Finite Group Databasebeta

← #29836391

Subgroups — #29836391

Order 15616.

496 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456391C211yesmaximal, characteristic, complemented
2#21456387C211yesmaximal, characteristic, complemented
3#21447982C211yesmaximal, characteristic, complemented
4#21447980C211yesmaximal, characteristic, complemented
5#21447975C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447931C211yesmaximal, characteristic, complemented
8#21447928C211yesmaximal, characteristic, complemented
9#21447921C211yesmaximal, characteristic
10#21447882C211yesmaximal, characteristic, complemented
11#21447880C211yesmaximal, characteristic, complemented
12#21447874C211yesmaximal, characteristic, complemented
13#21447873C211yesmaximal, characteristic, complemented
14#21447573C211yesmaximal, characteristic, complemented
15C61×#2228C211yesmaximal, characteristic, complemented
16#16840297—21no—
17#16840294—21no—
18C2×Dic488C2211yescharacteristic
19C2×#14519722C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519716C2211yescharacteristic
23C2×#14519716C2211yescharacteristic
24#16840163C2211yescharacteristic
25#16840146C2211yescharacteristic
26#16840145C2211yescharacteristic
27#16840119—21no—
28#16840115—21no—
29#16840004C2211yescharacteristic
30#16840004C2211yescharacteristic
31#16839999C2211yescharacteristic
32#16839995C2211yescharacteristic
33#16839992—21no—
34#16839990—21no—
35C4×Dic244C2211yescharacteristic
36#16839856C2211yescharacteristic
37#16839849—21no—
38C4×#11758200—21no—
39#16839844C2211yescharacteristic
40C4×C8×D61C2211yescharacteristic
41#16839589C2211yescharacteristic
42#16839587C2211yescharacteristic
43C4⋊#11758185C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839565C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839562C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C4×D8×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×Dic244C2311yescharacteristic
61C2×#11758190—21no—
62C2×#11758200—21no—
63C2×D488C2311yescharacteristic
64C2×C8×D61D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519772—41no—
67#14519770—41no—
68#14519768—21no—
69#14519765—21no—
70#14519764—21no—
71#14519762—41no—
72#14519759—41no—
73#14519757—21no—
74#14519757—21no—
75#14519752—21no—
76#14519748—21no—
77C4×Dic122—21no—
78#14519742—21no—
79#14519739—21no—
80#14519739—41no—
81C4×C22⋊D61—41no—
82#14519736—41no—
83#14519735—41no—
84C42×D61D411yescharacteristic
85Dic488—41no—
86#14519722—41no—
87#14519721—41no—
88#14519718—41no—
89#14519716—41no—
90#14519716—41no—
91C2×#11758185C2311yescharacteristic
92C2×#11758185C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519708—21no—
95#14519702—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519698—21no—
100#14519697—21no—
101#14519697—21no—
102C4×#92418D411yescharacteristic
103C8×Dic61D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106#125×C61—21no—
107C4×D4×C61—21no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#106×C61—21no—
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×D4×C61—21no—
117C22×C4×C61—41no—
118C2×C22⋊D61—41no—
119C2×D244—21no—
120C2×D244—41no—
121C2×C4×D61—41no—
122C2×C4×D61D811yescharacteristic
123C2×C4×D61D811yescharacteristic
124C2×C4×D61C2×D411yescharacteristic
125C2×Dic122—21no—
126C2×Dic122—21no—
127D8×C61—21no—
128D8×C61—21no—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2411yescharacteristic
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61C2×D411yescharacteristic
138C22⋊Dic61—41no—
139C22⋊Dic61—41no—
140C22⋊Dic61—41no—
141#11758200—42no—
142#11758198—41no—
143#11758198—41no—
144#11758198—41no—
145#11758198—41no—
146#11758198—41no—
147#11758198—41no—
148C4⋊Dic61—21no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196—41no—
155C4×Dic61—41no—
156C4×Dic61—41no—
157C4×Dic61D811yescharacteristic
158C4×Dic61D811yescharacteristic
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162Dic244—21no—
163Dic244—21no—
164Dic244—21no—
165Dic244—21no—
166D488—21no—
167D488—21no—
168#11758190—42no—
169C8×D61—21no—
170C8×D61—21no—
171C8×D61—41no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185—21no—
175#11758185—21no—
176C23×C61—41no—
177C22×D61—41no—
178C22×D61C2×D811yescharacteristic
179D4×C61—41no—
180D4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—82no—
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61C2×D811yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C22⋊D61—82no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61C2×D811yescharacteristic
193C2×Dic61C2×D811yescharacteristic
194C2×Dic61C2×D811yescharacteristic
195D244—41no—
196D244—41no—
197D244—82no—
198C4×D61—21no—
199C4×D61—21no—
200C4×D61—41no—
201C4×D61—41no—
202C4×D61—82no—
203Dic122—41no—
204Dic122—41no—
205Dic122—41no—
206Dic122—41no—
207C8×C61—21no—
208C8×C61H(5,𝔽2)11yescharacteristic
209C8×C612−1+411yescharacteristic
210#92418—21no—
211#92418—21no—
212#33822—611nomaximal, Sylow
213C22×C61—81no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61C22×D811yescharacteristic
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—21no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—81no—
226C4×C61—81no—
227C4×C61—81no—
228C4×C61#53711yescharacteristic
229C4×C61#57511yescharacteristic
230Dic61—21no—
231Dic61—21no—
232Dic61—21no—
233Dic61—81no—
234Dic61—81no—
235Dic61—81no—
236#3168—611no—
237#3168—611no—
238#2229—611no—
239#2229—611no—
240#2228—611no—
241#2228D6111yescharacteristic, complemented
242#2220—611no—
243#2220—611no—
244#2219—611no—
245#2218—611no—
246#2211—611no—
247#2211—611no—
248#2210—611no—
249#2210—611no—
250#2150—611no—
251C2×C61—161no—
252C2×C61#324511yescharacteristic
253C2×C61#341411yescharacteristic
254C2×C61#341411yescharacteristic
255D61—41no—
256D61—161no—
257#529—1221no—
258#529—1221no—
259C2×Q32—611no—
260C2×SD32—611no—
261C2×SD32—611no—
262C2×D16—611no—
263C2×M32—611no—
264C2×M32—611no—
265#499—611no—
266#499C2×D6111yescharacteristic
267#495—611no—
268#495—611no—
269#488—1221no—
270#488—1221no—
271#467—611no—
272#467—611no—
273#466—611no—
274#466—611no—
275#465—1221no—
276#465—1221no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×SD16—1221no—
280C4×SD16—1221no—
281C4×D8—611no—
282C4×D8C2×D6111yescharacteristic
283C2×C4×C8—611no—
284#367—611no—
285#367C2×D6111yescharacteristic
286#366—611no—
287#366C2×D6111yescharacteristic
288C4⋊C16—611no—
289C4⋊C16—611no—
290C4⋊C16—611no—
291C4⋊C16C2×D6111yescharacteristic
292#358—611no—
293#358—611no—
294#358—611no—
295#358—611no—
296#357—611no—
297#357—611no—
298#357C2×D6111yescharacteristic
299#357C2×D6111yescharacteristic
300C61#3382211yescharacteristic, complemented, Sylow
301C2×Q16—611no—
302C2×Q16—611no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—611no—
306C2×D8C22×D6111yescharacteristic
307C22×C8—611no—
308C22×C8—611no—
309#126—2441no—
310#126—2441no—
311#125—21no—
312#125—1221no—
313#124—1221no—
314#124—1221no—
315#123—2441no—
316#123—2441no—
317C22⋊Q8—1221no—
318C22⋊Q8—1221no—
319#121—1221no—
320#121—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×D4—21no—
324C4×D4—1221no—
325C4×D4—2441no—
326C4×D4—2441no—
327#117—2441no—
328#117—2441no—
329C2×C42—611no—
330Q32—2441no—
331SD32—2441no—
332SD32—2441no—
333D16—2441no—
334M32—2441no—
335M32—2441no—
336C2×C16—611no—
337C2×C16—611no—
338C2×C16C22×D6111yescharacteristic
339C2×C16C22×D6111yescharacteristic
340#107—611no—
341#107C22×D6111yescharacteristic
342#107C22×D6111yescharacteristic
343#107C22×D6111yescharacteristic
344#106—21no—
345#106—1221no—
346#103—1221no—
347#103—1221no—
348#103—1221no—
349#103—1221no—
350#102—21no—
351#102—1221no—
352#102—1221no—
353#102—1221no—
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8C22×D6111yescharacteristic
358C2×Q8—1221no—
359C2×Q8—1221no—
360C2×D4—21no—
361C2×D4—1221no—
362C2×D4—2441no—
363C2×D4—2441no—
364C22×C4—41no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—611no—
368C22×C4—2441no—
369Q16—1221no—
370Q16—1221no—
371Q16—1221no—
372Q16—1221no—
373SD16—2442no—
374SD16—2442no—
375D8—21no—
376D8—21no—
377D8—1221no—
378D8—1221no—
379C2×C8—611no—
380C2×C8—611no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—2441no—
384C2×C8D4×D6111yescharacteristic
385C2×C8C23×D6111yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—1221no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C22⋊C4—41no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C22⋊C4—2441no—
407C22⋊C4—2441no—
408C42—611no—
409C42—611no—
410C42—611no—
411C42—2441no—
412C42—2441no—
413C42D4×D6111yescharacteristic
414C16—21no—
415C16—21no—
416C16—1221no—
417C16—1221no—
418C23—41no—
419C23—611no—
420C23—2441no—
421Q8—2441no—
422Q8—2441no—
423Q8—2441no—
424Q8—2441no—
425D4—41no—
426D4—41no—
427D4—4882no—
428D4—4882no—
429D4—2441no—
430D4—2441no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—4882no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4D8×D6111yescharacteristic
447C2×C4D8×D6111yescharacteristic
448C2×C4C2×D4×D6111yescharacteristic
449C8—21no—
450C8—1221no—
451C8—1221no—
452C8#1451982311yescharacteristic
453C8#1451982511yescharacteristic
454C22—81no—
455C22—81no—
456C22—81no—
457C22—1221no—
458C22—1221no—
459C22—1221no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22C2×D8×D6111yescharacteristic
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—1221no—
470C4—1221no—
471C4—1221no—
472C4—4881no—
473C4—4881no—
474C4—4881no—
475C4#1684034211yescharacteristic
476C4#1684056111yescharacteristic
477C2—161no—
478C2—9761no—
479C2—2441no—
480C2#2145712811yescharacteristic
481C2#2145895111yescharacteristic
482C2#2145895211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2228C2
socleC22×C61C22×D8
prime radical#21447921C2
hypercenter#2228D61
hypocenterC61#33822
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #21456391C2semidirect
2. #21456387C2semidirect
3. #21447982C2semidirect
4. #21447980C2semidirect
5. #21447975C2semidirect
6. #21447933C2semidirect
7. #21447931C2semidirect
8. #21447928C2semidirect
10. #21447882C2semidirect
11. #21447880C2semidirect
12. #21447874C2semidirect
13. #21447873C2semidirect
14. #21447573C2semidirect
15. C61×#2228C2semidirect
241. #2228D61semidirect
300. C61#33822semidirect

Maximal subgroups

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

subgroupindex
#214563912
#214563872
#214479822
#214479802
#214479752
#214479332
#214479312
#214479282
#214479212
#214478822
#214478802
#214478742
#214478732
#214475732
C61×#22282
#3382261