Finite Group Databasebeta

← #29662500

Subgroups — #29662500

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456405C211yesmaximal, characteristic, complemented
2#21455891C212yesmaximal
3C8×#11758222C211yesmaximal, characteristic
4#21445250C212yesmaximal, complemented
5#21445243C212yesmaximal, complemented
6#21445214C211yesmaximal, characteristic, complemented
7#21445209C211yesmaximal, characteristic, complemented
8#21445190C211yesmaximal, characteristic, complemented
9#21445185C211yesmaximal, characteristic, complemented
10#21444483C212yesmaximal, complemented
11C61×#1564C211yesmaximal, characteristic, complemented
12#16840319C2212yes—
13C4×#11758222C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840142C2211yescharacteristic
16#16840113C2212yes—
17#16840104—22no—
18#16840048C2212yes—
19#16840043C2212yes—
20#16840003—21no—
21#16840003—21no—
22#16839973C2212yes—
23#16839968C2212yes—
24#16839926—22no—
25C8×Dic122C2211yescharacteristic
26C8×Dic122C2211yescharacteristic
27C4×Dic244—22no—
28C4×Dic244—21no—
29C4×Dic244—21no—
30C4×#11758190—22no—
31C8×D244C2211yescharacteristic
32C8×D244C2211yescharacteristic
33C8×C22⋊D61—21no—
34C8×C22⋊D61—21no—
35#16839815C2211yescharacteristic
36#16839813—21no—
37C4×C8×D61—21no—
38#16839509C2212yes—
39#16839509C2212yes—
40#16839507C2212yes—
41#16839507C2212yes—
42C61×#431C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#334C2212yes—
47C2×#11758222D411yescharacteristic
48C2×Dic244—22no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×#11758190—22no—
52C2×C8×D61—21no—
53C2×C8×D61—21no—
54#14519777C2312yes—
55#14519777C2312yes—
56#14519769—22no—
57#14519756—22no—
58#14519756—22no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×Dic122D411yescharacteristic
62C4×Dic122C2311yescharacteristic
63C4×D244D411yescharacteristic
64C4×D244C2311yescharacteristic
65C4×C22⋊D61—21no—
66C4×C22⋊D61—21no—
67#14519736—22no—
68#14519736D411yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71#14519711—21no—
72#14519711—21no—
73#14519711—22no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519702—22no—
83#14519702—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418—21no—
89C8×Dic61—21no—
90C8×Dic61—21no—
91C22×C8×C61D411yescharacteristic
92C22×C8×C61D411yescharacteristic
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61C2312yes—
95C4⋊C8×C61C2312yes—
96C22⋊C8×C61—22no—
97C4×C8×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106#11758222—22no—
107#11758222—22no—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×Dic122—22no—
113C2×Dic122—22no—
114C2×Dic122C2×D411yescharacteristic
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61—41no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127C22⋊Dic61—21no—
128C22⋊Dic61—42no—
129#11758198—21no—
130#11758198—21no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61C2×D411yescharacteristic
136C4⋊Dic61C2×D411yescharacteristic
137#11758196—21no—
138#11758196—21no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61—42no—
145C2×#92418—21no—
146C2×#92418—21no—
147Dic244—42no—
148Dic244—44no—
149Dic244—42no—
150#11758190—44no—
151C8×D61—42no—
152C8×D61—42no—
153C23×C61#12111yescharacteristic
154C22×D61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—22no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—24no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172D244—21no—
173D244#13512yes—
174C4×D61—24no—
175C4×D61—41no—
176C4×D61—41no—
177C4×D61—42no—
178Dic122—21no—
179Dic122—42no—
180Dic122—42no—
181Dic122—42no—
182Dic122—42no—
183Dic122#13512yes—
184C8×C61—21no—
185C8×C61—21no—
186C8×C61—21no—
187C8×C61—42no—
188C8×C61—42no—
189C8×C61#13512yes—
190#92418—41no—
191#92418—41no—
192#19129—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61C2×#12111yescharacteristic
197C2×D61—22no—
198C2×D61—41no—
199C2×D61—41no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61#47312yes—
204C4×C61C2×#13511yescharacteristic
205C4×C61C2×#13511yescharacteristic
206C4×C61C2×#13511yescharacteristic
207C4×C61#57511yescharacteristic
208Dic61—22no—
209Dic61—41no—
210Dic61—41no—
211Dic61—82no—
212Dic61—82no—
213#3170—611no—
214#3107—612no—
215C8×#41—611no—
216#1680—612no—
217#1679—612no—
218#1674—611no—
219#1673—611no—
220#1670—611no—
221#1669—611no—
222#1564D6111yescharacteristic, complemented
223#1544—612no—
224C2×C61—41no—
225C2×C61#303511yescharacteristic
226C2×C61#305011yescharacteristic
227C2×C61C2×#47311yescharacteristic
228D61—42no—
229#533—612no—
230C4×#41—611no—
231#500—611no—
232#494—611no—
233#487—612no—
234#485—1222no—
235#475—612no—
236#474—612no—
237#467—1221no—
238#467—1221no—
239#462—612no—
240#461—612no—
241#452—1222no—
242C8×Q8—611no—
243C8×Q8—611no—
244C4×Q16—1222no—
245C4×Q16—1221no—
246C4×Q16—1221no—
247C4×SD16—1222no—
248C8×D4—611no—
249C8×D4—611no—
250C8×D4—1221no—
251C8×D4—1221no—
252#431—611no—
253#431—1221no—
254#431C2×D6112yes—
255C2×C4×C8—1221no—
256C2×C4×C8C2×D6111yescharacteristic
257#335C2×D6111yescharacteristic
258#335C2×D6111yescharacteristic
259#334C2×D6112yes—
260#332—612no—
261#332—612no—
262#332—612no—
263#332—612no—
264C61#1912911yescharacteristic, complemented, Sylow
265C2×#41—611no—
266C2×Q16—1222no—
267C2×Q16—1221no—
268C2×Q16—1221no—
269C2×SD16—1222no—
270C22×C8—1221no—
271C22×C8—1221no—
272C22×C8D24411yescharacteristic
273C22×C8D24411yescharacteristic
274C4⋊Q8—612no—
275C4⋊Q8—612no—
276#125—1222no—
277C22⋊Q8—1222no—
278C22⋊Q8—1222no—
279C4×Q8—611no—
280C4×Q8—611no—
281C4×Q8—1222no—
282C4×Q8—1222no—
283C4×D4—611no—
284C4×D4—611no—
285C4×D4—1221no—
286C4×D4—1221no—
287#117—611no—
288#117—1221no—
289#117—1222no—
290C2×C42—1221no—
291C2×C42C22×D6111yescharacteristic
292#107—1222no—
293#107—1221no—
294#107—1221no—
295#106—1222no—
296C4⋊C8—611no—
297C4⋊C8—611no—
298C4⋊C8—1221no—
299C4⋊C8—1221no—
300C4⋊C8C22×D6112yes—
301C4⋊C8C22×D6112yes—
302#103—1222no—
303#103—1222no—
304#103—1222no—
305#103—1222no—
306#103—1222no—
307#102—1222no—
308C22⋊C8—22no—
309C22⋊C8—1221no—
310C22⋊C8—1221no—
311C22⋊C8—1221no—
312C4×C8—22no—
313C4×C8—1221no—
314C4×C8—1221no—
315C4×C8—1221no—
316C4×C8D24411yescharacteristic
317C4×C8D24411yescharacteristic
318C4×C8C22×D6111yescharacteristic
319C4×C8C22×D6111yescharacteristic
320#41—1222no—
321#41—1222no—
322C2×Q8—611no—
323C2×Q8—1222no—
324C2×Q8—1222no—
325C2×D4—611no—
326C2×D4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4C2×D24411yescharacteristic
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822611yescharacteristic
333Q16—2442no—
334Q16—2444no—
335Q16—2442no—
336SD16—2444no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—2442no—
344C2×C8—2442no—
345C2×C8C2×D24411yescharacteristic
346C2×C8C2×D24411yescharacteristic
347C2×C8C2×D24411yescharacteristic
348C2×C8C2×D24411yescharacteristic
349C4⋊C4—611no—
350C4⋊C4—611no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1222no—
354C4⋊C4—1222no—
355C4⋊C4—1222no—
356C4⋊C4—1222no—
357C4⋊C4—2442no—
358C4⋊C4—2442no—
359C22⋊C4—1221no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—2442no—
363C42—1221no—
364C42—1221no—
365C42—1221no—
366C42—2442no—
367C42C2×D24411yescharacteristic
368C42D4×D6111yescharacteristic
369C42#1175822611yescharacteristic
370C42C23×D6111yescharacteristic
371C23—1221no—
372C23#1451975211yescharacteristic
373Q8—612no—
374Q8—1221no—
375Q8—2442no—
376Q8—2442no—
377Q8—2442no—
378Q8—2442no—
379D4—612no—
380D4—1224no—
381D4—1221no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—1224no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2441no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4#1451975211yescharacteristic
397C2×C4#1451975211yescharacteristic
398C2×C4#1451975211yescharacteristic
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C22×D24411yescharacteristic
401C2×C4C2×#1175822611yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8—42no—
406C8—42no—
407C8—2441no—
408C8—2441no—
409C8#1451979912yes—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—1222no—
414C22—2441no—
415C22—2441no—
416C22C2×#1451975211yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—1222no—
421C4—4882no—
422C4—4882no—
423C4—2441no—
424C4—2441no—
425C4#1684003812yes—
426C4#135×D6111yescharacteristic
427C4#135×D6111yescharacteristic
428C4C2×#1451979911yescharacteristic
429C4#1684056211yescharacteristic
430C2—41no—
431C2—2442no—
432C2#2145527611yescharacteristic
433C2#2145540111yescharacteristic
434C2C2×#1684003811yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1564D61
hypocenterC61#19129
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #21456405C2semidirect
4. #21445250C2semidirect
5. #21445243C2semidirect
6. #21445214C2semidirect
7. #21445209C2semidirect
8. #21445190C2semidirect
9. #21445185C2semidirect
10. #21444483C2semidirect
11. C61×#1564C2semidirect
222. #1564D61semidirect
264. C61#19129semidirect

Maximal subgroups

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

subgroupindex
#214564052
#214558912
C8×#117582222
#214452502
#214452432
#214452142
#214452092
#214451902
#214451852
#214444832
C61×#15642
#1912961