Finite Group Databasebeta

← #29573193

Subgroups — #29573193

Order 15616.

432 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454785C211yesmaximal, characteristic
2#21445840C212yesmaximal
3#21444578C211yesmaximal, characteristic, complemented
4#21443722C212yesmaximal, complemented
5C4×#106×C61C211yesmaximal, characteristic, complemented
6C4×#11758222C2211yescharacteristic
7#16839892C411yescharacteristic, complemented
8#16839892C411yescharacteristic, complemented
9#16839841C411yescharacteristic, complemented
10#16839841C411yescharacteristic, complemented
11#16839833—21no—
12#16839833—21no—
13#16839815C2211yescharacteristic
14#16839813—21no—
15#16839786C2212yes—
16C2×#14519705—22no—
17C4×#11758189—21no—
18#16839522—22no—
19#16839515—21no—
20#16839515—21no—
21C2×#106×C61—21no—
22C2×C4×C8×C61C2211yescharacteristic
23C4×C4⋊C4×C61C2212yes—
24C61×#340—21no—
25C2×#11758222D411yescharacteristic
26C2×#11758194—22no—
27C2×#11758189—21no—
28C2×#11758189—21no—
29C4×Dic122C2×C411yescharacteristic
30C4×Dic122Q811yescharacteristic
31C4×D244C2×C411yescharacteristic
32C4×D244Q811yescharacteristic
33C4×C22⋊D61—21no—
34C4×C22⋊D61—21no—
35#14519736D411yescharacteristic
36#14519734—21no—
37C42×D61—21no—
38C4⋊#92418—22no—
39C4⋊#92418—22no—
40C4⋊#92418C2×C411yescharacteristic
41C4⋊#92418C2×C411yescharacteristic
42#14519706—21no—
43#14519706—21no—
44#14519705—44no—
45C22⋊#92418—21no—
46#14519689—21no—
47#14519689—21no—
48#14519687—21no—
49#14519687—21no—
50C4×#92418—21no—
51C22×C8×C61D411yescharacteristic
52C22×C8×C61D411yescharacteristic
53C2×C4⋊C4×C61—22no—
54C2×C42×C61—22no—
55C2×C42×C61C2311yescharacteristic
56#106×C61—24no—
57C4×C8×C61C2×C411yescharacteristic
58C4×C8×C61C2×C411yescharacteristic
59C4×C8×C61D411yescharacteristic
60C4×C8×C61D411yescharacteristic
61#95×C61—22no—
62C22×C4×C61—42no—
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61#4111yescharacteristic
66C2×C22⋊D61—21no—
67#11758222—22no—
68#11758222—22no—
69C2×D244C4⋊C411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×Dic122C4⋊C411yescharacteristic
74C2×C8×C61—22no—
75C2×C8×C61—41no—
76C2×C8×C61C22⋊C411yescharacteristic
77C2×C8×C61C22⋊C411yescharacteristic
78C2×C8×C61C22⋊C411yescharacteristic
79C2×C8×C61C22⋊C411yescharacteristic
80C4⋊C4×C61—24no—
81C4⋊C4×C61—42no—
82C42×C61—42no—
83C42×C61—42no—
84C42×C61C22×C411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61C2×Q811yescharacteristic
87C42×C61#4111yescharacteristic
88C22⋊Dic61—21no—
89#11758198—21no—
90#11758198—21no—
91C4⋊Dic61C4⋊C411yescharacteristic
92C4⋊Dic61C4⋊C411yescharacteristic
93#11758196—21no—
94#11758196—21no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98#11758194—42no—
99#11758194—42no—
100C2×#92418—21no—
101C2×#92418—21no—
102C2×#92418—42no—
103#11758189—42no—
104#11758189—42no—
105C23×C61#12111yescharacteristic
106C22×D61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—44no—
111C2×C4×C61—82no—
112C2×C4×C61—82no—
113C2×C4×C61C2×C22⋊C411yescharacteristic
114C2×C4×C61C2×C4⋊C411yescharacteristic
115C2×C4×C61#11711yescharacteristic
116C2×C4×C61#12111yescharacteristic
117C2×C4×C61C22⋊Q811yescharacteristic
118C2×C4×C61C22⋊Q811yescharacteristic
119C2×C4×C61AGL(1,ℤ/8)12yes—
120C2×C4×C61#13712yes—
121C22⋊D61—42no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125D244—22no—
126C4×D61—42no—
127C4×D61—41no—
128C4×D61—41no—
129C4×D61—41no—
130C4×D61—41no—
131Dic122—22no—
132C8×C61—21no—
133C8×C61—21no—
134C8×C61—21no—
135C8×C61#10412yescomplemented
136#92418—41no—
137#92418—41no—
138#92418—82no—
139#8669—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#38011yescharacteristic
143C22×C61#46812yes—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—41no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—84no—
151C4×C61C2×#10411yescharacteristic
152C4×C61#42111yescharacteristic
153C4×C61#42812yes—
154C4×C61#46911yescharacteristic
155C4×C61#47011yescharacteristic
156Dic61—41no—
157Dic61—41no—
158Dic61—41no—
159#2965—611no—
160#1806—612no—
161C4×#106D6111yescharacteristic, complemented
162#1567—611no—
163#1303—612no—
164C2×C61—22no—
165C2×C61#180611yescharacteristic
166C2×C61#180911yescharacteristic
167C2×C61#181411yescharacteristic
168D61—81no—
169C4×#41—611no—
170#446—611no—
171#446—611no—
172#436—611no—
173#436—611no—
174#435—1221no—
175#435—1221no—
176#431—611no—
177#431—1221no—
178C2×#106—21no—
179#423—612no—
180C2×#104—1222no—
181C4×M16—1221no—
182C2×C4×C8C2×D6111yescharacteristic
183C4×C4⋊C4C2×D6112yes—
184#340—21no—
185#339—1222no—
186#335—1221no—
187#335—1221no—
188C61#866911yescharacteristic, complemented, Sylow
189C2×#41—611no—
190C2×M16—1221no—
191C2×M16—1221no—
192C2×M16—1222no—
193C22×C8C22⋊D6111yescharacteristic
194C22×C8C22⋊D6111yescharacteristic
195C4×Q8—611no—
196C4×Q8—611no—
197C4×D4—611no—
198C4×D4—611no—
199C4×D4—1221no—
200C4×D4—1221no—
201#117—611no—
202#117—1221no—
203C2×C4⋊C4—22no—
204C2×C42—22no—
205C2×C42—1221no—
206C2×C42C22×D6111yescharacteristic
207#106—24no—
208C4⋊C8—611no—
209C4⋊C8—611no—
210C4⋊C8—1221no—
211C4⋊C8—1221no—
212C4⋊C8—1222no—
213C4⋊C8—1222no—
214#104—2444no—
215C22⋊C8—1221no—
216C22⋊C8—1221no—
217C22⋊C8—1221no—
218#97—1221no—
219#97—1221no—
220C4×C8—1221no—
221C4×C8C4×D6111yescharacteristic
222C4×C8C4×D6111yescharacteristic
223C4×C8D24411yescharacteristic
224C4×C8D24411yescharacteristic
225#95—22no—
226#41—1222no—
227#41—1222no—
228C2×Q8—611no—
229C2×D4—611no—
230C2×D4—1221no—
231C22×C4—42no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4D4×D6111yescharacteristic
236C22×C4#1175822411yescharacteristic
237C22×C4C2×C22⋊D6111yescharacteristic
238M16—2442no—
239M16—2442no—
240M16—2442no—
241M16—2442no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—1221no—
245C2×C8—1221no—
246C2×C8—2442no—
247C2×C8#1175819811yescharacteristic
248C2×C8#1175819811yescharacteristic
249C2×C8#1175819811yescharacteristic
250C2×C8#1175819811yescharacteristic
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—611no—
254C4⋊C4—611no—
255C4⋊C4—1221no—
256C4⋊C4—1221no—
257C22⋊C4—1221no—
258C22⋊C4—1221no—
259C22⋊C4—1221no—
260C42—42no—
261C42—42no—
262C42—1221no—
263C42—1221no—
264C42—1221no—
265C42C2×C4×D6111yescharacteristic
266C42C2×D24411yescharacteristic
267C42Q8×D6111yescharacteristic
268C42#1175822611yescharacteristic
269C23—1221no—
270C23#1451974811yescharacteristic
271Q8—1222no—
272D4—1222no—
273D4—2442no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—2442no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4C2×#1175819811yescharacteristic
289C2×C4C4⋊C4×D6111yescharacteristic
290C2×C4#1451973511yescharacteristic
291C2×C4#1451975211yescharacteristic
292C2×C4#1451975311yescharacteristic
293C2×C4#1451975711yescharacteristic
294C2×C4#1451980412yes—
295C2×C4#1451980612yes—
296C8—21no—
297C8—21no—
298C8—21no—
299C8—4882no—
300C8—2441no—
301C8—2441no—
302C8#1451970512yescomplemented
303C22—21no—
304C22—21no—
305C22—2441no—
306C22—2441no—
307C22—2441no—
308C22#1683961911yescharacteristic
309C22#1684000512yes—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—84no—
314C4—2441no—
315C4—2441no—
316C4—2441no—
317C4C2×#1451970511yescharacteristic
318C4#1683978011yescharacteristic
319C4#1683980112yes—
320C4#1684001611yescharacteristic
321C4#1684002111yescharacteristic
322C2—22no—
323C2—4881no—
324C2#2144584011yescharacteristic
325C2#2144584911yescharacteristic
326C2#2144587011yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC4×#106×C61C2
socleC22×C61#380
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenterC4×#106D61
hypocenterC61#8669
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
3. #21444578C2semidirect
4. #21443722C2semidirect
5. C4×#106×C61C2semidirect
7. #16839892C4semidirect
8. #16839892C4semidirect
9. #16839841C4semidirect
10. #16839841C4semidirect
135. C8×C61#104semidirect
161. C4×#106D61semidirect
188. C61#8669semidirect
302. C8#14519705semidirect

Maximal subgroups

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

subgroupindex
#214547852
#214458402
#214445782
#214437222
C4×#106×C612
#866961