Finite Group Databasebeta

← #29662772

Subgroups — #29662772

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456405C211yesmaximal, characteristic, complemented
2#21455891C212yesmaximal
3#21454785C211yesmaximal, characteristic
4#21445374C212yesmaximal, complemented
5#21445367C212yesmaximal, complemented
6#21445338C211yesmaximal, characteristic, complemented
7#21445333C211yesmaximal, characteristic, complemented
8#21445314C211yesmaximal, characteristic, complemented
9#21445309C211yesmaximal, 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—
25#16839892C2211yescharacteristic
26#16839892C2211yescharacteristic
27C4×Dic244—22no—
28C4×Dic244—21no—
29C4×Dic244—21no—
30C4×#11758190—22no—
31#16839841C2211yescharacteristic
32#16839841C2211yescharacteristic
33#16839833—21no—
34#16839833—21no—
35#16839815C2211yescharacteristic
36#16839813—21no—
37C4×#11758189—21no—
38#16839509C2212yes—
39#16839509C2212yes—
40#16839507C2212yes—
41#16839507C2212yes—
42C61×#431C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2212yes—
45C61×#334C2211yescharacteristic
46C61×#334C2211yescharacteristic
47C2×#11758222D411yescharacteristic
48C2×Dic244—22no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×#11758190—22no—
52C2×#11758189—21no—
53C2×#11758189—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—
88#14519687—21no—
89#14519687—21no—
90C4×#92418—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—42no—
107C2×D244C2×D411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×Dic122—22no—
112C2×Dic122—22no—
113C2×Dic122C2×D411yescharacteristic
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61—41no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61#4111yescharacteristic
125C42×C61C2411yescharacteristic
126C22⋊Dic61—21no—
127C22⋊Dic61—42no—
128#11758198—21no—
129#11758198—21no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61C2×D411yescharacteristic
135C4⋊Dic61C2×D411yescharacteristic
136#11758196—21no—
137#11758196—21no—
138#11758196—42no—
139#11758196—42no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—42no—
144C2×#92418—21no—
145C2×#92418—21no—
146Dic244—42no—
147Dic244—44no—
148Dic244—42no—
149#11758190—44no—
150#11758189—42no—
151#11758189—42no—
152C23×C61#12111yescharacteristic
153C22×D61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C22⋊D61—42no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170D244—22no—
171C4×D61—42no—
172C4×D61—41no—
173C4×D61—41no—
174C4×D61—42no—
175Dic122—22no—
176Dic122—42no—
177Dic122—42no—
178Dic122—42no—
179Dic122—42no—
180C8×C61—21no—
181C8×C61—21no—
182C8×C61—21no—
183C8×C61—42no—
184C8×C61—42no—
185C8×C61#13512yes—
186#92418—41no—
187#92418—41no—
188#19161—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61C2×#12111yescharacteristic
193C2×D61—41no—
194C2×D61—41no—
195C2×D61—41no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61#47312yes—
200C4×C61C2×#13511yescharacteristic
201C4×C61#57512yes—
202C4×C61#57511yescharacteristic
203Dic61—41no—
204Dic61—41no—
205Dic61—41no—
206Dic61—82no—
207Dic61—82no—
208#3170—611no—
209#3107—612no—
210#2965—611no—
211#1700—612no—
212#1699—612no—
213#1694—611no—
214#1693—611no—
215#1690—611no—
216#1689—611no—
217#1564D6111yescharacteristic, complemented
218#1544—612no—
219C2×C61—41no—
220C2×C61#303511yescharacteristic
221C2×C61#305411yescharacteristic
222C2×C61C2×#47311yescharacteristic
223D61—81no—
224#533—612no—
225C4×#41—611no—
226#500—611no—
227#494—611no—
228#487—612no—
229#485—1222no—
230#475—612no—
231#474—612no—
232#467—1221no—
233#467—1221no—
234#462—612no—
235#461—612no—
236#452—1222no—
237#446—611no—
238#446—611no—
239C4×Q16—1222no—
240C4×Q16—1221no—
241C4×Q16—1221no—
242C4×SD16—1222no—
243#436—611no—
244#436—611no—
245#435—1221no—
246#435—1221no—
247#431—611no—
248#431—1221no—
249#431C2×D6112yes—
250C4×M16—1221no—
251C2×C4×C8C2×D6111yescharacteristic
252#335C2×D6112yes—
253#334C2×D6111yescharacteristic
254#334C2×D6111yescharacteristic
255#332—612no—
256#332—612no—
257#332—612no—
258#332—612no—
259C61#1916111yescharacteristic, complemented, Sylow
260C2×#41—611no—
261C2×Q16—1222no—
262C2×Q16—1221no—
263C2×Q16—1221no—
264C2×SD16—1222no—
265C2×M16—1221no—
266C2×M16—1221no—
267C22×C8D24411yescharacteristic
268C22×C8D24411yescharacteristic
269C4⋊Q8—612no—
270C4⋊Q8—612no—
271#125—1222no—
272C22⋊Q8—1222no—
273C22⋊Q8—1222no—
274C4×Q8—611no—
275C4×Q8—611no—
276C4×Q8—1222no—
277C4×Q8—1222no—
278C4×D4—611no—
279C4×D4—611no—
280C4×D4—1221no—
281C4×D4—1221no—
282#117—611no—
283#117—1221no—
284#117—1222no—
285C2×C42—1221no—
286C2×C42C22×D6111yescharacteristic
287#107—1222no—
288#107—1221no—
289#107—1221no—
290#106—1222no—
291C4⋊C8—611no—
292C4⋊C8—611no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8C22×D6112yes—
296C4⋊C8C22×D6112yes—
297#103—1222no—
298#103—1222no—
299#103—1222no—
300#103—1222no—
301#103—1222no—
302#102—1222no—
303C22⋊C8—22no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307#97—1221no—
308#97—1221no—
309C4×C8—22no—
310C4×C8—1221no—
311C4×C8D24411yescharacteristic
312C4×C8D24411yescharacteristic
313C4×C8C22×D6111yescharacteristic
314C4×C8C22×D6111yescharacteristic
315#41—2442no—
316C2×Q8—611no—
317C2×Q8—1222no—
318C2×Q8—1222no—
319C2×D4—611no—
320C2×D4—1221no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4C2×D24411yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C22×C4#1175822611yescharacteristic
327Q16—2442no—
328Q16—2444no—
329Q16—2442no—
330SD16—2444no—
331M16—2442no—
332M16—2442no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—1221no—
338C2×C8—1221no—
339C2×C8C2×D24411yescharacteristic
340C2×C8C2×D24411yescharacteristic
341C2×C8C2×D24411yescharacteristic
342C2×C8C2×D24411yescharacteristic
343C4⋊C4—611no—
344C4⋊C4—611no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—1222no—
350C4⋊C4—1222no—
351C4⋊C4—2442no—
352C4⋊C4—2442no—
353C22⋊C4—1221no—
354C22⋊C4—1221no—
355C22⋊C4—1221no—
356C22⋊C4—2442no—
357C42—1221no—
358C42—1221no—
359C42—1221no—
360C42—2442no—
361C42C2×D24411yescharacteristic
362C42D4×D6111yescharacteristic
363C42#1175822611yescharacteristic
364C42C23×D6111yescharacteristic
365C23—1221no—
366C23#1451975211yescharacteristic
367Q8—1222no—
368Q8—2442no—
369Q8—2442no—
370Q8—2442no—
371Q8—2442no—
372D4—1222no—
373D4—2442no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4—2441no—
387C2×C4#1451975211yescharacteristic
388C2×C4#1451975211yescharacteristic
389C2×C4#1451975211yescharacteristic
390C2×C4C2×D4×D6111yescharacteristic
391C2×C4C22×D24411yescharacteristic
392C2×C4C2×#1175822611yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—42no—
397C8—42no—
398C8—2441no—
399C8—2441no—
400C8#1451979912yes—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—2441no—
405C22—2441no—
406C22—2441no—
407C22C2×#1451975211yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—4882no—
412C4—4882no—
413C4—2441no—
414C4—2441no—
415C4—2441no—
416C4#1684003812yes—
417C4C2×#1451979911yescharacteristic
418C4#1684055912yes—
419C4#1684056211yescharacteristic
420C2—41no—
421C2—4881no—
422C2#2145527611yescharacteristic
423C2#2145542911yescharacteristic
424C2C2×#1684003811yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1564D61
hypocenterC61#19161
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. #21445374C2semidirect
5. #21445367C2semidirect
6. #21445338C2semidirect
7. #21445333C2semidirect
8. #21445314C2semidirect
9. #21445309C2semidirect
10. #21444483C2semidirect
11. C61×#1564C2semidirect
217. #1564D61semidirect
259. C61#19161semidirect

Maximal subgroups

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

subgroupindex
#214564052
#214558912
#214547852
#214453742
#214453672
#214453382
#214453332
#214453142
#214453092
#214444832
C61×#15642
#1916161