22no—
54C22⋊#92418C2314yes—
55#14519684C2312yes—
56C22×Q8×C61—21no—
57C22×Q8×C61D412yescomplemented
58C22×D4×C61D411yescharacteristic
59C23×C4×C61D411yescharacteristic
60C23×C4×C61C2311yescharacteristic
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—24no—
64#125×C61—22no—
65#125×C61—24no—
66C22⋊Q8×C61—24no—
67C22⋊Q8×C61—24no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—24no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—24no—
76C22⋊Q8×C61D414yescomplemented
77C22⋊Q8×C61C2314yes—
78C4×Q8×C61—22no—
79C4×Q8×C61—22no—
80C4×D4×C61—24no—
81C4×D4×C61—24no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84#117×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61D412yescomplemented
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61D411yescharacteristic
93C2×C22⋊C4×C61C2311yescharacteristic
94C2×C42×C61D411yescharacteristic
95C24×C61C2×D411yescharacteristic
96C24×C61C2×D411yescharacteristic
97C23×D61—22no—
98C23×D61—44no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—82no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—44no—
107C2×Q8×C61—82no—
108C2×Q8×C61C2×D414yes—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61SD1614yescomplemented
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C22×Dic61—22no—
131C2×D244—28no—
132C2×D244—24no—
133C2×D244—24no—
134C2×D244—44no—
135C2×D244—48no—
136C2×D244—44no—
137C2×D244—42no—
138C2×D244—41no—
139C2×D244—41no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D414yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—22no—
156C42×C61—24no—
157C42×C61—21no—
158C42×C61—21no—
159C22⋊Dic61—42no—
160#11758201—416no—
161#11758198—44no—
162#11758198—44no—
163#11758198—82no—
164C4⋊Dic61—42no—
165C2×#92418—24no—
166C2×#92418—44no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61#12012yes—
174C23×C61C2×SD1612yes—
175C23×C61C2×SD1612yescomplemented
176C23×C61#13712yes—
177C23×C61C22×D411yescharacteristic
178C22×D61—24no—
179C22×D61—42no—
180C22×D61—48no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—88no—
187C22×D61—88no—
188C22×D61—84no—
189C22×D61—84no—
190Q8×C61—44no—
191Q8×C61—44no—
192Q8×C61—82no—
193Q8×C61—82no—
194D4×C61—28no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—21no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—81no—
209C2×C4×C61—82no—
210C2×C4×C61—82no—
211C2×C4×C61—82no—
212C2×C4×C61—82no—
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61C2×SD1612yescomplemented
216C2×C4×C61C22×D412yes—
217C22⋊D61—88no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221D244—28no—
222D244—44no—
223D244—44no—
224D244—44no—
225#92418—44no—
226#18196—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—22no—
229C22×C61—21no—
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—81no—
238C22×C61#44814yes—
239C22×C61C2×#12011yescharacteristic
240C22×C61C22×SD1611yescharacteristic
241C22×C61C2×#13711yescharacteristic
242C2×D61—44no—
243C2×D61—44no—
244C2×D61—44no—
245C2×D61—82no—
246C2×D61—82no—
247C2×D61—816no—
248C2×D61—82no—
249C2×D61—82no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—21no—
253C4×C61—41no—
254C4×C61—44no—
255C4×C61—41no—
256C4×C61—44no—
257C4×C61B(4,𝔽2)12yes—
258Dic61—82no—
259#3501D6111yescharacteristic, complemented
260C2×#450—612no—
261#2408—611no—
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×C61C2×#44811yescharacteristic
271C2×C61#302111yescharacteristic
272C2×C61C2×B(4,𝔽2)11yescharacteristic
273D61—84no—
274C23×D4—611no—
275C22×SD16—1222no—
276#554—24no—
277#554—22no—
278D4×Q8—22no—
279#544C2×D6112yescomplemented
280#543—21no—
281#543C2×D6112yescomplemented
282#533—21no—
283C2×C22⋊Q8—21no—
284C2×C22⋊Q8C2×D6112yes—
285C2×#121—1221no—
286C2×#120—1221no—
287#518—21no—
288C2×C4×D4C2×D6111yescharacteristic
289#450—618no—
290#450—1224no—
291C2×#102—1222no—
292C2×C22⋊C8—612no—
293#394—612no—
294#392—612no—
295#390—1221no—
296#380—611no—
297#379—1221no—
298#328—618no—
299#324—612no—
300#323—612no—
301C61#1819611yescharacteristic, complemented, Sylow
302C25—1221no—
303C22×Q8—21no—
304C22×Q8C22⋊D6112yescomplemented
305C22×D4—612no—
306C22×D4—1222no—
307C22×D4—1222no—
308C22×D4—1221no—
309C22×D4—1222no—
310C22×D4—2444no—
311C22×D4C22⋊D6111yescharacteristic
312C23×C4C22⋊D6111yescharacteristic
313C23×C4C22×D6111yescharacteristic
314C2×SD16—1228no—
315C2×SD16—2448no—
316C22×C8—1222no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—24no—
320#125—22no—
321#125—24no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332C22⋊Q8C22⋊D6114yescomplemented
333C22⋊Q8C22×D6114yes—
334#121—2444no—
335#120—2444no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×D4—24no—
339C4×D4—24no—
340C4×D4—22no—
341C4×D4—22no—
342#117—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4C22⋊D6112yescomplemented
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1221no—
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#95—612no—
362C24—1222no—
363C24—4882no—
364C24—2444no—
365C24—2442no—
366C24—2442no—
367C24—2442no—
368C24C2×C22⋊D6111yescharacteristic
369C24C2×C22⋊D6111yescharacteristic
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8C2×C22⋊D6114yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—1228no—
379C2×D4—1224no—
380C2×D4—1224no—
381C2×D4—4882no—
382C2×D4—2442no—
383C2×D4—2444no—
384C2×D4—2444no—
385C2×D4—2448no—
386C2×D4—2444no—
387C2×D4—2444no—
388C2×D4—2441no—
389C2×D4—2441no—
390C2×D4#1175820114yescomplemented
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—1222no—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6112yes—
406C22×C4C23×D6111yescharacteristic
407SD16—24416no—
408C2×C8—1224no—
409C2×C8—2444no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—2442no—
421C4⋊C4C2×C22⋊D6114yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—4882no—
428C22⋊C4—2444no—
429C22⋊C4—2442no—
430C22⋊C4—2444no—
431C42—24no—
432C42—21no—
433C42—21no—
434C23—21no—
435C23—21no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—1224no—
441C23—4884no—
442C23—4888no—
443C23—4888no—
444C23—4884no—
445C23—2442no—
446C23—2448no—
447C23—2442no—
448C23—2442no—
449C23—2442no—
450C23—2442no—
451C23—2442no—
452C23#1451974612yes—
453C23C2×#1175820112yes—
454C23C2×#1175820112yescomplemented
455C23#1451980812yes—
456C23C22×C22⋊D6111yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—82no—
460Q8—82no—
461D4—28no—
462D4—1228no—
463D4—4888no—
464D4—2444no—
465D4—2444no—
466D4—2444no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4#1451974611yescharacteristic
489C2×C4#1451974611yescharacteristic
490C2×C4C2×#1175820112yescomplemented
491C2×C4C22×C22⋊D6112yes—
492C8—2444no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—48816no—
505C22—4882no—
506C22—4882no—
507C22—4882no—
508C22—4882no—
509C22—2444no—
510C22—2444no—
511C22—2444no—
512C22#1683990114yes—
513C22C2×#1451974611yescharacteristic
514C22C22×#1175820111yescharacteristic
515C22C2×#1451980811yescharacteristic
516C4—22no—
517C4—21no—
518C4—22no—
519C4—41no—
520C4—44no—
521C4—41no—
522C4—44no—
523C4—4882no—
524C4#1683994912yes—
525C2—21no—
526C2—21no—
527C2—24no—
528C2—42no—
529C2—4884no—
530C2C2×#1683990111yescharacteristic
531C2#2145519111yescharacteristic
532C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21449175C2
hypercenter#3501D61
hypocenterC61#18196
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. C2×#16839912C2semidirect
3. #21444759C2semidirect
4. #21444742C2semidirect
5. #21444391C2semidirect
6. #21444137C2semidirect
7. C61×#3501C2semidirect
21. #16839499C22semidirect
22. #16839488C22semidirect
23. #16839486C22semidirect
27. C61×#544C22semidirect
29. C61×#543C22semidirect
57. C22×Q8×C61D4semidirect
76. C22⋊Q8×C61D4semidirect
88. C2×C4⋊C4×C61D4semidirect
111. C2×D4×C61SD16semidirect
175. C23×C61C2×SD16semidirect
215. C2×C4×C61C2×SD16semidirect
259. #3501D61semidirect
279. #544C2×D61semidirect
281. #543C2×D61semidirect
301. C61#18196semidirect
304. C22×Q8C22⋊D61semidirect
332. C22⋊Q8C22⋊D61semidirect
347. C2×C4⋊C4C22⋊D61semidirect
390. C2×D4#11758201semidirect
454. C23C2×#11758201semidirect
490. C2×C4C2×#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399122
#214491752
#214447592
#214447422
#214443912
#214441372
C61×#35012
#1819661