Finite Group Databasebeta

← #29834759

Subgroups — #29834759

Order 15616.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454737C211yesmaximal, characteristic
2#21447992C214yesmaximal, complemented
3#21447987C214yesmaximal, complemented
4#21447591C211yesmaximal, characteristic, complemented
5C2×#16839584C211yesmaximal, characteristic, complemented
6#21447585C211yesmaximal, characteristic, complemented
7#21447554C211yesmaximal, characteristic
8C2×#16839571C211yesmaximal, characteristic
9C61×#3163C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11#16840528—21no—
12C2×#14519717C2211yescharacteristic
13C2×#14519717C2211yescharacteristic
14#16839885—22no—
15#16839880C2214yes—
16C2×#14519713C2211yescharacteristic
17#16839780—21no—
18#16839584C2212yescomplemented
19#16839584C2214yescomplemented
20#16839584C2212yescomplemented
21#16839583C2214yescomplemented
22#16839571C2212yes—
23#16839571C2214yes—
24#16839571C2212yes—
25C61×#544—21no—
26C61×#501—22no—
27C61×#499C2212yescomplemented
28C61×#498C2212yescomplemented
29C61×#428—21no—
30C61×#427C2211yescharacteristic
31C2×#106×C61C2211yescharacteristic
32C61×#405C2211yescharacteristic
33C2×#11758222—21no—
34#14519799—21no—
35#14519799—21no—
36#14519799Q812yescomplemented
37#14519799C2312yes—
38C2×Dic244Q811yescharacteristic, complemented
39C2×#11758190—21no—
40C2×D488Q811yescharacteristic, complemented
41#14519785—42no—
42#14519783—24no—
43C2×#11758194—21no—
44M16×D61—42no—
45#14519717—21no—
46#14519717—21no—
47#14519717C2312yes—
48#14519717C2312yes—
49C2×#11758185—21no—
50C2×#11758185—21no—
51#14519713—21no—
52#14519713C2312yes—
53#14519705—24no—
54C2×M16×C61D411yescharacteristic
55C22×C8×C61C2311yescharacteristic
56C4⋊Q8×C61—22no—
57C4⋊Q8×C61—41no—
58#125×C61—22no—
59#125×C61—41no—
60C22⋊Q8×C61—42no—
61#117×C61—21no—
62#117×C61D411yescharacteristic
63C2×C4⋊C4×C61—21no—
64#107×C61—21no—
65#107×C61—22no—
66#106×C61—21no—
67#106×C61—22no—
68#106×C61C2312yes—
69#106×C61C2311yescharacteristic
70#106×C61C2311yescharacteristic
71#97×C61D412yes—
72C4×C8×C61C2312yes—
73C2×Q8×C61—42no—
74C22×C4×C61—41no—
75C22×C4×C61C2×D411yescharacteristic
76C2×C22⋊D61—41no—
77#11758222—22no—
78#11758222—22no—
79#11758222—41no—
80#11758222—41no—
81C2×D244—21no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84M16×C61SD1612yes—
85M16×C61SD1612yes—
86C2×C8×C61C2×D411yescharacteristic
87C2×C8×C61C2×D412yes—
88C2×C8×C61C2×Q811yescharacteristic
89C2×C8×C61C2×Q811yescharacteristic
90C2×C8×C61#4111yescharacteristic
91C2×C8×C61#4111yescharacteristic
92C2×C8×C61C2411yescharacteristic
93C4⋊C4×C61—21no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—42no—
98C4⋊C4×C61—42no—
99C4⋊C4×C61—42no—
100C4⋊C4×C61—41no—
101C22⋊C4×C61—21no—
102C22⋊C4×C61—41no—
103C42×C61—41no—
104C42×C61C2×D412yes—
105#11758194—22no—
106#11758194—41no—
107C2×#92418—41no—
108Dic244—21no—
109Dic244C2×Q812yes—
110D488—21no—
111D488C2×Q812yes—
112#11758190—22no—
113#11758190—22no—
114#11758189—44no—
115C8×D61—44no—
116#11758185—22no—
117#11758185—22no—
118C23×C61AGL(1,ℤ/8)11yescharacteristic
119C22×D61—41no—
120Q8×C61—162no—
121C2×C4×C61—22no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—161no—
126C2×C4×C61C22⋊Q811yescharacteristic
127C2×C4×C61C22⋊Q811yescharacteristic
128C2×C4×C61C2×SD1611yescharacteristic
129C2×C4×C61C2×SD1611yescharacteristic
130C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C61C22×D411yescharacteristic
132C22⋊D61—42no—
133C22⋊D61—42no—
134C2×Dic61—41no—
135D244—22no—
136D244—41no—
137C4×D61—42no—
138C4×D61—42no—
139Dic122—22no—
140Dic122—41no—
141C8×C61C22⋊Q811yescharacteristic
142C8×C61C22⋊Q811yescharacteristic
143C8×C61C2×SD1614yes—
144C8×C61C22×Q811yescharacteristic
145C8×C61C2×#4111yescharacteristic
146#92418—42no—
147#33638—611nomaximal, Sylow
148C22×C61—41no—
149C22×C61#47611yescharacteristic
150C22×C61#47611yescharacteristic
151C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
152C2×D61—42no—
153C2×D61—161no—
154C4×C61—42no—
155C4×C61—81no—
156C4×C61—82no—
157C4×C61—81no—
158C4×C61#47611yescharacteristic
159C4×C61#47611yescharacteristic
160C4×C61C2×C22⋊Q811yescharacteristic
161C4×C61C22×SD1611yescharacteristic
162Dic61—42no—
163#3163D6111yescharacteristic, complemented
164#2957—611no—
165#2231—614no—
166#2230—614no—
167#2155—611no—
168C2×#365—611no—
169#2153—611no—
170#2145—611no—
171C2×#360—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61C2×#47611yescharacteristic
176D61—82no—
177C2×#135—611no—
178#568—1221no—
179#544—21no—
180C2×M32—611no—
181C2×M32—611no—
182#501—22no—
183#499C2×D6112yescomplemented
184#498C2×D6112yescomplemented
185#444—1222no—
186#443—614no—
187C2×#108—611no—
188#428—21no—
189#427C2×D6111yescharacteristic
190C2×#106C2×D6111yescharacteristic
191#421—1221no—
192#405C2×D6111yescharacteristic
193#365—614no—
194#365—612no—
195#365—612no—
196#364—614no—
197#360—614no—
198#360—612no—
199#360—612no—
200C61#3363811yescharacteristic, complemented, Sylow
201C2×#41—1221no—
202#135—612no—
203#135—612no—
204#135—1221no—
205#135—1221no—
206C2×Q16—611no—
207C2×SD16—1221no—
208C2×D8—611no—
209#131—1224no—
210#131—2442no—
211C2×M16—1221no—
212C2×M16—2442no—
213C2×M16C22⋊D6111yescharacteristic
214C22×C8C22×D6111yescharacteristic
215C4⋊Q8—22no—
216C4⋊Q8—41no—
217#125—22no—
218#125—41no—
219C22⋊Q8—42no—
220#117—21no—
221#117C22⋊D6111yescharacteristic
222C2×C4⋊C4—21no—
223M32—612no—
224M32—612no—
225M32—1221no—
226M32—1221no—
227C2×C16—1221no—
228C2×C16—1221no—
229#108—612no—
230#108—1221no—
231#107—21no—
232#107—22no—
233#106—21no—
234#106—22no—
235#106C22×D6112yes—
236#106C22×D6111yescharacteristic
237#106C22×D6111yescharacteristic
238#104—1224no—
239#97C22⋊D6112yes—
240C4×C8C22×D6112yes—
241#41—1222no—
242#41—1222no—
243#41—2441no—
244#41—2441no—
245C2×Q8—42no—
246C2×Q8—1221no—
247C2×D4—1221no—
248C2×D4—2441no—
249C22×C4—41no—
250C22×C4—2441no—
251C22×C4C2×C22⋊D6111yescharacteristic
252Q16—612no—
253Q16—1221no—
254SD16—1222no—
255SD16—1222no—
256D8—612no—
257D8—1221no—
258M16—1222no—
259M16—2441no—
260M16—2444no—
261M16#1175820112yes—
262M16#1175820112yes—
263C2×C8—2441no—
264C2×C8—2444no—
265C2×C8Q8×D6111yescharacteristic
266C2×C8Q8×D6111yescharacteristic
267C2×C8#1175822611yescharacteristic
268C2×C8#1175822611yescharacteristic
269C2×C8C2×C22⋊D6111yescharacteristic
270C2×C8C2×C22⋊D6112yes—
271C2×C8C23×D6111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C22⋊C4—21no—
281C22⋊C4—41no—
282C42—41no—
283C42C2×C22⋊D6112yes—
284C16—1222no—
285C16—1222no—
286C23—2441no—
287C23#1451980311yescharacteristic
288Q8—162no—
289Q8—1222no—
290Q8—2441no—
291D4—1222no—
292D4—2442no—
293D4—2442no—
294D4—2441no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—161no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4#1451975311yescharacteristic
304C2×C4#1451975311yescharacteristic
305C2×C4C2×#1175820111yescharacteristic
306C2×C4C2×#1175820111yescharacteristic
307C2×C4#1451980311yescharacteristic
308C2×C4C22×C22⋊D6111yescharacteristic
309C8—2442no—
310C8#1451975311yescharacteristic
311C8#1451975311yescharacteristic
312C8C2×#1175820114yes—
313C8C2×Q8×D6111yescharacteristic
314C8C2×#1175822611yescharacteristic
315C22—41no—
316C22—9761no—
317C22—2442no—
318C22#1684004911yescharacteristic
319C22#1684004911yescharacteristic
320C22C2×#1451980311yescharacteristic
321C4—42no—
322C4—81no—
323C4—82no—
324C4—81no—
325C4—2442no—
326C4#1684004911yescharacteristic
327C4#1684004911yescharacteristic
328C4C2×#1451975311yescharacteristic
329C4C22×#1175820111yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—4882no—
334C2C2×#1684004911yescharacteristic
subgroupG/H
centerC2C2×#16840049
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3163C2
socleC2×C61C2×#476
prime radicalC2×#14519799C22
hypercenter#3163D61
hypocenterC61#33638
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #3163

Complements

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

NH ≅ G/Ntype
2. #21447992C2semidirect
3. #21447987C2semidirect
4. #21447591C2semidirect
5. C2×#16839584C2semidirect
6. #21447585C2semidirect
9. C61×#3163C2semidirect
18. #16839584C22semidirect
19. #16839584C22semidirect
20. #16839584C22semidirect
21. #16839583C22semidirect
27. C61×#499C22semidirect
28. C61×#498C22semidirect
36. #14519799Q8semidirect
38. C2×Dic244Q8semidirect
40. C2×D488Q8semidirect
163. #3163D61semidirect
183. #499C2×D61semidirect
184. #498C2×D61semidirect
200. C61#33638semidirect

Maximal subgroups

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

subgroupindex
#214547372
#214479922
#214479872
#214475912
C2×#168395842
#214475852
#214475542
C2×#168395712
C61×#31632
#3363861