Finite Group Databasebeta

← #29708503

Subgroups — #29708503

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519807C212yesmaximal, complemented
2#21448535C211yesmaximal, characteristic, complemented
3#21447166C211yesmaximal, characteristic, complemented
4#21447045C211yesmaximal, characteristic
5#21447033C212yesmaximal, complemented
6#21446977C211yesmaximal, characteristic, complemented
7#21446866C211yesmaximal, characteristic
8#21446853C212yesmaximal, complemented
9#21445629C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C61×#3461C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C2×#14519775C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16839867—22no—
16#16839862—22no—
17C4×#11758202—22no—
18C4×#11758200—22no—
19#16839805C2212yes—
20C2×#14519704C2212yes—
21C2×#14519704C2212yes—
22#16839770C2212yes—
23#16839763C2212yes—
24C2×#14519696C2212yes—
25C4×#11758194C2212yes—
26#16839669C2211yescharacteristic
27#16839667C2211yescharacteristic
28#16839646—22no—
29#16839643—22no—
30#16839627C2211yescharacteristic
31#16839608C2211yescharacteristic
32#16839531C2211yescharacteristic
33#16839531C2211yescharacteristic
34#16839521C2212yes—
35#16839521C2212yes—
36C61×#544C2211yescharacteristic
37C61×#542—22no—
38C61×#541—22no—
39C61×#539—22no—
40C61×#537C2211yescharacteristic
41C61×#536C2212yes—
42C61×#533—22no—
43C61×#529—22no—
44C61×#529—22no—
45C4×#41×C61C2212yes—
46C2×C4×Q8×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48#14519807—24no—
49#14519807—24no—
50C2×#11758202—22no—
51C2×#11758200—22no—
52C2×#11758194C2312yes—
53C2×#11758194C2312yes—
54#14519775—22no—
55#14519775—22no—
56C4×Dic122—22no—
57C4×Dic122—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758196—22no—
60C2×#11758196—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61C2311yescharacteristic
63C2×C4×Dic61C2311yescharacteristic
64#14519710—22no—
65#14519708—22no—
66#14519704—24no—
67#14519704—24no—
68#14519701—22no—
69#14519700—22no—
70#14519700—22no—
71#14519698—22no—
72#14519696—24no—
73#14519688—22no—
74C4×#92418—22no—
75#14519684—21no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80C2×#41×C61C2312yes—
81C22×Q8×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84#126×C61—24no—
85#126×C61—24no—
86#125×C61—24no—
87#125×C61—22no—
88#125×C61—22no—
89#124×C61—22no—
90#124×C61—22no—
91#123×C61—24no—
92#123×C61—24no—
93C22⋊Q8×C61—24no—
94C22⋊Q8×C61—22no—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98#121×C61—22no—
99#121×C61—22no—
100C4×Q8×C61—22no—
101C4×Q8×C61—22no—
102C4×Q8×C61—22no—
103C4×Q8×C61D412yes—
104C4×Q8×C61D412yes—
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61D412yes—
109C4×D4×C61D412yes—
110#117×C61—24no—
111#117×C61—22no—
112#117×C61C2312yes—
113C2×C4⋊C4×C61D412yes—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61—22no—
116C2×C42×C61D412yes—
117C2×C42×C61C2311yescharacteristic
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61C2×D412yes—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61C2×D412yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—21no—
146C2×Dic122#4112yes—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—22no—
167C42×C61—22no—
168C42×C61—22no—
169C42×C61C2×D412yes—
170C42×C61C2×D412yes—
171#11758202—28no—
172#11758200—28no—
173C4⋊Dic61—22no—
174#11758196—28no—
175#11758196—41no—
176#11758196—41no—
177C4×Dic61—21no—
178C4×Dic61—22no—
179C4×Dic61—42no—
180C4×Dic61#4112yes—
181#11758194—22no—
182#11758194—22no—
183#11758194—22no—
184#11758194#4114yes—
185C2×#92418—22no—
186C2×#92418—22no—
187C23×C61—22no—
188C23×C61C22×D411yescharacteristic
189Q8×C61—22no—
190Q8×C61—22no—
191Q8×C61—42no—
192Q8×C61—42no—
193Q8×C61—82no—
194D4×C61—22no—
195D4×C61—22no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—82no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—44no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—82no—
213C2×C4×C61#12012yes—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61C22×D412yes—
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C61C2×#4111yescharacteristic
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231Dic122—24no—
232Dic122—42no—
233#92418—24no—
234#92418—42no—
235#23594—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61C2×#12011yescharacteristic
242C22×C61#52511yescharacteristic
243C22×C61#53211yescharacteristic
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61#52512yes—
253C4×C61#53212yes—
254Dic61—42no—
255Dic61—42no—
256Dic61—81no—
257Dic61—81no—
258#3461D6111yescharacteristic
259C4×#137—612no—
260#2327—611no—
261#2061—611no—
262#2040—611no—
263#2038—612no—
264#2028—611no—
265#2007—611no—
266#2005—612no—
267#1759—612no—
268C4×#101—611no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—82no—
272C2×C61#238811yescharacteristic
273C2×C61#302011yescharacteristic
274C2×C61#302211yescharacteristic
275C2×#137—612no—
276#544C2×D6111yescharacteristic
277#542—22no—
278#541—22no—
279#539—22no—
280#537C2×D6111yescharacteristic
281#536C2×D6112yes—
282#533—22no—
283C2×C4⋊Q8—611no—
284#529—22no—
285#529—22no—
286C4×#41C2×D6112yes—
287C2×C4×Q8—611no—
288C2×C4×Q8C2×D6111yescharacteristic
289#441—1222no—
290#440—1222no—
291C4×Q16—1222no—
292C4×SD16—1222no—
293#428—612no—
294C2×#104—612no—
295C2×#104—612no—
296#419—612no—
297#417—612no—
298C2×#101—612no—
299C4×M16—612no—
300#391—611no—
301#391—611no—
302#387—1222no—
303#387—1222no—
304#382—611no—
305#376—611no—
306#343—611no—
307#343—611no—
308#339—612no—
309#339—612no—
310C61#2359411yescharacteristic, complemented, Sylow
311C2×#41C22×D6112yes—
312C22×Q8—611no—
313C22×Q8C22×D6111yescharacteristic
314#137—1224no—
315#137—1224no—
316C2×Q16—1222no—
317C2×SD16—1222no—
318C2×M16—612no—
319C2×M16—612no—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324#126—24no—
325#126—24no—
326#125—22no—
327#125—24no—
328#125—22no—
329#124—22no—
330#124—22no—
331#123—24no—
332#123—24no—
333C22⋊Q8—24no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338#121—22no—
339#121—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—1222no—
344C4×Q8—1222no—
345C4×Q8C22⋊D6112yes—
346C4×Q8C22⋊D6112yes—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4C22⋊D6112yes—
351C4×D4C22⋊D6112yes—
352#117—24no—
353#117—22no—
354#117C22×D6112yes—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4C22⋊D6112yes—
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C42—22no—
361C2×C42—611no—
362C2×C42—611no—
363C2×C42—1221no—
364C2×C42C22⋊D6112yes—
365C2×C42C22×D6111yescharacteristic
366#107—1222no—
367#106—1222no—
368#104—1224no—
369#104—1224no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#102—1222no—
374#101—1224no—
375#97—1222no—
376C4×C8—1222no—
377#95—1221no—
378#95—1221no—
379#95—1222no—
380#95—1222no—
381#95—1222no—
382#41—22no—
383#41—22no—
384#41—22no—
385#41—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—612no—
392C2×Q8—1221no—
393C2×Q8—1222no—
394C2×Q8—1222no—
395C2×Q8C2×C22⋊D6112yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4C2×C22⋊D6112yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6112yes—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411Q16—1228no—
412SD16—1228no—
413M16—614no—
414M16—1222no—
415M16—1222no—
416M16—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—1228no—
429C4⋊C4—1222no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C4⋊C4C2×C22⋊D6112yes—
433C4⋊C4C2×C22⋊D6112yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—612no—
446C42—1221no—
447C42—1222no—
448C42—2442no—
449C42C2×C22⋊D6112yes—
450C42C2×C22⋊D6112yes—
451C23—22no—
452C23C22×C22⋊D6111yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—42no—
456Q8—42no—
457Q8—82no—
458Q8—1224no—
459Q8—2442no—
460D4—22no—
461D4—22no—
462D4—42no—
463D4—42no—
464D4—82no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4#1451974611yescharacteristic
491C2×C4#1451974611yescharacteristic
492C2×C4#1451974612yes—
493C2×C4C22×C22⋊D6112yes—
494C2×C4#41×D6111yescharacteristic
495C2×C4#41×D6112yes—
496C2×C4#41×D6111yescharacteristic
497C8—1224no—
498C8—2442no—
499C22—21no—
500C22—21no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22C2×#1451974611yescharacteristic
505C22#1684026911yescharacteristic
506C22#1684030811yescharacteristic
507C4—21no—
508C4—21no—
509C4—22no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—4881no—
516C4—4881no—
517C4—2442no—
518C4—2442no—
519C4#1684026912yes—
520C4#1684030812yes—
521C2—21no—
522C2—21no—
523C2—82no—
524C2#2144903211yescharacteristic
525C2#2145518311yescharacteristic
526C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3461C2
socleC22×C61C2×#120
prime radicalC61×#537C22
hypercenter#3461D61
hypocenterC61#23594
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. C4×#14519807C2semidirect
2. #21448535C2semidirect
3. #21447166C2semidirect
5. #21447033C2semidirect
6. #21446977C2semidirect
8. #21446853C2semidirect
9. #21445629C2semidirect
10. C4×#14519696C2semidirect
310. C61#23594semidirect

Maximal subgroups

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

subgroupindex
C4×#145198072
#214485352
#214471662
#214470452
#214470332
#214469772
#214468662
#214468532
#214456292
C4×#145196962
C61×#34612
#2359461