Finite Group Databasebeta

← #29835565

Subgroups — #29835565

Order 15616.

526 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#21456386C211yesmaximal, characteristic, complemented
3#21448095C211yesmaximal, characteristic, complemented
4#21448078C211yesmaximal, characteristic, complemented
5#21448051C211yesmaximal, characteristic, complemented
6#21447918C211yesmaximal, characteristic, complemented
7#21447914C211yesmaximal, characteristic, complemented
8#21447909C211yesmaximal, characteristic, complemented
9#21447904C211yesmaximal, characteristic
10#21447675C211yesmaximal, characteristic, complemented
11#21447674C211yesmaximal, characteristic, complemented
12#21447668C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14C4×C16×D61C211yesmaximal, characteristic, complemented
15C61×#2242C211yesmaximal, characteristic, complemented
16#16840297—21no—
17#16840291—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic
23C2×C16×D61C2211yescharacteristic
24#16840164C2211yescharacteristic
25#16840146C2211yescharacteristic
26#16840119—21no—
27#16840116—21no—
28#16840004C2211yescharacteristic
29#16840000C2211yescharacteristic
30#16839999C2211yescharacteristic
31#16839995C2211yescharacteristic
32#16839992—21no—
33#16839988—21no—
34Q16×Dic61C2211yescharacteristic
35#16839856C2211yescharacteristic
36SD16×Dic61—21no—
37#16839849—21no—
38D8×Dic61C2211yescharacteristic
39#16839844C2211yescharacteristic
40C4×C8×D61C2211yescharacteristic
41#16839589C2211yescharacteristic
42#16839589C2211yescharacteristic
43#16839587C2211yescharacteristic
44#16839587C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839564C2211yescharacteristic
49C4×#11758185C2211yescharacteristic
50C16×Dic61C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C61×#495C2211yescharacteristic
54C61×#358C2211yescharacteristic
55C61×#358C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C4×C16×C61C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758190—21no—
61C2×D488C2311yescharacteristic
62C2×C8×D61D411yescharacteristic
63C2×C8×D61C2311yescharacteristic
64#14519772—41no—
65#14519770—41no—
66#14519769—21no—
67#14519765—21no—
68#14519762—41no—
69#14519759—41no—
70#14519757—21no—
71#14519753—21no—
72#14519752—21no—
73#14519748—21no—
74Q8×Dic61—21no—
75#14519742—21no—
76D4×Dic61—21no—
77#14519739—21no—
78#14519739—41no—
79#14519737—41no—
80#14519735—41no—
81#14519733—41no—
82C42×D61D411yescharacteristic
83#14519723—22no—
84#14519721—22no—
85#14519720—22no—
86#14519718—22no—
87C2×#11758185C2311yescharacteristic
88C2×#11758185C2311yescharacteristic
89C16×D61—22no—
90C16×D61—22no—
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519709—21no—
94#14519701—21no—
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519697—21no—
101#14519697—21no—
102C4×#92418D411yescharacteristic
103C8×Dic61D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×Q16×C61C2311yescharacteristic
106C2×SD16×C61—21no—
107C2×D8×C61C2311yescharacteristic
108#125×C61—21no—
109#124×C61—21no—
110C2×C16×C61C2311yescharacteristic
111C2×C16×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C2×C22⋊D61—41no—
119C22×Dic61—41no—
120C2×D244—21no—
121C2×D244—41no—
122C2×C4×D61—41no—
123C2×C4×D61D811yescharacteristic
124C2×C4×D61D811yescharacteristic
125C2×C4×D61C2×D411yescharacteristic
126C2×Dic122—21no—
127Q16×C61—21no—
128Q16×C61—21no—
129SD16×C61—42no—
130D8×C61—21no—
131D8×C61—21no—
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2411yescharacteristic
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C42×C61C2×D411yescharacteristic
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143#11758198—41no—
144#11758198—41no—
145#11758198—41no—
146#11758198—41no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196—41no—
156#11758196—41no—
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61D811yescharacteristic
160C4×Dic61D811yescharacteristic
161C4×Dic61C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164Dic244—21no—
165Dic244—21no—
166D488—21no—
167D488—21no—
168#11758190—42no—
169C8×D61—22no—
170C8×D61—21no—
171C8×D61—21no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185#4112yes—
175#11758185#4112yes—
176C23×C61—41no—
177C22×D61—41no—
178C22×D61C2×D811yescharacteristic
179Q8×C61—41no—
180Q8×C61—41no—
181D4×C61—41no—
182D4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61C2×D811yescharacteristic
187C2×C4×C61C2×D811yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C22⋊D61—82no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—82no—
194C2×Dic61C2×D811yescharacteristic
195C2×Dic61C2×D811yescharacteristic
196C2×Dic61C2×D811yescharacteristic
197D244—41no—
198D244—41no—
199D244—82no—
200C4×D61—22no—
201C4×D61—21no—
202C4×D61—21no—
203C4×D61—22no—
204C4×D61—82no—
205Dic122—41no—
206Dic122—41no—
207C8×C61#13512yes—
208C8×C61C2×#4111yescharacteristic
209C8×C61C2×#4111yescharacteristic
210#92418—21no—
211#92418—21no—
212#33735—611nomaximal, Sylow
213C22×C61—81no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61C22×D811yescharacteristic
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61#50812yes—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—81no—
226C4×C61—81no—
227C4×C61—81no—
228C4×C61#53211yescharacteristic
229C4×C61C2×#13511yescharacteristic
230Dic61—21no—
231Dic61—21no—
232Dic61—81no—
233Dic61—81no—
234Dic61—81no—
235Dic61#50812yes—
236#3168—611no—
237#3168—611no—
238#2253—611no—
239#2249—611no—
240#2242—611no—
241#2242D6111yescharacteristic, complemented
242#2217—611no—
243#2217—611no—
244#2216—611no—
245#2215—611no—
246C4×Q32—611no—
247C4×SD32—611no—
248C4×SD32—611no—
249C4×D16—611no—
250C2×C4×C16—611no—
251C2×C61—161no—
252C2×C61#316911yescharacteristic
253C2×C61C2×#50811yescharacteristic
254C2×C61C2×#50811yescharacteristic
255D61—22no—
256D61—161no—
257#529—1221no—
258#529—1221no—
259C2×Q32—611no—
260C2×SD32—611no—
261C2×SD32—611no—
262C2×D16—611no—
263C22×C16—611no—
264C22×C16—611no—
265#499—611no—
266#499C2×D6111yescharacteristic
267#495—611no—
268#495C2×D6111yescharacteristic
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×D8—611no—
283C2×C4×C8—611no—
284#367—611no—
285#367—611no—
286#366—611no—
287#366—611no—
288#358—611no—
289#358—611no—
290#358C2×D6111yescharacteristic
291#358C2×D6111yescharacteristic
292#357—611no—
293#357—611no—
294#357C2×D6111yescharacteristic
295#357C2×D6111yescharacteristic
296C4×C16—611no—
297C4×C16—611no—
298C4×C16—611no—
299C4×C16C2×D6111yescharacteristic
300C61#3373511yescharacteristic, complemented, Sylow
301C2×Q16—611no—
302C2×Q16C22×D6111yescharacteristic
303C2×SD16—21no—
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—21no—
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—1221no—
324C4×D4—1221no—
325C4×D4—2441no—
326C4×D4—2441no—
327#117—2441no—
328#117—2441no—
329C2×C42—611no—
330Q32—1222no—
331SD32—1222no—
332SD32—1222no—
333D16—1222no—
334C2×C16—611no—
335C2×C16—611no—
336C2×C16—1222no—
337C2×C16—1222no—
338C2×C16C22×D6111yescharacteristic
339C2×C16C22×D6111yescharacteristic
340#107—611no—
341#107—611no—
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—1221no—
351#102—1221no—
352#102—1221no—
353#102—1221no—
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8C22×D6111yescharacteristic
358C2×Q8—21no—
359C2×Q8—1221no—
360C2×D4—21no—
361C2×D4—1221no—
362C2×D4—2441no—
363C2×D4—2441no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—2441no—
368C22×C4—2441no—
369Q16—21no—
370Q16—21no—
371Q16—1221no—
372Q16—1221no—
373SD16—42no—
374SD16—2442no—
375D8—21no—
376D8—21no—
377D8—1221no—
378D8—1221no—
379C2×C8—611no—
380C2×C8—611no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—1222no—
384C2×C8D4×D6111yescharacteristic
385C2×C8C23×D6111yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—1221no—
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—41no—
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—612no—
417C16—612no—
418C23—41no—
419C23—611no—
420C23—2441no—
421Q8—41no—
422Q8—41no—
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—611no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—4882no—
442C2×C4—4882no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4D8×D6111yescharacteristic
447C2×C4D8×D6111yescharacteristic
448C2×C4C2×D4×D6111yescharacteristic
449C8—1221no—
450C8—1221no—
451C8#1451979812yes—
452C8#41×D6111yescharacteristic
453C8#41×D6111yescharacteristic
454C22—81no—
455C22—81no—
456C22—81no—
457C22—612no—
458C22—1221no—
459C22—1221no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22C2×D8×D6111yescharacteristic
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—612no—
470C4—1221no—
471C4—1221no—
472C4—4881no—
473C4—4881no—
474C4—4881no—
475C4#1684031111yescharacteristic
476C4C2×#1451979811yescharacteristic
477C2—161no—
478C2—1222no—
479C2—9761no—
480C2#2145639711yescharacteristic
481C2#508×D6111yescharacteristic
482C2#508×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2242C2
socleC22×C61C22×D8
prime radical#21447904C2
hypercenter#2242D61
hypocenterC61#33735
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2242

Complements

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

NH ≅ G/Ntype
1. #21456391C2semidirect
2. #21456386C2semidirect
3. #21448095C2semidirect
4. #21448078C2semidirect
5. #21448051C2semidirect
6. #21447918C2semidirect
7. #21447914C2semidirect
8. #21447909C2semidirect
10. #21447675C2semidirect
11. #21447674C2semidirect
12. #21447668C2semidirect
13. #21447663C2semidirect
14. C4×C16×D61C2semidirect
15. C61×#2242C2semidirect
241. #2242D61semidirect
300. C61#33735semidirect

Maximal subgroups

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

subgroupindex
#214563912
#214563862
#214480952
#214480782
#214480512
#214479182
#214479142
#214479092
#214479042
#214476752
#214476742
#214476682
#214476632
C4×C16×D612
C61×#22422
#3373561