Finite Group Databasebeta

← #29834717

Subgroups — #29834717

Order 15616.

534 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#21456352C211yesmaximal, characteristic, complemented
2#21454689C211yesmaximal, characteristic
3#21447984C211yesmaximal, characteristic, complemented
4#21447982C211yesmaximal, characteristic, complemented
5#21447978C211yesmaximal, characteristic, complemented
6#21447976C211yesmaximal, characteristic, complemented
7#21447970C211yesmaximal, characteristic, complemented
8#21447968C211yesmaximal, characteristic, complemented
9#21447954C211yesmaximal, characteristic, complemented
10#21447604C211yesmaximal, characteristic, complemented
11#366×D61C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic
14#21447523C211yesmaximal, characteristic
15C61×#2224C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17#16840389—21no—
18#16840241—21no—
19C2×#14519716C2211yescharacteristic
20C2×C16×D61C2211yescharacteristic
21#16840170—21no—
22#16840168—21no—
23#16840166C2211yescharacteristic
24#16840164C2211yescharacteristic
25#16840163C2211yescharacteristic
26#16839862—21no—
27#16839859—21no—
28C4×Dic244C2211yescharacteristic
29#16839856C2211yescharacteristic
30D8×Dic61C2211yescharacteristic
31#16839803—21no—
32#16839799C2211yescharacteristic
33#107×D61C2211yescharacteristic
34#107×D61C2211yescharacteristic
35#16839760—21no—
36#16839716C2211yescharacteristic
37#16839590C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839587C2211yescharacteristic
42#16839587C2211yescharacteristic
43C4⋊#11758185C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839567C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839565C2211yescharacteristic
50#16839563C2211yescharacteristic
51#16839563C2211yescharacteristic
52C61×#500C2211yescharacteristic
53C4×D8×C61C2211yescharacteristic
54C61×#366C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×#11758224—21no—
60#14519796—22no—
61#14519796Q814yescomplemented
62C2×Dic244C2311yescharacteristic
63C2×#11758200—21no—
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519776—21no—
67#14519775—41no—
68#14519769—21no—
69#14519768—21no—
70#14519767—41no—
71#14519755—41no—
72#14519753—41no—
73C4×Dic122—21no—
74#14519742—21no—
75D4×Dic61—21no—
76C4×C22⋊D61—41no—
77#14519737—41no—
78#14519735—21no—
79#14519734D411yescharacteristic
80#14519733—41no—
81C2×#11758196—41no—
82C4⋊C4×D61—21no—
83C4⋊C4×D61—21no—
84#14519716—22no—
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C16×D61—22no—
88#14519711C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519709—21no—
95#14519708—21no—
96#14519708—21no—
97#14519702—21no—
98#14519700—21no—
99#14519698—21no—
100#14519688D411yescharacteristic
101#14519687D411yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×D8×C61C2311yescharacteristic
104C4⋊Q8×C61—21no—
105C4×D4×C61—21no—
106C2×C16×C61C2311yescharacteristic
107C2×C16×C61C2311yescharacteristic
108#107×C61—21no—
109#107×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#102×C61—21no—
113C4×C8×C61C2311yescharacteristic
114C2×Q8×C61—41no—
115C2×D4×C61—21no—
116C22×C4×C61—41no—
117C2×C22⋊D61—41no—
118C22×Dic61—41no—
119C22×Dic61—41no—
120#11758224—24no—
121#11758224—42no—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61C2×D411yescharacteristic
125C2×Dic122—21no—
126C2×Dic122—41no—
127D8×C61—21no—
128D8×C61C2×Q812yes—
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—
138#11758200—24no—
139#11758198—41no—
140#11758198D811yescharacteristic
141#11758198D811yescharacteristic
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—41no—
146#11758196—41no—
147#11758196—41no—
148#11758196—41no—
149#11758196—41no—
150#11758196—41no—
151#11758196—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196D811yescharacteristic
155#11758196D811yescharacteristic
156C4×Dic61—41no—
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162Dic244—21no—
163Dic244C2×Q812yes—
164#11758189—22no—
165C8×D61C2×Q812yes—
166C8×D61#4112yes—
167C16×C61—21no—
168C16×C61—21no—
169#11758185—21no—
170#11758185—21no—
171C23×C61—41no—
172C22×D61C2×D811yescharacteristic
173Q8×C61—161no—
174D4×C61—22no—
175D4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—161no—
180C2×C4×C61C2×D811yescharacteristic
181C2×C4×C61C2×D811yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C22⋊D61—44no—
184C2×Dic61—41no—
185C2×Dic61—44no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—161no—
190C2×Dic61C2×D811yescharacteristic
191C2×Dic61C2×D811yescharacteristic
192C2×Dic61C2×D811yescharacteristic
193C4×D61—161no—
194C4×D61—161no—
195C4×D61C22⋊Q812yes—
196C4×D61#13712yes—
197Dic122—22no—
198Dic122—41no—
199Dic122—161no—
200C8×C61—21no—
201C8×C61C22×Q811yescharacteristic
202C8×C61C2×#4111yescharacteristic
203#92418—21no—
204#92418C22⋊Q812yes—
205#33633—611nomaximal, Sylow
206C22×C61—42no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61C22×D811yescharacteristic
210C2×D61—21no—
211C2×D61—21no—
212C2×D61#47412yes—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61C2×C22⋊Q811yescharacteristic
219C4×C61C2×#13711yescharacteristic
220Dic61—21no—
221Dic61—21no—
222Dic61—42no—
223Dic61—81no—
224Dic61—81no—
225Dic61—81no—
226Dic61#47412yes—
227#3164—611no—
228#2951—611no—
229#2229—611no—
230#2229—611no—
231#2228—611no—
232#2228—611no—
233#2227—611no—
234#2227—611no—
235#2224—611no—
236#2224D6111yescharacteristic, complemented
237#2159—611no—
238C2×#366—611no—
239#2151—611no—
240#2139—611no—
241#2138—611no—
242C2×C61—82no—
243C2×C61C2×#47411yescharacteristic
244C2×C61#341511yescharacteristic
245C2×C61#341711yescharacteristic
246D61—22no—
247C2×#135—611no—
248#544—1221no—
249#520—1221no—
250C2×M32—611no—
251C22×C16—611no—
252#501—1221no—
253#501—1221no—
254#500—611no—
255#500C2×D6111yescharacteristic
256#499—611no—
257#499—611no—
258#440—1221no—
259#440—1221no—
260C4×Q16—611no—
261C4×Q16—611no—
262C4×D8—611no—
263C4×D8C2×D6111yescharacteristic
264#428—1221no—
265#427—611no—
266C2×#107—611no—
267C2×#107—611no—
268#417—1221no—
269#405—611no—
270#367—611no—
271#367—611no—
272#366—611no—
273#366—611no—
274#366—611no—
275#366—611no—
276#366C2×D6111yescharacteristic
277#366C2×D6111yescharacteristic
278C4⋊C16—611no—
279C4⋊C16—611no—
280C4⋊C16—611no—
281C4⋊C16C2×D6111yescharacteristic
282#358—611no—
283#358—611no—
284#358—611no—
285#358—611no—
286#357—611no—
287#357—611no—
288#357C2×D6111yescharacteristic
289#357C2×D6111yescharacteristic
290C61#3363311yescharacteristic, complemented, Sylow
291C2×#41—1221no—
292#135—614no—
293#135—1222no—
294C2×Q16—611no—
295C2×SD16—1221no—
296C2×D8C22×D6111yescharacteristic
297C2×M16—611no—
298C22×C8—611no—
299C4⋊Q8—21no—
300C4⋊Q8—1221no—
301C4⋊Q8—2441no—
302#125—1221no—
303#125—1221no—
304#125—2441no—
305C22⋊Q8—2441no—
306C22⋊Q8—2441no—
307C4×Q8—1221no—
308C4×Q8—1221no—
309C4×D4—21no—
310C4×D4—1221no—
311C4×D4—2441no—
312C4×D4—2441no—
313#117—611no—
314#117—1221no—
315#117—2441no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—2441no—
319M32—1222no—
320C2×C16—611no—
321C2×C16—611no—
322C2×C16—1222no—
323C2×C16C22×D6111yescharacteristic
324C2×C16C22×D6111yescharacteristic
325#107—21no—
326#107—611no—
327#107—611no—
328#107—611no—
329#107—1221no—
330#107—1221no—
331#107—1221no—
332#107C22×D6111yescharacteristic
333#107C22×D6111yescharacteristic
334#107C22×D6111yescharacteristic
335#106—1221no—
336#106—1221no—
337#106—1221no—
338#103—1221no—
339#103—1221no—
340#102—21no—
341#102—1221no—
342#97—611no—
343#97—611no—
344C4×C8—611no—
345C4×C8C22×D6111yescharacteristic
346#41—1224no—
347#41—2442no—
348C2×Q8—41no—
349C2×Q8—1221no—
350C2×Q8—2441no—
351C2×D4—21no—
352C2×D4—2441no—
353C22×C4—41no—
354C22×C4—611no—
355C22×C4—2441no—
356C22×C4—2441no—
357C22×C4—2441no—
358C22×C4—2441no—
359Q16—612no—
360Q16—1221no—
361SD16—1224no—
362D8—21no—
363D8Q8×D6112yes—
364M16—1222no—
365C2×C8—611no—
366C2×C8—612no—
367C2×C8—612no—
368C2×C8—611no—
369C2×C8D4×D6111yescharacteristic
370C2×C8C23×D6111yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—611no—
376C4⋊C4—611no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C4⋊C4—2441no—
383C4⋊C4—2441no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C22⋊C4—41no—
390C22⋊C4—611no—
391C22⋊C4—611no—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C42—611no—
395C42—2441no—
396C42—2441no—
397C42—2441no—
398C42D4×D6111yescharacteristic
399C16—21no—
400C16—21no—
401C16—1221no—
402C16—1221no—
403C23—41no—
404C23—611no—
405Q8—161no—
406Q8—1222no—
407Q8—9761no—
408Q8—2441no—
409D4—22no—
410D4—41no—
411D4—2444no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—161no—
416C2×C4—611no—
417C2×C4—612no—
418C2×C4—612no—
419C2×C4—611no—
420C2×C4—611no—
421C2×C4—9761no—
422C2×C4—9761no—
423C2×C4—9761no—
424C2×C4—2444no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4D8×D6111yescharacteristic
430C2×C4D8×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C8—21no—
433C8—612no—
434C8—1221no—
435C8C2×Q8×D6111yescharacteristic
436C8#41×D6111yescharacteristic
437C22—42no—
438C22—81no—
439C22—81no—
440C22—612no—
441C22—1221no—
442C22—1221no—
443C22C2×D8×D6111yescharacteristic
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—612no—
450C4—1221no—
451C4—1221no—
452C4—4881no—
453C4—4881no—
454C4—4881no—
455C4—2442no—
456C4C22⋊Q8×D6111yescharacteristic
457C4#137×D6111yescharacteristic
458C2—82no—
459C2—1222no—
460C2#474×D6111yescharacteristic
461C2#2145895611yescharacteristic
462C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2224C2
socleC22×C61C22×D8
prime radicalC2×#14519796C22
hypercenter#2224D61
hypocenterC61#33633
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. #21456352C2semidirect
3. #21447984C2semidirect
4. #21447982C2semidirect
5. #21447978C2semidirect
6. #21447976C2semidirect
7. #21447970C2semidirect
8. #21447968C2semidirect
9. #21447954C2semidirect
10. #21447604C2semidirect
11. #366×D61C2semidirect
12. #21447576C2semidirect
15. C61×#2224C2semidirect
61. #14519796Q8semidirect
236. #2224D61semidirect
290. C61#33633semidirect

Maximal subgroups

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

subgroupindex
#214563522
#214546892
#214479842
#214479822
#214479782
#214479762
#214479702
#214479682
#214479542
#214476042
#366×D612
#214475762
#214475282
#214475232
C61×#22242
#3363361