Finite Group Databasebeta

← #29575249

Subgroups — #29575249

Order 15616.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic
2#21454565C211yesmaximal, characteristic
3#21446763C211yesmaximal, characteristic, complemented
4#21444315C211yesmaximal, characteristic
5#21444223C211yesmaximal, characteristic, complemented
6#21443659C211yesmaximal, characteristic, complemented
7C61×#1278C211yesmaximal, characteristic, complemented
8C2×#14519785—21no—
9#16840319C2211yescharacteristic
10#16840312C2211yescharacteristic
11#16840160C411yescharacteristic, complemented
12#16840135C411yescharacteristic, complemented
13#16840113C411yescharacteristic, complemented
14#16840109C411yescharacteristic, complemented
15#16839991—22no—
16C4×#11758190—22no—
17#16839841—21no—
18#16839835—22no—
19C8×D244—21no—
20C2×C4⋊#92418C2211yescharacteristic
21#16839730—21no—
22C4×C4⋊Dic61C2211yescharacteristic
23#16839523—21no—
24#16839523—21no—
25#16839497—22no—
26#16839494—21no—
27#16839494—21no—
28C61×#431C2211yescharacteristic
29C4×M16×C61C2211yescharacteristic
30C2×C4×C8×C61C2211yescharacteristic
31C2×#11758222—21no—
32C2×#11758190—22no—
33#14519785—42no—
34#14519785—42no—
35C2×#11758189—22no—
36C22×#92418—21no—
37C2×C8×D61—22no—
38#14519777C2×C411yescharacteristic
39#14519777C2×C411yescharacteristic
40#14519777C2×C411yescharacteristic
41#14519774C2×C411yescharacteristic
42#14519769—21no—
43#14519765—21no—
44#14519756—21no—
45#14519756—21no—
46#14519751—21no—
47#14519751D412yes—
48C4×Dic122—21no—
49C4×Dic122—21no—
50C4×D244—21no—
51C4×D244D412yes—
52#14519736—21no—
53C2×C4⋊Dic61—21no—
54C2×C4×Dic61—21no—
55#14519709—22no—
56C4⋊#92418—21no—
57C4⋊#92418D411yescharacteristic
58C4⋊#92418D411yescharacteristic
59#14519702—22no—
60#14519699—22no—
61#14519689—22no—
62#14519689—22no—
63#14519684—21no—
64C2×M16×C61D411yescharacteristic
65C2×M16×C61D411yescharacteristic
66C22×C8×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70#97×C61—21no—
71C4×C8×C61—21no—
72C4×C8×C61—21no—
73C4×C8×C61—21no—
74C4×C8×C61C2×C411yescharacteristic
75C4×C8×C61C2×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61#4111yescharacteristic
79C2×C22⋊D61—22no—
80C22×Dic61—41no—
81#11758222—41no—
82#11758222—41no—
83#11758222—41no—
84#11758222—41no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244C22⋊C412yes—
88C2×C4×D61—22no—
89C2×Dic122—21no—
90C2×Dic122—21no—
91C2×Dic122—21no—
92M16×C61—21no—
93M16×C61—21no—
94M16×C61—21no—
95M16×C61SD1612yescomplemented
96C2×C8×C61—21no—
97C2×C8×C61—21no—
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C2×C8×C61—42no—
101C2×C8×C61—41no—
102C42×C61C22×C411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C22⋊Dic61—41no—
107#11758198—22no—
108#11758198—22no—
109C4⋊Dic61—21no—
110C4⋊Dic61—22no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—41no—
116C4⋊Dic61C22⋊C412yes—
117#11758196—41no—
118#11758196—41no—
119C4×Dic61—41no—
120C4×Dic61—41no—
121C4×Dic61—41no—
122C2×#92418—21no—
123C2×#92418—21no—
124C2×#92418—41no—
125C2×#92418—41no—
126#11758190—44no—
127#11758189—44no—
128C8×D61—44no—
129C23×C61#12111yescharacteristic
130C22×D61—22no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—41no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12311yescharacteristic
140C2×C4×C61C2×SD1611yescharacteristic
141C2×C4×C61#13511yescharacteristic
142C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C22⋊D61—42no—
145C22⋊D61—42no—
146C2×Dic61—22no—
147C2×Dic61—42no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—81no—
151C2×Dic61—81no—
152D244—42no—
153D244—41no—
154D244—41no—
155D244—41no—
156D244—41no—
157C4×D61—42no—
158C4×D61—42no—
159Dic122—41no—
160Dic122—41no—
161Dic122—42no—
162Dic122—41no—
163Dic122—41no—
164C8×C61—22no—
165C8×C61—22no—
166C8×C61—41no—
167C8×C61—41no—
168C8×C61—81no—
169#92418—41no—
170#92418—41no—
171#9055—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61#38611yescharacteristic
175C22×C61#46111yescharacteristic
176C22×C61#46311yescharacteristic
177C2×D61—42no—
178C2×D61—42no—
179C2×D61—42no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61C4×SD1611yescharacteristic
184C4×C61#44011yescharacteristic
185C4×C61#44311yescharacteristic
186C4×C61#44411yescharacteristic
187C4×C61#45011yescharacteristic
188C4×C61#45211yescharacteristic
189Dic61—42no—
190Dic61—82no—
191Dic61—81no—
192Dic61—81no—
193#3167—611no—
194#2935—611no—
195#1986—611no—
196#1502—611no—
197#1481—611no—
198#1279—611no—
199#1278D6111yescharacteristic, complemented
200C2×C61—21no—
201C2×C61—21no—
202C2×C61#187611yescharacteristic
203C2×C61#187711yescharacteristic
204C2×C61#188411yescharacteristic
205D61—82no—
206C2×#131—1221no—
207#533—611no—
208#532—611no—
209#498—611no—
210#492—611no—
211#487—611no—
212#486—611no—
213#465—1222no—
214C4×SD16—1222no—
215#436—1221no—
216#435—1222no—
217C8×D4—1221no—
218#431C2×D6111yescharacteristic
219C2×C4⋊C8—611no—
220#408—1221no—
221C4×M16C2×D6111yescharacteristic
222C2×C4×C8C2×D6111yescharacteristic
223C4×C4⋊C4—611no—
224#339—1221no—
225#339—1221no—
226#327—1222no—
227Q8⋊C8—1221no—
228Q8⋊C8—1221no—
229C61#905511yescharacteristic, complemented, Sylow
230C2×#41—1221no—
231C2×SD16—1222no—
232#131—2442no—
233#131—2442no—
234C2×M16—1222no—
235C2×M16D24411yescharacteristic
236C2×M16D24411yescharacteristic
237C22×C8—21no—
238C22×C8—1222no—
239C22×C8—1221no—
240C4⋊Q8—611no—
241C4⋊Q8—611no—
242C4⋊Q8—611no—
243#127—611no—
244#125—1221no—
245#124—1221no—
246C22⋊Q8—1221no—
247C22⋊Q8—1221no—
248#121—612no—
249#121—1221no—
250C4×Q8—1221no—
251C4×Q8—1221no—
252C4×D4—612no—
253C4×D4—1221no—
254#117—1221no—
255C2×C4⋊C4—1221no—
256C2×C42—1221no—
257C2×C42C22×D6111yescharacteristic
258#106—1222no—
259C4⋊C8—21no—
260C4⋊C8—611no—
261C4⋊C8—611no—
262C4⋊C8—1221no—
263#103—1222no—
264#102—1222no—
265C22⋊C8—21no—
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268#97—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8C4×D6111yescharacteristic
273C4×C8C4×D6111yescharacteristic
274#95—1221no—
275#41—2441no—
276#41—2441no—
277#41—2441no—
278#41—2441no—
279C2×Q8—1221no—
280C2×Q8—1221no—
281C2×Q8—1221no—
282C2×D4—612no—
283C2×D4—1222no—
284C2×D4—1221no—
285C2×D4—1221no—
286C22×C4—1222no—
287C22×C4—2441no—
288C22×C4C2×D24411yescharacteristic
289C22×C4D4×D6111yescharacteristic
290C22×C4#1175822611yescharacteristic
291SD16—2444no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—2444no—
296M16#1175819012yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—1221no—
304C2×C8—1221no—
305C2×C8—2444no—
306C2×C8—2441no—
307C2×C8—2441no—
308C4⋊C4—612no—
309C4⋊C4—1222no—
310C4⋊C4—1221no—
311C4⋊C4—1221no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C22⋊C4—1222no—
319C22⋊C4—1222no—
320C22⋊C4—2441no—
321C42—2441no—
322C42—2441no—
323C42—2441no—
324C42C2×C4×D6111yescharacteristic
325C42#1175822211yescharacteristic
326C42D4×D6111yescharacteristic
327C42D4×D6111yescharacteristic
328C23—1222no—
329C23#1451975211yescharacteristic
330Q8—2442no—
331Q8—2441no—
332Q8—2441no—
333Q8—2441no—
334Q8—2441no—
335D4—2442no—
336D4—2442no—
337D4—2442no—
338D4—2441no—
339D4—2441no—
340D4—2441no—
341D4—2441no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—1222no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4C22⋊C4×D6111yescharacteristic
354C2×C4#1451973911yescharacteristic
355C2×C4C4×D24411yescharacteristic
356C2×C4#1451974711yescharacteristic
357C2×C4#1451975011yescharacteristic
358C2×C4#1451976211yescharacteristic
359C2×C4C2×#1175819011yescharacteristic
360C2×C4#1451979911yescharacteristic
361C2×C4#1451980511yescharacteristic
362C2×C4#1451981011yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C8—81no—
368C8—2441no—
369C8—2441no—
370C22—21no—
371C22—21no—
372C22—2442no—
373C22—2442no—
374C22—2442no—
375C22#1683964011yescharacteristic
376C22#1683996811yescharacteristic
377C22#1683997811yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—4882no—
382C4—4881no—
383C4—4881no—
384C4—2442no—
385C4C4×#1175819011yescharacteristic
386C4#1683986411yescharacteristic
387C4#1683987811yescharacteristic
388C4#1683988311yescharacteristic
389C4#1683991511yescharacteristic
390C4#1683992611yescharacteristic
391C2—21no—
392C2—21no—
393C2—4882no—
394C2#2144614711yescharacteristic
395C2#2144615211yescharacteristic
396C2#2144620111yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1278C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1278D61
hypocenterC61#9055
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
3. #21446763C2semidirect
5. #21444223C2semidirect
6. #21443659C2semidirect
7. C61×#1278C2semidirect
11. #16840160C4semidirect
12. #16840135C4semidirect
13. #16840113C4semidirect
14. #16840109C4semidirect
95. M16×C61SD16semidirect
199. #1278D61semidirect
229. C61#9055semidirect
296. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214545652
#214467632
#214443152
#214442232
#214436592
C61×#12782
#905561