Finite Group Databasebeta

← #29596976

Subgroups — #29596976

Order 15616.

512 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840032C211yesmaximal, characteristic
2#21454419C211yesmaximal, characteristic, complemented
3#21447536C211yesmaximal, characteristic, complemented
4C2×#16839564C211yesmaximal, characteristic, complemented
5#21443928C211yesmaximal, characteristic, complemented
6#21443915C211yesmaximal, characteristic, complemented
7C2×C61×#350C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C22×D488C2211yescharacteristic
10#16840245—21no—
11#16840032—22no—
12#16840032C414yescomplemented
13C2×#14519713C2211yescharacteristic
14#16839804—22no—
15#16839804—21no—
16#16839800—21no—
17#16839800C2211yescharacteristic
18C2×#14519711—21no—
19C2×#14519711C2211yescharacteristic
20C2×#14519693—21no—
21#16839564—22no—
22#16839564—21no—
23#16839564—21no—
24C2×#131×C61C2211yescharacteristic
25C2×M32×C61C2211yescharacteristic
26C22×C16×C61C2211yescharacteristic
27C61×#350—22no—
28C22×D244—21no—
29#14519802—24no—
30#14519802—42no—
31C2×#11758201—41no—
32C2×D488—21no—
33C2×D488—21no—
34C2×D488—21no—
35C2×D488—21no—
36C2×D488—21no—
37C2×D488C2×C412yescomplemented
38C2×#11758199—41no—
39C2×#11758194—21no—
40Q8×Dic61—41no—
41D4×Dic61—42no—
42D4×Dic61—41no—
43#14519736—22no—
44#14519736—21no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—21no—
47#14519713—21no—
48#14519713C2×C412yescomplemented
49#14519711—22no—
50#14519711—21no—
51#14519711D411yescharacteristic
52#14519711D412yes—
53#14519711D411yescharacteristic
54#14519709—22no—
55#14519709—21no—
56#14519693—24no—
57C2×#41×C61D411yescharacteristic
58#131×C61—21no—
59#131×C61—21no—
60#131×C61C2×C412yes—
61#131×C61D412yes—
62C2×M16×C61—21no—
63C2×M16×C61D411yescharacteristic
64C22×C8×C61—21no—
65C22×C8×C61C2311yescharacteristic
66M32×C61—21no—
67M32×C61—21no—
68C2×C16×C61—21no—
69C2×C16×C61—21no—
70C2×C16×C61—21no—
71C2×C16×C61—21no—
72C23×D61—41no—
73#41×C61—21no—
74#41×C61—21no—
75#41×C61C22⋊C412yes—
76#41×C61SD1612yes—
77C2×Q8×C61D811yescharacteristic
78C2×D4×C61—21no—
79C2×D4×C61D811yescharacteristic
80C22×C4×C61—21no—
81C22×C4×C61C2×D411yescharacteristic
82C22×Dic61—42no—
83C22×Dic61—41no—
84C2×D244—22no—
85C2×D244—41no—
86C2×D244—41no—
87C2×D244—41no—
88C2×D244—41no—
89C2×D244—41no—
90M16×C61—21no—
91M16×C61—22no—
92M16×C61—22no—
93M16×C61C22⋊C412yes—
94C2×C8×C61—21no—
95C2×C8×C61—22no—
96C2×C8×C61—22no—
97C2×C8×C61C22×C411yescharacteristic
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61#4111yescharacteristic
102C2×C8×C61#4111yescharacteristic
103C22⋊Dic61—42no—
104C22⋊Dic61—41no—
105#11758201—44no—
106#11758199—44no—
107C4⋊Dic61—21no—
108C4⋊Dic61—22no—
109C4⋊Dic61—21no—
110C4⋊Dic61—42no—
111C4⋊Dic61—41no—
112C4×Dic61—42no—
113C4×Dic61—41no—
114#11758194—22no—
115#11758194—41no—
116C2×#92418—41no—
117D488—22no—
118D488—22no—
119D488—41no—
120D488—41no—
121C16×C61—41no—
122C16×C61—41no—
123C23×C61—21no—
124C23×C61AGL(1,ℤ/8)11yescharacteristic
125C22×D61—42no—
126C22×D61—81no—
127C22×D61—81no—
128C22×D61—161no—
129C22×D61—161no—
130Q8×C61—21no—
131Q8×C61#10212yes—
132D4×C61—22no—
133D4×C61—22no—
134D4×C61—21no—
135D4×C61#10212yes—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61C2×C22⋊C411yescharacteristic
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×C4×C61C2×D811yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61#13711yescharacteristic
145C2×Dic61—41no—
146C2×Dic61—42no—
147C2×Dic61—41no—
148C2×Dic61—161no—
149C2×Dic61—162no—
150D244—42no—
151D244—42no—
152D244—81no—
153D244—81no—
154C8×C61—22no—
155C8×C61C4×D411yescharacteristic
156C8×C61C4×D411yescharacteristic
157C8×C61#12111yescharacteristic
158C8×C61#12111yescharacteristic
159#92418—42no—
160#12996—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61#41711yescharacteristic
167C22×C61#44711yescharacteristic
168C22×C61#48111yescharacteristic
169C2×D61—82no—
170C2×D61—162no—
171C2×D61—161no—
172C2×D61—161no—
173C2×D61—162no—
174C4×C61—22no—
175C4×C61#38611yescharacteristic
176C4×C61C2×#10211yescharacteristic
177C4×C61#44811yescharacteristic
178C4×C61#48011yescharacteristic
179Dic61—82no—
180Dic61—81no—
181Dic61—81no—
182C2×#472—611no—
183#2913—611no—
184#2140—611no—
185C2×#357—611no—
186C2×#350D6111yescharacteristic, complemented
187#1387—611no—
188#1382—611no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61—42no—
192C2×C61#187411yescharacteristic
193C2×C61#219211yescharacteristic
194C2×C61#219311yescharacteristic
195D61—162no—
196C2×AGL(1,ℤ/8)—1221no—
197C22×D8—611no—
198C2×#131C2×D6111yescharacteristic
199#520—1221no—
200C2×M32C2×D6111yescharacteristic
201C22×C16C2×D6111yescharacteristic
202#472—614no—
203#472—1222no—
204C2×#108—611no—
205#428—1222no—
206#428—1221no—
207#427—611no—
208#427—1221no—
209C2×#107—611no—
210C2×#107—1221no—
211C2×#100—1221no—
212#357—1222no—
213#357—1221no—
214#357—1221no—
215#350—22no—
216C61#1299611yescharacteristic, complemented, Sylow
217C2×#41D24411yescharacteristic
218C22×D4—1221no—
219AGL(1,ℤ/8)—1224no—
220AGL(1,ℤ/8)—2442no—
221C2×SD16—2441no—
222C2×D8—612no—
223C2×D8—1221no—
224C2×D8—1221no—
225C2×D8—1221no—
226C2×D8—1221no—
227C2×D8—1221no—
228C2×D8—2441no—
229#131—21no—
230#131—21no—
231#131C4×D6112yes—
232#131C22⋊D6112yes—
233C2×M16—21no—
234C2×M16—1221no—
235C2×M16D24411yescharacteristic
236C22×C8—21no—
237C22×C8C22×D6111yescharacteristic
238C4×Q8—2441no—
239C4×D4—2442no—
240C4×D4—2441no—
241#117—1222no—
242#117—1221no—
243C2×C4⋊C4—1222no—
244C2×C4⋊C4—1221no—
245M32—21no—
246M32—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251#108—612no—
252#108—1221no—
253#107—612no—
254#107—611no—
255#107—611no—
256#107—1222no—
257#107—1221no—
258#106—1222no—
259#106—1221no—
260#100—1224no—
261C24—2441no—
262#41—21no—
263#41—21no—
264#41#1175819012yes—
265#41#1175819812yes—
266C2×Q8D48811yescharacteristic
267C2×D4—21no—
268C2×D4—1222no—
269C2×D4—2441no—
270C2×D4—2441no—
271C2×D4—2441no—
272C2×D4—2441no—
273C2×D4—2441no—
274C2×D4D48811yescharacteristic
275C22×C4—21no—
276C22×C4—2442no—
277C22×C4—2441no—
278C22×C4C2×D24411yescharacteristic
279SD16—2444no—
280D8—1222no—
281D8—1222no—
282D8—2444no—
283D8—2441no—
284D8—2441no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—1222no—
289M16—2441no—
290M16#1175819812yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—2441no—
295C2×C8C2×C4×D6111yescharacteristic
296C2×C8D4×D6111yescharacteristic
297C2×C8D4×D6111yescharacteristic
298C2×C8#1175822411yescharacteristic
299C2×C8#1175822411yescharacteristic
300C2×C8C2×C22⋊D6111yescharacteristic
301C4⋊C4—1221no—
302C4⋊C4—1222no—
303C4⋊C4—1221no—
304C4⋊C4—2442no—
305C4⋊C4—2441no—
306C22⋊C4—2442no—
307C22⋊C4—2441no—
308C42—2442no—
309C42—2441no—
310C16—41no—
311C16—41no—
312C23—21no—
313C23—9761no—
314C23—9761no—
315C23—4881no—
316C23—4881no—
317C23—2442no—
318C23#1451980511yescharacteristic
319Q8—21no—
320Q8#1451969912yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—4881no—
325D4—4881no—
326D4—2442no—
327D4—2442no—
328D4#1451969912yes—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—9761no—
333C2×C4—9762no—
334C2×C4—2441no—
335C2×C4—2442no—
336C2×C4—2441no—
337C2×C4C2×#1175819811yescharacteristic
338C2×C4#1451974711yescharacteristic
339C2×C4#1451976111yescharacteristic
340C2×C4C2×D48811yescharacteristic
341C2×C4C2×#1175819011yescharacteristic
342C2×C4#1451981011yescharacteristic
343C8—22no—
344C8—2442no—
345C8#1451973711yescharacteristic
346C8#1451973711yescharacteristic
347C8#1451974911yescharacteristic
348C8#1451974911yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—9762no—
355C22—9762no—
356C22—9761no—
357C22—9761no—
358C22—4882no—
359C22#1683976411yescharacteristic
360C22#1683989811yescharacteristic
361C22#1684008111yescharacteristic
362C4—22no—
363C4—4882no—
364C4—4881no—
365C4—4881no—
366C4#1683963911yescharacteristic
367C4C2×#1451969911yescharacteristic
368C4#1683990311yescharacteristic
369C4#1684007611yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—9762no—
374C2#2144613411yescharacteristic
375C2#2144777811yescharacteristic
376C2#2144778511yescharacteristic
subgroupG/H
centerC22#16839898
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#350C2
socleC22×C61#447
prime radicalC2×#16840032C2
cosocleC22×C8×C61C23
hypercenterC2×#350D61
hypocenterC61#12996
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. #21454419C2semidirect
3. #21447536C2semidirect
4. C2×#16839564C2semidirect
5. #21443928C2semidirect
6. #21443915C2semidirect
7. C2×C61×#350C2semidirect
12. #16840032C4semidirect
37. C2×D488C2×C4semidirect
48. #14519713C2×C4semidirect
186. C2×#350D61semidirect
216. C61#12996semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214544192
#214475362
C2×#168395642
#214439282
#214439152
C2×C61×#3502
#1299661