Finite Group Databasebeta

← #29834655

Subgroups — #29834655

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456331C211yesmaximal, characteristic, complemented
2#135×Dic61C211yesmaximal, characteristic
3#21447985C211yesmaximal, characteristic, complemented
4#21447981C211yesmaximal, characteristic, complemented
5#21447978C211yesmaximal, characteristic, complemented
6#21447974C211yesmaximal, characteristic, complemented
7#21447969C211yesmaximal, characteristic, complemented
8#21447964C211yesmaximal, characteristic, complemented
9#21447959C211yesmaximal, characteristic, complemented
10#21447954C211yesmaximal, characteristic, complemented
11#21447603C211yesmaximal, characteristic, complemented
12#21447601C211yesmaximal, characteristic, complemented
13C2×#16839580C211yesmaximal, characteristic, complemented
14#21447526C211yesmaximal, characteristic
15C61×#2138C211yesmaximal, characteristic
16#16840315—21no—
17#41×Dic61—21no—
18C22×#11758185C2211yescharacteristic
19#16840166C2211yescharacteristic
20#16840166C2211yescharacteristic
21#16840163C2211yescharacteristic
22#16840163C2211yescharacteristic
23#16840162—21no—
24#16840159—21no—
25Q16×Dic61C2211yescharacteristic
26Q16×Dic61C2211yescharacteristic
27SD16×Dic61—21no—
28SD16×Dic61—21no—
29D8×Dic61C2211yescharacteristic
30D8×Dic61C2211yescharacteristic
31#16839800C2211yescharacteristic
32#16839800C2211yescharacteristic
33#16839798—21no—
34#16839758—21no—
35C2×C8×Dic61C2211yescharacteristic
36#16839590C2211yescharacteristic
37#16839590C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839589C2211yescharacteristic
40#16839588C2211yescharacteristic
41#16839588C2211yescharacteristic
42#16839587C2211yescharacteristic
43#16839587C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839580C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839566C2211yescharacteristic
49#16839566C2211yescharacteristic
50#16839563C2211yescharacteristic
51#16839563C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C22×C16×C61C2211yescharacteristic
54C61×#427C2211yescharacteristic
55C61×#358C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C22×#92418D411yescharacteristic
60#14519776—21no—
61#14519776—21no—
62#14519768—21no—
63#14519768—21no—
64#14519754—41no—
65#14519754—41no—
66Q8×Dic61—21no—
67Q8×Dic61—21no—
68C4×Dic122—41no—
69C4×Dic122—41no—
70D4×Dic61—21no—
71D4×Dic61—21no—
72D4×Dic61—41no—
73D4×Dic61—41no—
74#14519736—21no—
75#14519736—21no—
76#14519736—41no—
77C2×C4×Dic61—41no—
78C2×C4×Dic61D411yescharacteristic
79C2×#11758185—22no—
80C2×#11758185C2311yescharacteristic
81C2×#11758185C2311yescharacteristic
82#14519711C2311yescharacteristic
83#14519711C2311yescharacteristic
84#14519711C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519710—21no—
87#14519710—21no—
88#14519709—21no—
89#14519709—21no—
90#14519708—21no—
91#14519708—21no—
92#14519701—21no—
93#14519701—21no—
94#14519698—21no—
95#14519698—21no—
96C8×Dic61D411yescharacteristic
97C8×Dic61D411yescharacteristic
98C8×Dic61C2311yescharacteristic
99C8×Dic61C2311yescharacteristic
100C2×#41×C61—21no—
101#135×C61—22no—
102#135×C61Q814yescomplemented
103C2×Q16×C61C2311yescharacteristic
104C2×SD16×C61—21no—
105C2×D8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107#117×C61—21no—
108C2×C16×C61—22no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#106×C61—21no—
114#41×C61—24no—
115#41×C61—42no—
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61C2×D411yescharacteristic
121C22×Dic61—41no—
122C22×Dic61—41no—
123C22×Dic61D811yescharacteristic
124C22×Dic61D811yescharacteristic
125C2×Dic122—41no—
126C2×Dic122—41no—
127Q16×C61—21no—
128Q16×C61C2×Q812yes—
129SD16×C61—24no—
130D8×C61—21no—
131D8×C61C2×Q812yes—
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×Q812yes—
134C2×C8×C61#4112yes—
135C2×C8×C61C2411yescharacteristic
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140C22⋊Dic61—41no—
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—41no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196—41no—
156C4×Dic61—41no—
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61—41no—
160C4×Dic61D811yescharacteristic
161C4×Dic61D811yescharacteristic
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164C2×#92418—22no—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C16×C61—21no—
168C16×C61—21no—
169#11758185—21no—
170#11758185—21no—
171C23×C61—41no—
172C23×C61C2×D811yescharacteristic
173Q8×C61—22no—
174Q8×C61—41no—
175D4×C61—22no—
176D4×C61—44no—
177D4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—44no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61C22⋊Q812yes—
183C2×C4×C61C2×D811yescharacteristic
184C2×C4×C61#13712yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—82no—
193C2×Dic61—82no—
194C2×Dic61C2×D811yescharacteristic
195C2×Dic61C2×D811yescharacteristic
196C2×Dic61C2×D811yescharacteristic
197C2×Dic61C2×D811yescharacteristic
198Dic122—82no—
199Dic122—82no—
200C8×C61C22⋊Q812yes—
201C8×C61C22×Q811yescharacteristic
202C8×C61C2×#4111yescharacteristic
203#92418—21no—
204#92418—21no—
205#33626—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—42no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61#47412yes—
212C22×C61C22×D811yescharacteristic
213C4×C61—42no—
214C4×C61—81no—
215C4×C61—81no—
216C4×C61#47412yes—
217C4×C61C2×C22⋊Q811yescharacteristic
218C4×C61C2×#13711yescharacteristic
219Dic61—21no—
220Dic61—21no—
221Dic61—21no—
222Dic61—81no—
223Dic61—81no—
224Dic61—81no—
225Dic61—81no—
226Dic61#50812yes—
227#3160—611no—
228C4×#135—611no—
229#2229—611no—
230#2229—611no—
231#2228—611no—
232#2228—611no—
233#2227—611no—
234#2226—611no—
235#2225—611no—
236#2224—611no—
237#2158—611no—
238#2158—611no—
239C2×C4⋊C16—611no—
240#2138—611no—
241#2138D6111yescharacteristic
242C2×C61—22no—
243C2×C61—82no—
244C2×C61C2×#47411yescharacteristic
245C2×C61C2×#50811yescharacteristic
246C2×C61#341411yescharacteristic
247C2×#135C2×D6111yescharacteristic
248#533—1221no—
249C4×#41—1221no—
250C22×C16—611no—
251C22×C16C2×D6111yescharacteristic
252#500—611no—
253#500—611no—
254#499—611no—
255#499—611no—
256#499—1221no—
257#498—1221no—
258C4×Q16—611no—
259C4×Q16—611no—
260C4×SD16—1221no—
261C4×SD16—1221no—
262C4×D8—611no—
263C4×D8—611no—
264#427—611no—
265#427—611no—
266#427—1221no—
267#427C2×D6111yescharacteristic
268#416—1221no—
269C2×C4×C8—611no—
270#367—611no—
271#367—611no—
272#367—611no—
273#367—611no—
274#366—611no—
275#366—611no—
276#366—611no—
277#366—611no—
278C4⋊C16—611no—
279C4⋊C16—611no—
280C4⋊C16—611no—
281C4⋊C16—611no—
282#358—611no—
283#358—611no—
284#358C2×D6111yescharacteristic
285#358C2×D6111yescharacteristic
286#357—611no—
287#357—611no—
288#357C2×D6111yescharacteristic
289#357C2×D6111yescharacteristic
290C61#3362611yescharacteristic, complemented, Sylow
291C2×#41—21no—
292#135—22no—
293#135Dic12214yescomplemented
294C2×Q16C22×D6111yescharacteristic
295C2×SD16—21no—
296C2×D8C22×D6111yescharacteristic
297C22×C8—611no—
298C22×C8C22×D6111yescharacteristic
299C4⋊Q8—1221no—
300C4⋊Q8—1221no—
301#125—1221no—
302#125—1221no—
303C22⋊Q8—2441no—
304C22⋊Q8—2441no—
305C4×Q8—1221no—
306C4×Q8—1221no—
307C4×Q8—2441no—
308C4×Q8—2441no—
309C4×D4—1221no—
310C4×D4—1221no—
311C4×D4—2441no—
312C4×D4—2441no—
313#117—21no—
314#117—1221no—
315#117—1221no—
316#117—2441no—
317C2×C42—611no—
318C2×C42—2441no—
319C2×C16—22no—
320C2×C16—611no—
321C2×C16—611no—
322C2×C16—1222no—
323C2×C16C22×D6111yescharacteristic
324C2×C16C22×D6111yescharacteristic
325#107—611no—
326#107—611no—
327#107—611no—
328#107—611no—
329#107—1221no—
330#107—1221no—
331#107C22×D6111yescharacteristic
332#107C22×D6111yescharacteristic
333#106—21no—
334#106—1221no—
335#106—1221no—
336#106—1221no—
337#106—1221no—
338#103—1221no—
339#103—1221no—
340#102—1221no—
341#102—1221no—
342C4×C8—611no—
343C4×C8—611no—
344C4×C8—611no—
345C4×C8—611no—
346#41—24no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—2441no—
350C2×Q8—2441no—
351C2×D4—21no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—2441no—
357C22×C4—2441no—
358C22×C4D4×D6111yescharacteristic
359Q16—21no—
360Q16C2×Dic12212yes—
361SD16—24no—
362D8—21no—
363D8C2×Dic12212yes—
364C2×C8—611no—
365C2×C8—611no—
366C2×C8—1222no—
367C2×C8C2×Dic12212yes—
368C2×C8D4×D6111yescharacteristic
369C2×C8#1175822412yes—
370C2×C8C23×D6111yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C4⋊C4—2441no—
383C4⋊C4—2441no—
384C4⋊C4—2441no—
385C22⋊C4—41no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C42—41no—
391C42—611no—
392C42—611no—
393C42—611no—
394C42—611no—
395C42—2441no—
396C42—2441no—
397C42—2441no—
398C42—2441no—
399C16—21no—
400C16—21no—
401C16—1221no—
402C16—1221no—
403C23—41no—
404C23D8×D6111yescharacteristic
405Q8—22no—
406Q8—41no—
407Q8—4882no—
408Q8—4882no—
409D4—22no—
410D4—44no—
411D4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—611no—
417C2×C4—611no—
418C2×C4—611no—
419C2×C4—611no—
420C2×C4—1222no—
421C2×C4—4882no—
422C2×C4—4882no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4#1451975412yes—
429C2×C4D8×D6111yescharacteristic
430C2×C4#1451980612yes—
431C2×C4C2×D4×D6111yescharacteristic
432C8—1221no—
433C8—1221no—
434C8#1451975412yes—
435C8C22×Dic12211yescharacteristic
436C8C2×#1175822411yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22#1684004012yes—
443C22C2×D8×D6111yescharacteristic
444C4—42no—
445C4—81no—
446C4—81no—
447C4—612no—
448C4—1221no—
449C4—1221no—
450C4—1221no—
451C4—4881no—
452C4—4881no—
453C4—4881no—
454C4—4881no—
455C4#1684004012yes—
456C4C2×#1451975411yescharacteristic
457C4C2×#1451980611yescharacteristic
458C2—22no—
459C2—82no—
460C2C2×#1684004011yescharacteristic
461C2#508×D6111yescharacteristic
462C2#2145895211yescharacteristic
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radicalC2×#135×C61C22
hypercenter#2138D61
hypocenterC61#33626
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21456331C2semidirect
3. #21447985C2semidirect
4. #21447981C2semidirect
5. #21447978C2semidirect
6. #21447974C2semidirect
7. #21447969C2semidirect
8. #21447964C2semidirect
9. #21447959C2semidirect
10. #21447954C2semidirect
11. #21447603C2semidirect
12. #21447601C2semidirect
13. C2×#16839580C2semidirect
102. #135×C61Q8semidirect
290. C61#33626semidirect
293. #135Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563312
#135×Dic612
#214479852
#214479812
#214479782
#214479742
#214479692
#214479642
#214479592
#214479542
#214476032
#214476012
C2×#168395802
#214475262
C61×#21382
#3362661