Finite Group Databasebeta

← #29657293

Subgroups — #29657293

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456937C211yesmaximal, characteristic, complemented
2C2×#16840112C211yesmaximal, characteristic, complemented
3#21455306C211yesmaximal, characteristic, complemented
4#21445047C214yesmaximal, complemented
5#21445033C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic
10C2×#14519807—21no—
11C2×#14519777C2211yescharacteristic
12#16840165—21no—
13#16840112—22no—
14#16840112C2214yes—
15#16840108—21no—
16#16840073—22no—
17#16840042C2214yes—
18#16839972—21no—
19#16839972C2212yes—
20#16839967—21no—
21#16839967C2212yes—
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839770—21no—
25C2×#14519701—22no—
26C2×C4×#92418C2211yescharacteristic
27#16839511C2214yes—
28Q8⋊#92418C2212yes—
29D4⋊#92418C2212yes—
30D4⋊#92418C2214yes—
31C61×#556—21no—
32C61×#555C2212yes—
33C61×#554C2212yes—
34C61×#553—22no—
35C61×#548—22no—
36C61×#547—21no—
37C61×#542—22no—
38C61×#537—21no—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520—21no—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46#14519807—44no—
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758194—21no—
50C22×#92418—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57#14519710—24no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—28no—
62#14519700—22no—
63#14519700—22no—
64#14519698—24no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
69C2×#41×C61—22no—
70C2×#41×C61—21no—
71C2×#41×C61D411yescharacteristic
72C22×Q8×C61D412yes—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61D412yes—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61C2312yes—
78#127×C61D411yescharacteristic
79#126×C61—24no—
80#125×C61—22no—
81#124×C61—22no—
82#124×C61—24no—
83#124×C61—21no—
84#123×C61—24no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—24no—
89#121×C61—22no—
90#121×C61—24no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—21no—
94#121×C61D412yes—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61D411yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C4×D4×C61—22no—
100C4×D4×C61—24no—
101C4×D4×C61—22no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61D411yescharacteristic
106C4×D4×C61C2312yes—
107C4×D4×C61C2311yescharacteristic
108#117×C61—22no—
109#117×C61—22no—
110#117×C61—21no—
111#117×C61D411yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—22no—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117#41×C61—22no—
118#41×C61—22no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—84no—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61—44no—
125C2×Q8×C61C2×D411yescharacteristic
126C2×Q8×C61C2×D414yes—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C2×D412yes—
133C2×D4×C61C2×D411yescharacteristic
134C22×C4×C61—24no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×Dic61—41no—
144C2×Dic122—22no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×Dic122—41no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C2×D411yescharacteristic
158C4⋊C4×C61C2×D411yescharacteristic
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—21no—
164C22⋊C4×C61—21no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C42×C61—24no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2411yescharacteristic
176#11758202—44no—
177#11758200—44no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—41no—
181C4⋊Dic61—41no—
182#11758194—42no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—42no—
187C23×C61—21no—
188C23×C61—22no—
189C23×C61#12011yescharacteristic
190Q8×C61—21no—
191Q8×C61—21no—
192Q8×C61—44no—
193Q8×C61—42no—
194Q8×C61—44no—
195D4×C61—21no—
196D4×C61—21no—
197D4×C61—24no—
198D4×C61—42no—
199D4×C61—44no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—84no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—24no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—82no—
216C2×C4×C61—84no—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61#13512yes—
221C2×C4×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×Dic61—42no—
226C2×Dic61—82no—
227Dic122—42no—
228Dic122—42no—
229Dic122—82no—
230#92418—42no—
231#92418—42no—
232#92418—42no—
233#18636—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—42no—
240C22×C61—41no—
241C22×C61—42no—
242C22×C61#44912yes—
243C22×C61C2×#12011yescharacteristic
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—41no—
247C4×C61—42no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—44no—
251C4×C61#44912yes—
252C4×C61#45512yes—
253C4×C61C2×#13511yescharacteristic
254C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C61#57812yes—
256Dic61—82no—
257#3511D6111yescharacteristic
258#3226—611no—
259C2×#487—611no—
260#3039—611no—
261#1646—614no—
262#1644—614no—
263#1548—611no—
264#1492—611no—
265#1476—611no—
266C2×C61—22no—
267C2×C61—81no—
268C2×C61—82no—
269C2×C61C2×#44911yescharacteristic
270C2×C61#301111yescharacteristic
271C2×C61C2×#45511yescharacteristic
272C2×#137—1221no—
273#556—21no—
274#555C2×D6112yes—
275#554C2×D6112yes—
276#553—22no—
277#548—22no—
278#547—21no—
279#542—22no—
280#537—21no—
281#533—21no—
282#532—21no—
283#532C2×D6111yescharacteristic
284C2×C4⋊Q8—611no—
285C2×C4⋊Q8C2×D6111yescharacteristic
286#520—21no—
287C4×#41C2×D6111yescharacteristic
288#500—1221no—
289#487—614no—
290#487—1222no—
291#486—1221no—
292#480—1222no—
293#474—614no—
294#462—612no—
295#462—1221no—
296#461—612no—
297#461—1221no—
298#431—611no—
299#423—611no—
300#419—1221no—
301C2×#103—1222no—
302C2×C4×C8—611no—
303#333—614no—
304Q8⋊C8—612no—
305D4⋊C8—614no—
306D4⋊C8—612no—
307C61#1863611yescharacteristic, complemented, Sylow
308C2×#41—22no—
309C2×#41—21no—
310C2×#41C22⋊D6111yescharacteristic
311C22×Q8—1221no—
312C22×Q8C22⋊D6112yes—
313#137—2444no—
314C2×Q16—1222no—
315C2×SD16—1222no—
316C2×M16—1221no—
317C22×C8—1221no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—612no—
321C4⋊Q8—1221no—
322C4⋊Q8—1221no—
323C4⋊Q8—1221no—
324C4⋊Q8C22⋊D6112yes—
325C4⋊Q8C22⋊D6111yescharacteristic
326C4⋊Q8C22×D6112yes—
327#127C22⋊D6111yescharacteristic
328#126—24no—
329#125—22no—
330#124—22no—
331#124—24no—
332#124—21no—
333#123—24no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—24no—
338#121—22no—
339#121—24no—
340#121—22no—
341#121—22no—
342#121—21no—
343#121C22⋊D6112yes—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8C22⋊D6111yescharacteristic
347C4×Q8C22×D6111yescharacteristic
348C4×D4—22no—
349C4×D4—24no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4C22⋊D6111yescharacteristic
355C4×D4C22×D6111yescharacteristic
356C4×D4C22×D6112yes—
357#117—22no—
358#117—22no—
359#117—21no—
360#117C22⋊D6111yescharacteristic
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C42—21no—
367C2×C42C22×D6111yescharacteristic
368#107—1224no—
369C4⋊C8—612no—
370C4⋊C8—612no—
371C4⋊C8—1221no—
372#103—1228no—
373#103—1222no—
374#103—1222no—
375#102—1224no—
376C22⋊C8—1221no—
377C4×C8—612no—
378C4×C8—1221no—
379C4×C8—1221no—
380#41—22no—
381#41—22no—
382#41—42no—
383#41—42no—
384#41—84no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8—1222no—
389C2×Q8—2442no—
390C2×Q8—2442no—
391C2×Q8—2441no—
392C2×Q8C2×C22⋊D6111yescharacteristic
393C2×Q8C2×C22⋊D6114yes—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4C2×C22⋊D6112yes—
400C2×D4C2×C22⋊D6111yescharacteristic
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—2441no—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411Q16—2444no—
412SD16—2444no—
413M16—2442no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—2442no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C4⋊C4C2×C22⋊D6111yescharacteristic
432C4⋊C4C2×C22⋊D6111yescharacteristic
433C4⋊C4C2×C22⋊D6112yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C42—24no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42C2×C22⋊D6111yescharacteristic
447C42C2×C22⋊D6111yescharacteristic
448C42C2×C22⋊D6111yescharacteristic
449C42C23×D6111yescharacteristic
450C23—21no—
451C23—22no—
452C23#1451974611yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—44no—
456Q8—44no—
457Q8—42no—
458Q8—4882no—
459Q8—2442no—
460Q8—2442no—
461D4—21no—
462D4—21no—
463D4—24no—
464D4—42no—
465D4—44no—
466D4—42no—
467D4—42no—
468D4—84no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—4882no—
484C2×C4—2442no—
485C2×C4#1451974611yescharacteristic
486C2×C4#1451974611yescharacteristic
487C2×C4#1451974611yescharacteristic
488C2×C4#1451979712yes—
489C2×C4#1451980212yes—
490C2×C4C22×C22⋊D6111yescharacteristic
491C2×C4C22×C22⋊D6111yescharacteristic
492C2×C4C22×C22⋊D6111yescharacteristic
493C8—2442no—
494C8—2442no—
495C8—2442no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22#1683990712yes—
505C22C2×#1451974611yescharacteristic
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—44no—
511C4—41no—
512C4—42no—
513C4—4882no—
514C4#1683990712yes—
515C4#1683994012yes—
516C4C2×#1451979711yescharacteristic
517C4C2×#1451980211yescharacteristic
518C4#1684057712yes—
519C2—22no—
520C2—81no—
521C2—82no—
522C2C2×#1683990711yescharacteristic
523C2#2145511411yescharacteristic
524C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#18636
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456937C2semidirect
2. C2×#16840112C2semidirect
3. #21455306C2semidirect
4. #21445047C2semidirect
5. #21445033C2semidirect
6. #21444500C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
307. C61#18636semidirect

Maximal subgroups

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

subgroupindex
#214569372
C2×#168401122
#214553062
#214450472
#214450332
#214445002
#214442702
#214442012
C61×#35112
#1863661