Finite Group Databasebeta

← #16858777

Subgroups — #16858777

Order 3968.

368 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#14521784C211yesmaximal, characteristic, complemented
2#14521441C211yesmaximal, characteristic, complemented
3#14521439C211yesmaximal, characteristic
4#14521431C211yesmaximal, characteristic, complemented
5#14521271C211yesmaximal, characteristic, complemented
6#14521268C211yesmaximal, characteristic, complemented
7#14521264C211yesmaximal, characteristic, complemented
8#14521261C211yesmaximal, characteristic, complemented
9#14521251C211yesmaximal, characteristic, complemented
10#14521249C211yesmaximal, characteristic, complemented
11#14521180C211yesmaximal, characteristic, complemented
12#14521159C211yesmaximal, characteristic, complemented
13#14521154C211yesmaximal, characteristic, complemented
14C4×#92481C211yesmaximal, characteristic, complemented
15C31×#556C211yesmaximal, characteristic, complemented
16C31×C4⋊Q8C2211yescharacteristic
17C31×#126—21no—
18C31×#126—21no—
19C31×#124C2211yescharacteristic
20C31×#124C2211yescharacteristic
21C31×C22⋊Q8—21no—
22C4×Q8×C31C2211yescharacteristic
23C4×Q8×C31C2211yescharacteristic
24C4×Q8×C31C2211yescharacteristic
25C4×D4×C31C2211yescharacteristic
26C31×#117—21no—
27#11758524—21no—
28C2×#92482C2211yescharacteristic
29C2×#92482C2211yescharacteristic
30C2×#92481C2211yescharacteristic
31C2×#92481C2211yescharacteristic
32C2×#92481C2211yescharacteristic
33C2×#92481C2211yescharacteristic
34C2×#92480C2211yescharacteristic
35C2×#92479C2211yescharacteristic
36C2×#92479C2211yescharacteristic
37#11758471—21no—
38#11758471—21no—
39#11758470—21no—
40#11758452C2211yescharacteristic
41#11758452C2211yescharacteristic
42#11758451C2211yescharacteristic
43C4×D124C2211yescharacteristic
44C4×D124C2211yescharacteristic
45C4×Dic62C2211yescharacteristic
46#11758423C2211yescharacteristic
47#11758423C2211yescharacteristic
48#11758420C2211yescharacteristic
49#11758398C2211yescharacteristic
50#11758397C2211yescharacteristic
51#11758397C2211yescharacteristic
52#11758396C2211yescharacteristic
53C4⋊(C31⋊C8)C2211yescharacteristic
54C4⋊(C31⋊C8)C2211yescharacteristic
55C4⋊(C31⋊C8)C2211yescharacteristic
56#11758391C2211yescharacteristic
57#11758391C2211yescharacteristic
58C4×C31⋊C8C2211yescharacteristic
59C2×Q8×C31D411yescharacteristic
60C2×Q8×C31C2311yescharacteristic
61C2×Q8×C31C2311yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C2×D124—21no—
65C2×D124—21no—
66C2×D124C2311yescharacteristic
67C2×D124C2311yescharacteristic
68C2×C4×D31—21no—
69C2×C4×D31—21no—
70C2×Dic62C2311yescharacteristic
71C4⋊C4×C31—21no—
72C4⋊C4×C31—21no—
73C4⋊C4×C31—21no—
74C4⋊C4×C31—21no—
75C4⋊C4×C31D411yescharacteristic
76C4⋊C4×C31D411yescharacteristic
77C4⋊C4×C31D411yescharacteristic
78C4⋊C4×C31C2311yescharacteristic
79C22⋊C4×C31—21no—
80C22⋊C4×C31—21no—
81C22⋊C4×C31—21no—
82C22⋊C4×C31—21no—
83C22⋊C4×C31—21no—
84C42×C31—21no—
85C42×C31—21no—
86C42×C31C2311yescharacteristic
87C42×C31C2311yescharacteristic
88C42×C31C2311yescharacteristic
89#92482—22no—
90#92482—41no—
91#92481—22no—
92#92481—41no—
93#92481—41no—
94#92481—41no—
95#92480—41no—
96#92479—22no—
97#92479—41no—
98(C2×D31)⋊C4—21no—
99(C2×D31)⋊C4—21no—
100(C2×D31)⋊C4—21no—
101(C2×D31)⋊C4—21no—
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—21no—
104C4⋊Dic31C2311yescharacteristic
105Dic31⋊C4—21no—
106C4×Dic31—21no—
107C2×C31⋊C8C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110C2×C31⋊C8C2311yescharacteristic
111C23×C31—21no—
112C22×D31—21no—
113C22×D31—21no—
114Q8×C31—21no—
115Q8×C31—21no—
116Q8×C31—21no—
117Q8×C31—21no—
118Q8×C31—41no—
119D4×C31—21no—
120D4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—42no—
125C2×C4×C31C2×D411yescharacteristic
126C2×C4×C31C2×D411yescharacteristic
127C2×C4×C31C2×D411yescharacteristic
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2×D411yescharacteristic
131C2×C4×C31C2411yescharacteristic
132C2×Dic31—21no—
133C2×Dic31—21no—
134D124—21no—
135D124—21no—
136D124—21no—
137D124—21no—
138D124—42no—
139D124—81no—
140C4×D31—81no—
141C4×D31—81no—
142Dic62—21no—
143Dic62—21no—
144C31⋊C8—21no—
145C31⋊C8—21no—
146C31⋊C8—21no—
147C31⋊C8#4112yes—
148#3401—311nomaximal, Sylow
149C22×C31—41no—
150C22×C31—41no—
151C22×C31—41no—
152C22×C31C22×D411yescharacteristic
153C2×D31—41no—
154C2×D31—41no—
155C2×D31—41no—
156C2×D31—41no—
157C2×D31—41no—
158C2×D31—41no—
159C4×C31—21no—
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—21no—
164C4×C31—21no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31C2×#4111yescharacteristic
169C4×C31H(5,𝔽2)11yescharacteristic
170Dic31—41no—
171Dic31—41no—
172#559—311no—
173#556D3111yescharacteristic, complemented
174#497—311no—
175#496—311no—
176#495—311no—
177#464—311no—
178#464—311no—
179#463—311no—
180#463—311no—
181#461—311no—
182#460—311no—
183#446—311no—
184#442—311no—
185#441—311no—
186C4×SD16—311no—
187C2×C31—81no—
188C2×C31#55011yescharacteristic
189C2×C31#57511yescharacteristic
190C2×C31#57711yescharacteristic
191D31—81no—
192D31—81no—
193C2×Q16—311no—
194C2×Q16—311no—
195C2×SD16—311no—
196C2×SD16—311no—
197C2×SD16—311no—
198C2×SD16—311no—
199C2×SD16—311no—
200C2×D8—311no—
201C2×D8—311no—
202C4⋊Q8C2×D3111yescharacteristic
203#127—311no—
204#126—21no—
205#126—21no—
206#124—311no—
207#124—311no—
208#124—621no—
209#124C2×D3111yescharacteristic
210#124C2×D3111yescharacteristic
211#123—621no—
212#123—621no—
213C22⋊Q8—21no—
214C4×Q8—311no—
215C4×Q8C2×D3111yescharacteristic
216C4×Q8C2×D3111yescharacteristic
217C4×Q8C2×D3111yescharacteristic
218C4×D4—311no—
219C4×D4—311no—
220C4×D4—621no—
221C4×D4C2×D3111yescharacteristic
222#117—21no—
223#106—311no—
224C4⋊C8—311no—
225C4⋊C8—311no—
226C4⋊C8—311no—
227#103—311no—
228#103—311no—
229#103—311no—
230#102—311no—
231#102—311no—
232#102—311no—
233#97—311no—
234#97—311no—
235C4×C8—311no—
236C31#340111yescharacteristic, complemented, Sylow
237C2×Q8—311no—
238C2×Q8C22⋊D3111yescharacteristic
239C2×Q8C22×D3111yescharacteristic
240C2×Q8C22×D3111yescharacteristic
241C2×D4—311no—
242C2×D4—311no—
243C2×D4—621no—
244C2×D4—621no—
245C2×D4C22×D3111yescharacteristic
246C22×C4—21no—
247C22×C4—621no—
248C22×C4—621no—
249Q16—622no—
250Q16—1241no—
251SD16—622no—
252SD16—1241no—
253SD16—1241no—
254SD16—1241no—
255SD16—1241no—
256D8—622no—
257D8—1241no—
258C2×C8—311no—
259C2×C8—311no—
260C2×C8—311no—
261C2×C8—311no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—311no—
267C4⋊C4—621no—
268C4⋊C4C22⋊D3111yescharacteristic
269C4⋊C4C22⋊D3111yescharacteristic
270C4⋊C4C22⋊D3111yescharacteristic
271C4⋊C4C22×D3111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—621no—
278C22⋊C4—621no—
279C22⋊C4—621no—
280C22⋊C4—621no—
281C22⋊C4—621no—
282C22⋊C4—621no—
283C42—21no—
284C42—21no—
285C42—621no—
286C42C22×D3111yescharacteristic
287C42C22×D3111yescharacteristic
288C42C22×D3111yescharacteristic
289C23—21no—
290C23—621no—
291C23—621no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—621no—
298Q8—621no—
299D4—21no—
300D4—21no—
301D4—621no—
302D4—621no—
303D4—621no—
304D4—621no—
305D4—1242no—
306D4—2481no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—621no—
312C2×C4—621no—
313C2×C4—2481no—
314C2×C4—2481no—
315C2×C4C2×C22⋊D3111yescharacteristic
316C2×C4C2×C22⋊D3111yescharacteristic
317C2×C4C2×C22⋊D3111yescharacteristic
318C2×C4C2×C22⋊D3111yescharacteristic
319C2×C4C2×C22⋊D3111yescharacteristic
320C2×C4C2×C22⋊D3111yescharacteristic
321C2×C4C23×D3111yescharacteristic
322C8—312no—
323C8—621no—
324C8—621no—
325C8—621no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—1241no—
330C22—1241no—
331C22—1241no—
332C22—1241no—
333C22—1241no—
334C22—1241no—
335C22C22×C22⋊D3111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—1241no—
346C4—1241no—
347C4#41×D3111yescharacteristic
348C4#1175856611yescharacteristic
349C2—81no—
350C2—2481no—
351C2—2481no—
352C2#1452171811yescharacteristic
353C2#1452184711yescharacteristic
354C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#556C2
socleC22×C31C22×D4
prime radical#14521439C2
hypercenter#556D31
hypocenterC31#3401
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #14521784C2semidirect
2. #14521441C2semidirect
4. #14521431C2semidirect
5. #14521271C2semidirect
6. #14521268C2semidirect
7. #14521264C2semidirect
8. #14521261C2semidirect
9. #14521251C2semidirect
10. #14521249C2semidirect
11. #14521180C2semidirect
12. #14521159C2semidirect
13. #14521154C2semidirect
14. C4×#92481C2semidirect
15. C31×#556C2semidirect
173. #556D31semidirect
236. C31#3401semidirect

Maximal subgroups

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

subgroupindex
#145217842
#145214412
#145214392
#145214312
#145212712
#145212682
#145212642
#145212612
#145212512
#145212492
#145211802
#145211592
#145211542
C4×#924812
C31×#5562
#340131