Finite Group Databasebeta

← #29663925

Subgroups — #29663925

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456247C211yesmaximal, characteristic, complemented
2#21455971C212yesmaximal, complemented
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21445190C211yesmaximal, characteristic, complemented
5#21445185C211yesmaximal, characteristic, complemented
6#21445180C211yesmaximal, characteristic, complemented
7#21445175C211yesmaximal, characteristic, complemented
8#21445168C212yesmaximal, complemented
9#21445161C212yesmaximal, complemented
10#21444525C212yesmaximal, complemented
11C61×#1568C211yesmaximal, characteristic, complemented
12C2×#14519799—21no—
13C2×#14519799—21no—
14#16840312C2212yes—
15C4×#11758222C2211yescharacteristic
16#16840146—21no—
17#16840146—21no—
18#16840142C2211yescharacteristic
19#16840139C2211yescharacteristic
20#16840135C2212yes—
21#16840087—22no—
22#16840048C2212yes—
23#16840043C2212yes—
24#16839963C2212yes—
25#16839958C2212yes—
26#16839909—22no—
27C8×Dic122C2211yescharacteristic
28C8×Dic122C2211yescharacteristic
29#16839875—22no—
30#16839864—22no—
31C8×D244C2211yescharacteristic
32C8×D244C2211yescharacteristic
33C8×C22⋊D61—21no—
34C8×C22⋊D61—21no—
35#16839815C2211yescharacteristic
36#16839813—21no—
37C4×C8×D61—21no—
38#16839507C2212yes—
39#16839507C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C61×#432C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#334C2212yes—
47C2×#11758222—22no—
48C2×#11758222D411yescharacteristic
49#14519799—44no—
50#14519799—44no—
51C2×Dic244—21no—
52C2×Dic244—21no—
53C2×#11758190—22no—
54C2×#11758190—22no—
55C2×#11758190—22no—
56C2×D488—22no—
57C2×D488—21no—
58C2×D488—21no—
59C2×C8×D61—21no—
60C2×C8×D61—21no—
61#14519777C2312yes—
62#14519774C2312yes—
63#14519765—22no—
64#14519751—22no—
65#14519751—22no—
66C4×Dic122D411yescharacteristic
67C4×Dic122C2311yescharacteristic
68C4×D244—22no—
69C4×D244—22no—
70C4×D244D411yescharacteristic
71C4×D244C2311yescharacteristic
72C4×C22⋊D61—21no—
73C4×C22⋊D61—21no—
74#14519736D411yescharacteristic
75#14519734—21no—
76C42×D61—21no—
77#14519711—22no—
78#14519709—22no—
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2311yescharacteristic
81#14519706—21no—
82#14519706—21no—
83#14519702—22no—
84#14519699—22no—
85#14519699—22no—
86#14519699—22no—
87C22⋊#92418—21no—
88#14519689—21no—
89#14519689—21no—
90C4×#92418—21no—
91C8×Dic61—21no—
92C8×Dic61—21no—
93C22×C8×C61D411yescharacteristic
94C22×C8×C61D411yescharacteristic
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61C2312yes—
97C4⋊C8×C61C2312yes—
98C22⋊C8×C61—22no—
99#97×C61—22no—
100C4×C8×C61D411yescharacteristic
101C4×C8×C61D411yescharacteristic
102C4×C8×C61C2311yescharacteristic
103C4×C8×C61C2311yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—21no—
108C2×C22⋊D61—42no—
109#11758222—22no—
110#11758222—22no—
111#11758222—44no—
112#11758222—44no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244C2×D411yescharacteristic
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—42no—
121C2×Dic122—22no—
122C2×Dic122C2×D411yescharacteristic
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61#4111yescharacteristic
134C42×C61C2411yescharacteristic
135C22⋊Dic61—21no—
136#11758198—21no—
137#11758198—21no—
138#11758198—42no—
139#11758198—42no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61C2×D411yescharacteristic
143C4⋊Dic61C2×D411yescharacteristic
144#11758196—21no—
145#11758196—21no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C4×Dic61—21no—
149C2×#92418—21no—
150C2×#92418—21no—
151Dic244—42no—
152Dic244—42no—
153D488—42no—
154D488—44no—
155D488—42no—
156#11758190—44no—
157#11758190—44no—
158#11758190—44no—
159C8×D61—42no—
160C8×D61—42no—
161C23×C61#12111yescharacteristic
162C22×D61—21no—
163C22×D61—42no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—41no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C22⋊D61—24no—
176C22⋊D61—84no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—42no—
181D244—21no—
182D244—42no—
183D244—42no—
184D244—42no—
185D244—42no—
186D244—82no—
187D244#13512yescomplemented
188C4×D61—24no—
189C4×D61—42no—
190C4×D61—41no—
191C4×D61—41no—
192C4×D61—84no—
193Dic122—21no—
194Dic122—42no—
195Dic122—42no—
196Dic122#13512yescomplemented
197C8×C61—21no—
198C8×C61—21no—
199C8×C61—42no—
200C8×C61—42no—
201C8×C61AGL(1,ℤ/8)12yes—
202C8×C61AGL(1,ℤ/8)12yes—
203#92418—41no—
204#92418—41no—
205#19269—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61C2×#12111yescharacteristic
210C2×D61—22no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—82no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61C2×#13511yescharacteristic
221C4×C61C2×#13511yescharacteristic
222C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
224Dic61—22no—
225Dic61—41no—
226Dic61—41no—
227Dic61—82no—
228#3147—611no—
229#3115—612no—
230C8×#41—611no—
231#1670—611no—
232#1669—611no—
233#1668—611no—
234#1667—611no—
235#1666—612no—
236#1665—612no—
237#1568D6111yescharacteristic, complemented
238#1553—612no—
239C2×C61—41no—
240C2×C61#303811yescharacteristic
241C2×C61#305011yescharacteristic
242C2×C61#306711yescharacteristic
243D61—42no—
244D61—162no—
245C2×#135—1221no—
246C2×#135—1221no—
247#532—612no—
248C4×#41—611no—
249#495—1221no—
250#495—1221no—
251#494—611no—
252#493—611no—
253#492—612no—
254#482—1222no—
255#475—612no—
256#474—612no—
257#460—612no—
258#459—612no—
259#449—1222no—
260C8×Q8—611no—
261C8×Q8—611no—
262#442—1222no—
263#440—1222no—
264C8×D4—611no—
265C8×D4—611no—
266C8×D4—1221no—
267C8×D4—1221no—
268#432C2×D6112yes—
269#431—611no—
270#431—1221no—
271C2×C4×C8—1221no—
272C2×C4×C8C2×D6111yescharacteristic
273#335C2×D6111yescharacteristic
274#335C2×D6111yescharacteristic
275#334C2×D6112yes—
276#332—612no—
277#332—612no—
278#331—612no—
279#331—612no—
280C61#1926911yescharacteristic, complemented, Sylow
281C2×#41—611no—
282C2×#41—1222no—
283#135—2444no—
284#135—2444no—
285C2×Q16—1221no—
286C2×Q16—1221no—
287C2×SD16—1222no—
288C2×SD16—1222no—
289C2×SD16—1222no—
290C2×D8—1222no—
291C2×D8—1221no—
292C2×D8—1221no—
293C22×C8—1221no—
294C22×C8—1221no—
295C22×C8D24411yescharacteristic
296C22×C8D24411yescharacteristic
297C4⋊Q8—612no—
298#127—612no—
299#124—1222no—
300#121—1222no—
301#121—1222no—
302C4×Q8—611no—
303C4×Q8—611no—
304C4×D4—611no—
305C4×D4—611no—
306C4×D4—1221no—
307C4×D4—1221no—
308C4×D4—1222no—
309C4×D4—1222no—
310#117—611no—
311#117—1221no—
312C2×C42—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1222no—
315#106—1222no—
316C4⋊C8—611no—
317C4⋊C8—611no—
318C4⋊C8—1221no—
319C4⋊C8—1221no—
320C4⋊C8C22×D6112yes—
321C4⋊C8C22×D6112yes—
322#103—1222no—
323#102—1222no—
324#102—1222no—
325#102—1222no—
326C22⋊C8—22no—
327C22⋊C8—1221no—
328C22⋊C8—1221no—
329C22⋊C8—1221no—
330#97—22no—
331C4×C8—1221no—
332C4×C8—1221no—
333C4×C8—1221no—
334C4×C8D24411yescharacteristic
335C4×C8D24411yescharacteristic
336C4×C8C22×D6111yescharacteristic
337C4×C8C22×D6111yescharacteristic
338#41—1222no—
339#41—1222no—
340#41—2444no—
341#41—2444no—
342C2×Q8—611no—
343C2×Q8—1222no—
344C2×D4—611no—
345C2×D4—1221no—
346C2×D4—1222no—
347C2×D4—1222no—
348C2×D4—1222no—
349C2×D4—2442no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—2442no—
354C22×C4C2×D24411yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4#1175822611yescharacteristic
357Q16—2442no—
358Q16—2442no—
359SD16—2444no—
360SD16—2444no—
361SD16—2444no—
362D8—2442no—
363D8—2444no—
364D8—2442no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—2442no—
372C2×C8—2442no—
373C2×C8C2×D24411yescharacteristic
374C2×C8C2×D24411yescharacteristic
375C2×C8C2×D24411yescharacteristic
376C2×C8C2×D24411yescharacteristic
377C4⋊C4—611no—
378C4⋊C4—611no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1222no—
382C4⋊C4—1222no—
383C22⋊C4—1221no—
384C22⋊C4—1221no—
385C22⋊C4—1221no—
386C22⋊C4—2442no—
387C22⋊C4—2442no—
388C42—1221no—
389C42—1221no—
390C42—1221no—
391C42C2×D24411yescharacteristic
392C42D4×D6111yescharacteristic
393C42#1175822611yescharacteristic
394C42C23×D6111yescharacteristic
395C23—1221no—
396C23—2442no—
397C23#1451975211yescharacteristic
398Q8—612no—
399Q8—1221no—
400Q8—2442no—
401Q8—2442no—
402D4—612no—
403D4—1224no—
404D4—1221no—
405D4—4882no—
406D4—4884no—
407D4—2442no—
408D4—2442no—
409D4—2442no—
410D4—2442no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—1224no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—4884no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4#1451975211yescharacteristic
426C2×C4#1451975211yescharacteristic
427C2×C4#1451975211yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C22×D24411yescharacteristic
430C2×C4C2×#1175822611yescharacteristic
431C8—21no—
432C8—21no—
433C8—42no—
434C8—42no—
435C8—2441no—
436C8—2441no—
437C8#1451980512yes—
438C8#1451980512yes—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—1222no—
443C22—4882no—
444C22—4882no—
445C22—4882no—
446C22—2441no—
447C22—2441no—
448C22C2×#1451975211yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—1222no—
454C4—4882no—
455C4—2441no—
456C4—2441no—
457C4#135×D6111yescharacteristic
458C4#135×D6111yescharacteristic
459C4C2×#1451980511yescharacteristic
460C4C2×#1451980511yescharacteristic
461C2—41no—
462C2—9762no—
463C2—2442no—
464C2#2145530011yescharacteristic
465C2#2145540111yescharacteristic
466C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1568C2
socleC22×C61C2×#121
hypercenter#1568D61
hypocenterC61#19269
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1568

Complements

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

NH ≅ G/Ntype
1. #21456247C2semidirect
2. #21455971C2semidirect
3. C8×#11758222C2semidirect
4. #21445190C2semidirect
5. #21445185C2semidirect
6. #21445180C2semidirect
7. #21445175C2semidirect
8. #21445168C2semidirect
9. #21445161C2semidirect
10. #21444525C2semidirect
11. C61×#1568C2semidirect
187. D244#135semidirect
196. Dic122#135semidirect
237. #1568D61semidirect
280. C61#19269semidirect

Maximal subgroups

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

subgroupindex
#214562472
#214559712
C8×#117582222
#214451902
#214451852
#214451802
#214451752
#214451682
#214451612
#214445252
C61×#15682
#1926961