Finite Group Databasebeta

← #29722470

Subgroups — #29722470

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#21455475C211yesmaximal, characteristic, complemented
3#21455369C212yesmaximal
4C2×#16839877C211yesmaximal, characteristic
5#21447183C212yesmaximal, complemented
6#21447038C212yesmaximal, complemented
7#21446678C211yesmaximal, characteristic, complemented
8#21446568C211yesmaximal, characteristic, complemented
9#21446188C212yesmaximal, complemented
10#21445884C211yesmaximal, characteristic, complemented
11C61×#2913C211yesmaximal, characteristic, complemented
12C2×#14519807C2212yes—
13C2×#14519797C2211yescharacteristic
14C22×Dic244C2211yescharacteristic
15C2×#14519783C2211yescharacteristic
16#16840338C2212yes—
17#16840037—24no—
18#16840021—22no—
19#16840004—21no—
20#16840004—21no—
21#16839984—22no—
22#16839977—22no—
23#16839972—22no—
24#16839967—22no—
25#16839877—24no—
26C2×#14519713C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839789C2212yes—
29C2×#14519704C2211yescharacteristic
30C2×#14519703C2211yescharacteristic
31#16839770C2212yes—
32#16839761C2212yes—
33C2×#14519700C2212yes—
34C2×#14519695C2212yes—
35C2×C8×Dic61C2211yescharacteristic
36#16839667C2212yes—
37#16839531C2212yes—
38#16839521C2212yes—
39C2×#131×C61C2211yescharacteristic
40C61×#520C2212yes—
41C61×#428—22no—
42C61×#428C2212yes—
43C61×#427—21no—
44C61×#427C2211yescharacteristic
45C2×#107×C61—21no—
46C2×#107×C61C2211yescharacteristic
47C2×#11758222C2311yescharacteristic
48C22×Dic122C2312yes—
49#14519807—24no—
50#14519807—24no—
51#14519797—22no—
52#14519797—22no—
53C2×Dic244—21no—
54C2×Dic244—21no—
55C2×Dic244—22no—
56C2×Dic244—22no—
57C2×#11758202—22no—
58C2×#11758202D411yescharacteristic
59C2×#11758200—22no—
60C2×#11758200D411yescharacteristic
61C2×#11758201D411yescharacteristic
62C2×#11758199D411yescharacteristic
63#14519783—22no—
64#14519783—22no—
65C2×#11758194C2312yes—
66C2×#11758194C2311yescharacteristic
67C2×#11758189—21no—
68C22×#92418C2311yescharacteristic
69C2×C8×D61—21no—
70#14519777—22no—
71#14519765—22no—
72#14519761—42no—
73#14519757—22no—
74#14519757—22no—
75C2×C4×Dic61C2311yescharacteristic
76#14519713—22no—
77#14519712—22no—
78C4⋊#92418—22no—
79C4⋊#92418—22no—
80#14519704—22no—
81#14519703—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85#14519700—22no—
86#14519700—22no—
87#14519697—22no—
88#14519695—24no—
89C8×Dic61—22no—
90#14519684—42no—
91C2×#41×C61C2311yescharacteristic
92#131×C61—22no—
93#131×C61D414yes—
94C2×M16×C61—21no—
95C2×M16×C61C2311yescharacteristic
96C22×C8×C61—21no—
97C22×C8×C61C2311yescharacteristic
98C4×Q8×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—42no—
101#117×C61—22no—
102#117×C61C2312yes—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61C2312yes—
105#107×C61—21no—
106#107×C61—21no—
107#107×C61—22no—
108#107×C61D411yescharacteristic
109#107×C61D411yescharacteristic
110#106×C61—22no—
111#106×C61D412yes—
112#41×C61—21no—
113#41×C61—21no—
114#41×C61C2×D412yes—
115#41×C61#4112yes—
116C2×Q8×C61C2×D411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C2×D411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126#11758222—21no—
127#11758222—21no—
128#11758222—21no—
129#11758222—21no—
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122C2×D411yescharacteristic
139M16×C61—21no—
140M16×C61—24no—
141M16×C61C2×D412yes—
142C2×C8×C61—21no—
143C2×C8×C61—24no—
144C2×C8×C61C2×D412yes—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61#4112yes—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—42no—
155C42×C61—22no—
156C42×C61—42no—
157#11758202—22no—
158#11758202—44no—
159#11758201—22no—
160#11758200—22no—
161#11758200—44no—
162#11758199—22no—
163#11758198—42no—
164#11758198—42no—
165C4⋊Dic61—42no—
166C4⋊Dic61—42no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169#11758194—22no—
170#11758194—21no—
171#11758194—21no—
172#11758194—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418C2×D411yescharacteristic
178C2×#92418C2×D411yescharacteristic
179Dic244—42no—
180Dic244—42no—
181#11758189—42no—
182C8×D61—42no—
183C23×C61—21no—
184C23×C61H(5,𝔽2)11yescharacteristic
185C22×D61—21no—
186Q8×C61—21no—
187Q8×C61#12112yes—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—21no—
191D4×C61#12112yes—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61—82no—
199C2×C4×C61—162no—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C22⋊D61—41no—
207C22⋊D61—41no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215D244—21no—
216D244—21no—
217C4×D61—41no—
218C4×D61—41no—
219Dic122—21no—
220Dic122—21no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—42no—
225C8×C61—22no—
226C8×C61#12112yes—
227C8×C61#12112yes—
228#92418—21no—
229#92418—21no—
230#92418—41no—
231#92418—42no—
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—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61C2×#12111yescharacteristic
249C4×C61C2×#12111yescharacteristic
250C4×C61#54711yescharacteristic
251C4×C61#54811yescharacteristic
252Dic61—42no—
253Dic61—41no—
254Dic61—82no—
255C2×#473—611no—
256#3059—611no—
257#3046—612no—
258C2×#443—611no—
259#2913D6111yescharacteristic, complemented
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×#137—612no—
274C2×#135—611no—
275C22×Q16—611no—
276C2×#131—611no—
277C2×#131C2×D6111yescharacteristic
278#536—612no—
279#520C2×D6112yes—
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—1224no—
289C2×#108—611no—
290C2×#108—611no—
291#428—22no—
292#428C2×D6112yes—
293#427—21no—
294#427C2×D6111yescharacteristic
295C2×#107—21no—
296C2×#107C2×D6111yescharacteristic
297#424—612no—
298C2×#104—611no—
299C2×#104—611no—
300#419—612no—
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×Q8—612no—
312#137—1224no—
313#137—1224no—
314#135—1222no—
315#135—1222no—
316C2×Q16—611no—
317C2×Q16—1222no—
318C2×Q16—1222no—
319C2×Q16—1222no—
320C2×Q16—1221no—
321C2×Q16—1221no—
322C2×SD16—611no—
323C2×SD16—611no—
324C2×SD16—1222no—
325C2×D8—611no—
326#131—22no—
327#131—1222no—
328#131—1222no—
329#131D24414yes—
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—1222no—
340#124—1222no—
341#123—2442no—
342C22⋊Q8—1222no—
343C22⋊Q8—1222no—
344C4×Q8—22no—
345C4×D4—22no—
346C4×D4—42no—
347#117—22no—
348#117C22×D6112yes—
349C2×C4⋊C4—22no—
350C2×C4⋊C4C22×D6112yes—
351C2×C42—611no—
352#108—1222no—
353#108—1222no—
354#107—21no—
355#107—21no—
356#107—22no—
357#107C22⋊D6111yescharacteristic
358#107C22⋊D6111yescharacteristic
359#106—22no—
360#106C22⋊D6112yes—
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—1224no—
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#41C2×D24412yes—
381#41#1175822612yes—
382C2×Q8—611no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8D4×D6111yescharacteristic
390C2×D4—21no—
391C2×D4—611no—
392C2×D4—1221no—
393C2×D4D4×D6111yescharacteristic
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4C23×D6111yescharacteristic
401Q16—1222no—
402Q16—2444no—
403Q16—2442no—
404Q16—2442no—
405SD16—1222no—
406SD16—1222no—
407SD16—2444no—
408D8—1222no—
409M16—21no—
410M16—24no—
411M16—1222no—
412M16—1221no—
413M16—1221no—
414M16—1222no—
415M16—2442no—
416M16C2×D24412yes—
417C2×C8—21no—
418C2×C8—24no—
419C2×C8—611no—
420C2×C8—611no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1222no—
424C2×C8—1221no—
425C2×C8—2442no—
426C2×C8C2×D24412yes—
427C2×C8#1175822212yes—
428C2×C8C2×C22⋊D6111yescharacteristic
429C2×C8C2×C22⋊D6111yescharacteristic
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—2442no—
434C4⋊C4—2442no—
435C4⋊C4C2×C22⋊D6112yes—
436C4⋊C4C2×C22⋊D6112yes—
437C22⋊C4—22no—
438C22⋊C4—42no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C42—22no—
442C42—42no—
443C42—1221no—
444C42—1221no—
445C23—21no—
446C23—1221no—
447C23#1451982211yescharacteristic
448Q8—21no—
449Q8—1221no—
450Q8—1221no—
451Q8—2442no—
452Q8—2442no—
453Q8—2442no—
454Q8—2442no—
455Q8#1451975212yes—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—1221no—
460D4—1221no—
461D4—2441no—
462D4—2441no—
463D4#1451975212yes—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—162no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4C22×D24411yescharacteristic
484C2×C4C2×#1175822211yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982511yescharacteristic
487C8—22no—
488C8—1221no—
489C8—1221no—
490C8—2441no—
491C8—2442no—
492C8#1451975112yes—
493C8#1451975112yes—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—2441no—
500C22—2441no—
501C22—2441no—
502C22#1684034311yescharacteristic
503C22D4×C22⋊D6111yescharacteristic
504C22#1684046011yescharacteristic
505C4—22no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—4882no—
510C4—2442no—
511C4—2441no—
512C4C2×#1451975111yescharacteristic
513C4C2×#1451975211yescharacteristic
514C4#1684041311yescharacteristic
515C4#1684042311yescharacteristic
516C2—21no—
517C2—21no—
518C2—42no—
519C2—4881no—
520C2#2144925311yescharacteristic
521C2#2145760111yescharacteristic
522C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2913C2
socleC22×C61D42
prime radicalC2×#14519797C22
hypercenter#2913D61
hypocenterC61#24798
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
2. #21455475C2semidirect
5. #21447183C2semidirect
6. #21447038C2semidirect
7. #21446678C2semidirect
8. #21446568C2semidirect
9. #21446188C2semidirect
10. #21445884C2semidirect
11. C61×#2913C2semidirect
259. #2913D61semidirect
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
#214554752
#214553692
C2×#168398772
#214471832
#214470382
#214466782
#214465682
#214461882
#214458842
C61×#29132
#2479861