Finite Group Databasebeta

← #16855430

Subgroups — #16855430

Order 3968.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521849C211yesmaximal, characteristic, complemented
2#14521847C211yesmaximal, characteristic, complemented
3C2×#11758527C211yesmaximal, characteristic, complemented
4C2×#11758484C211yesmaximal, characteristic, complemented
5#14521323C212yesmaximal, complemented
6#14521318C212yesmaximal, complemented
7#14521315C212yesmaximal, complemented
8#14521311C212yesmaximal, complemented
9C2×#11758410C211yesmaximal, characteristic, complemented
10#14521031C211yesmaximal, characteristic, complemented
11C31×#413C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#92495C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C2×M16×C31C2211yescharacteristic
16C2×M16×C31C2211yescharacteristic
17#11758528—21no—
18#11758527—22no—
19#11758527—21no—
20#11758527—21no—
21#11758527—21no—
22#11758527—21no—
23#11758526—21no—
24C2×#92482C2211yescharacteristic, complemented
25#11758519C2212yes—
26C2×#92481C2211yescharacteristic, complemented
27C2×#92480C2211yescharacteristic, complemented
28#11758509C2212yes—
29C2×#92479C2211yescharacteristic, complemented
30C2×#92474C2211yescharacteristic
31C22×C31⋊C8C2211yescharacteristic
32#11758486—21no—
33#11758486C2212yes—
34#11758485—21no—
35#11758485C2212yes—
36#11758484—21no—
37#11758484—21no—
38#11758484C2212yes—
39#11758484C2212yes—
40#11758481—22no—
41C2×#92469—21no—
42C2×C8×D31—21no—
43C31×#101C2212yescomplemented
44C31×#100C2212yescomplemented
45#11758412C2212yescomplemented
46#11758411C2212yescomplemented
47#11758410C2212yescomplemented
48#11758410C2212yescomplemented
49#41×C31—21no—
50#41×C31—21no—
51#41×C31—21no—
52#41×C31—21no—
53C2×Q8×C31C2311yescharacteristic
54C2×D4×C31—21no—
55C2×D4×C31C2311yescharacteristic
56C22×C4×C31—21no—
57C22×C4×C31C2311yescharacteristic
58C2×C22⋊D31—21no—
59C2×C22⋊D31—21no—
60#92495—21no—
61#92495—21no—
62#92495—21no—
63#92495—21no—
64#92495—21no—
65#92495—21no—
66#92495D412yes—
67#92495C2312yes—
68C2×D124D411yescharacteristic
69C2×D124C2311yescharacteristic
70C2×C4×D31—21no—
71C2×C4×D31—21no—
72C2×Dic62D411yescharacteristic
73C2×Dic62C2311yescharacteristic
74M16×C31—21no—
75M16×C31—21no—
76M16×C31C2312yes—
77M16×C31C2312yes—
78C2×C8×C31—21no—
79C2×C8×C31—21no—
80#92482—22no—
81#92482—21no—
82#92482—21no—
83#92481—22no—
84#92481—21no—
85#92481—21no—
86#92480—22no—
87#92480—21no—
88#92480—21no—
89#92479—22no—
90#92479—21no—
91#92479—21no—
92#92474—21no—
93#92474C2312yes—
94C2×C31⋊C8—21no—
95C2×C31⋊C8D411yescharacteristic
96C2×C31⋊C8D411yescharacteristic
97C2×C31⋊C8D411yescharacteristic
98C2×C31⋊C8D411yescharacteristic
99C2×C31⋊C8C2311yescharacteristic
100C2×C31⋊C8C2311yescharacteristic
101Dic124—22no—
102D248—22no—
103#92470—22no—
104#92470—22no—
105#92469—22no—
106#92469—22no—
107C8×D31—22no—
108C8×D31—22no—
109C23×C31—21no—
110C23×C31#4111yescharacteristic
111C22×D31—21no—
112C22×D31—21no—
113Q8×C31—21no—
114Q8×C31—21no—
115D4×C31—21no—
116D4×C31—21no—
117D4×C31—41no—
118D4×C31—41no—
119C2×C4×C31—21no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31C2×D411yescharacteristic
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31#4111yescharacteristic
127C2×C4×C31C2411yescharacteristic
128C22⋊D31—22no—
129C22⋊D31—22no—
130C22⋊D31—41no—
131C22⋊D31—41no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134D124—21no—
135D124—21no—
136D124—21no—
137D124C2×D412yes—
138C4×D31—22no—
139C4×D31—22no—
140C4×D31—41no—
141C4×D31—41no—
142Dic62—21no—
143Dic62—21no—
144Dic62—21no—
145Dic62C2×D412yes—
146C8×C31—22no—
147C8×C31—22no—
148C31⋊C8—22no—
149C31⋊C8C2×D411yescharacteristic
150C31⋊C8C2×D411yescharacteristic
151C31⋊C8C2×D411yescharacteristic
152C31⋊C8C2×D411yescharacteristic
153#3067—311nomaximal, Sylow
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—81no—
157C22×C31#12111yescharacteristic
158C22×C31#12111yescharacteristic
159C22×C31C2×#4111yescharacteristic
160C2×D31—22no—
161C2×D31—42no—
162C2×D31—41no—
163C2×D31—81no—
164C4×C31—41no—
165C4×C31#12111yescharacteristic
166C4×C31#12111yescharacteristic
167C4×C31C22×D411yescharacteristic
168C4×C31C22×D411yescharacteristic
169Dic31—22no—
170Dic31—41no—
171#575—311no—
172#575—311no—
173C2×#135—311no—
174C2×#131—311no—
175#473—312no—
176#472—312no—
177#471—312no—
178#471—312no—
179C2×#108—311no—
180#413—311no—
181#413D3111yescharacteristic, complemented
182C2×C31—21no—
183C2×C31—21no—
184C2×C31—21no—
185C2×C31—81no—
186C2×C31C2×#12111yescharacteristic
187D31—42no—
188D31—81no—
189C2×#41—311no—
190C2×#41—311no—
191C2×#41C2×D3111yescharacteristic
192#137—312no—
193#137—312no—
194#137—621no—
195#137—621no—
196AGL(1,ℤ/8)—312no—
197AGL(1,ℤ/8)—312no—
198AGL(1,ℤ/8)—621no—
199AGL(1,ℤ/8)—621no—
200#135—622no—
201#135—621no—
202#135—621no—
203#135—621no—
204#135—621no—
205#135—622no—
206C2×Q16—311no—
207C2×SD16—311no—
208C2×SD16—311no—
209C2×D8—311no—
210#131—312no—
211#131—312no—
212#131—621no—
213#131—621no—
214C2×M16—311no—
215C2×M16—621no—
216C2×M16C2×D3111yescharacteristic
217C2×M16C2×D3111yescharacteristic
218C22×C8—311no—
219C22×C8—621no—
220#108—312no—
221#108—312no—
222#101—312no—
223#101C2×D3112yescomplemented
224#100—312no—
225#100C2×D3112yescomplemented
226C31#306711yescharacteristic, complemented, Sylow
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—312no—
232#41—312no—
233#41—621no—
234#41—621no—
235#41—621no—
236#41—621no—
237#41—621no—
238#41—621no—
239C2×Q8—311no—
240C2×Q8—311no—
241C2×Q8C22×D3111yescharacteristic
242C2×D4—21no—
243C2×D4—311no—
244C2×D4—311no—
245C2×D4—621no—
246C2×D4—621no—
247C2×D4C22×D3111yescharacteristic
248C22×C4—21no—
249C22×C4—621no—
250C22×C4—621no—
251C22×C4C22×D3111yescharacteristic
252Q16—622no—
253Q16—621no—
254Q16—621no—
255Q16—622no—
256SD16—622no—
257SD16—622no—
258SD16—621no—
259SD16—621no—
260SD16—621no—
261SD16—621no—
262SD16—622no—
263SD16—622no—
264D8—622no—
265D8—621no—
266D8—621no—
267D8—622no—
268M16—21no—
269M16—21no—
270M16—312no—
271M16—622no—
272M16—622no—
273M16—621no—
274M16C22×D3112yes—
275M16C22×D3112yes—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—311no—
279C2×C8—311no—
280C2×C8—311no—
281C2×C8—311no—
282C2×C8—311no—
283C2×C8—311no—
284C2×C8—622no—
285C2×C8—622no—
286C2×C8—621no—
287C23—21no—
288C23—621no—
289C23—621no—
290C23#9249511yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—312no—
294Q8—621no—
295Q8—621no—
296Q8—621no—
297D4—21no—
298D4—21no—
299D4—41no—
300D4—41no—
301D4—312no—
302D4—622no—
303D4—622no—
304D4—621no—
305D4—621no—
306D4—621no—
307D4—1241no—
308D4—1241no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—621no—
313C2×C4—622no—
314C2×C4—622no—
315C2×C4—621no—
316C2×C4—1241no—
317C2×C4—1241no—
318C2×C4#9249511yescharacteristic
319C2×C4D4×D3111yescharacteristic
320C2×C4D4×D3111yescharacteristic
321C2×C4D4×D3111yescharacteristic
322C2×C4D4×D3111yescharacteristic
323C2×C4C23×D3111yescharacteristic
324C8—22no—
325C8—22no—
326C8—311no—
327C8—311no—
328C8—311no—
329C8—311no—
330C8—622no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—622no—
335C22—1242no—
336C22—1241no—
337C22—2481no—
338C22#1175846111yescharacteristic
339C22#1175846111yescharacteristic
340C22C2×#9249511yescharacteristic
341C4—41no—
342C4—622no—
343C4—1241no—
344C4#1175846111yescharacteristic
345C4#1175846111yescharacteristic
346C4C2×D4×D3111yescharacteristic
347C4C2×D4×D3111yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—81no—
352C2—1242no—
353C2—2481no—
354C2C2×#1175846111yescharacteristic
subgroupG/H
centerC4#11758461
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#413C2
socleC2×C31C2×#121
hypercenter#413D31
hypocenterC31#3067
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #14521849C2semidirect
2. #14521847C2semidirect
3. C2×#11758527C2semidirect
4. C2×#11758484C2semidirect
5. #14521323C2semidirect
6. #14521318C2semidirect
7. #14521315C2semidirect
8. #14521311C2semidirect
9. C2×#11758410C2semidirect
10. #14521031C2semidirect
11. C31×#413C2semidirect
24. C2×#92482C22semidirect
26. C2×#92481C22semidirect
27. C2×#92480C22semidirect
29. C2×#92479C22semidirect
43. C31×#101C22semidirect
44. C31×#100C22semidirect
45. #11758412C22semidirect
46. #11758411C22semidirect
47. #11758410C22semidirect
48. #11758410C22semidirect
181. #413D31semidirect
223. #101C2×D31semidirect
225. #100C2×D31semidirect
226. C31#3067semidirect

Maximal subgroups

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

subgroupindex
#145218492
#145218472
C2×#117585272
C2×#117584842
#145213232
#145213182
#145213152
#145213112
C2×#117584102
#145210312
C31×#4132
#306731