Finite Group Databasebeta

← #16855428

Subgroups — #16855428

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#14521847C211yesmaximal, characteristic, complemented
2C2×#11758527C211yesmaximal, characteristic, complemented
3C2×#11758525C211yesmaximal, characteristic, complemented
4#14521322C212yesmaximal, complemented
5#14521317C212yesmaximal, complemented
6#14521314C212yesmaximal, complemented
7#14521310C212yesmaximal, complemented
8C2×#11758410C211yesmaximal, characteristic, complemented
9#14521033C211yesmaximal, characteristic, complemented
10#14521031C211yesmaximal, characteristic, complemented
11C31×#575C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C2×#41×C31C2211yescharacteristic
14C2×#92495C2211yescharacteristic
15C31×#137—21no—
16C31×#137C2212yes—
17C31×AGL(1,ℤ/8)—21no—
18C31×AGL(1,ℤ/8)C2212yes—
19C31×#135—22no—
20C2×M16×C31C2211yescharacteristic
21#11758528C2212yes—
22#11758527—22no—
23#11758527—21no—
24#11758527—21no—
25#11758527—21no—
26#11758527—21no—
27#11758526C2212yes—
28#11758525—21no—
29#11758525—21no—
30#11758525C2212yes—
31#11758525C2212yes—
32C2×#92482C2211yescharacteristic, complemented
33#11758519—21no—
34C2×#92481C2211yescharacteristic, complemented
35C2×#92480C2211yescharacteristic, complemented
36#11758509—21no—
37C2×#92479C2211yescharacteristic, complemented
38C2×#92474—21no—
39C2×#92474C2211yescharacteristic
40C2×#92474C2211yescharacteristic
41C22×C31⋊C8—21no—
42C22×C31⋊C8C2211yescharacteristic
43#11758424C2212yescomplemented
44#11758421C2212yescomplemented
45#11758412C2212yescomplemented
46#11758411C2212yescomplemented
47#11758410C2212yescomplemented
48#11758410C2212yescomplemented
49#41×C31—21no—
50#41×C31—21no—
51#41×C31—21no—
52#41×C31—21no—
53#41×C31—21no—
54#41×C31—21no—
55#41×C31D412yes—
56#41×C31C2312yes—
57C2×Q8×C31D411yescharacteristic
58C2×Q8×C31C2311yescharacteristic
59C2×D4×C31—21no—
60C2×D4×C31—21no—
61C2×D4×C31D411yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31C2311yescharacteristic
66C2×C22⋊D31—21no—
67#92495—21no—
68#92495—21no—
69#92495—21no—
70#92495—21no—
71C2×D124C2311yescharacteristic
72C2×C4×D31—21no—
73C2×Dic62C2311yescharacteristic
74Q16×C31—22no—
75SD16×C31—22no—
76SD16×C31—22no—
77D8×C31—22no—
78M16×C31—21no—
79M16×C31C2312yes—
80C2×C8×C31—21no—
81#92482—22no—
82#92482—21no—
83#92482—21no—
84#92481—22no—
85#92481—21no—
86#92481—21no—
87#92480—22no—
88#92480—21no—
89#92480—21no—
90#92479—22no—
91#92479—21no—
92#92479—21no—
93#92474—21no—
94#92474—21no—
95#92474—22no—
96#92474—22no—
97#92474C2312yes—
98#92474C2312yes—
99C2×C31⋊C8—21no—
100C2×C31⋊C8—21no—
101C2×C31⋊C8—22no—
102C2×C31⋊C8—22no—
103C2×C31⋊C8D411yescharacteristic
104C2×C31⋊C8D411yescharacteristic
105C2×C31⋊C8D411yescharacteristic
106C2×C31⋊C8D411yescharacteristic
107C2×C31⋊C8C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C23×C31—21no—
110C23×C31—21no—
111C23×C31#4111yescharacteristic
112C22×D31—21no—
113Q8×C31—21no—
114Q8×C31—21no—
115Q8×C31—21no—
116Q8×C31C2×D412yes—
117D4×C31—22no—
118D4×C31—22no—
119D4×C31—21no—
120D4×C31—21no—
121D4×C31—21no—
122D4×C31—41no—
123D4×C31—41no—
124D4×C31C2×D412yes—
125C2×C4×C31—21no—
126C2×C4×C31—22no—
127C2×C4×C31—22no—
128C2×C4×C31—21no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D411yescharacteristic
133C2×C4×C31C2×D411yescharacteristic
134C2×C4×C31C2×D411yescharacteristic
135C2×C4×C31#4111yescharacteristic
136C2×C4×C31C2411yescharacteristic
137C22⋊D31—41no—
138C22⋊D31—41no—
139C2×Dic31—21no—
140D124—21no—
141D124—21no—
142C4×D31—41no—
143C4×D31—41no—
144Dic62—21no—
145Dic62—21no—
146C8×C31—22no—
147C31⋊C8—22no—
148C31⋊C8—22no—
149C31⋊C8C2×D411yescharacteristic
150C31⋊C8C2×D411yescharacteristic
151C31⋊C8C2×D411yescharacteristic
152C31⋊C8C2×D411yescharacteristic
153#3067—311nomaximal, Sylow
154C22×C31—22no—
155C22×C31—41no—
156C22×C31—42no—
157C22×C31—41no—
158C22×C31—81no—
159C22×C31#12111yescharacteristic
160C22×C31#12111yescharacteristic
161C22×C31C2×#4111yescharacteristic
162C2×D31—41no—
163C2×D31—81no—
164C4×C31—22no—
165C4×C31—41no—
166C4×C31#12111yescharacteristic
167C4×C31#12111yescharacteristic
168C4×C31C22×D411yescharacteristic
169C4×C31C22×D411yescharacteristic
170Dic31—41no—
171#575—311no—
172#575D3111yescharacteristic, complemented
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#413—311no—
182C2×C31—21no—
183C2×C31—21no—
184C2×C31—21no—
185C2×C31—42no—
186C2×C31—81no—
187C2×C31C2×#12111yescharacteristic
188D31—81no—
189C2×#41—311no—
190C2×#41C2×D3111yescharacteristic
191C2×#41C2×D3111yescharacteristic
192#137—21no—
193#137—312no—
194#137—621no—
195#137C2×D3112yes—
196AGL(1,ℤ/8)—21no—
197AGL(1,ℤ/8)—312no—
198AGL(1,ℤ/8)—621no—
199AGL(1,ℤ/8)C2×D3112yes—
200#135—22no—
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—311no—
216C2×M16—621no—
217C2×M16C2×D3111yescharacteristic
218C22×C8—311no—
219C22×C8—621no—
220#108—312no—
221#108—312no—
222#101—312no—
223#101—312no—
224#100—312no—
225#100—312no—
226C31#306711yescharacteristic, complemented, Sylow
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—621no—
234#41—621no—
235#41—621no—
236#41—621no—
237#41C22⋊D3112yes—
238#41C22×D3112yes—
239C2×Q8—311no—
240C2×Q8C22⋊D3111yescharacteristic
241C2×Q8C22×D3111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—311no—
245C2×D4—621no—
246C2×D4C22⋊D3111yescharacteristic
247C2×D4C22×D3111yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—621no—
251C22×C4C22×D3111yescharacteristic
252Q16—22no—
253Q16—621no—
254Q16—621no—
255Q16—622no—
256SD16—22no—
257SD16—22no—
258SD16—621no—
259SD16—621no—
260SD16—621no—
261SD16—621no—
262SD16—622no—
263SD16—622no—
264D8—22no—
265D8—621no—
266D8—621no—
267D8—622no—
268M16—21no—
269M16—312no—
270M16—312no—
271M16—622no—
272M16—622no—
273M16—621no—
274M16—621no—
275M16C22×D3112yes—
276C2×C8—21no—
277C2×C8—311no—
278C2×C8—311no—
279C2×C8—311no—
280C2×C8—311no—
281C2×C8—311no—
282C2×C8—311no—
283C2×C8—622no—
284C2×C8—622no—
285C2×C8—621no—
286C2×C8—621no—
287C23—21no—
288C23—21no—
289C23—621no—
290C23#9249711yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—621no—
295Q8—621no—
296Q8C2×C22⋊D3112yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—621no—
305D4—621no—
306D4—1241no—
307D4—1241no—
308D4C2×C22⋊D3112yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—621no—
316C2×C4—1241no—
317C2×C4—1241no—
318C2×C4D4×D3111yescharacteristic
319C2×C4D4×D3111yescharacteristic
320C2×C4#9249711yescharacteristic
321C2×C4C2×C22⋊D3111yescharacteristic
322C2×C4C2×C22⋊D3111yescharacteristic
323C2×C4C23×D3111yescharacteristic
324C8—22no—
325C8—311no—
326C8—311no—
327C8—311no—
328C8—311no—
329C8—622no—
330C8—622no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—1241no—
337C22—2481no—
338C22#1175851611yescharacteristic
339C22#1175851611yescharacteristic
340C22C2×#9249711yescharacteristic
341C4—22no—
342C4—41no—
343C4—1241no—
344C4#1175851611yescharacteristic
345C4#1175851611yescharacteristic
346C4C2×D4×D3111yescharacteristic
347C4C22×C22⋊D3111yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—2481no—
354C2C2×#1175851611yescharacteristic
subgroupG/H
centerC4#11758516
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#575C2
socleC2×C31C2×#121
hypercenter#575D31
hypocenterC31#3067
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #14521847C2semidirect
2. C2×#11758527C2semidirect
3. C2×#11758525C2semidirect
4. #14521322C2semidirect
5. #14521317C2semidirect
6. #14521314C2semidirect
7. #14521310C2semidirect
8. C2×#11758410C2semidirect
9. #14521033C2semidirect
10. #14521031C2semidirect
11. C31×#575C2semidirect
32. C2×#92482C22semidirect
34. C2×#92481C22semidirect
35. C2×#92480C22semidirect
37. C2×#92479C22semidirect
43. #11758424C22semidirect
44. #11758421C22semidirect
45. #11758412C22semidirect
46. #11758411C22semidirect
47. #11758410C22semidirect
48. #11758410C22semidirect
172. #575D31semidirect
226. C31#3067semidirect

Maximal subgroups

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

subgroupindex
#145218472
C2×#117585272
C2×#117585252
#145213222
#145213172
#145213142
#145213102
C2×#117584102
#145210332
#145210312
C31×#5752
#306731