Finite Group Databasebeta

← #29681639

Subgroups — #29681639

Order 15616.

1974 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2C2×#16839736C211yesmaximal, characteristic, complemented
3#21448198C212yesmaximal, complemented
4#21446232C214yesmaximal, complemented
5#21446227C214yesmaximal, complemented
6#21445552C212yesmaximal, complemented
7C61×#3428C211yesmaximal, characteristic, complemented
8C23×C4×D61C2211yescharacteristic
9#16840269—24no—
10#16840269C418yescomplemented
11C2×#14519758C2212yes—
12C2×#14519753—21no—
13C2×#14519748—21no—
14C2×#14519745C2212yes—
15C2×C4×C22⋊D61—22no—
16C2×#14519736C2212yes—
17C2×#14519736C2211yescharacteristic
18#16839736—24no—
19C2×#14519692C2214yes—
20#16839623C2214yes—
21#16839619—24no—
22#16839617C2214yes—
23C4×#11758198—24no—
24#16839529C2218yes—
25C22×#41×C61C2211yescharacteristic
26C61×#520—24no—
27C61×#519—21no—
28C61×#518—21no—
29C61×#518C2214yes—
30C4×#41×C61—24no—
31C2×#117×C61—22no—
32C2×#117×C61C2212yes—
33C24×D61C2311yescharacteristic
34C22×C22⋊D61—22no—
35C23×Dic61C2311yescharacteristic
36C22×C4×D61—21no—
37C22×C4×D61—21no—
38C22×C4×D61—22no—
39C22×C4×D61—22no—
40C22×C4×D61—22no—
41C22×C4×D61—24no—
42C22×C4×D61C2×C414yescomplemented
43#14519758—24no—
44#14519758C2×C418yescomplemented
45#14519753—28no—
46#14519748—28no—
47#14519745—24no—
48#14519745C2×C418yescomplemented
49C4×C22⋊D61—216no—
50#14519736—22no—
51#14519736—24no—
52#14519736—24no—
53#14519736C2×C414yescomplemented
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—22no—
56C2×#11758196—22no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61C2314yes—
59C2×C22⋊Dic61C2312yes—
60C2×#11758198—22no—
61C2×#11758198—28no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—22no—
64#14519692—28no—
65#14519684—28no—
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—24no—
69C2×#41×C61C2×C414yes—
70C2×#41×C61D414yes—
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61D411yescharacteristic
73C22×D4×C61C2312yes—
74C23×C4×C61D412yes—
75C23×C4×C61C2311yescharacteristic
76C4×Q8×C61—28no—
77C4×D4×C61—28no—
78C4×D4×C61—216no—
79#117×C61—24no—
80#117×C61—24no—
81#117×C61—22no—
82#117×C61—24no—
83#117×C61—22no—
84#117×C61—22no—
85#117×C61—22no—
86C2×C4⋊C4×C61—24no—
87C2×C4⋊C4×C61D414yescomplemented
88C2×C22⋊C4×C61—24no—
89C2×C22⋊C4×C61C2314yes—
90C2×C42×C61—24no—
91C2×C42×C61D414yescomplemented
92C24×C61C2×D412yes—
93C24×C61C2411yescharacteristic
94C23×D61—21no—
95C23×D61—22no—
96C23×D61—21no—
97C23×D61—22no—
98C23×D61—44no—
99C23×D61C22×C412yes—
100#41×C61—28no—
101#41×C61—28no—
102#41×C61—28no—
103#41×C61—28no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61C22⋊C414yescomplemented
107C2×D4×C61—24no—
108C2×D4×C61—24no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61C22⋊C414yescomplemented
112C2×D4×C61C22×C414yes—
113C2×D4×C61C2×D414yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—28no—
118C22×C4×C61C22⋊C418yescomplemented
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D418yes—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61#4114yes—
128C2×C22⋊D61—28no—
129C2×C22⋊D61—48no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—24no—
136C22×Dic61C22×C412yes—
137C2×C4×D61—24no—
138C2×C4×D61—24no—
139C2×C4×D61—28no—
140C2×C4×D61—28no—
141C2×C4×D61—42no—
142C2×C4×D61—41no—
143C2×C4×D61—41no—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C4⋊C4×C61—28no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C42×C61—28no—
155C42×C61—24no—
156C42×C61—24no—
157C22⋊Dic61—28no—
158C22⋊Dic61—22no—
159C22⋊Dic61—24no—
160C22⋊Dic61—24no—
161C22⋊Dic61C22×C414yes—
162#11758198—28no—
163#11758198—416no—
164C4⋊Dic61—24no—
165C4⋊Dic61—44no—
166#11758196—28no—
167C4×Dic61—24no—
168C4×Dic61—44no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—42no—
172C23×C61—44no—
173C23×C61C2×C22⋊C414yes—
174C23×C61C23×C411yescharacteristic
175C23×C61C22×D411yescharacteristic
176C23×C61C22×D412yes—
177C23×C61C22×D411yescharacteristic
178C23×C61C2×#4112yes—
179C22×D61—28no—
180C22×D61—24no—
181C22×D61—22no—
182C22×D61—24no—
183C22×D61—44no—
184C22×D61—44no—
185C22×D61—48no—
186C22×D61—44no—
187C22×D61—42no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—41no—
191Q8×C61—28no—
192D4×C61—28no—
193D4×C61—28no—
194D4×C61—28no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—28no—
198C2×C4×C61—28no—
199C2×C4×C61—28no—
200C2×C4×C61—48no—
201C2×C4×C61—88no—
202C2×C4×C61C2×C22⋊C414yes—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C4×D414yes—
206C2×C4×C61C4×D414yes—
207C2×C4×C61#12012yes—
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12114yes—
211C2×C4×C61#12114yes—
212C2×C4×C61#12312yes—
213C2×C4×C61#12312yes—
214C22⋊D61—416no—
215C2×Dic61—24no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—44no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—48no—
223C2×Dic61—44no—
224C4×D61—44no—
225C4×D61—44no—
226C4×D61—48no—
227#20959—611nomaximal, Sylow
228C22×C61—24no—
229C22×C61—22no—
230C22×C61—24no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—42no—
234C22×C61—44no—
235C22×C61—44no—
236C22×C61—82no—
237C22×C61C22×C22⋊C411yescharacteristic
238C22×C61C2×C4×D412yes—
239C22×C61C2×#12011yescharacteristic
240C22×C61C2×#12112yes—
241C22×C61C2×#12311yescharacteristic
242C2×D61—24no—
243C2×D61—28no—
244C2×D61—42no—
245C2×D61—48no—
246C2×D61—44no—
247C2×D61—44no—
248C2×D61—44no—
249C2×D61—42no—
250C4×C61—24no—
251C4×C61—48no—
252C4×C61#38612yes—
253C4×C61#38612yes—
254C4×C61#38614yes—
255Dic61—44no—
256Dic61—42no—
257Dic61—84no—
258C2×#525—611no—
259#3428D6111yescharacteristic, complemented
260C2×#410—611no—
261#2280—612no—
262#1890—614no—
263#1889—614no—
264#1740—612no—
265C2×C61—21no—
266C2×C61—22no—
267C2×C61—22no—
268C2×C61—21no—
269C2×C61—44no—
270C2×C61C2×#38611yescharacteristic
271C2×C61#302512yes—
272D61—44no—
273C22×#41C2×D6111yescharacteristic
274C24×C4—611no—
275#525—618no—
276#525—1224no—
277C2×#123—612no—
278C2×C22⋊Q8—1221no—
279C2×#121—1221no—
280C2×#120—612no—
281#520—24no—
282#519—21no—
283#518—21no—
284#518C2×D6114yes—
285C4×#41—24no—
286C2×C4×D4—1222no—
287C2×#117—22no—
288C2×#117—611no—
289C2×#117—612no—
290C2×#117C2×D6112yes—
291#410—1224no—
292C2×#99—614no—
293#381—614no—
294#380—1224no—
295#379—614no—
296C4×C22⋊C4—1224no—
297#342—618no—
298C61#2095911yescharacteristic, complemented, Sylow
299C25—611no—
300C2×#41—24no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41C4×D6114yes—
304C2×#41D24414yes—
305C22×Q8C22⋊D6111yescharacteristic
306C22×D4—1222no—
307C22×D4C22⋊D6111yescharacteristic
308C22×D4C22×D6112yes—
309C23×C4—611no—
310C23×C4—614no—
311C23×C4—1222no—
312C23×C4—1224no—
313C23×C4—1221no—
314C23×C4—1221no—
315C23×C4—1222no—
316C23×C4—1222no—
317C23×C4C22⋊D6112yes—
318C23×C4C22×D6111yescharacteristic
319#123—618no—
320#123—1224no—
321C22⋊Q8—1228no—
322#121—1228no—
323#120—618no—
324#120—1224no—
325C4×Q8—28no—
326C4×D4—28no—
327C4×D4—216no—
328C4×D4—12216no—
329#117—24no—
330#117—24no—
331#117—22no—
332#117—24no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—614no—
337#117—1222no—
338#117—1224no—
339#117—1224no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4C22⋊D6114yescomplemented
345C2×C22⋊C4—24no—
346C2×C22⋊C4—612no—
347C2×C22⋊C4—614no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1228no—
351C2×C22⋊C4C22×D6114yes—
352C2×C42—24no—
353C2×C42—1221no—
354C2×C42—1222no—
355C2×C42C22⋊D6114yescomplemented
356#99—1228no—
357#95—1228no—
358C24—612no—
359C24—1222no—
360C24—1221no—
361C24—1221no—
362C24—1222no—
363C24—2444no—
364C24C2×C22⋊D6112yes—
365C24C23×D6111yescharacteristic
366#41—28no—
367#41—28no—
368#41—28no—
369#41—28no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8#1175819814yescomplemented
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—1228no—
378C2×D4—2448no—
379C2×D4#1175819814yescomplemented
380C2×D4C2×C4×D6114yes—
381C2×D4C2×D24414yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—612no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1222no—
392C22×C4—1224no—
393C22×C4—1224no—
394C22×C4—1228no—
395C22×C4—1228no—
396C22×C4—1224no—
397C22×C4—2442no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4—2444no—
401C22×C4—2444no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4#1175819818yescomplemented
405C22×C4C2×C4×D6112yes—
406C22×C4C2×D24412yes—
407C22×C4#1175822214yes—
408C22×C4C2×C22⋊D6112yes—
409C22×C4C2×C22⋊D6118yes—
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6112yes—
413C22×C4C2×C22⋊D6112yes—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—1228no—
418C4⋊C4—1224no—
419C4⋊C4—2444no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—614no—
424C22⋊C4—1222no—
425C22⋊C4—1228no—
426C22⋊C4—1228no—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C22⋊C4—24416no—
430C42—28no—
431C42—24no—
432C42—24no—
433C42—1224no—
434C42—2444no—
435C23—22no—
436C23—22no—
437C23—42no—
438C23—44no—
439C23—1228no—
440C23—1224no—
441C23—1222no—
442C23—1224no—
443C23—2444no—
444C23—2442no—
445C23—2444no—
446C23—2444no—
447C23—2448no—
448C23—2441no—
449C23—2442no—
450C23—2441no—
451C23C2×#1175819814yes—
452C23C22×C4×D6111yescharacteristic
453C23C22×C22⋊D6111yescharacteristic
454C23C22×C22⋊D6112yes—
455C23C22×D24411yescharacteristic
456C23C2×#1175822212yes—
457Q8—28no—
458D4—28no—
459D4—28no—
460D4—28no—
461D4—24416no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—48no—
468C2×C4—88no—
469C2×C4—1224no—
470C2×C4—1222no—
471C2×C4—1224no—
472C2×C4—2448no—
473C2×C4—2444no—
474C2×C4—2444no—
475C2×C4—2441no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2444no—
479C2×C4—2444no—
480C2×C4—2448no—
481C2×C4C2×#1175819814yes—
482C2×C4C2×#1175819812yes—
483C2×C4C2×#1175819812yes—
484C2×C4C4×C22⋊D6114yes—
485C2×C4C4×C22⋊D6114yes—
486C2×C4#1451974612yes—
487C2×C4#1451974611yescharacteristic
488C2×C4#1451974611yescharacteristic
489C2×C4#1451975114yes—
490C2×C4#1451975114yes—
491C2×C4#1451976012yes—
492C2×C4#1451976012yes—
493C22—24no—
494C22—22no—
495C22—24no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—82no—
502C22—1224no—
503C22—1228no—
504C22—2448no—
505C22—2442no—
506C22—2444no—
507C22—2444no—
508C22—2442no—
509C22—2444no—
510C22C22×#1175819811yescharacteristic
511C22C2×C4×C22⋊D6112yes—
512C22C2×#1451974611yescharacteristic
513C22C2×#1451975112yes—
514C22C2×#1451976011yescharacteristic
515C4—24no—
516C4—48no—
517C4—4884no—
518C4—2444no—
519C4—2442no—
520C4#1683964112yes—
521C4#1683964112yes—
522C4#1683964114yes—
523C2—22no—
524C2—21no—
525C2—21no—
526C2—22no—
527C2—44no—
528C2—2444no—
529C2C2×#1683964111yescharacteristic
530C2#2145521112yes—
subgroupG/H
centerC2×C4#14519746
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#3428C2
socleC22×C61C2×#120
prime radicalC2×#16840269C2
cosocleC24×C61C24
hypercenter#3428D61
hypocenterC61#20959
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. C2×#16839736C2semidirect
3. #21448198C2semidirect
4. #21446232C2semidirect
5. #21446227C2semidirect
6. #21445552C2semidirect
7. C61×#3428C2semidirect
10. #16840269C4semidirect
42. C22×C4×D61C2×C4semidirect
44. #14519758C2×C4semidirect
48. #14519745C2×C4semidirect
53. #14519736C2×C4semidirect
87. C2×C4⋊C4×C61D4semidirect
91. C2×C42×C61D4semidirect
106. C2×Q8×C61C22⋊C4semidirect
111. C2×D4×C61C22⋊C4semidirect
118. C22×C4×C61C22⋊C4semidirect
259. #3428D61semidirect
298. C61#20959semidirect
344. C2×C4⋊C4C22⋊D61semidirect
355. C2×C42C22⋊D61semidirect
372. C2×Q8#11758198semidirect
379. C2×D4#11758198semidirect
404. C22×C4#11758198semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
C2×#168397362
#214481982
#214462322
#214462272
#214455522
C61×#34282
#2095961