Finite Group Databasebeta

← #29737041

Subgroups — #29737041

Order 15616.

1490 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449471C212yesmaximal, complemented
2#21449175C211yesmaximal, characteristic
3#21446982C214yesmaximal, complemented
4#21446805C212yesmaximal, complemented
5#21446227C214yesmaximal, complemented
6C4×#14519692C211yesmaximal, characteristic, complemented
7C61×#3501C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic
9C2×#14519766—22no—
10C2×#14519751—21no—
11C2×#14519745C2212yes—
12C2×#14519692C2212yes—
13#16839679—21no—
14#16839679—24no—
15#16839671—21no—
16#16839664—21no—
17#16839663—22no—
18#16839622C2211yescharacteristic
19#16839617C2212yes—
20#16839617C2218yes—
21C4×C22⋊Dic61C2212yes—
22#16839529C2212yes—
23#16839529C2218yes—
24C61×#554—22no—
25C61×#554—24no—
26D4×Q8×C61—22no—
27C61×#544—21no—
28C61×#543—21no—
29C61×#543C2212yes—
30C61×#533—21no—
31C2×C22⋊Q8×C61—21no—
32C2×C22⋊Q8×C61C2212yes—
33C61×#518C2212yes—
34C2×C4×D4×C61C2211yescharacteristic
35C24×D61C2312yes—
36C22×D244—21no—
37C22×D244—22no—
38C22×D244—21no—
39C22×D244—22no—
40C22×D244—44no—
41C22×C22⋊D61—22no—
42#14519766—48no—
43#14519751—44no—
44#14519745—28no—
45C2×C4⋊Dic61—21no—
46C2×C22⋊Dic61C2314yes—
47C2×C22⋊Dic61C2312yes—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×#11758198—28no—
51C2×#11758198—28no—
52C2×#11758198—22no—
53C2×C4×Dic61—22no—
54#14519692—24no—
55#14519684—22no—
56#14519684—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61D412yescomplemented
59C22×D4×C61C2311yescharacteristic
60C23×C4×C61D411yescharacteristic
61C23×C4×C61C2311yescharacteristic
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—24no—
65#125×C61—22no—
66#125×C61—24no—
67C22⋊Q8×C61—24no—
68C22⋊Q8×C61—24no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—24no—
76C22⋊Q8×C61—24no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—22no—
79C4×Q8×C61—22no—
80C4×Q8×C61—22no—
81C4×D4×C61—22no—
82C4×D4×C61—24no—
83C4×D4×C61—22no—
84C4×D4×C61—24no—
85#117×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61D411yescharacteristic
89C2×C4⋊C4×C61D412yescomplemented
90C2×C4⋊C4×C61D414yescomplemented
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61D411yescharacteristic
93C2×C22⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61C2314yes—
95C2×C42×C61D411yescharacteristic
96C24×C61C2×D411yescharacteristic
97C24×C61C2411yescharacteristic
98C23×D61—22no—
99C23×D61—24no—
100C23×D61—24no—
101C23×D61—24no—
102C23×D61—44no—
103C23×D61—82no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—44no—
109C2×Q8×C61—82no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61#4114yes—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D414yes—
125C22×C4×C61C2×D414yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—82no—
131C22×Dic61—22no—
132C22×Dic61—24no—
133C2×D244—24no—
134C2×D244—28no—
135C2×D244—44no—
136C2×D244—42no—
137C2×D244—48no—
138C2×D244—44no—
139C2×D244—41no—
140C2×D244—42no—
141C2×D244—41no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—22no—
158C42×C61—24no—
159C42×C61—21no—
160C42×C61—21no—
161C22⋊Dic61—24no—
162C22⋊Dic61—24no—
163C22⋊Dic61—24no—
164#11758198—44no—
165#11758198—416no—
166#11758198—44no—
167#11758198—88no—
168#11758198—82no—
169C4⋊Dic61—42no—
170C4×Dic61—44no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61#12412yescomplemented
178C23×C61C22×D411yescharacteristic
179C23×C61C22×D412yes—
180C23×C61C2×#4112yes—
181C23×C612−1+412yes—
182C22×D61—28no—
183C22×D61—24no—
184C22×D61—48no—
185C22×D61—42no—
186C22×D61—44no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—48no—
193C22×D61—88no—
194C22×D61—88no—
195Q8×C61—44no—
196Q8×C61—82no—
197Q8×C61—82no—
198Q8×C61—82no—
199D4×C61—28no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—24no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—21no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—44no—
213C2×C4×C61—44no—
214C2×C4×C61—41no—
215C2×C4×C61—81no—
216C2×C4×C61—82no—
217C2×C4×C61—82no—
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12012yes—
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12412yescomplemented
222C22⋊D61—88no—
223C2×Dic61—44no—
224C2×Dic61—42no—
225C2×Dic61—44no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—44no—
229D244—44no—
230D244—48no—
231D244—44no—
232#25947—611nomaximal, Sylow
233C22×C61—22no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—21no—
237C22×C61—22no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—81no—
244C22×C61C2×#12011yescharacteristic
245C22×C61C2×#12411yescharacteristic
246C22×C61#53611yescharacteristic
247C22×C61#54814yes—
248C2×D61—44no—
249C2×D61—416no—
250C2×D61—44no—
251C2×D61—44no—
252C2×D61—82no—
253C2×D61—88no—
254C2×D61—82no—
255C2×D61—82no—
256C2×D61—82no—
257C4×C61—22no—
258C4×C61—21no—
259C4×C61—22no—
260C4×C61—41no—
261C4×C61—44no—
262C4×C61—44no—
263C4×C61—41no—
264C4×C61B(4,𝔽2)12yes—
265Dic61—84no—
266Dic61—82no—
267#3501D6111yescharacteristic, complemented
268#2435—612no—
269#2408—611no—
270#2029—614no—
271#1996—612no—
272#1889—614no—
273C4×#99—611no—
274C2×C61—21no—
275C2×C61—24no—
276C2×C61—21no—
277C2×C61—42no—
278C2×C61#243611yescharacteristic
279C2×C61C2×B(4,𝔽2)11yescharacteristic
280C2×C61#302711yescharacteristic
281D61—84no—
282C23×D4—611no—
283#554—22no—
284#554—24no—
285D4×Q8—22no—
286#544—21no—
287#543—21no—
288#543C2×D6112yes—
289#533—21no—
290C2×#124—1222no—
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8C2×D6112yes—
293C2×#121—1221no—
294C2×#120—612no—
295#518C2×D6112yes—
296C2×C4×D4C2×D6111yescharacteristic
297C2×#99—612no—
298#394—1221no—
299#394—1224no—
300#392—1221no—
301#390—1222no—
302#390—1221no—
303#380—611no—
304#379—618no—
305#379—612no—
306C4×C22⋊C4—612no—
307#342—618no—
308#342—612no—
309C61#2594711yescharacteristic, complemented, Sylow
310C25—612no—
311C22×Q8—21no—
312C22×Q8C22⋊D6112yescomplemented
313C22×D4—1222no—
314C22×D4—1221no—
315C22×D4—1222no—
316C22×D4—1221no—
317C22×D4—1222no—
318C22×D4—2444no—
319C22×D4C22×D6111yescharacteristic
320C23×C4C22⋊D6111yescharacteristic
321C23×C4C22×D6111yescharacteristic
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—24no—
325#125—22no—
326#125—24no—
327#124—2448no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340#121—2444no—
341#120—1228no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—24no—
347C4×D4—24no—
348#117—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4C22⋊D6111yescharacteristic
353C2×C4⋊C4C22⋊D6112yescomplemented
354C2×C4⋊C4C22⋊D6114yescomplemented
355C2×C22⋊C4—22no—
356C2×C22⋊C4—612no—
357C2×C22⋊C4—614no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1228no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1228no—
363C2×C22⋊C4C22⋊D6111yescharacteristic
364C2×C22⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4C22×D6114yes—
366C2×C42—1222no—
367C2×C42C22⋊D6111yescharacteristic
368#99—1224no—
369#95—1222no—
370#95—1221no—
371C24—1224no—
372C24—1222no—
373C24—1224no—
374C24—1224no—
375C24—4882no—
376C24—2444no—
377C24C2×C22⋊D6111yescharacteristic
378C24C23×D6111yescharacteristic
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—44no—
384C2×Q8—82no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—1224no—
388C2×D4—1228no—
389C2×D4—4882no—
390C2×D4—2444no—
391C2×D4—2442no—
392C2×D4—2448no—
393C2×D4—2444no—
394C2×D4—2444no—
395C2×D4—2444no—
396C2×D4—2441no—
397C2×D4—2442no—
398C2×D4—2441no—
399C2×D4#1175822614yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—1224no—
407C22×C4—1222no—
408C22×C4C2×C22⋊D6112yes—
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6114yes—
414C22×C4C2×C22⋊D6114yes—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6112yes—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—2442no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—4888no—
438C22⋊C4—4882no—
439C22⋊C4—2444no—
440C22⋊C4—24416no—
441C22⋊C4—2444no—
442C42—24no—
443C42—21no—
444C42—21no—
445C42—2444no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—1228no—
453C23—1224no—
454C23—4888no—
455C23—4888no—
456C23—2448no—
457C23—2442no—
458C23—2442no—
459C23—2444no—
460C23—2442no—
461C23—2442no—
462C23—2442no—
463C23—2442no—
464C23—2448no—
465C23#1451976612yescomplemented
466C23C22×C22⋊D6111yescharacteristic
467C23C22×C22⋊D6112yes—
468C23C2×#1175822612yes—
469C23#1451982512yes—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474D4—28no—
475D4—4888no—
476D4—2444no—
477D4—2448no—
478D4—2444no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—2444no—
498C2×C4—2444no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2444no—
503C2×C4#1451974611yescharacteristic
504C2×C4#1451974612yes—
505C2×C4#1451974611yescharacteristic
506C2×C4#1451976612yescomplemented
507C22—22no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—81no—
518C22—4882no—
519C22—4888no—
520C22—4882no—
521C22—4882no—
522C22—4882no—
523C22—2444no—
524C22—24416no—
525C22—2444no—
526C22—2444no—
527C22C2×#1451974611yescharacteristic
528C22C2×#1451976611yescharacteristic
529C22#1684033611yescharacteristic
530C22#1684042314yes—
531C4—22no—
532C4—21no—
533C4—22no—
534C4—41no—
535C4—44no—
536C4—44no—
537C4—41no—
538C4—4884no—
539C4—4882no—
540C4#1683994912yes—
541C2—21no—
542C2—24no—
543C2—21no—
544C2—42no—
545C2—4884no—
546C2#2144947311yescharacteristic
547C2C2×#1683994911yescharacteristic
548C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radical#21449175C2
hypercenter#3501D61
hypocenterC61#25947
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21449471C2semidirect
3. #21446982C2semidirect
4. #21446805C2semidirect
5. #21446227C2semidirect
6. C4×#14519692C2semidirect
7. C61×#3501C2semidirect
58. C22×Q8×C61D4semidirect
89. C2×C4⋊C4×C61D4semidirect
90. C2×C4⋊C4×C61D4semidirect
177. C23×C61#124semidirect
221. C2×C4×C61#124semidirect
267. #3501D61semidirect
309. C61#25947semidirect
312. C22×Q8C22⋊D61semidirect
353. C2×C4⋊C4C22⋊D61semidirect
354. C2×C4⋊C4C22⋊D61semidirect
465. C23#14519766semidirect
506. C2×C4#14519766semidirect

Maximal subgroups

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

subgroupindex
#214494712
#214491752
#214469822
#214468052
#214462272
C4×#145196922
C61×#35012
#2594761