Finite Group Databasebeta

← #29688994

Subgroups — #29688994

Order 15616.

1446 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2#407×D61C212yesmaximal, complemented
3#21446284C214yesmaximal, complemented
4#21446279C214yesmaximal, complemented
5#21445900C212yesmaximal, complemented
6#21445747C211yesmaximal, characteristic, complemented
7C61×#1785C211yesmaximal, characteristic, complemented
8C23×C4×D61C2211yescharacteristic
9C2×M16×D61C2214yes—
10#16840269—22no—
11#16840269—22no—
12#16840269—22no—
13#16840269—22no—
14C2×#14519758C412yes—
15C2×#14519753C2212yes—
16C2×#14519748C2212yes—
17C2×#14519745C412yes—
18C2×C4×C22⋊D61C414yes—
19C2×#14519736C2211yescharacteristic
20C2×#14519696C2212yescomplemented
21C2×#14519694C2212yescomplemented
22#16839726C2212yes—
23#16839725C2214yes—
24#16839724—22no—
25#16839724—22no—
26C22⋊C8×D61—22no—
27C22⋊C8×D61—22no—
28#16839530C2218yescomplemented
29C22×#41×C61C2211yescharacteristic
30C2×#101×C61C2212yescomplemented
31C2×#100×C61C2212yescomplemented
32C61×#407C2212yes—
33C24×D61C2×C411yescharacteristic
34C22×C22⋊D61C2×C414yes—
35C23×Dic61C2×C411yescharacteristic
36C22×C4×D61—42no—
37C22×C4×D61—42no—
38C22×C4×D61C2×C412yes—
39C22×C4×D61D412yes—
40C22×C4×D61D412yes—
41C22×C4×D61D412yes—
42C22×C4×D61D412yes—
43C22×C4×D61C2312yes—
44C2×#11758194C2314yes—
45C2×#11758189—24no—
46M16×D61—44no—
47M16×D61—44no—
48C2×C8×D61—24no—
49#14519758—22no—
50#14519758—24no—
51#14519758—22no—
52#14519753—22no—
53#14519753—22no—
54#14519753—22no—
55#14519753—22no—
56#14519748—22no—
57#14519748—22no—
58#14519748—22no—
59#14519748—22no—
60#14519745—22no—
61#14519745—24no—
62#14519745—22no—
63C4×C22⋊D61—24no—
64C4×C22⋊D61—24no—
65C4×C22⋊D61—24no—
66C4×C22⋊D61—24no—
67#14519736—22no—
68#14519736—22no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196C2×C414yes—
71C2×C22⋊Dic61C2×C412yes—
72C2×C22⋊Dic61C2×C414yes—
73C2×#11758198C2×C414yes—
74C2×C4×Dic61C2×C412yes—
75#14519696—42no—
76#14519694—42no—
77C22⋊#92418—22no—
78C22⋊#92418—22no—
79#14519689—24no—
80#14519689—24no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—42no—
84C2×#41×C61—42no—
85C22×Q8×C61C2311yescharacteristic
86C22×D4×C61C2×C412yes—
87C22×D4×C61C2311yescharacteristic
88C23×C4×C61C2×C412yes—
89C23×C4×C61C2311yescharacteristic
90C2×M16×C61C2314yes—
91#101×C61—42no—
92#100×C61—42no—
93C22⋊C8×C61—22no—
94C22⋊C8×C61—22no—
95C24×C61C22×C412yes—
96C24×C61C22×C411yescharacteristic
97C23×D61—44no—
98C23×D61—41no—
99C23×D61—41no—
100C23×D61C22⋊C411yescharacteristic
101C23×D61C22⋊C412yes—
102C23×D61C22⋊C411yescharacteristic
103C23×D61C22×C412yes—
104#41×C61—48no—
105#41×C61—48no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—24no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—48no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61C22⋊C411yescharacteristic
132C22×Dic61C22⋊C412yes—
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C22×C412yes—
135C22×Dic61C22×C414yes—
136C2×C4×D61—24no—
137C2×C4×D61—24no—
138C2×C4×D61—44no—
139C2×C4×D61—44no—
140C2×C4×D61—44no—
141C2×C4×D61—44no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61C22⋊C412yes—
145C2×C4×D61C22⋊C412yes—
146C2×C4×D61C22⋊C412yes—
147C2×C4×D61C22⋊C412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D412yes—
151C2×C4×D61C2×D412yes—
152M16×C61—24no—
153M16×C61—24no—
154C2×C8×C61—24no—
155C22⋊Dic61—22no—
156C22⋊Dic61—22no—
157C22⋊Dic61—24no—
158C22⋊Dic61—24no—
159#11758198—24no—
160#11758198—24no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—24no—
166#11758194—24no—
167#11758194—24no—
168C2×#92418—24no—
169#11758189—48no—
170C8×D61—48no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—42no—
177C23×C61—44no—
178C23×C61C2×C22⋊C411yescharacteristic
179C23×C61C2×C22⋊C411yescharacteristic
180C23×C61C23×C411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—24no—
184C22×D61—24no—
185C22×D61—24no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—48no—
193C22×D61—48no—
194C22×D61C2×C22⋊C412yes—
195C22×D61C2×C22⋊C412yes—
196Q8×C61—44no—
197D4×C61—44no—
198D4×C61—44no—
199D4×C61—44no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—48no—
208C2×C4×C61C2×C22⋊C412yes—
209C2×C4×C61C22×D412yes—
210C2×C4×C61H(5,𝔽2)12yes—
211C2×C4×C61H(5,𝔽2)12yes—
212C22⋊D61—48no—
213C22⋊D61—48no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—24no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—44no—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C4×D61—24no—
224C4×D61—24no—
225C4×D61—24no—
226C4×D61—24no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C8×C61—44no—
232#92418—44no—
233#21754—611nomaximal, Sylow
234C22×C61—22no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—44no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61C22×C22⋊C411yescharacteristic
246C22×C61#51811yescharacteristic
247C22×C61#51811yescharacteristic
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—22no—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—44no—
254C2×D61—44no—
255C2×D61—44no—
256C2×D61—48no—
257C2×D61—44no—
258C2×D61#41314yes—
259C4×C61—22no—
260C4×C61—42no—
261C4×C61#53414yes—
262Dic61—22no—
263Dic61—22no—
264Dic61—44no—
265C2×#525—611no—
266C2×#407—612no—
267#1901—614no—
268#1900—614no—
269#1822—612no—
270#1785—611no—
271#1785D6111yescharacteristic, complemented
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—41no—
275C2×C61—41no—
276C2×C61—82no—
277C2×C61#230811yescharacteristic
278C2×C61C2×#41312yes—
279D61—24no—
280D61—42no—
281C22×#41C2×D6111yescharacteristic
282C24×C4—611no—
283C22×M16—614no—
284#525—1222no—
285#525—1222no—
286#525—1222no—
287#525—1222no—
288C2×#123—612no—
289C2×C22⋊Q8—612no—
290C2×#121—612no—
291C2×#120—612no—
292C2×C4×D4—614no—
293C2×#117—611no—
294C2×#101—612no—
295C2×#101C2×D6112yescomplemented
296C2×#100—612no—
297C2×#100C2×D6112yescomplemented
298#407—612no—
299#407—614no—
300#407—1222no—
301#407—1222no—
302#407C2×D6112yes—
303C2×C22⋊C8—1222no—
304C2×C22⋊C8—1222no—
305#343—618no—
306C61#2175411yescharacteristic, complemented, Sylow
307C25—611no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41—42no—
311C2×#41—42no—
312C22×Q8C22×D6111yescharacteristic
313C22×D4—614no—
314C22×D4C4×D6112yes—
315C22×D4C22×D6111yescharacteristic
316C23×C4—611no—
317C23×C4—612no—
318C23×C4—612no—
319C23×C4—612no—
320C23×C4—612no—
321C23×C4—612no—
322C23×C4—612no—
323C23×C4—2442no—
324C23×C4—2442no—
325C23×C4C4×D6112yes—
326C23×C4C22×D6111yescharacteristic
327C2×M16—614no—
328C2×M16—1224no—
329C2×M16—2444no—
330C2×M16—2444no—
331C2×M16C22×D6114yes—
332C22×C8—1224no—
333#123—1222no—
334#123—1224no—
335#123—1222no—
336C22⋊Q8—1222no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C22⋊Q8—1222no—
340#121—1222no—
341#121—1222no—
342#121—1222no—
343#121—1222no—
344#120—1222no—
345#120—1224no—
346#120—1222no—
347C4×D4—1224no—
348C4×D4—1224no—
349C4×D4—1224no—
350C4×D4—1224no—
351#117—1222no—
352#117—1222no—
353C2×C4⋊C4—614no—
354C2×C4⋊C4—612no—
355C2×C22⋊C4—612no—
356C2×C22⋊C4—614no—
357C2×C22⋊C4—614no—
358C2×C42—612no—
359#101—42no—
360#101—2442no—
361#100—42no—
362#100—2442no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8—1224no—
366C22⋊C8—1224no—
367C22⋊C8—1222no—
368C22⋊C8—1222no—
369C24—612no—
370C24—611no—
371C24—612no—
372C24—611no—
373C24—2441no—
374C24—2441no—
375C24—2444no—
376C24C2×C4×D6112yes—
377C24C2×C4×D6111yescharacteristic
378#41—48no—
379#41—48no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—1224no—
391C2×D4—1224no—
392C2×D4—2448no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—612no—
399C22×C4—611no—
400C22×C4—612no—
401C22×C4—611no—
402C22×C4—612no—
403C22×C4—612no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—614no—
411C22×C4—1224no—
412C22×C4—1224no—
413C22×C4—2444no—
414C22×C4—2444no—
415C22×C4—2444no—
416C22×C4—2444no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4—2441no—
420C22×C4—2441no—
421C22×C4C2×C4×D6112yes—
422C22×C4C2×C4×D6111yescharacteristic
423C22×C4D4×D6112yes—
424C22×C4D4×D6112yes—
425C22×C4C23×D6111yescharacteristic
426M16—24no—
427M16—24no—
428M16—1224no—
429M16—1224no—
430M16—2448no—
431C2×C8—24no—
432C2×C8—1224no—
433C2×C8—2448no—
434C4⋊C4—1224no—
435C4⋊C4—1224no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C22⋊C4—1224no—
439C22⋊C4—1224no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1222no—
443C22⋊C4—1222no—
444C42—1224no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—44no—
452C23—612no—
453C23—612no—
454C23—1224no—
455C23—1224no—
456C23—1224no—
457C23—1222no—
458C23—1222no—
459C23—2441no—
460C23—2442no—
461C23—2441no—
462C23—2442no—
463C23—2442no—
464C23—2442no—
465C23—2448no—
466C23—2448no—
467C23C22⋊C4×D6111yescharacteristic
468C23C22⋊C4×D6111yescharacteristic
469C23C22×C4×D6111yescharacteristic
470Q8—44no—
471D4—44no—
472D4—44no—
473D4—44no—
474D4—2448no—
475D4—2448no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—612no—
485C2×C4—612no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1224no—
489C2×C4—1224no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1224no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2441no—
500C2×C4—2444no—
501C2×C4C22⋊C4×D6112yes—
502C2×C4C2×D4×D6112yes—
503C2×C4#1451982212yes—
504C2×C4#1451982212yes—
505C8—44no—
506C8—2444no—
507C22—22no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—44no—
516C22—82no—
517C22—82no—
518C22—614no—
519C22—1222no—
520C22—1222no—
521C22—1222no—
522C22—1222no—
523C22—1222no—
524C22—2444no—
525C22—2444no—
526C22—2444no—
527C22—2448no—
528C22—2444no—
529C22C2×C22⋊C4×D6111yescharacteristic
530C22#1684023511yescharacteristic
531C22#1684023511yescharacteristic
532C4—22no—
533C4—42no—
534C4—1222no—
535C4—1222no—
536C4—2444no—
537C4#1684032214yes—
538C2—21no—
539C2—21no—
540C2—41no—
541C2—41no—
542C2—82no—
543C2—1224no—
544C2—2442no—
545C2#2144839511yescharacteristic
546C2#413×D6112yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1785C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840269C2
cosocleC22×C4×C61C24
hypercenter#1785D61
hypocenterC61#21754
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #407×D61C2semidirect
3. #21446284C2semidirect
4. #21446279C2semidirect
5. #21445900C2semidirect
6. #21445747C2semidirect
7. C61×#1785C2semidirect
20. C2×#14519696C22semidirect
21. C2×#14519694C22semidirect
28. #16839530C22semidirect
30. C2×#101×C61C22semidirect
31. C2×#100×C61C22semidirect
271. #1785D61semidirect
295. C2×#101C2×D61semidirect
297. C2×#100C2×D61semidirect
306. C61#21754semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
#407×D612
#214462842
#214462792
#214459002
#214457472
C61×#17852
#2175461