Finite Group Databasebeta

← #16856301

Subgroups — #16856301

Order 3968.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521594C211yesmaximal, characteristic, complemented
2#14521580C211yesmaximal, characteristic, complemented
3#14521456C211yesmaximal, characteristic, complemented
4#14521437C211yesmaximal, characteristic, complemented
5#14521412C211yesmaximal, characteristic, complemented
6#14521407C211yesmaximal, characteristic, complemented
7#14521300C211yesmaximal, characteristic, complemented
8#14521295C211yesmaximal, characteristic, complemented
9#14521293C211yesmaximal, characteristic, complemented
10#14521140C211yesmaximal, characteristic, complemented
11#14521136C211yesmaximal, characteristic, complemented
12D8×Dic31C211yesmaximal, characteristic, complemented
13#14521129C211yesmaximal, characteristic, complemented
14M16×Dic31C211yesmaximal, characteristic, complemented
15C31×#469C211yesmaximal, characteristic, complemented
16C2×#92497—21no—
17C2×D8×C31C2211yescharacteristic
18C2×M16×C31C2211yescharacteristic
19C31×#121C2211yescharacteristic
20C31×#121C2211yescharacteristic
21#11758517C2211yescharacteristic
22#11758516—21no—
23#11758513C2211yescharacteristic
24#11758512—21no—
25D4×Dic31—21no—
26D4×Dic31C2211yescharacteristic
27D4×Dic31C2211yescharacteristic
28C2×#92480C2211yescharacteristic
29C2×#92479C2211yescharacteristic
30#11758507C2211yescharacteristic
31#11758501C2211yescharacteristic
32C2×C4×Dic31C2211yescharacteristic
33C2×#92474C2211yescharacteristic
34C2×#92470C2211yescharacteristic
35#11758470C2211yescharacteristic
36#11758467C2211yescharacteristic
37#11758465C2211yescharacteristic
38#11758464C2211yescharacteristic
39#11758462—21no—
40(C22⋊D31)⋊C4—21no—
41#11758456—21no—
42#11758454—21no—
43C31×#106C2211yescharacteristic
44C31×#102C2211yescharacteristic
45C31×#102C2211yescharacteristic
46#11758420C2211yescharacteristic
47#11758420C2211yescharacteristic
48#11758409C2211yescharacteristic
49#11758406C2211yescharacteristic
50#11758404C2211yescharacteristic
51#11758403C2211yescharacteristic
52#11758403C2211yescharacteristic
53C8×Dic31C2211yescharacteristic
54C8×Dic31C2211yescharacteristic
55#11758398C2211yescharacteristic
56#11758397C2211yescharacteristic
57#11758396C2211yescharacteristic
58#11758395C2211yescharacteristic
59C2×D4×C31—21no—
60C2×D4×C31—21no—
61C2×D4×C31C2311yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C2×C22⋊D31—21no—
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67C22×Dic31—21no—
68C22×Dic31—21no—
69C22×Dic31D411yescharacteristic
70C22×Dic31D411yescharacteristic
71#92497—44no—
72C2×D124C2311yescharacteristic
73C2×C4×D31—21no—
74C2×Dic62—21no—
75C2×Dic62C2311yescharacteristic
76D8×C31—22no—
77M16×C31—22no—
78C2×C8×C31C2311yescharacteristic
79C2×C8×C31C2311yescharacteristic
80C4⋊C4×C31C2311yescharacteristic
81C4⋊C4×C31C2311yescharacteristic
82C22⋊C4×C31—21no—
83C22⋊C4×C31—21no—
84C22⋊Dic31—21no—
85C22⋊Dic31—21no—
86C22⋊Dic31—21no—
87C22⋊Dic31—21no—
88C22⋊Dic31—21no—
89#92480—22no—
90#92479—22no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93C4⋊Dic31—21no—
94C4⋊Dic31C2311yescharacteristic
95C4⋊Dic31C2311yescharacteristic
96Dic31⋊C4—21no—
97Dic31⋊C4—21no—
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100C4×Dic31—21no—
101C4×Dic31D411yescharacteristic
102C4×Dic31D411yescharacteristic
103C4×Dic31C2311yescharacteristic
104C4×Dic31C2311yescharacteristic
105#92474—22no—
106C2×C31⋊C8C2311yescharacteristic
107C2×C31⋊C8C2311yescharacteristic
108#92470—22no—
109C23×C31—21no—
110C23×C31—21no—
111C23×C31C2×D411yescharacteristic
112C22×D31—21no—
113D4×C31—21no—
114D4×C31—21no—
115D4×C31—21no—
116D4×C31—21no—
117D4×C31—42no—
118D4×C31—42no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2411yescharacteristic
125C22⋊D31—42no—
126C22⋊D31—42no—
127C22⋊D31—42no—
128C2×Dic31—22no—
129C2×Dic31—21no—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—41no—
133C2×Dic31—42no—
134C2×Dic31—42no—
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137C2×Dic31C2×D411yescharacteristic
138C2×Dic31C2×D411yescharacteristic
139D124—21no—
140D124—21no—
141C4×D31—42no—
142Dic62—21no—
143Dic62—21no—
144Dic62—42no—
145C8×C31—21no—
146C8×C31#4112yescomplemented
147C31⋊C8—21no—
148C31⋊C8#4112yescomplemented
149#3171—311nomaximal, Sylow
150C22×C31—21no—
151C22×C31—21no—
152C22×C31—21no—
153C22×C31—41no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31C22×D411yescharacteristic
160C2×D31—41no—
161C2×D31—41no—
162C2×D31—41no—
163C4×C31—21no—
164C4×C31—41no—
165C4×C31—41no—
166C4×C31C2×#4111yescharacteristic
167C4×C31C2×#4111yescharacteristic
168Dic31—21no—
169Dic31—21no—
170Dic31—21no—
171Dic31—41no—
172Dic31—41no—
173Dic31—41no—
174Dic31AGL(1,ℤ/8)12yes—
175#532—311no—
176#529—311no—
177#501—311no—
178#496—311no—
179#490—311no—
180#489—311no—
181#469—311no—
182#469D3111yescharacteristic, complemented
183#468—311no—
184#468—311no—
185C4×SD16—311no—
186C4×SD16—311no—
187C4×D8—311no—
188C4×D8—311no—
189C4×M16—311no—
190C2×C31—41no—
191C2×C31—81no—
192C2×C31—81no—
193C2×C31#53211yescharacteristic
194C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
195C2×C31#57511yescharacteristic
196D31—81no—
197C2×#41—621no—
198C2×SD16—311no—
199C2×SD16—311no—
200C2×D8—311no—
201C2×D8C2×D3111yescharacteristic
202C2×M16—311no—
203C2×M16C2×D3111yescharacteristic
204C4⋊Q8—311no—
205#127—311no—
206#126—621no—
207#125—311no—
208#124—311no—
209#124—621no—
210#123—621no—
211C22⋊Q8—311no—
212#121—311no—
213#121—621no—
214#121C2×D3111yescharacteristic
215#121C2×D3111yescharacteristic
216C4×Q8—311no—
217C4×D4—311no—
218C4×D4—311no—
219C4×D4—311no—
220C4×D4—621no—
221C4×D4—621no—
222#117—621no—
223C2×C42—311no—
224#107—311no—
225#107—311no—
226#106—311no—
227#106C2×D3111yescharacteristic
228#103—311no—
229#103—311no—
230#102—311no—
231#102—311no—
232#102—311no—
233#102—311no—
234#102C2×D3111yescharacteristic
235#102C2×D3111yescharacteristic
236#97—311no—
237#97—311no—
238C4×C8—311no—
239C4×C8—311no—
240C31#317111yescharacteristic, complemented, Sylow
241#41—1244no—
242C2×Q8—311no—
243C2×Q8—621no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—311no—
247C2×D4—621no—
248C2×D4—621no—
249C2×D4—621no—
250C2×D4C22×D3111yescharacteristic
251C2×D4C22×D3111yescharacteristic
252C22×C4—311no—
253C22×C4—311no—
254C22×C4—621no—
255C22×C4—621no—
256C22×C4—621no—
257C22×C4C22×D3111yescharacteristic
258SD16—622no—
259SD16—622no—
260D8—22no—
261D8—622no—
262M16—22no—
263M16—622no—
264C2×C8—311no—
265C2×C8—311no—
266C2×C8C22×D3111yescharacteristic
267C2×C8C22×D3111yescharacteristic
268C4⋊C4—311no—
269C4⋊C4—311no—
270C4⋊C4—621no—
271C4⋊C4—621no—
272C4⋊C4—621no—
273C4⋊C4—621no—
274C4⋊C4—621no—
275C4⋊C4C22×D3111yescharacteristic
276C4⋊C4C22×D3111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—621no—
280C22⋊C4—621no—
281C22⋊C4—621no—
282C22⋊C4—621no—
283C22⋊C4—621no—
284C22⋊C4—621no—
285C22⋊C4—621no—
286C42—311no—
287C42—311no—
288C42—311no—
289C42—311no—
290C42—621no—
291C23—21no—
292C23—21no—
293C23—621no—
294C23D4×D3111yescharacteristic
295Q8—621no—
296Q8—621no—
297Q8—1242no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—42no—
304D4—621no—
305D4—621no—
306D4—1242no—
307D4—1242no—
308D4—1242no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—311no—
314C2×C4—311no—
315C2×C4—311no—
316C2×C4—311no—
317C2×C4—622no—
318C2×C4—621no—
319C2×C4—621no—
320C2×C4—621no—
321C2×C4—1242no—
322C2×C4—1242no—
323C2×C4—1242no—
324C2×C4—1241no—
325C2×C4D4×D3111yescharacteristic
326C2×C4C23×D3111yescharacteristic
327C8—21no—
328C8—312no—
329C8—621no—
330C8#9249712yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—1241no—
341C22—1241no—
342C22—1241no—
343C22C2×D4×D3111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—312no—
348C4—621no—
349C4—621no—
350C4—621no—
351C4—1241no—
352C4—1241no—
353C4—1241no—
354C4C2×#9249711yescharacteristic
355C4#41×D3111yescharacteristic
356C2—41no—
357C2—81no—
358C2—81no—
359C2—2481no—
360C2#1452159411yescharacteristic
361C2AGL(1,ℤ/8)×D3111yescharacteristic
362C2#1452184511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#469C2
socleC22×C31C22×D4
hypercenter#469D31
hypocenterC31#3171
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #14521594C2semidirect
2. #14521580C2semidirect
3. #14521456C2semidirect
4. #14521437C2semidirect
5. #14521412C2semidirect
6. #14521407C2semidirect
7. #14521300C2semidirect
8. #14521295C2semidirect
9. #14521293C2semidirect
10. #14521140C2semidirect
11. #14521136C2semidirect
12. D8×Dic31C2semidirect
13. #14521129C2semidirect
14. M16×Dic31C2semidirect
15. C31×#469C2semidirect
146. C8×C31#41semidirect
148. C31⋊C8#41semidirect
182. #469D31semidirect
240. C31#3171semidirect
330. C8#92497semidirect

Maximal subgroups

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

subgroupindex
#145215942
#145215802
#145214562
#145214372
#145214122
#145214072
#145213002
#145212952
#145212932
#145211402
#145211362
D8×Dic312
#145211292
M16×Dic312
C31×#4692
#317131