Finite Group Databasebeta

← #29606932

Subgroups — #29606932

Order 15616.

1310 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839692C211yesmaximal, characteristic
2C2×#16839660C211yesmaximal, characteristic
3C2×#16839646C212yesmaximal
4#21444086C211yesmaximal, characteristic
5#21444079C212yesmaximal
6#21444075C211yesmaximal, characteristic
7#21444072C211yesmaximal, characteristic
8#21444069C212yesmaximal
9#21444052C212yesmaximal
10#21444046C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C61C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758196C2212yes—
16#16839692C418yescomplemented
17#16839660—24no—
18#16839646—28no—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C22×C42×C61C2211yescharacteristic
36C4×C4⋊C4×C61—24no—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—42no—
47C2×C4⋊Dic61C2×C414yes—
48C2×#11758196—24no—
49C2×#11758196—24no—
50C2×#11758196—24no—
51C2×#11758196—24no—
52C2×#11758196—24no—
53C2×#11758196C2×C418yes—
54#14519684—24no—
55#14519684—24no—
56#14519684—24no—
57#14519684—24no—
58#14519684—24no—
59#14519684—24no—
60#14519684—22no—
61#14519684—42no—
62#14519684—42no—
63#14519684—42no—
64#14519684C2×C414yes—
65#14519684C2×C418yes—
66C23×C4×C61C2311yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C23×C4×C61C2311yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61C2×C414yes—
79C2×C42×C61—22no—
80C2×C42×C61—22no—
81C2×C42×C61—22no—
82C2×C42×C61—42no—
83C2×C42×C61—42no—
84C2×C42×C61D414yes—
85C2×C42×C61D414yes—
86#95×C61—22no—
87#95×C61—22no—
88C24×C61C2411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C22×C412yes—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61C22×C414yes—
153C22×Dic61C22×C414yes—
154C4⋊C4×C61—28no—
155C4⋊C4×C61—28no—
156C42×C61—28no—
157C42×C61—44no—
158C42×C61—44no—
159C4⋊Dic61—28no—
160C4⋊Dic61—44no—
161#11758196—216no—
162#11758196—48no—
163C23×C61C23×C411yescharacteristic
164C23×C61C22×D411yescharacteristic
165C23×C61C22×D411yescharacteristic
166C23×C61C2×#4111yescharacteristic
167C23×C61C2×#4111yescharacteristic
168C23×C61C2×#4111yescharacteristic
169C23×C61C2×#4111yescharacteristic
170C23×C61H(5,𝔽2)12yes—
171C23×C612−1+412yes—
172C23×C612−1+412yes—
173C23×C612−1+412yes—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—24no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61C4×D414yes—
209C2×C4×C61C4×D414yes—
210C2×C4×C61#12114yes—
211C2×Dic61—24no—
212C2×Dic61—24no—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239#14314—611nomaximal, Sylow
240C22×C61C2×C4×D411yescharacteristic
241C22×C61C2×C4×D411yescharacteristic
242C22×C61C4×#4111yescharacteristic
243C22×C61#51912yes—
244C22×C61#52012yes—
245C22×C61C2×#12111yescharacteristic
246C22×C61#52511yescharacteristic
247C22×C61#52911yescharacteristic
248C22×C61#53211yescharacteristic
249C22×C61#53411yescharacteristic
250C22×C61#53611yescharacteristic
251C22×C61#53611yescharacteristic
252C22×C61#53611yescharacteristic
253C22×C61#53612yes—
254C22×C61#53712yes—
255C22×C61#53912yes—
256C22×C61#53912yes—
257C22×C61#53912yes—
258C22×C61#54112yes—
259C22×C61#54112yes—
260C22×C61#54112yes—
261C22×C61#54112yes—
262C22×C61#54212yes—
263C4×C61—24no—
264C4×C61—24no—
265C4×C61—24no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—42no—
270Dic61—44no—
271Dic61—44no—
272Dic61—82no—
273Dic61—82no—
274C2×#398—611no—
275C2×#389—611no—
276C2×#387—612no—
277C2×C4×C4⋊C4D6111yescharacteristic
278#1445—611no—
279#1443—612no—
280#1442—611no—
281#1441—612no—
282#1441—611no—
283#1436—612no—
284#1434—611no—
285C2×C61C4×#12111yescharacteristic
286C2×C61#232211yescharacteristic
287C2×C61#232312yes—
288C2×C61#232711yescharacteristic
289C2×C61#233612yes—
290C2×C61#261811yescharacteristic
291C2×C61#262111yescharacteristic
292C2×C61#263411yescharacteristic
293C2×C61#263811yescharacteristic
294C2×C61#265512yes—
295C2×C61#266412yes—
296C22×C4⋊C4—612no—
297C22×C4⋊C4—611no—
298C22×C4⋊C4—612no—
299C22×C4⋊C4—611no—
300C22×C4⋊C4C2×D6111yescharacteristic
301C22×C4⋊C4C2×D6111yescharacteristic
302C22×C42C2×D6111yescharacteristic
303C22×C42C2×D6111yescharacteristic
304C22×C42C2×D6111yescharacteristic
305#398—618no—
306#389—1224no—
307#387—1228no—
308C4×C4⋊C4—24no—
309C2×#95—611no—
310C2×#95—612no—
311C2×#95—612no—
312C2×#95—611no—
313C2×#95—612no—
314C2×#95—612no—
315C2×#95—612no—
316C2×#95—612no—
317C2×#95—612no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95—612no—
321C2×#95C2×D6111yescharacteristic
322C2×#95C2×D6111yescharacteristic
323C61#1431411yescharacteristic, complemented, Sylow
324C23×C4—612no—
325C23×C4—612no—
326C23×C4—612no—
327C23×C4—612no—
328C23×C4C22×D6111yescharacteristic
329C23×C4C22×D6111yescharacteristic
330C23×C4C22×D6111yescharacteristic
331C23×C4C22×D6111yescharacteristic
332C23×C4C22×D6111yescharacteristic
333C23×C4C22×D6111yescharacteristic
334C23×C4C22×D6111yescharacteristic
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—618no—
341C2×C4⋊C4—614no—
342C2×C4⋊C4—1224no—
343C2×C4⋊C4—1224no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1224no—
347C2×C4⋊C4—1224no—
348C2×C4⋊C4—1224no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—2442no—
351C2×C4⋊C4C4×D6114yes—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42D24414yes—
358C2×C42C22⋊D6114yes—
359#95—22no—
360#95—22no—
361#95—614no—
362#95—618no—
363#95—1224no—
364#95—1224no—
365#95—1224no—
366#95—1224no—
367#95—1224no—
368#95—1222no—
369#95—1224no—
370#95—2442no—
371#95—2442no—
372#95—2442no—
373C24C23×D6111yescharacteristic
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—614no—
399C22×C4—614no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4C2×C4×D6112yes—
427C22×C4C2×C4×D6112yes—
428C22×C4C2×C4×D6112yes—
429C22×C4C2×D24412yes—
430C22×C4C2×D24412yes—
431C22×C4C2×D24412yes—
432C22×C4#1175822212yes—
433C22×C4#1175822212yes—
434C22×C4#1175822212yes—
435C22×C4#1175822212yes—
436C22×C4C2×C22⋊D6112yes—
437C22×C4C2×C22⋊D6112yes—
438C22×C4C2×C22⋊D6112yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—1228no—
442C4⋊C4—12216no—
443C4⋊C4—2444no—
444C4⋊C4—2448no—
445C42—28no—
446C42—44no—
447C42—44no—
448C23C22×C4×D6111yescharacteristic
449C23C22×C22⋊D6111yescharacteristic
450C23C22×D24411yescharacteristic
451C23C2×#1175822211yescharacteristic
452C23C2×#1175822211yescharacteristic
453C23C2×#1175822211yescharacteristic
454C23C2×#1175822211yescharacteristic
455C23#1451982212yes—
456C23#1451982412yes—
457C23#1451982412yes—
458C23#1451982512yes—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—1224no—
494C2×C4—1224no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—1224no—
498C2×C4—1224no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4C4×C22⋊D6114yes—
522C2×C4C4×D24414yes—
523C2×C4#1451975114yes—
524C22C2×C4×C22⋊D6111yescharacteristic
525C22C2×C4×D24411yescharacteristic
526C22C4×#1175822211yescharacteristic
527C22#1684023812yes—
528C22#1684024112yes—
529C22C2×#1451975111yescharacteristic
530C22#1684027311yescharacteristic
531C22#1684029611yescharacteristic
532C22#1684031211yescharacteristic
533C22#1684032111yescharacteristic
534C22#1684033311yescharacteristic
535C22#1684033311yescharacteristic
536C22#1684033611yescharacteristic
537C22#1684033812yes—
538C22#1684034312yes—
539C22#1684035412yes—
540C22#1684035412yes—
541C22#1684035912yes—
542C22#1684037012yes—
543C22#1684037012yes—
544C22#1684037012yes—
545C22#1684037412yes—
546C22#1684037712yes—
547C4—24no—
548C4—24no—
549C4—24no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—4882no—
555C4—4882no—
556C4—2444no—
557C4—2444no—
558C2C4×#1451975111yescharacteristic
559C2#2144848411yescharacteristic
560C2#2144850212yes—
561C2#2144853511yescharacteristic
562C2#2144861812yes—
563C2#2145156111yescharacteristic
564C2#2145158511yescharacteristic
565C2#2145169511yescharacteristic
566C2#2145173611yescharacteristic
567C2#2145190612yes—
568C2#2145197812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C4⋊C4×C61C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×C4×C4⋊C4D61
hypocenterC61#14314
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
16. #16839692C4semidirect
323. C61#14314semidirect

Maximal subgroups

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

subgroupindex
C2×#168396922
C2×#168396602
C2×#168396462
#214440862
#214440792
#214440752
#214440722
#214440692
#214440522
#214440462
C2×C4×C4⋊C4×C612
#1431461