Finite Group Databasebeta

← #16854549

Subgroups — #16854549

Order 3968.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758495C211yesmaximal, characteristic
2#14521531C211yesmaximal, characteristic
3#14521525C211yesmaximal, characteristic, complemented
4#14521156C211yesmaximal, characteristic, complemented
5#14521154C211yesmaximal, characteristic, complemented
6#14521148C211yesmaximal, characteristic, complemented
7#14521147C211yesmaximal, characteristic, complemented
8#14521141C211yesmaximal, characteristic, complemented
9C4×#92470C211yesmaximal, characteristic, complemented
10#14521134C211yesmaximal, characteristic, complemented
11#14521090C211yesmaximal, characteristic, complemented
12#14521044C211yesmaximal, characteristic
13#103×D31C211yesmaximal, characteristic, complemented
14#14521003C211yesmaximal, characteristic, complemented
15C4×Q16×C31C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic
18C2×Q16×C31C2211yescharacteristic
19C4×Q8×C31C2211yescharacteristic
20C4×Q8×C31C2211yescharacteristic
21Q8×Dic31C2211yescharacteristic
22Q8×Dic31C2211yescharacteristic
23C2×#92482C2211yescharacteristic
24C2×#92481C2211yescharacteristic
25#11758495C418yescomplemented
26C2×#92470C2211yescharacteristic
27C2×#92469C2211yescharacteristic
28C2×C8×D31C2211yescharacteristic
29#11758470—21no—
30#11758470C2211yescharacteristic
31#11758469—21no—
32#11758469C2211yescharacteristic
33C4⋊C4×D31—21no—
34C4⋊C4×D31C2211yescharacteristic
35#11758464—21no—
36#11758464C2211yescharacteristic
37C4×D124—21no—
38C4×D124C2211yescharacteristic
39#11758449—21no—
40#11758449—21no—
41#11758449C2211yescharacteristic
42C4×Dic62—21no—
43C4×Dic62C2211yescharacteristic
44C31×#107C2211yescharacteristic
45C31×#103C2211yescharacteristic
46C31×#103C2211yescharacteristic
47C4×C8×C31C2211yescharacteristic
48#11758423C2211yescharacteristic
49#11758423C2211yescharacteristic
50#11758409C2211yescharacteristic
51#11758405C2211yescharacteristic
52#11758404C2211yescharacteristic
53#11758403C2211yescharacteristic
54C8×Dic31C2211yescharacteristic
55#11758398C2211yescharacteristic
56#11758397C2211yescharacteristic
57#11758396C2211yescharacteristic
58#11758395C2211yescharacteristic
59#11758391C2211yescharacteristic
60C2×Q8×C31C2311yescharacteristic
61C2×Q8×C31C2311yescharacteristic
62#92499—22no—
63#92499C2×C414yes—
64Q8×D31—22no—
65Q8×D31C2×C414yescomplemented
66C2×D124—21no—
67C2×D124C2311yescharacteristic
68C2×C4×D31—21no—
69C2×C4×D31—21no—
70C2×C4×D31—21no—
71C2×C4×D31—21no—
72C2×C4×D31—21no—
73C2×C4×D31C2311yescharacteristic
74C2×Dic62—21no—
75C2×Dic62C2311yescharacteristic
76Q16×C31C2×C414yescomplemented
77C2×C8×C31C2311yescharacteristic
78C2×C8×C31C2311yescharacteristic
79C4⋊C4×C31—21no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31C2311yescharacteristic
82C4⋊C4×C31C2311yescharacteristic
83C42×C31—21no—
84C42×C31—21no—
85C42×C31C2311yescharacteristic
86#92482C2×C414yescomplemented
87#92481C2×C414yescomplemented
88(C2×D31)⋊C4—21no—
89(C2×D31)⋊C4—21no—
90(C2×D31)⋊C4—21no—
91(C2×D31)⋊C4D411yescharacteristic
92(C2×D31)⋊C4D411yescharacteristic
93C4⋊Dic31—21no—
94C4⋊Dic31—21no—
95C4⋊Dic31C2311yescharacteristic
96C4⋊Dic31C2311yescharacteristic
97Dic31⋊C4—21no—
98Dic31⋊C4—21no—
99Dic31⋊C4—21no—
100Dic31⋊C4D411yescharacteristic
101Dic31⋊C4D411yescharacteristic
102C4×Dic31—21no—
103C4×Dic31—21no—
104C4×Dic31—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110#92470C2×C414yescomplemented
111#92469C2×C414yescomplemented
112C8×D31—22no—
113C22×D31—21no—
114C22×D31C2×D411yescharacteristic
115Q8×C31—21no—
116Q8×C31—21no—
117Q8×C31C22×C412yes—
118Q8×C31C22×C412yes—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31C2×D411yescharacteristic
124C2×C4×C31C2×D411yescharacteristic
125C2×C4×C31C2411yescharacteristic
126C2×Dic31—21no—
127C2×Dic31—21no—
128C2×Dic31—21no—
129C2×Dic31C2×D411yescharacteristic
130C2×Dic31C2×D411yescharacteristic
131C2×Dic31C2×D411yescharacteristic
132D124—24no—
133D124—21no—
134D124C22×C412yes—
135C4×D31—24no—
136C4×D31—24no—
137C4×D31—81no—
138C4×D31—81no—
139C4×D31—81no—
140C4×D31C22×C412yes—
141C4×D31#4112yes—
142Dic62—21no—
143Dic62—24no—
144Dic62C22×C412yes—
145C8×C31—21no—
146C8×C31C22×C412yes—
147C31⋊C8—21no—
148C31⋊C8C22×C412yes—
149#2947—311nomaximal, Sylow
150C22×C31C22×D411yescharacteristic
151C2×D31—21no—
152C2×D31—21no—
153C2×D31—22no—
154C2×D31—41no—
155C2×D31—41no—
156C2×D31C4×D412yes—
157C4×C31—22no—
158C4×C31—22no—
159C4×C31—21no—
160C4×C31—21no—
161C4×C31—41no—
162C4×C31—41no—
163C4×C31C23×C411yescharacteristic
164C4×C31C2×#4111yescharacteristic
165Dic31—22no—
166Dic31—21no—
167Dic31—21no—
168Dic31—41no—
169Dic31—41no—
170Dic31C4×D412yes—
171C2×#137—311no—
172#520—311no—
173#519—311no—
174#441—311no—
175#441—311no—
176#440—311no—
177#440—311no—
178C4×Q16—311no—
179C4×Q16D3111yescharacteristic, complemented
180C4×SD16—311no—
181C4×SD16—311no—
182#428—311no—
183#416—311no—
184C2×#103—311no—
185#405—311no—
186C2×C31C2×C4×D411yescharacteristic
187C2×C31#57711yescharacteristic
188C2×C31#57811yescharacteristic
189D31—22no—
190D31—42no—
191C2×#41—311no—
192C22×Q8—311no—
193#137—318no—
194C2×Q16—311no—
195C2×Q16C2×D3111yescharacteristic
196C2×SD16—311no—
197C2×SD16—311no—
198C2×M16—311no—
199C22×C8—311no—
200C4×Q8—311no—
201C4×Q8—311no—
202C4×Q8—311no—
203C4×Q8—311no—
204C4×Q8—621no—
205C4×Q8—621no—
206C4×Q8C2×D3111yescharacteristic
207C4×Q8C2×D3111yescharacteristic
208C4×D4—311no—
209C4×D4—311no—
210C4×D4—621no—
211C4×D4—621no—
212#117—311no—
213#117—311no—
214#117—621no—
215#117—621no—
216#117—621no—
217C2×C4⋊C4—311no—
218C2×C4⋊C4—621no—
219#107—311no—
220#107C2×D3111yescharacteristic
221#106—311no—
222#106—311no—
223#103—311no—
224#103—311no—
225#103—311no—
226#103—311no—
227#103C2×D3111yescharacteristic
228#103C2×D3111yescharacteristic
229#102—311no—
230#102—311no—
231#97—311no—
232#97—311no—
233C4×C8—311no—
234C4×C8C2×D3111yescharacteristic
235C31#294711yescharacteristic, complemented, Sylow
236#41—314no—
237#41—622no—
238C2×Q8—314no—
239C2×Q8—311no—
240C2×Q8—621no—
241C2×Q8—622no—
242C2×Q8C22×D3111yescharacteristic
243C2×Q8C22×D3111yescharacteristic
244C2×D4—311no—
245C2×D4—621no—
246C22×C4—311no—
247C22×C4—621no—
248C22×C4—621no—
249C22×C4—621no—
250C22×C4—621no—
251C22×C4—621no—
252Q16—314no—
253Q16C4×D3114yescomplemented
254SD16—314no—
255SD16—314no—
256M16—314no—
257C2×C8—311no—
258C2×C8—311no—
259C2×C8—622no—
260C2×C8C22×D3111yescharacteristic
261C2×C8C22×D3111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—311no—
265C4⋊C4—311no—
266C4⋊C4—311no—
267C4⋊C4—311no—
268C4⋊C4—621no—
269C4⋊C4—621no—
270C4⋊C4—621no—
271C4⋊C4—621no—
272C4⋊C4—621no—
273C4⋊C4C22×D3111yescharacteristic
274C4⋊C4C22×D3111yescharacteristic
275C22⋊C4—311no—
276C22⋊C4—311no—
277C22⋊C4—621no—
278C22⋊C4—621no—
279C22⋊C4—621no—
280C42—21no—
281C42—21no—
282C42—311no—
283C42—621no—
284C42—621no—
285C42—621no—
286C42—621no—
287C42—621no—
288C42C22×D3111yescharacteristic
289C23—311no—
290C23—621no—
291Q8—21no—
292Q8—21no—
293Q8—312no—
294Q8—624no—
295Q8—621no—
296Q8C2×C4×D3112yes—
297Q8C2×C4×D3112yes—
298D4—312no—
299D4—624no—
300D4—621no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—311no—
306C2×C4—312no—
307C2×C4—312no—
308C2×C4—311no—
309C2×C4—311no—
310C2×C4—624no—
311C2×C4—624no—
312C2×C4—621no—
313C2×C4—621no—
314C2×C4—621no—
315C2×C4—2481no—
316C2×C4—2481no—
317C2×C4—2481no—
318C2×C4D4×D3111yescharacteristic
319C2×C4D4×D3111yescharacteristic
320C2×C4C23×D3111yescharacteristic
321C8—21no—
322C8—312no—
323C8—621no—
324C8C2×C4×D3112yes—
325C22—312no—
326C22—621no—
327C22—621no—
328C22—622no—
329C22—1241no—
330C22—1241no—
331C22C2×D4×D3111yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—312no—
339C4—621no—
340C4—621no—
341C4—622no—
342C4—1241no—
343C4—1241no—
344C4C22×C4×D3111yescharacteristic
345C4#41×D3111yescharacteristic
346C2—622no—
347C2—1242no—
348C2C4×D4×D3111yescharacteristic
349C2#1452185611yescharacteristic
350C2#1452186211yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutatorC4×C31C23×C4
FrattiniC2×C4C23×D31
FittingC4×Q16×C31C2
socleC22×C31C22×D4
prime radicalC2×#92499C22
cosocleC2×C4×C31C24
hypercenterC4×Q16D31
hypocenterC31#2947
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #14521525C2semidirect
4. #14521156C2semidirect
5. #14521154C2semidirect
6. #14521148C2semidirect
7. #14521147C2semidirect
8. #14521141C2semidirect
9. C4×#92470C2semidirect
10. #14521134C2semidirect
11. #14521090C2semidirect
13. #103×D31C2semidirect
14. #14521003C2semidirect
15. C4×Q16×C31C2semidirect
25. #11758495C4semidirect
65. Q8×D31C2×C4semidirect
76. Q16×C31C2×C4semidirect
86. #92482C2×C4semidirect
87. #92481C2×C4semidirect
110. #92470C2×C4semidirect
111. #92469C2×C4semidirect
179. C4×Q16D31semidirect
235. C31#2947semidirect
253. Q16C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584952
#145215312
#145215252
#145211562
#145211542
#145211482
#145211472
#145211412
C4×#924702
#145211342
#145210902
#145210442
#103×D312
#145210032
C4×Q16×C312
#294731