Finite Group Databasebeta

← #29722472

Subgroups — #29722472

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×#16840034C211yesmaximal, characteristic, complemented
2#21455365C212yesmaximal
3C2×#16839877C211yesmaximal, characteristic
4#21454416C211yesmaximal, characteristic, complemented
5#21447186C212yesmaximal, complemented
6#21447033C212yesmaximal, complemented
7#21446681C211yesmaximal, characteristic, complemented
8#21446568C211yesmaximal, characteristic, complemented
9#21446185C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C61×#3059C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519797C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840242C2212yes—
16#16840034—24no—
17#16839981—22no—
18#16839974—22no—
19#16839969—22no—
20#16839964—22no—
21#16839877—24no—
22C2×#14519713C2211yescharacteristic
23C2×#14519712C2211yescharacteristic
24#16839805C2212yes—
25#16839801—22no—
26#16839799—21no—
27C2×#14519711C2211yescharacteristic
28#107×D61—21no—
29#16839789C2212yes—
30C2×#14519704C2211yescharacteristic
31C2×#14519703C2211yescharacteristic
32#16839763C2212yes—
33C2×#14519701C2212yes—
34C2×#14519696C2212yes—
35C2×C8×Dic61C2211yescharacteristic
36#16839669C2212yes—
37#16839530C2212yes—
38#16839521C2212yes—
39C22×Q16×C61C2211yescharacteristic
40C2×#131×C61C2211yescharacteristic
41C61×#536C2212yes—
42C61×#470—22no—
43C61×#467—21no—
44C61×#467—21no—
45C61×#427C2211yescharacteristic
46C61×#419C2212yes—
47C2×#11758222C2311yescharacteristic
48#14519808—24no—
49#14519808—24no—
50#14519797—22no—
51#14519797—22no—
52C2×#11758202—22no—
53C2×#11758202D411yescharacteristic
54C2×#11758200D411yescharacteristic
55C2×#11758201—22no—
56C2×#11758201D411yescharacteristic
57C2×#11758199D411yescharacteristic
58#14519783—22no—
59#14519783D414yes—
60C2×#11758194C2312yes—
61C2×#11758194C2311yescharacteristic
62C2×#11758189—21no—
63C22×#92418C2311yescharacteristic
64C2×C8×D61—21no—
65C4×Dic122—22no—
66C4×D244—22no—
67#14519737—42no—
68#14519735—22no—
69C2×C4⋊Dic61C2312yes—
70C4⋊C4×D61—22no—
71C2×C4×Dic61C2311yescharacteristic
72#14519713—22no—
73#14519712—22no—
74#14519711—21no—
75#14519711—21no—
76#14519710—22no—
77#14519708—22no—
78C4⋊#92418—22no—
79C4⋊#92418—22no—
80#14519704—22no—
81#14519703—22no—
82#14519701—22no—
83#14519701—22no—
84#14519701—22no—
85#14519698—22no—
86#14519696—24no—
87C8×Dic61—22no—
88#14519684—42no—
89C2×#41×C61C2311yescharacteristic
90C22×Q8×C61C2312yes—
91C2×Q16×C61—22no—
92C2×Q16×C61—21no—
93C2×Q16×C61—21no—
94C2×Q16×C61—22no—
95#131×C61—22no—
96#131×C61—22no—
97C2×M16×C61—21no—
98C2×M16×C61C2311yescharacteristic
99C22×C8×C61—21no—
100C22×C8×C61C2311yescharacteristic
101C4⋊Q8×C61—22no—
102#124×C61—22no—
103#123×C61—42no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106#117×C61C2312yes—
107#107×C61D411yescharacteristic
108#107×C61D411yescharacteristic
109#106×C61D412yes—
110#103×C61—22no—
111#103×C61—22no—
112#41×C61—21no—
113#41×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
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—
131#11758222—21no—
132#11758222—21no—
133#11758222C2×D412yes—
134#11758222#4112yes—
135C2×D244C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—42no—
138C2×Dic122C2×D411yescharacteristic
139Q16×C61—42no—
140Q16×C61—42no—
141M16×C61—21no—
142M16×C61—21no—
143M16×C61—42no—
144C2×C8×C61—21no—
145C2×C8×C61—42no—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61#4112yes—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C42×C61—22no—
158#11758202—22no—
159#11758202—44no—
160#11758201—22no—
161#11758201—44no—
162#11758200—22no—
163#11758199—22no—
164#11758198—42no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167#11758196—42no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—42no—
171#11758194—21no—
172#11758194—22no—
173#11758194—22no—
174#11758194C2×D412yes—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418C2×D411yescharacteristic
180C2×#92418C2×D411yescharacteristic
181#11758189—24no—
182C8×D61—24no—
183C23×C61—21no—
184C23×C61H(5,𝔽2)11yescharacteristic
185C22×D61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191Q8×C61—42no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C22⋊D61—22no—
211C22⋊D61—22no—
212C2×Dic61—21no—
213C2×Dic61—22no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—82no—
218D244—21no—
219D244#12112yes—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—162no—
223Dic122—21no—
224Dic122#12112yes—
225C8×C61—41no—
226C8×C61#12112yes—
227C8×C61#12112yes—
228#92418—22no—
229#92418—21no—
230#92418—21no—
231#92418—42no—
232#24798—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61#53711yescharacteristic
239C22×C61D4211yescharacteristic
240C22×C61#55311yescharacteristic
241C2×D61—22no—
242C2×D61—41no—
243C2×D61—41no—
244C4×C61—41no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61C2×#12111yescharacteristic
249C4×C61C2×#12111yescharacteristic
250C4×C61#54711yescharacteristic
251C4×C61#54811yescharacteristic
252Dic61—22no—
253Dic61—42no—
254Dic61—82no—
255C2×#473—611no—
256#3059D6111yescharacteristic, complemented
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—81no—
269C2×C61#241611yescharacteristic
270C2×C61#329011yescharacteristic
271C2×C61#329311yescharacteristic
272D61—42no—
273C2×#137—612no—
274C2×#135—611no—
275C22×Q16C2×D6111yescharacteristic
276C2×#131—611no—
277C2×#131C2×D6111yescharacteristic
278#536C2×D6112yes—
279#520—612no—
280#473—1224no—
281#470—22no—
282#467—21no—
283#467—21no—
284#464—1222no—
285#463—1222no—
286#462—1222no—
287#461—1222no—
288#443—1224no—
289C2×#108—611no—
290C2×#108—611no—
291#428—612no—
292#428—1222no—
293#427—1221no—
294#427C2×D6111yescharacteristic
295C2×#107—611no—
296C2×#107—1221no—
297#424—612no—
298C2×#104—611no—
299C2×#104—611no—
300#419C2×D6112yes—
301#417—612no—
302C2×#103—612no—
303C2×#101—612no—
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—1224no—
313#137—1224no—
314#135—1222no—
315#135—1222no—
316C2×Q16—22no—
317C2×Q16—22no—
318C2×Q16—21no—
319C2×Q16—21no—
320C2×Q16—611no—
321C2×Q16—1222no—
322C2×SD16—611no—
323C2×SD16—611no—
324C2×SD16—1222no—
325C2×D8—611no—
326#131—22no—
327#131—22no—
328#131—614no—
329#131—1222no—
330C2×M16—21no—
331C2×M16—611no—
332C2×M16—612no—
333C2×M16—1221no—
334C2×M16C22×D6111yescharacteristic
335C22×C8—21no—
336C22×C8—611no—
337C22×C8—1221no—
338C22×C8C22×D6111yescharacteristic
339C4⋊Q8—22no—
340#124—22no—
341#123—42no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C4×Q8—1222no—
345C4×D4—1222no—
346C4×D4—2442no—
347#117—1222no—
348#117C22×D6112yes—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1222no—
351C2×C42—611no—
352#108—1222no—
353#108—1222no—
354#107—1221no—
355#107—1221no—
356#107—1222no—
357#107C22⋊D6111yescharacteristic
358#107C22⋊D6111yescharacteristic
359#106—1222no—
360#106C22⋊D6112yes—
361C4⋊C8—1222no—
362C4⋊C8—1222no—
363#104—1222no—
364#104—1222no—
365#103—22no—
366#103—22no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#102—1222no—
371#101—1224no—
372C4×C8—1222no—
373#95—2442no—
374#41—21no—
375#41—21no—
376#41—21no—
377#41—21no—
378#41—612no—
379#41—612no—
380#41—1221no—
381#41—1221no—
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—42no—
402Q16—42no—
403Q16—1222no—
404Q16—2444no—
405SD16—1222no—
406SD16—1222no—
407SD16—2444no—
408D8—1222no—
409M16—21no—
410M16—21no—
411M16—42no—
412M16—612no—
413M16—1221no—
414M16—1224no—
415M16—1222no—
416M16—1222no—
417C2×C8—21no—
418C2×C8—42no—
419C2×C8—611no—
420C2×C8—611no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1224no—
425C2×C8—1222no—
426C2×C8D4×D6112yes—
427C2×C8#1175822412yes—
428C2×C8C2×C22⋊D6111yescharacteristic
429C2×C8C2×C22⋊D6111yescharacteristic
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—2442no—
435C4⋊C4C2×C22⋊D6112yes—
436C4⋊C4C2×C22⋊D6112yes—
437C22⋊C4—22no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—2442no—
441C42—22no—
442C42—1221no—
443C42—1221no—
444C42—2442no—
445C23—21no—
446C23—1221no—
447C23#1451982211yescharacteristic
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—612no—
455Q8—1221no—
456D4—21no—
457D4—21no—
458D4—41no—
459D4—41no—
460D4—612no—
461D4—1222no—
462D4—1222no—
463D4—1221no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—1221no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—9762no—
477C2×C4—4882no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C2×C4#41×D6111yescharacteristic
485C2×C4C2×#1175822411yescharacteristic
486C2×C4#1451982411yescharacteristic
487C8—41no—
488C8—1221no—
489C8—1221no—
490C8—1222no—
491C8—2442no—
492C8#1451974812yes—
493C8#1451974812yes—
494C22—21no—
495C22—21no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—1222no—
500C22—2441no—
501C22—2441no—
502C22#1684034111yescharacteristic
503C22D4×C22⋊D6111yescharacteristic
504C22#1684045911yescharacteristic
505C4—41no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—1222no—
510C4—4882no—
511C4—2442no—
512C4#121×D6111yescharacteristic
513C4C2×#1451974811yescharacteristic
514C4#1684041211yescharacteristic
515C4#1684041911yescharacteristic
516C2—21no—
517C2—21no—
518C2—81no—
519C2—2442no—
520C2#2144925511yescharacteristic
521C2#2145760511yescharacteristic
522C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3059C2
socleC22×C61D42
prime radicalC2×#14519797C22
hypercenter#3059D61
hypocenterC61#24798
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3059

Complements

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

NH ≅ G/Ntype
1. C2×#16840034C2semidirect
4. #21454416C2semidirect
5. #21447186C2semidirect
6. #21447033C2semidirect
7. #21446681C2semidirect
8. #21446568C2semidirect
9. #21446185C2semidirect
10. #21445883C2semidirect
11. C61×#3059C2semidirect
256. #3059D61semidirect
308. C61#24798semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214553652
C2×#168398772
#214544162
#214471862
#214470332
#214466812
#214465682
#214461852
#214458832
C61×#30592
#2479861