Finite Group Databasebeta

← #29722477

Subgroups — #29722477

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
1#21455470C211yesmaximal, characteristic, complemented
2#21455373C212yesmaximal
3C2×#16839877C211yesmaximal, characteristic
4#21454419C211yesmaximal, characteristic, complemented
5#21447182C212yesmaximal, complemented
6#21447037C212yesmaximal, complemented
7#21446681C211yesmaximal, characteristic, complemented
8#21446565C211yesmaximal, characteristic, complemented
9#21446190C212yesmaximal, complemented
10#21445880C211yesmaximal, characteristic, complemented
11C2×C61×#473C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840336C2212yes—
15#16840245C2212yes—
16#16840017—22no—
17#16840000—21no—
18#16840000—21no—
19#16839986—22no—
20#16839979—22no—
21#16839970—22no—
22#16839965—22no—
23#16839877—24no—
24C2×#14519712C2211yescharacteristic
25#16839804—22no—
26#16839804C2212yes—
27#16839800—21no—
28#16839800C2211yescharacteristic
29C2×#14519711—21no—
30C2×#14519711C2211yescharacteristic
31#16839790C2212yes—
32C2×#14519704C2211yescharacteristic
33C2×#14519703C2211yescharacteristic
34#16839771C2212yes—
35#16839764C2212yes—
36C2×#14519701C2212yes—
37C2×C8×Dic61C2211yescharacteristic
38#16839669C2212yes—
39#16839530C2212yes—
40#16839523C2212yes—
41C2×#137×C61C2212yes—
42C22×Q16×C61C2211yescharacteristic
43C2×#131×C61C2211yescharacteristic
44C61×#473—24no—
45C2×#108×C61C2211yescharacteristic
46C2×#101×C61C2212yes—
47C2×#11758222C2311yescharacteristic
48#14519797—22no—
49#14519797—22no—
50C2×#11758202D411yescharacteristic
51C2×#11758200D411yescharacteristic
52C2×#11758201D411yescharacteristic
53C2×#11758199D411yescharacteristic
54#14519783—22no—
55#14519783—22no—
56C2×#11758194C2311yescharacteristic
57C2×#11758189—21no—
58C22×#92418C2311yescharacteristic
59C2×C8×D61—21no—
60#14519775—22no—
61#14519766—22no—
62#14519760—42no—
63#14519753—22no—
64#14519753—22no—
65Q8×Dic61—22no—
66D4×Dic61—22no—
67D4×Dic61—42no—
68#14519736—22no—
69#14519736C2312yes—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2312yes—
72C2×C4×Dic61C2311yescharacteristic
73#14519712—22no—
74#14519711—21no—
75#14519711—21no—
76#14519711—22no—
77#14519711D411yescharacteristic
78#14519711D411yescharacteristic
79#14519709—22no—
80#14519709D412yes—
81#14519706—22no—
82#14519706—22no—
83#14519704—22no—
84#14519703—22no—
85#14519702—22no—
86#14519701—22no—
87#14519701—22no—
88#14519701—22no—
89#14519701—22no—
90#14519699—22no—
91C8×Dic61—22no—
92#14519684—42no—
93C2×#41×C61C2311yescharacteristic
94C22×Q8×C61C2312yes—
95#137×C61—24no—
96#137×C61—24no—
97C2×Q16×C61—22no—
98C2×Q16×C61—21no—
99C2×Q16×C61—21no—
100C2×Q16×C61—22no—
101C2×Q16×C61—22no—
102C2×SD16×C61—22no—
103#131×C61—22no—
104#131×C61D414yes—
105C2×M16×C61—21no—
106C2×M16×C61C2312yes—
107C2×M16×C61C2311yescharacteristic
108C22×C8×C61—21no—
109C22×C8×C61C2311yescharacteristic
110#108×C61—22no—
111#101×C61—24no—
112#41×C61—21no—
113#41×C61—21no—
114#41×C61C2×D412yes—
115#41×C61#4112yes—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C2×D411yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61C2×D411yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—42no—
132#11758222—21no—
133#11758222—21no—
134#11758222—21no—
135#11758222—21no—
136C2×D244C2×D411yescharacteristic
137C2×C4×D61—21no—
138C2×Dic122C2×D411yescharacteristic
139Q16×C61—44no—
140Q16×C61—42no—
141Q16×C61—42no—
142SD16×C61—44no—
143M16×C61—21no—
144M16×C61—24no—
145M16×C61—22no—
146M16×C61—22no—
147M16×C61C2×D412yes—
148C2×C8×C61—21no—
149C2×C8×C61—24no—
150C2×C8×C61—22no—
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2×D412yes—
154C2×C8×C61#4112yes—
155C22⋊Dic61—22no—
156C22⋊Dic61—42no—
157#11758202—22no—
158#11758201—22no—
159#11758200—22no—
160#11758199—22no—
161#11758198—42no—
162#11758198—42no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—42no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D412yes—
168#11758196—42no—
169#11758196—42no—
170C4×Dic61—22no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—42no—
174#11758194—21no—
175#11758194—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418C2×D411yescharacteristic
180C2×#92418C2×D411yescharacteristic
181#11758189—42no—
182C8×D61—42no—
183C23×C61—21no—
184C23×C61H(5,𝔽2)11yescharacteristic
185C22×D61—21no—
186Q8×C61—21no—
187Q8×C61—42no—
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191Q8×C61#12112yes—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—21no—
195D4×C61#12112yes—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C22⋊D61—41no—
209C22⋊D61—41no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—82no—
218C2×Dic61—162no—
219D244—21no—
220D244—21no—
221C4×D61—41no—
222C4×D61—41no—
223Dic122—21no—
224Dic122—21no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61#12112yes—
228C8×C61#12112yes—
229#92418—21no—
230#92418—21no—
231#92418—41no—
232#24798—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61#53711yescharacteristic
239C22×C61D4211yescharacteristic
240C22×C61#55311yescharacteristic
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C4×C61—22no—
245C4×C61—82no—
246C4×C61C2×#12111yescharacteristic
247C4×C61C2×#12111yescharacteristic
248C4×C61#54711yescharacteristic
249C4×C61#54811yescharacteristic
250Dic61—42no—
251Dic61—41no—
252Dic61—42no—
253Dic61—42no—
254Dic61—82no—
255C2×#473D6111yescharacteristic, complemented
256#3059—611no—
257#3046—612no—
258C2×#443—611no—
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#241611yescharacteristic
270C2×C61#329011yescharacteristic
271C2×C61#329311yescharacteristic
272D61—81no—
273C2×#137C2×D6112yes—
274C2×#135—611no—
275C22×Q16C2×D6111yescharacteristic
276C2×#131—611no—
277C2×#131C2×D6111yescharacteristic
278#536—612no—
279#520—612no—
280#473—24no—
281#470—1222no—
282#467—1221no—
283#467—1221no—
284#464—1222no—
285#463—1222no—
286#462—1222no—
287#461—1222no—
288#443—1224no—
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×#104—611no—
299C2×#104—611no—
300#419—612no—
301#417—612no—
302C2×#103—612no—
303C2×#101C2×D6112yes—
304C2×C4×C8—611no—
305#391—612no—
306#343—612no—
307#339—612no—
308C61#2479811yescharacteristic, complemented, Sylow
309C2×#41—611no—
310C2×#41C22×D6111yescharacteristic
311C22×Q8C22×D6112yes—
312#137—24no—
313#137—24no—
314#135—1222no—
315#135—1222no—
316C2×Q16—22no—
317C2×Q16—21no—
318C2×Q16—21no—
319C2×Q16—22no—
320C2×Q16—22no—
321C2×Q16—611no—
322C2×SD16—22no—
323C2×SD16—611no—
324C2×SD16—611no—
325C2×D8—611no—
326#131—22no—
327#131—1222no—
328#131—1222no—
329#131C22⋊D6114yes—
330C2×M16—21no—
331C2×M16—611no—
332C2×M16—1221no—
333C2×M16C22×D6112yes—
334C2×M16C22×D6111yescharacteristic
335C22×C8—21no—
336C22×C8—611no—
337C22×C8—1221no—
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×C42—611no—
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—1222no—
364#104—1222no—
365#103—1222no—
366#103—1222no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#102—1222no—
371#101—24no—
372C4×C8—1222no—
373#95—2442no—
374#41—21no—
375#41—21no—
376#41—1221no—
377#41—1221no—
378#41—1221no—
379#41—1221no—
380#41#1175822412yes—
381#41C2×C22⋊D6112yes—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—611no—
389C2×Q8D4×D6111yescharacteristic
390C2×D4—21no—
391C2×D4—611no—
392C2×D4—1221no—
393C2×D4D4×D6111yescharacteristic
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4C23×D6111yescharacteristic
401Q16—44no—
402Q16—42no—
403Q16—42no—
404Q16—1222no—
405SD16—44no—
406SD16—1222no—
407SD16—1222no—
408D8—1222no—
409M16—21no—
410M16—24no—
411M16—22no—
412M16—22no—
413M16—1221no—
414M16—1221no—
415M16—2442no—
416M16C2×C22⋊D6112yes—
417C2×C8—21no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—611no—
421C2×C8—611no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—2442no—
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—1222no—
442C42—1221no—
443C42—1221no—
444C42—2442no—
445C23—21no—
446C23—1221no—
447C23#1451982311yescharacteristic
448Q8—21no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—1221no—
454Q8—1221no—
455Q8#1451974812yes—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—1221no—
460D4—1221no—
461D4—2441no—
462D4—2441no—
463D4#1451974812yes—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—9762no—
475C2×C4—4882no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C2×C4C2×#1175822411yescharacteristic
485C2×C4C2×#1175822411yescharacteristic
486C2×C4#1451982411yescharacteristic
487C8—22no—
488C8—42no—
489C8—1221no—
490C8—1221no—
491C8—2441no—
492C8#1451974912yes—
493C8#1451974912yes—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—2441no—
500C22—2441no—
501C22—2441no—
502C22#1684034011yescharacteristic
503C22#1684039611yescharacteristic
504C22#1684046411yescharacteristic
505C4—22no—
506C4—82no—
507C4—4882no—
508C4—2442no—
509C4—2441no—
510C4—2442no—
511C4—2442no—
512C4C2×#1451974811yescharacteristic
513C4C2×#1451974911yescharacteristic
514C4#1684041211yescharacteristic
515C4#1684041811yescharacteristic
516C2—21no—
517C2—21no—
518C2—42no—
519C2—4881no—
520C2#2144926011yescharacteristic
521C2#2145760811yescharacteristic
522C2#2145762211yescharacteristic
subgroupG/H
centerC22#16840396
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#473C2
socleC22×C61D42
prime radicalC2×#14519797C22
hypercenterC2×#473D61
hypocenterC61#24798
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#473

Complements

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

NH ≅ G/Ntype
1. #21455470C2semidirect
4. #21454419C2semidirect
5. #21447182C2semidirect
6. #21447037C2semidirect
7. #21446681C2semidirect
8. #21446565C2semidirect
9. #21446190C2semidirect
10. #21445880C2semidirect
11. C2×C61×#473C2semidirect
255. C2×#473D61semidirect
308. C61#24798semidirect

Maximal subgroups

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

subgroupindex
#214554702
#214553732
C2×#168398772
#214544192
#214471822
#214470372
#214466812
#214465652
#214461902
#214458802
C2×C61×#4732
#2479861