Finite Group Databasebeta

← #16857513

Subgroups — #16857513

Order 3968.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521864C212yesmaximal, complemented
2#14521863C211yesmaximal, characteristic, complemented
3#14521427C211yesmaximal, characteristic, complemented
4#14521324C211yesmaximal, characteristic, complemented
5#14521322C212yesmaximal, complemented
6#14521321C211yesmaximal, characteristic, complemented
7#14521232C212yesmaximal, complemented
8#14521230C212yesmaximal, complemented
9#14521166C211yesmaximal, characteristic, complemented
10#14521164C211yesmaximal, characteristic, complemented
11C31×#578C211yesmaximal, characteristic, complemented
12C31×2−1+4C2212yes—
13#11758567C2212yes—
14C31×#137—22no—
15C31×#137C2212yes—
16C31×#135—21no—
17C31×#135C2211yescharacteristic
18C2×Q16×C31—21no—
19C2×Q16×C31C2211yescharacteristic
20C31×#131C2211yescharacteristic
21#11758528—22no—
22#11758528C2212yes—
23#11758528C2212yes—
24#11758527—22no—
25#11758527C2211yescharacteristic
26#11758525C2212yes—
27#11758521C2212yes—
28C2×#92482—22no—
29C2×#92482C2212yes—
30#11758519—22no—
31#11758519C2212yes—
32Q16×D31—21no—
33#11758492—22no—
34#11758489—21no—
35C2×Dic124C2211yescharacteristic
36#11758478C2211yescharacteristic
37#11758425C2211yescharacteristic
38#11758425C2212yes—
39#11758424C2212yes—
40#11758413C2211yescharacteristic
41#11758412C2212yes—
42#11758410C2211yescharacteristic
43#11758407C2211yescharacteristic
44C8×Dic31C2211yescharacteristic
45#41×C31—22no—
46#41×C31—22no—
47#41×C31—42no—
48#41×C31C2312yes—
49#41×C31C2311yescharacteristic
50C2×Q8×C31—22no—
51C2×Q8×C31—22no—
52C2×Q8×C31C2312yes—
53Q8×D31—22no—
54#92497—22no—
55#92497—42no—
56#92495—22no—
57#92495C2311yescharacteristic
58C2×Dic62—22no—
59C2×Dic62C2312yes—
60Q16×C31—22no—
61Q16×C31—21no—
62Q16×C31—21no—
63Q16×C31D411yescharacteristic
64SD16×C31—22no—
65SD16×C31D412yes—
66D8×C31D411yescharacteristic
67M16×C31—21no—
68M16×C31C2311yescharacteristic
69C2×C8×C31—21no—
70C2×C8×C31C2311yescharacteristic
71#92482—22no—
72#92482—22no—
73#92482—22no—
74#92482—22no—
75#92482—22no—
76#92482D411yescharacteristic
77#92481—22no—
78#92481D411yescharacteristic
79#92480—22no—
80#92480—22no—
81#92480D411yescharacteristic
82#92479D411yescharacteristic
83Dic31⋊C4—42no—
84C4×Dic31C2311yescharacteristic
85#92474—22no—
86#92474C2312yes—
87#92474C2311yescharacteristic
88C2×C31⋊C8—22no—
89C2×C31⋊C8C2311yescharacteristic
90Dic124—21no—
91Dic124—21no—
92#92469—21no—
93C8×D31—21no—
94Q8×C31—22no—
95Q8×C31—22no—
96Q8×C31—42no—
97Q8×C31C2×D411yescharacteristic
98Q8×C31C2×D412yes—
99D4×C31—22no—
100D4×C31—21no—
101D4×C31—42no—
102D4×C31C2×D411yescharacteristic
103D4×C31C2×D412yes—
104C2×C4×C31—22no—
105C2×C4×C31—21no—
106C2×C4×C31—22no—
107C2×C4×C31—42no—
108C2×C4×C31—42no—
109C2×C4×C31C2411yescharacteristic
110C22⋊D31—21no—
111C22⋊D31—42no—
112C2×Dic31—21no—
113C2×Dic31—22no—
114C2×Dic31—42no—
115D124C2×D411yescharacteristic
116C4×D31—21no—
117C4×D31—42no—
118Dic62—22no—
119Dic62—22no—
120Dic62—42no—
121Dic62C2×D411yescharacteristic
122C8×C31—21no—
123C8×C31C2×D411yescharacteristic
124C8×C31C2×D411yescharacteristic
125C31⋊C8—21no—
126C31⋊C8—22no—
127C31⋊C8C2×D411yescharacteristic
128C31⋊C8C2×D411yescharacteristic
129#3293—311nomaximal, Sylow
130C22×C31—22no—
131C22×C31—21no—
132C22×C31H(5,𝔽2)11yescharacteristic
133C2×D31—21no—
134C4×C31—22no—
135C4×C31—21no—
136C4×C31—42no—
137C4×C31C22×D411yescharacteristic
138C4×C31C22×D411yescharacteristic
139Dic31—22no—
140Dic31—21no—
141Dic31—42no—
142#578—312no—
143#578—311no—
144#578D3111yescharacteristic, complemented
145#494—311no—
146#473—312no—
147#473—311no—
148#473—311no—
149#456—312no—
150#456—312no—
151#443—311no—
152#443—311no—
153C2×C31—21no—
154C2×C31—42no—
155C2×C31—41no—
156C2×C31D4211yescharacteristic
157D31—41no—
1582−1+4—312no—
1592−1+4C2×D3112yes—
160#137—22no—
161#137—312no—
162#137—312no—
163#137—312no—
164#137—622no—
165#137—622no—
166#137—622no—
167#137C2×D3112yes—
168#135—21no—
169#135—311no—
170#135—622no—
171#135—621no—
172#135C2×D3111yescharacteristic
173C2×Q16—21no—
174C2×Q16—312no—
175C2×Q16—311no—
176C2×Q16—622no—
177C2×Q16—621no—
178C2×Q16C2×D3111yescharacteristic
179#131—312no—
180#131—311no—
181#131C2×D3111yescharacteristic
182C4⋊Q8—312no—
183#108—311no—
184#108—311no—
185#104—311no—
186#104—312no—
187#104—311no—
188#101—312no—
189#101—312no—
190C4×C8—311no—
191C31#329311yescharacteristic, complemented, Sylow
192#41—22no—
193#41—22no—
194#41—42no—
195#41—311no—
196#41—622no—
197#41—622no—
198#41—1242no—
199#41C22×D3111yescharacteristic
200#41C22×D3112yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—312no—
204C2×Q8—622no—
205C2×Q8—622no—
206C2×Q8C22×D3112yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—311no—
211Q16—622no—
212Q16—622no—
213Q16—622no—
214Q16—622no—
215Q16—622no—
216Q16—621no—
217Q16—621no—
218Q16C22⋊D3111yescharacteristic
219SD16—22no—
220SD16—311no—
221SD16—311no—
222SD16—622no—
223SD16—622no—
224SD16—622no—
225SD16C22⋊D3112yes—
226D8—311no—
227D8C22⋊D3111yescharacteristic
228M16—21no—
229M16—312no—
230M16—311no—
231M16—622no—
232M16—621no—
233M16C22×D3111yescharacteristic
234C2×C8—21no—
235C2×C8—311no—
236C2×C8—622no—
237C2×C8—621no—
238C2×C8C22×D3111yescharacteristic
239C4⋊C4—1242no—
240C42—311no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—311no—
245Q8—622no—
246Q8—622no—
247Q8—1242no—
248Q8D4×D3111yescharacteristic
249Q8C2×C22⋊D3112yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—311no—
254D4—621no—
255D4—1242no—
256D4D4×D3111yescharacteristic
257D4C2×C22⋊D3112yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—621no—
264C2×C4—621no—
265C2×C4—622no—
266C2×C4—1242no—
267C2×C4—1242no—
268C2×C4C23×D3111yescharacteristic
269C8—21no—
270C8—311no—
271C8—311no—
272C8—622no—
273C8—621no—
274C8C2×C22⋊D3111yescharacteristic
275C8C2×C22⋊D3111yescharacteristic
276C22—21no—
277C22—22no—
278C22—621no—
279C22#1175856111yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—622no—
284C4—621no—
285C4—1242no—
286C4C2×D4×D3111yescharacteristic
287C4C22×C22⋊D3111yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—1241no—
292C2D4×C22⋊D3111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#578C2
socleC2×C31D42
hypercenter#578D31
hypocenterC31#3293
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #14521864C2semidirect
2. #14521863C2semidirect
3. #14521427C2semidirect
4. #14521324C2semidirect
5. #14521322C2semidirect
6. #14521321C2semidirect
7. #14521232C2semidirect
8. #14521230C2semidirect
9. #14521166C2semidirect
10. #14521164C2semidirect
11. C31×#578C2semidirect
144. #578D31semidirect
191. C31#3293semidirect

Maximal subgroups

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

subgroupindex
#145218642
#145218632
#145214272
#145213242
#145213222
#145213212
#145212322
#145212302
#145211662
#145211642
C31×#5782
#329331