Finite Group Databasebeta

← #29835249

Subgroups — #29835249

Order 15616.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454730C211yesmaximal, characteristic
2#21448036C212yesmaximal, complemented
3#21447987C212yesmaximal, complemented
4#21447897C212yesmaximal, complemented
5#21447769C212yesmaximal
6#21447708C214yesmaximal, complemented
7#21447631C211yesmaximal, characteristic, complemented
8C61×#3185C211yesmaximal, characteristic, complemented
9#16840562C2211yescharacteristic
10#16840528—21no—
11#16840203C2214yes—
12#16839885—21no—
13#16839885C2212yes—
14#16839880—21no—
15#16839880C2212yes—
16#16839811C2211yescharacteristic
17C2×#14519705—21no—
18#16839584C2214yescomplemented
19#16839583C2212yescomplemented
20#16839578C2214yescomplemented
21#16839571C2214yes—
22#16839544C2212yes—
23#16839540C2212yescomplemented
24C61×#533—21no—
25C61×#498C2211yescharacteristic, complemented
26C61×#494C2211yescharacteristic, complemented
27C61×#487—21no—
28C61×#470—21no—
29C61×#470C2212yes—
30C61×#441—21no—
31C61×#441C2212yescomplemented
32C4×M16×C61C2211yescharacteristic
33C2×#11758222—21no—
34#14519810—22no—
35#14519805—22no—
36#14519799—21no—
37#14519799C2312yes—
38#14519785—42no—
39#14519783—22no—
40#14519783—22no—
41C2×#11758194—21no—
42M16×D61—42no—
43#14519723—24no—
44#14519720—24no—
45#14519717C2314yes—
46#14519713—21no—
47#14519713C2312yes—
48#14519705—22no—
49#14519705—22no—
50C2×Q16×C61—21no—
51C2×Q16×C61C2312yes—
52C2×M16×C61D411yescharacteristic
53C2×M16×C61C2311yescharacteristic
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61—21no—
56#125×C61—41no—
57C22⋊Q8×C61—22no—
58C22⋊Q8×C61—41no—
59C4×Q8×C61—22no—
60C4×Q8×C61—41no—
61#117×C61—41no—
62C2×C42×C61D411yescharacteristic
63#106×C61—21no—
64#106×C61C2312yes—
65#103×C61—22no—
66#103×C61—22no—
67#97×C61D411yescharacteristic
68#97×C61C2311yescharacteristic
69C4×C8×C61D411yescharacteristic
70C4×C8×C61C2311yescharacteristic
71C2×Q8×C61—22no—
72C2×Q8×C61—41no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C2×C22⋊D61—41no—
76#11758222—22no—
77#11758222—22no—
78#11758222—41no—
79#11758222—41no—
80C2×D244—21no—
81C2×C4×D61—41no—
82C2×Dic122—21no—
83Q16×C61—24no—
84Q16×C61—42no—
85M16×C61—22no—
86M16×C61—22no—
87C2×C8×C61C2×D411yescharacteristic
88C2×C8×C61C2×D411yescharacteristic
89C2×C8×C61C2×D411yescharacteristic
90C2×C8×C61C2411yescharacteristic
91C4⋊C4×C61—22no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—42no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—42no—
97C22⋊C4×C61—42no—
98C42×C61—21no—
99C42×C61—42no—
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D411yescharacteristic
102#11758194—22no—
103#11758194—41no—
104C2×#92418—41no—
105Dic244—21no—
106Dic244#4112yes—
107D488—21no—
108D488#4112yes—
109#11758190—22no—
110#11758190—22no—
111#11758189—42no—
112#11758189—42no—
113C8×D61—42no—
114C8×D61—42no—
115#11758185—24no—
116C23×C61AGL(1,ℤ/8)11yescharacteristic
117C22×D61—41no—
118Q8×C61—42no—
119Q8×C61—42no—
120Q8×C61—82no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C61C22×D411yescharacteristic
130C22⋊D61—42no—
131C22⋊D61—42no—
132C2×Dic61—41no—
133D244—22no—
134D244—41no—
135C4×D61—42no—
136C4×D61—42no—
137Dic122—22no—
138Dic122—41no—
139C8×C61—21no—
140C8×C61—21no—
141C8×C61AGL(1,ℤ/8)12yes—
142C8×C61C2×#4112yes—
143#92418—42no—
144#33689—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—41no—
147C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
148C2×D61—42no—
149C2×D61—161no—
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—41no—
153C4×C61—82no—
154C4×C61—82no—
155C4×C61#52511yescharacteristic
156C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
157Dic61—42no—
158#3185D6111yescharacteristic, complemented
159#2956—611no—
160#2239—612no—
161#2230—612no—
162#2214—612no—
163#2191—612no—
164#2180—614no—
165#2165—611no—
166C2×C61—21no—
167C2×C61—41no—
168C2×C61#310911yescharacteristic
169D61—82no—
170#575—611no—
171#568—1221no—
172#533—21no—
173#510—614no—
174#498C2×D6111yescharacteristic, complemented
175#494C2×D6111yescharacteristic, complemented
176#487—21no—
177#470—21no—
178#470C2×D6112yes—
179#444—612no—
180#444—1221no—
181#443—612no—
182#443—1221no—
183#441—21no—
184#441C2×D6112yescomplemented
185#430—611no—
186C2×#104—1221no—
187C4×M16C2×D6111yescharacteristic
188#365—614no—
189#364—612no—
190#362—614no—
191#360—614no—
192#349—612no—
193#347—612no—
194C61#3368911yescharacteristic, complemented, Sylow
195C2×#41—1221no—
196#137—1222no—
197AGL(1,ℤ/8)—1222no—
198#135—612no—
199#135—1221no—
200C2×Q16—21no—
201C2×Q16C22×D6112yes—
202#131—1222no—
203#131—1222no—
204#131—2442no—
205C2×M16—1221no—
206C2×M16—2442no—
207C2×M16C22⋊D6111yescharacteristic
208C2×M16C22×D6111yescharacteristic
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211#125—41no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C4×Q8—22no—
215C4×Q8—41no—
216#117—41no—
217C2×C42C22⋊D6111yescharacteristic
218Q32—1224no—
219SD32—1224no—
220M32—614no—
221#108—612no—
222#108—1221no—
223#106—21no—
224#106C22×D6112yes—
225#104—1222no—
226#104—1222no—
227#103—22no—
228#103—22no—
229#97C22⋊D6111yescharacteristic
230#97C22×D6111yescharacteristic
231C4×C8C22⋊D6111yescharacteristic
232C4×C8C22×D6111yescharacteristic
233#41—1222no—
234#41—1222no—
235#41—2441no—
236#41—2441no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—1221no—
240C2×D4—1221no—
241C2×D4—2441no—
242C22×C4—21no—
243C22×C4—2441no—
244C22×C4C2×C22⋊D6111yescharacteristic
245Q16—24no—
246Q16—42no—
247Q16—612no—
248Q16—1221no—
249SD16—1222no—
250SD16—1222no—
251D8—612no—
252D8—1221no—
253M16—22no—
254M16—22no—
255M16—1222no—
256M16—2442no—
257M16—2441no—
258M16—2442no—
259C2×C8—2442no—
260C2×C8—2441no—
261C2×C8—2442no—
262C2×C8C2×C22⋊D6111yescharacteristic
263C2×C8C2×C22⋊D6111yescharacteristic
264C2×C8C2×C22⋊D6111yescharacteristic
265C2×C8C23×D6111yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C22⋊C4—42no—
273C42—21no—
274C42—42no—
275C42C2×C22⋊D6111yescharacteristic
276C42C2×C22⋊D6111yescharacteristic
277C16—1224no—
278C23—2441no—
279C23#1451980311yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—82no—
283Q8—1222no—
284Q8—2441no—
285D4—1222no—
286D4—2441no—
287D4—2442no—
288D4—2442no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—2442no—
297C2×C4—2441no—
298C2×C4—2442no—
299C2×C4#1451980311yescharacteristic
300C2×C4C22×C22⋊D6111yescharacteristic
301C8—21no—
302C8—21no—
303C8—2442no—
304C8#1451980212yes—
305C8#41×D6112yes—
306C22—21no—
307C22—41no—
308C22—9761no—
309C22—2442no—
310C22C2×#1451980311yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—82no—
315C4—82no—
316C4—2442no—
317C4#1684026911yescharacteristic
318C4C2×#1451980211yescharacteristic
319C2—21no—
320C2—41no—
321C2—4882no—
322C2#2145590611yescharacteristic
subgroupG/H
centerC2#21455906
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3185C2
socleC2×C61#3109
prime radical#16840562C22
hypercenter#3185D61
hypocenterC61#33689
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
2. #21448036C2semidirect
3. #21447987C2semidirect
4. #21447897C2semidirect
6. #21447708C2semidirect
7. #21447631C2semidirect
8. C61×#3185C2semidirect
18. #16839584C22semidirect
19. #16839583C22semidirect
20. #16839578C22semidirect
23. #16839540C22semidirect
25. C61×#498C22semidirect
26. C61×#494C22semidirect
31. C61×#441C22semidirect
158. #3185D61semidirect
174. #498C2×D61semidirect
175. #494C2×D61semidirect
184. #441C2×D61semidirect
194. C61#33689semidirect

Maximal subgroups

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

subgroupindex
#214547302
#214480362
#214479872
#214478972
#214477692
#214477082
#214476312
C61×#31852
#3368961