Finite Group Databasebeta

← #29722476

Subgroups — #29722476

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840037C211yesmaximal, characteristic, complemented
2#21455474C211yesmaximal, characteristic, complemented
3#21455367C212yesmaximal
4#21454419C211yesmaximal, characteristic, complemented
5#21447181C212yesmaximal, complemented
6#21447035C212yesmaximal, complemented
7#21446680C211yesmaximal, characteristic, complemented
8#21446567C211yesmaximal, characteristic, complemented
9#21446186C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C2×C61×#443C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C22×Dic244C2211yescharacteristic
14#16840333C2212yes—
15#16840245C2212yes—
16#16840037—24no—
17#16840020—22no—
18#16840003—21no—
19#16840003—21no—
20#16839982—22no—
21#16839975—22no—
22#16839970—22no—
23#16839965—22no—
24C2×#14519713C2211yescharacteristic
25#16839804—22no—
26#16839804C2212yes—
27#16839800—21no—
28#16839800C2211yescharacteristic
29C2×#14519711—21no—
30C2×#14519711C2211yescharacteristic
31#16839788C2212yes—
32#16839772C2212yes—
33#16839763C2212yes—
34C2×#14519702C2212yes—
35C2×#14519695C2212yes—
36#16839668C2212yes—
37#16839530C2212yes—
38#16839522C2212yes—
39C2×#135×C61C2211yescharacteristic
40C2×#131×C61C2211yescharacteristic
41C2×#131×C61C2211yescharacteristic
42C61×#443—24no—
43C2×#108×C61C2211yescharacteristic
44C2×#104×C61C2211yescharacteristic
45C2×#104×C61C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48#14519807—24no—
49#14519807—24no—
50C2×Dic244—21no—
51C2×Dic244—21no—
52C2×Dic244—22no—
53C2×Dic244—22no—
54C2×#11758202—22no—
55C2×#11758200—22no—
56C2×#11758194C2312yes—
57#14519775—22no—
58#14519764—22no—
59#14519758—42no—
60#14519756—22no—
61#14519756—22no—
62Q8×Dic61—22no—
63D4×Dic61—22no—
64D4×Dic61—42no—
65#14519736—22no—
66#14519736C2312yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2312yes—
69#14519713—22no—
70#14519711—21no—
71#14519711—21no—
72#14519711—22no—
73#14519711D411yescharacteristic
74#14519711D411yescharacteristic
75#14519709—22no—
76#14519709D412yes—
77#14519706—22no—
78#14519706—22no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519702—22no—
83#14519701—22no—
84#14519698—22no—
85#14519695—24no—
86#14519684—42no—
87C2×#41×C61C2311yescharacteristic
88C2×#41×C61C2311yescharacteristic
89#135×C61—22no—
90#135×C61—22no—
91C2×Q16×C61D411yescharacteristic
92C2×SD16×C61D411yescharacteristic
93C2×SD16×C61D411yescharacteristic
94C2×D8×C61D411yescharacteristic
95#131×C61—22no—
96#131×C61—22no—
97#131×C61—22no—
98#131×C61D414yes—
99C2×M16×C61—21no—
100C2×M16×C61—21no—
101C2×M16×C61C2311yescharacteristic
102C2×M16×C61C2311yescharacteristic
103C22×C8×C61—21no—
104C22×C8×C61—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#108×C61—22no—
109#104×C61—22no—
110#104×C61—22no—
111C4×C8×C61—22no—
112#41×C61—21no—
113#41×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118#41×C61C2×D412yes—
119#41×C61#4112yes—
120C2×Q8×C61C2×D411yescharacteristic
121C2×Q8×C61C2×D411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C2×D411yescharacteristic
125C2×D4×C61C2×D411yescharacteristic
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—42no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139Q16×C61—22no—
140SD16×C61—22no—
141SD16×C61—22no—
142D8×C61—22no—
143M16×C61—21no—
144M16×C61—24no—
145M16×C61—21no—
146M16×C61—21no—
147M16×C61—42no—
148M16×C61C2×D412yes—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—24no—
153C2×C8×C61—21no—
154C2×C8×C61—42no—
155C2×C8×C61C2×D411yescharacteristic
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D412yes—
160C2×C8×C61#4112yes—
161C42×C61—21no—
162C42×C61—21no—
163C22⋊Dic61—22no—
164C22⋊Dic61—42no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167#11758202—44no—
168#11758200—44no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61C2×D412yes—
173C4⋊Dic61C2×D412yes—
174#11758196—42no—
175#11758196—42no—
176C4×Dic61—22no—
177C4×Dic61—42no—
178#11758194—22no—
179#11758194—22no—
180C2×#92418—22no—
181Dic244—42no—
182Dic244—42no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61H(5,𝔽2)11yescharacteristic
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—21no—
189Q8×C61#12112yes—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—21no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61#12112yes—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—82no—
220C2×Dic61—162no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—42no—
225C8×C61—22no—
226C8×C61—21no—
227C8×C61—21no—
228C8×C61—41no—
229C8×C61#12112yes—
230C8×C61#12112yes—
231#92418—42no—
232#24798—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61#53711yescharacteristic
242C22×C61D4211yescharacteristic
243C22×C61#55311yescharacteristic
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—41no—
247C4×C61C2×#12111yescharacteristic
248C4×C61C2×#12111yescharacteristic
249C4×C61#54711yescharacteristic
250C4×C61#54811yescharacteristic
251Dic61—42no—
252Dic61—42no—
253Dic61—82no—
254Dic61—82no—
255C2×#473—611no—
256#3059—611no—
257#3046—612no—
258C2×#443D6111yescharacteristic
259#2913—611no—
260#2064—612no—
261#2038—612no—
262#1969—611no—
263#1945—611no—
264#1882—612no—
265#1816—611no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—42no—
269C2×C61—81no—
270C2×C61#241611yescharacteristic
271C2×C61#329011yescharacteristic
272C2×C61#329311yescharacteristic
273C2×#137—612no—
274C2×#135C2×D6111yescharacteristic
275C22×Q16—611no—
276C2×#131C2×D6111yescharacteristic
277C2×#131C2×D6111yescharacteristic
278#536—612no—
279#520—612no—
280#473—1224no—
281#470—1222no—
282#467—1221no—
283#467—1221no—
284#464—1222no—
285#463—1222no—
286#462—1222no—
287#461—1222no—
288#443—24no—
289C2×#108—611no—
290C2×#108C2×D6111yescharacteristic
291#428—612no—
292#428—1222no—
293#427—611no—
294#427—1221no—
295C2×#107—611no—
296C2×#107—1221no—
297#424—612no—
298C2×#104C2×D6111yescharacteristic
299C2×#104C2×D6111yescharacteristic
300#419—612no—
301#417—612no—
302C2×#103—612no—
303C2×#101—612no—
304C2×C4×C8C2×D6111yescharacteristic
305#391—612no—
306#343—612no—
307#339—612no—
308C61#2479811yescharacteristic, complemented, Sylow
309C2×#41C22×D6111yescharacteristic
310C2×#41C22×D6111yescharacteristic
311C22×Q8—612no—
312#137—1224no—
313#137—1224no—
314#135—22no—
315#135—22no—
316C2×Q16—1222no—
317C2×Q16—1221no—
318C2×Q16—1221no—
319C2×Q16—1222no—
320C2×Q16—1222no—
321C2×Q16D24411yescharacteristic
322C2×SD16—1222no—
323C2×SD16D24411yescharacteristic
324C2×SD16D24411yescharacteristic
325C2×D8D24411yescharacteristic
326#131—22no—
327#131—22no—
328#131—22no—
329#131C22⋊D6114yes—
330C2×M16—21no—
331C2×M16—21no—
332C2×M16—612no—
333C2×M16C22×D6111yescharacteristic
334C2×M16C22×D6111yescharacteristic
335C22×C8—21no—
336C22×C8—21no—
337C22×C8C22×D6111yescharacteristic
338C22×C8C22×D6111yescharacteristic
339C4⋊Q8—1222no—
340#124—1222no—
341#123—2442no—
342C22⋊Q8—1222no—
343C22⋊Q8—1222no—
344C4×Q8—1222no—
345C4×D4—1222no—
346C4×D4—2442no—
347#117—612no—
348#117—1222no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1222no—
351C2×C42C22×D6111yescharacteristic
352#108—22no—
353#108—1222no—
354#107—611no—
355#107—611no—
356#107—1222no—
357#107—1221no—
358#107—1221no—
359#106—612no—
360#106—1222no—
361C4⋊C8—1222no—
362C4⋊C8—1222no—
363#104—22no—
364#104—22no—
365#103—1222no—
366#103—1222no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#102—1222no—
371#101—1224no—
372C4×C8—22no—
373#95—2442no—
374#41—21no—
375#41—21no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—21no—
380#41#1175822212yes—
381#41C2×C22⋊D6112yes—
382C2×Q8—1222no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8C2×D24411yescharacteristic
389C2×Q8C2×D24411yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×D24411yescharacteristic
393C2×D4C2×D24411yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4C23×D6111yescharacteristic
401Q16—22no—
402Q16—2444no—
403Q16—2442no—
404Q16—2442no—
405SD16—22no—
406SD16—22no—
407SD16—2444no—
408D8—22no—
409M16—21no—
410M16—24no—
411M16—21no—
412M16—21no—
413M16—42no—
414M16—1222no—
415M16—1222no—
416M16C2×C22⋊D6112yes—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—24no—
421C2×C8—21no—
422C2×C8—42no—
423C2×C8—1222no—
424C2×C8C2×D24411yescharacteristic
425C2×C8C2×D24411yescharacteristic
426C2×C8D4×D6111yescharacteristic
427C2×C8D4×D6111yescharacteristic
428C2×C8#1175822412yes—
429C2×C8C2×C22⋊D6112yes—
430C4⋊C4—612no—
431C4⋊C4—612no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—2442no—
435C4⋊C4—2442no—
436C4⋊C4—2442no—
437C22⋊C4—1222no—
438C22⋊C4—2442no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C42—21no—
442C42—21no—
443C42—1222no—
444C42—2442no—
445C23—21no—
446C23—21no—
447C23#1451982311yescharacteristic
448Q8—21no—
449Q8—21no—
450Q8—21no—
451Q8—2442no—
452Q8—2442no—
453Q8—2442no—
454Q8—2442no—
455Q8#1451975112yes—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—21no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4#1451975112yes—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—9762no—
476C2×C4—4882no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4C22×D24411yescharacteristic
484C2×C4C2×#1175822411yescharacteristic
485C2×C4C2×#1175822211yescharacteristic
486C2×C4#1451982411yescharacteristic
487C8—22no—
488C8—21no—
489C8—21no—
490C8—41no—
491C8—2442no—
492C8#1451974912yes—
493C8#1451974912yes—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22#1684034011yescharacteristic
503C22D4×D24411yescharacteristic
504C22#1684045811yescharacteristic
505C4—22no—
506C4—42no—
507C4—41no—
508C4—4882no—
509C4—4882no—
510C4—2442no—
511C4—2442no—
512C4C2×#1451974911yescharacteristic
513C4C2×#1451975111yescharacteristic
514C4#1684041611yescharacteristic
515C4#1684042111yescharacteristic
516C2—21no—
517C2—21no—
518C2—42no—
519C2—81no—
520C2#2144925911yescharacteristic
521C2#2145760611yescharacteristic
522C2#2145762411yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#443C2
socleC22×C61D42
prime radicalC2×#135×C61C22
hypercenterC2×#443D61
hypocenterC61#24798
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
2. #21455474C2semidirect
4. #21454419C2semidirect
5. #21447181C2semidirect
6. #21447035C2semidirect
7. #21446680C2semidirect
8. #21446567C2semidirect
9. #21446186C2semidirect
10. #21445883C2semidirect
308. C61#24798semidirect

Maximal subgroups

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

subgroupindex
C2×#168400372
#214554742
#214553672
#214544192
#214471812
#214470352
#214466802
#214465672
#214461862
#214458832
C2×C61×#4432
#2479861