Finite Group Databasebeta

← #29836451

Subgroups — #29836451

Order 15616.

488 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#21456461C211yesmaximal, characteristic, complemented
2#21456454C211yesmaximal, characteristic, complemented
3#21447970C211yesmaximal, characteristic, complemented
4#21447968C211yesmaximal, characteristic, complemented
5#21447955C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447931C211yesmaximal, characteristic, complemented
8#21447911C211yesmaximal, characteristic, complemented
9#21447904C211yesmaximal, characteristic
10#21447882C211yesmaximal, characteristic, complemented
11#21447880C211yesmaximal, characteristic, complemented
12#21447874C211yesmaximal, characteristic, complemented
13#21447873C211yesmaximal, characteristic, complemented
14#21447576C211yesmaximal, characteristic, complemented
15C61×#2224C211yesmaximal, characteristic, complemented
16#16840386—21no—
17#16840377—21no—
18C2×Dic488C2211yescharacteristic
19C2×#14519722C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519716C2211yescharacteristic
23C2×C16×D61C2211yescharacteristic
24#16840166C2211yescharacteristic
25#16840146C2211yescharacteristic
26#16840145C2211yescharacteristic
27#16840124—21no—
28#16840120—21no—
29#16840009—21no—
30#16840007—21no—
31#16840004C2211yescharacteristic
32#16840004C2211yescharacteristic
33#16839999C2211yescharacteristic
34#16839995C2211yescharacteristic
35#16839862—21no—
36#16839861—21no—
37C4×Dic244C2211yescharacteristic
38#16839856C2211yescharacteristic
39#16839844C2211yescharacteristic
40#16839716C2211yescharacteristic
41#16839587C2211yescharacteristic
42#16839587C2211yescharacteristic
43C4⋊#11758185C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839565C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839562C2211yescharacteristic
52C61×#500C2211yescharacteristic
53C4×D8×C61C2211yescharacteristic
54C61×#366C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×Dic244C2311yescharacteristic
61C2×#11758190—21no—
62C2×#11758200—21no—
63C2×D488C2311yescharacteristic
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519776—21no—
67#14519772—41no—
68#14519770—41no—
69#14519766—41no—
70#14519765—21no—
71#14519764—21no—
72#14519763—41no—
73#14519762—41no—
74#14519760—41no—
75#14519757—21no—
76#14519757—21no—
77#14519756—41no—
78#14519753—41no—
79#14519752—21no—
80#14519748—21no—
81C4×Dic122—21no—
82#14519742—21no—
83#14519739—21no—
84#14519734D411yescharacteristic
85Dic488—41no—
86#14519722—41no—
87#14519721—41no—
88#14519718—41no—
89#14519716—41no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711C2311yescharacteristic
94#14519710—21no—
95#14519702—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519698—21no—
100#14519697—21no—
101#14519697—21no—
102#14519688D411yescharacteristic
103#14519687D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106C4⋊Q8×C61—21no—
107C4×D4×C61—21no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61—21no—
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—41no—
117C2×D4×C61—21no—
118C22×C4×C61—41no—
119C2×C22⋊D61—41no—
120C2×D244—21no—
121C2×D244—41no—
122C2×C4×D61—41no—
123C2×C4×D61C2×D411yescharacteristic
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—41no—
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—
135C22⋊C4×C61—41no—
136C42×C61C2×D411yescharacteristic
137C22⋊Dic61—41no—
138C22⋊Dic61—41no—
139C22⋊Dic61—41no—
140#11758200—42no—
141#11758198—41no—
142#11758198—41no—
143#11758198—41no—
144#11758198—41no—
145#11758198—41no—
146#11758198—41no—
147#11758198D811yescharacteristic
148#11758198D811yescharacteristic
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196D811yescharacteristic
156#11758196D811yescharacteristic
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162Dic244—21no—
163Dic244—21no—
164Dic244—21no—
165Dic244—21no—
166D488—21no—
167D488—21no—
168#11758190—42no—
169#11758189—41no—
170C8×D61—21no—
171C8×D61—21no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185—21no—
175#11758185—21no—
176C23×C61—41no—
177C22×D61—41no—
178C22×D61C2×D811yescharacteristic
179Q8×C61—161no—
180D4×C61—41no—
181D4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—161no—
186C2×C4×C61C2×D811yescharacteristic
187C2×C4×C61C2×D811yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C22⋊D61—161no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61C2×D811yescharacteristic
194C2×Dic61C2×D811yescharacteristic
195C2×Dic61C2×D811yescharacteristic
196D244—41no—
197D244—41no—
198D244—161no—
199C4×D61—21no—
200C4×D61—21no—
201C4×D61—161no—
202Dic122—41no—
203Dic122—41no—
204Dic122—41no—
205Dic122—41no—
206Dic122—161no—
207C8×C61—21no—
208C8×C61H(5,𝔽2)11yescharacteristic
209C8×C612−1+411yescharacteristic
210#92418—21no—
211#92418—21no—
212#33828—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#3175—611no—
237#3175—611no—
238#2227—611no—
239#2227—611no—
240#2224—611no—
241#2224D6111yescharacteristic, complemented
242#2220—611no—
243#2220—611no—
244#2216—611no—
245#2215—611no—
246#2211—611no—
247#2211—611no—
248#2210—611no—
249#2210—611no—
250#2151—611no—
251C2×C61—161no—
252C2×C61#324511yescharacteristic
253C2×C61#341511yescharacteristic
254C2×C61#341711yescharacteristic
255D61—41no—
256D61—161no—
257#542—1221no—
258#542—1221no—
259C2×Q32—611no—
260C2×SD32—611no—
261C2×SD32—611no—
262C2×D16—611no—
263C2×M32—611no—
264C22×C16—611no—
265#500—611no—
266#500C2×D6111yescharacteristic
267#495—611no—
268#495—611no—
269#489—1221no—
270#489—1221no—
271#468—1221no—
272#468—1221no—
273#467—611no—
274#467—611no—
275#466—611no—
276#466—611no—
277#440—1221no—
278#440—1221no—
279C4×Q16—611no—
280C4×Q16—611no—
281C4×D8—611no—
282C4×D8C2×D6111yescharacteristic
283#405—611no—
284#366—611no—
285#366—611no—
286#366C2×D6111yescharacteristic
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#3382811yescharacteristic, complemented, Sylow
301C2×Q16—611no—
302C2×Q16—611no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—611no—
306C2×D8C22×D6111yescharacteristic
307C2×M16—611no—
308C22×C8—611no—
309C4⋊Q8—21no—
310C4⋊Q8—1221no—
311#126—2441no—
312#126—2441no—
313#124—1221no—
314#124—1221no—
315#124—2441no—
316#124—2441no—
317#123—2441no—
318#123—2441no—
319C22⋊Q8—1221no—
320C22⋊Q8—1221no—
321C22⋊Q8—2441no—
322C22⋊Q8—2441no—
323#121—1221no—
324#121—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—21no—
328C4×D4—1221no—
329#117—611no—
330Q32—2441no—
331SD32—2441no—
332SD32—2441no—
333D16—2441no—
334M32—2441no—
335C2×C16—611no—
336C2×C16—611no—
337C2×C16—2441no—
338C2×C16C22×D6111yescharacteristic
339C2×C16C22×D6111yescharacteristic
340#107—21no—
341#107—611no—
342#107—1221no—
343#107C22×D6111yescharacteristic
344#107C22×D6111yescharacteristic
345#107C22×D6111yescharacteristic
346#103—1221no—
347#103—1221no—
348#103—1221no—
349#103—1221no—
350#102—21no—
351#102—1221no—
352#102—1221no—
353#102—1221no—
354#97—611no—
355#97—611no—
356C4×C8—611no—
357C4×C8C22×D6111yescharacteristic
358C2×Q8—41no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×Q8—2441no—
362C2×D4—21no—
363C2×D4—1221no—
364C2×D4—2441no—
365C2×D4—2441no—
366C22×C4—41no—
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—
379M16—2441no—
380C2×C8—611no—
381C2×C8—611no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8D4×D6111yescharacteristic
385C2×C8C23×D6111yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—611no—
391C4⋊C4—611no—
392C4⋊C4—1221no—
393C4⋊C4—2441no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C22⋊C4—41no—
399C22⋊C4—611no—
400C22⋊C4—611no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C22⋊C4—2441no—
407C22⋊C4—2441no—
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C42—611no—
411C42—2441no—
412C42—2441no—
413C42D4×D6111yescharacteristic
414C16—21no—
415C16—21no—
416C16—1221no—
417C16—1221no—
418C23—41no—
419C23—611no—
420C23—2441no—
421Q8—161no—
422Q8—9761no—
423Q8—2441no—
424Q8—2441no—
425Q8—2441no—
426Q8—2441no—
427D4—41no—
428D4—41no—
429D4—9761no—
430D4—9761no—
431D4—2441no—
432D4—2441no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—161no—
437C2×C4—611no—
438C2×C4—611no—
439C2×C4—611no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—9761no—
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#2145895611yescharacteristic
482C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2224C2
socleC22×C61C22×D8
prime radical#21447904C2
hypercenter#2224D61
hypocenterC61#33828
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2224

Complements

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

NH ≅ G/Ntype
1. #21456461C2semidirect
2. #21456454C2semidirect
3. #21447970C2semidirect
4. #21447968C2semidirect
5. #21447955C2semidirect
6. #21447933C2semidirect
7. #21447931C2semidirect
8. #21447911C2semidirect
10. #21447882C2semidirect
11. #21447880C2semidirect
12. #21447874C2semidirect
13. #21447873C2semidirect
14. #21447576C2semidirect
15. C61×#2224C2semidirect
241. #2224D61semidirect
300. C61#33828semidirect

Maximal subgroups

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

subgroupindex
#214564612
#214564542
#214479702
#214479682
#214479552
#214479332
#214479312
#214479112
#214479042
#214478822
#214478802
#214478742
#214478732
#214475762
C61×#22242
#3382861